Lectures on Applied Partial Differential Equations I. M. Sigal Dept of Mathematics, Univ of Toronto Spring 2019 Preface Partial differential equations (PDEs) come from physical sciences (e.g. the equation for the electric/gravitational potential created by a give charge/mass distribution - the Poisson equation) or mathematics (e.g. the Cauchy-Riemann equations for holomorphic functions). They are used in physical and life sciences, economics and social studies as well as in mathematics (recently, in the solution of the Poincar´ e conjecture using the Ricci flow). In this course, we concentrate mostly on the Laplace/Poisson, the reaction-diffusion, Schr¨ odinger and wave equations, which occupy a big chunk of work in PDEs. We will address the following questions: • Existence • Key solutions • Stability Contents 1 Examples of equations and general set-up 7 1.1 Important examples ............................... 7 1.2 Abstract evolution equations .......................... 9 1
Transcript
Lectures on Applied Partial Differential Equations
I. M. Sigal
Dept of Mathematics, Univ of Toronto
Spring 2019
Preface
Partial differential equations (PDEs) come from physical sciences (e.g. the equationfor the electric/gravitational potential created by a give charge/mass distribution - thePoisson equation) or mathematics (e.g. the Cauchy-Riemann equations for holomorphicfunctions). They are used in physical and life sciences, economics and social studies aswell as in mathematics (recently, in the solution of the Poincare conjecture using the Ricciflow).
In this course, we concentrate mostly on the Laplace/Poisson, the reaction-diffusion,Schrodinger and wave equations, which occupy a big chunk of work in PDEs. We willaddress the following questions:
In this course we will consider several key partial differential equations arising in physics,material science, biology and geometry. We develop an existence theory for these equa-tions, describe their key properties, isolate their most important solutions and studystability or instability of these solutions.
1.1 Important examples
Here are some examples of the equations we will be studying in the course:
The Allen-Cahn equation. This equation describes among other things motion ofinterfaces between different compounds and is given by
∂u
∂t= ε2∆u+ u− u3. (1.1)
Here u = u(x, t) is a real function of x ∈ Rd and t ∈ R+ := [0,∞) and ∆ is the Laplaceoperator (the Laplacian):
∆u :=d∑j=1
∂2u
∂x2j
.
This simple looking equation has a rich structure and is an object of an intensive inves-tigation in the last 50 years. It has an important set of static solutions, called kinks,χε(x) = tahn
(xd√2ε
), or more generally,
χε,x0,e(x) = tahn
((x− x0) · e√
2ε
), (1.2)
for any e, x0 ∈ Rd. The kink χε(x) = tahn(xd√2ε
)describes a flat interface at xd = 0.
(The kink (1.2) describes the flat interface which is shifted and rotated compared tothe previous one.) In fact, any slowly varying interface can be constructed by “gluing”together rotated and shifted kinks.
The Gross-Pitaevski/nonliear Schrodinger equation. or nonlinear Schrodingerequation. These describe dynamics of superfluids, Bose-Einstein condensates, nonlinearwaves in plasmas and on surface of a fluid:
i∂ψ
∂t= −∆ψ + λ|ψ|2ψ (1.3)
for some real constant λ. This equation has remarkable special solutions - the vortices,which for d = 2 are given by
ψ(n)(x) = f (n)(r)einθ, (1.4)
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where (r, θ) are the polar coordinates of x ∈ R2. Even more interestingly, the vortices canarrange themselves into lattices, giving new solutions for which A.A. Abrikosov receiveda few years back a Nobel prize in physics.
s1s1
s2 nt
Figure 1: Portraits of vortices
The Keller - Segel equations. The next system of equations models the chemotaxis,which is the directed movement of organisms in response to the concentration gradient ofan external chemical signal and is common in biology. The chemical signals either comefrom external sources or are secreted by the organisms themselves. The latter situationleads to aggregation of organisms and to the formation of patterns.
Consider organisms moving and interacting in a domain Ω ⊆ Rd, d = 1, 2 or 3.Assuming that the organism population is large and the individuals are small relative tothe domain Ω, Keller and Segel derived a system of reaction-diffusion equations governingthe organism density ρ : Ω × R+ → R+ and chemical concentration c : Ω × R+ → R+.The equations are of the form
∂tρ = Dρ∆ρ−∇ (f(ρ)∇c)∂tc = Dc∆c+ αρ− βc. (1.5)
Here Dρ, Dc, α, β are positive functions of x and t, ρ and c, and f(ρ) is a positive functionmodelling chemotaxis (positive chemotaxis).
Assuming α and β are constants, this system has the simple homogeneous staticsolution c = const and ρ = β/α =const. However this solution is unstable under smallperturbations. For initial conditions close to this static solution, ρ starts growing atsome point and becomes highly concentrated around this point. This is a chemotacticaggregation which leads to formation of fruition bodies (colonies formed by Escherichiacoli under starvation conditions, or multicellular structures of ∼ 105 cells, by single cellbacterivores, when challenged by adverse conditions). Similar phenomenon occurs indynamics of tumours.
Chemotaxis underlies many social activities of micro-organisms, e.g. social motility,fruiting body development, quorum sensing and biofilm formation.
A simplified version of these equations is used to model stellar collapse and crimepatterns.
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The Fisher-Kolmogorov-Petrovsky-Piskunov equation. This equation describespropagation of mutations in population biology and flames in combustion theory. It isgiven by
∂u
∂t= ∆u− (u− 1)u. (1.6)
It has interesting solutions, called traveling waves, which are of the form
u(x, t) = φ(x− vt), (1.7)
where ϕ(y) and v are independent of t and are called the traveling wave profile and thetraveling wave velocity.
PDEs of Quantum Physics.
1.2 Abstract evolution equations
We would like to address a general theory of evolution PDEs, i.e. equations of the form
∂tu = F (u), u|t=0 = u0, (1.8)
where t→ u(t) is a path in some space, F is a map defined on the same space, ∂tu = u = ∂u∂t
and we understand (1.8) as ∂tu(t) = F (u(t)). u0 is called the initial condition and (1.8),the initial value problem.
The situation we will be mostly interested in is when F is a partial differential operator,linear or nonlinear. An example of such an equation is the equation:
∂tu = ∆u+ g(u). (1.9)
In this example, F (u) = ∆u + g(u). This is the celebrated nonlinear heat or reaction-diffusion equation. The term ∆ is responsible for diffusion and the term, g(u), describesthe reaction in the system modelled.
Let us consider first several other examples of maps F :
1) F (u) = ∆u,
2) F (u) = f u for a given function f ,
3) F (u) = div( ∇u√1+|∇u|2
).
4) F (u) = ϕ ∗ u for given ϕ ∈ L1(Rn).
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Note that the maps in 1) and 4) are linear, while the map in 2) is linear if f is linear, andnonlinear otherwise and the map in 3) is nonlinear.
Sometimes (1.8) is called the dynamical system and F , the vector field (finite or infinitedimensional). Though we define F on a vector space, it can be also defined on a manifold(again, finite or infinite dimensional).
To solve equation (1.8), we have to choose a space, say X, to which the vector-functiont → u(t) belongs for every t considered. We have to make sure that F is defined on thisspace (and maps it into another space, say, Y ). Depending on the problem at hand, wechoose different spaces. For instance, for the examples 1) and 3) above, F maps the spaceX := Ck(Ω) into the space Ck−2(Ω), and in the examples 4), Lp(Rn) into itself. Moreabstractly, X could be some space with a norm, or a Banach space. For the definitionand review of Banach spaces see Subsection A.2 below.
We also have to choose a space, say V , to which the function t → u(t) belongs. Ifu(t) ∈ X for any t in some interval I, then we can choose
• V = C(I,X), where, say, I := [0, T ].
Here V = C(I,X) is the space of once differentiable vector - functions u : t → u(t) on Iwith values in X and with the norm
‖u‖V := supt∈I
(∥∥u(t)
∥∥X
+∥∥∂tu(t)
∥∥Y
).
If the map F (u) is linear, F (u) = Au, then the corresponding equation is also linear,
∂tu = Au and u|t=0 = u0. (1.10)
An important example of the linear evolution equation is the linear (free) heat equation
∂tu = ∆u and u|t=0 = u0, (1.11)
where u : Rnx×R+
t → R, R+t := t ∈ R : t ≥ 0, is an unknown function, and u0 : Rn → R
is a given initial condition.For linear equations, there are general situations where we can show the existence of
the global (all t’s) solutions for all reasonable initial conditions:
• A is a bounded operator on A;
• A is either self-adjoint (A∗ = A) and bounded above, A ≤ C for some C < ∞(〈u,Au〉 ≤ C‖u‖2) or anti-self-adjoint (A∗ = −A);
• A is a ‘constant coefficient pseudo-differential’ operator; more precisely, A = a(−i∇x),where a is some decent function.
Exercise 1.1. Review Appendix A.1 and do Exercise A.1.
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2 Key classes of solutions of PDEs and symmetries
In this section, X and Y are Banach space with X densely embedded in Y , and F : X →Y , a map from X to Y . We consider the dynamical system
∂u
∂t= F (u). (2.1)
To this we may add an initial condition u|t=0 = u0. To be specific, We assume that Y isa space of functions on Rn, or on an open subset it.
2.1 Key special solutions
We consider key solutions of dynamical system (2.1).
Static solutions. Static solutions are the solutions independent of time t. As the resultthey satisfy the equation F (u) = 0. Here are some examples.
Allen-Cahn equation. First, we consider the Allen-Cahn equation, which plays a cen-tral role in material science. It presents a basic model with many generalizations andextensions. It is a reaction diffusion equation of the form
∂u∂t
= ε2∆u− g(u), (2.2)
with the initial condition u|t=0 = u0. Here u = u(x, t) is a real function on the space timeRn × R+, with R+ = [0,∞), u0 = u0(x), ε is a small parameter and g(u) = u3 − u. . Wecan also consider x ∈ Ω, where Ω ⊆ Rn. Then we have to add a boundary condition on∂Ω, e.g. the Neumann boundary condition ∂u
∂n= 0, x ∈ ∂Ω.
More generally, g : R → R is the derivative, g = G′, of a double-well potential:G(u) ≥ 0 and has two non-degenerate global minima with the minimum value 0. G(u) isalso called a bistable potential (see Figure 2). In our case, G(u) = 1
4(u2 − 1)2.
ïG
ba
a
b
x
u
Figure 2: Hill-valley-hill structure for −G, and the kink for u.
We consider static solutions of Allen-Cahn equation. They satisfy the static Allen-Cahn equation,
− ε2∆u+ g(u) = 0. (2.3)
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For complex u and g(u) = |u|2u− u, (2.3) is the Ginzburg-Landau equation. It appearedfirst in condensed matter physics. For this equation G is of the form G1(|u|2), where G1
is the “half” of the double well potential.Equation (2.3) has two trivial solutions u = ±1, which descibe pure phases.We are interested in the phase separation phenomena when the solutions are close to
+1 or −1 (pure phases) in most of the space with sharp transitional interfaces separating+1 regions from −1 regions. We consider the simplest of planar interface.
We look for static solutions which vary only in a direction, e, transversal to a plane,x · e = 0, i.e. of the form
χε(x) = χ
(x · eε
), (2.4)
for any e ∈ Rd, where χ is a function of single variable. By rotational symmetry andscaling properties of (2.3), the function χ(s) satisfies the equation
χ′′ = g(χ). (2.5)
This is the Newton’s equation with the potential −G(φ), where G′(φ) = g(φ). Thepotential −G(φ) has equilibria at φ = ±φ0 and at φ = 0. Hence the equation (2.3) hasthe homogeneous solutions, φ = ±φ0 and φ = 0.
It is clear from this mechanical analogy that (2.3), besides these homogeneous solu-tions, has the solutions which go asymptotically to ±φ0 as s → ±∞ (see Figure 2). Toshow the existence of these solutions, we observe that by the conservation of mechanicalenergy, we have
1
2(φ′)2 −G(φ) = 0 (2.6)
(we take the mechanical energy to be 0), which can be solved for φ′ as φ′ = ±√
2G(φ),and then integrated to obtain φ. The solution with the + sign is called the kink and, withthe − sign, the anti-kink.
In our case, G(φ) = 14(φ2−1)2, i.e. g(u) = u3−u, and the integration can be performed
explicitly and gives
χε(x) = tahn
(x · e√
2ε
), (2.7)
for any e, x0 ∈ Rd.
Example 2.1. Let
G(ϕ) =
ωo(ϕ− ϕo)2, ϕ ≥ 0,ωw(ϕ− ϕw)2, ϕ ≤ 0,
and ωoϕ2o = ωwϕ
2w. (2.8)
Then equation (2.3) is piecewise linear, and we get
ϕk(z) =
ϕw(1− e−
√ωwz), z ≥ 0,
ϕo(1− e√ωoz), z ≤ 0.
(2.9)
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Exercise 2.1. Show (2.9).
One can show existence of kink (and anti-kink) solutions by dynamical systems theoryand variational techniques (see Appendix 2.2 and Section 6, respectively).
Lamellar phase. In this situation, layers of +1 and −1 phases (substances) coexist ina periodic array. One can prove by the variational or Lyapunov-Schmidt decompositiontechniques that one can construct a periodic solution (‘kink crystal’) corresponding to anarray of kinks and antikinks (see below for a discussion of an approach to this problem).
Stationary waves. Stationary or standing waves (or breathers) are solutions of theform
Ψ(x, t) = e−iλtφ(x), (2.10)
where φ and λ are time-independent. The function ϕ is called the profile of the stationarywave (2.10). As an example, consider the nonlinear Schrodinger or Gross - Pitaevskiiequation (NLS or GPE)
i∂tΨ = −∆Ψ + κ|Ψ|2Ψ, (2.11)
for Ψ : Rd → C and Ψ|t=0 = Ψ0. It has solutions of the form (2.10), where φ and λ ∈ Rsatisfy the equation
−∆φ+ κ|φ|2φ = λφ. (2.12)
This is the nonlinear eigenvalue problem.
Traveling waves. Traveling waves are solutions of the form
u(x, t) = φ(x− vt), (2.13)
where φ(y) and v are independent of t and are called the traveling wave profile and thetraveling wave velocity. Consider the reaction-diffusion equation
∂u
∂t= ∆u− g(u). (2.14)
Substituting (2.13) into (2.14) and passing to the variable y = x − vt (the moving coor-dinate frame), we obtain the following equation for φ(y) and v
∆φ− g(φ) + v · ∇φ = 0. (2.15)
Consider the one-dimensional case. Then this equation can be rewritten as
φ′′ = g(φ)− vφ′. (2.16)
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This is the Newton’s equation with the potential −G(φ), where G′(φ) = g(φ), and thefriction term vφ′.
Compute the change in mechanical energy:
∂x(1
2(φ′)2 −G(φ)
)= (φ′′ − g(φ))φ′ = −v(φ′)2 < 0, for v > 0. (2.17)
This implies that for v > 0 the particle descends to a (local) minimum of −G.To be specific we consider the Fisher-Kolmogorov-Petrovsky-Piscunov (FKPP) equa-
tion, which appears in population biology and combustion theory. This is Eqn. (2.14)with g(u) = u(u− 1):
∂u
∂t= ∆u− (u− 1)u. (2.18)
For this equation, G = 13u3 − 1
2u2, see Figure 3 (where the label G should be replaced by
−G).
g
G
p vn
vm
vm
o oo ph
Figure 3: Functions −G, g and φ (the labels G and o in the first figure should be replacedby −G and 1; the labels oo and ph in the second figure should be replaced by 1 and φ).
Using the mechanical analogy described above, we can solve the equation (2.16) forthe nonlinearities described. At the remote past (time y = −∞), the particle leaves thetop of the hill in −G(φ) at φ = 1 and moves to the left toward the wall loosing thealtitude due to the dissipation. Hitting the wall particle turns around and moves towardthe opposite wall and so forth until it relaxes to the bottom of the well at φ = 0. Thefront of the wave is at y at which φ(y) hits 0 the first time.
An analysis of the traveling wave equation (2.16) using elementary dynamical systemtheory, done in Appendix 2.2 shows that the solution is monotonically decreasing for v > 2(the over-damped case) and is oscillating for v < 2.
More generally we assume g(u) satisfies the conditions g(0) = g(1) = 0 together withg′(0) < 0 and g′(1) > 0, see Figure 3 (where the label G should be replaced by −G).
Another type of the the nonlinearity g(u), which is used in combustion theory is shownin Fig. 4.
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u1
T T
g −G
1
Figure 4: Ignition case
Exercise 2.2. Investigate (and find if possible) the traveling waves in the reaction-diffusionequation with 1) g(u) = (u − a)(u − 1)u with 0 < a < 1/2 (the a = 1/2 case gives theAllen-Cahn equation) 2) g(u) as in Figure 4 (flame propagation for the ignition nonlin-earity).
A more complicated situation takes place for the nonlinear Schrodinger or Gross -Pitaevskii equation (9.12). A simple computation shows that (9.12) has the solutions ofthe form
Ψ(x, t) := ei(12v·x− 1
4|v|2t−λt)φ(x− vt− x0) (2.19)
where e−iλtφ(x) is a stationary solution to (9.12). This solution is obviously a travelingwave.
As another example we consider the celebrated Korteweg-de Vries (KdV) equationdescribing water waves in shallow channels:
∂tu+ u∂xu+ ∂3xu = 0. (2.20)
The traveling waves u(x, t) = φ(x− vt) for this equation satisfy the equation
φ∂yφ+ ∂3yφ− v∂yφ = 0,
where we passed to the moving frame variables y = x − vt. Integrate this equation, toobtain 1
2φ2 + ∂2
yφ− vφ = c′, where c′ is a constant of integration. To solve this equation,we interpret it as a Newton’s equation,
∂2yφ = −1
2φ2 + vφ+ c′
(in time y) and write out the conservation of energy, i.e. multiply it by ∂yφ and integratethe result to obtain
1
2(∂yφ)2 = −1
6φ3 +
1
2vφ2 + c′φ+ c′′,
where c′′ is another constant of integration. The latter is the first order ODE. If φ, ∂yφ→0, as |y| → ∞, then c′ = c′′ = 0 and it can be rewritten as ∂yφ = ±
√−1
6φ3 + 1
2vφ2 and
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integrated as ∫dφ√
−16φ3 + 1
2vφ2
= ±y + c,
where c is a constant of integration. Taking above the minus sign and using the substitu-tion φ = v
2sech2 ψ (sech := 1/ cosh), one can compute (see e.g. [6]) φ(y) = v
2sech2(v
2(y−c))
which gives the solution
η(x, t) =v
2sech2(
v
2(x− vt− c)). (2.21)
This is the celebrated KdV soliton.
Exercise 2.3. Verify that (2.21) is a solution to the KdV (2.20) using Mathematicaprogram in https://young.physics.ucsc.edu/250/mathematica/soliton.nb.pdf.
Exercise 2.4. Review Appendices A.2 and A.3 and do Exercise A.2.
Self-similar solutions. Self-similar solutions are solutions of the form
u(x, t) = t−αφ(x/tβ), (2.22)
for some real α and β, where φ(y) is independent of t and is called the self-similar solutionwave profile. Consider the porous medium equation
∂u
∂t= ∆(uγ), (2.23)
for u ≥ 0 and γ > 1 and on Rn. Then (2.22) solves (2.23), provided α+ 1 = αγ + 2β andφ(y) solves the equation
∆(φγ) + αφ+ βy · ∇φ = 0. (2.24)
Note that we passed to the variable y = x/tβ (the moving (shrinking or expanding)coordinate frame).
Exercise 2.5. (see e.g. [6]) Verify that if α = nβ, then equation (2.24) has the one-parameter family of solutions
φ(y) = (b− γ − 1
2γβ|y|2)1/γ−1, (2.25)
where b > 0.
Recall that we imposed the conditions α + 1 = αγ + 2β and α = nβ, which togethergive
α =n
n(γ − 1) + 2, β =
1
n(γ − 1) + 2. (2.26)
(2.25) are called the Barenblatt-Kompaneetz-Zeldovich solutions.
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2.2 Appendix: A dynamical systems perspective
We investigate the static equation (2.5) and the traveling wave equation (2.16) also fromthe point of view of elementary dynamical system theory.
Rewrite (2.5) asχ′ = ψ, ψ′ = g(χ). (2.27)
This dynamical system has equilibria (φ0, 0), where φ0 solves g(φ) = 0. In our case,φ0 = ±1, 0, and therefore the equilibria are (±1, 0) and (0, 0).
The linearized vector field at an equilibrium (φ0, 0), where φ0 = ±1, 0, is(0 1
g′(φ0) 0
). (2.28)
The eigenvalue equation, λ2 − g′(φ0) = 0, gives λ = ±√g′(φ0). At φ0 = 0 we have
g′(0) < 0, which gives two purely imaginary eigenvalues. Moreover, g′(±1) > 0, whichgives one positive and one negative eigenvalue. Hence the stationary point (0, 0) is astable equilibrium (a stable focus) and (±1, 0) are unstable equilibria (saddle points).There is one trajectory going from (−1, 0) to (1, 0) and one from (1, 0) to (−1, 0). Theseare exactly the kinks and anti-kinks (or fronts) mentioned above (see Figure 2).
−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
X
Y
Separatrix
Figure 5: v > 2 there exists a separatrix. Plotted is arrows with direction dψ/dφ, atX-coordinate φ and Y -coordinate ψ, note that dψ/dφ = ψ−1φ(φ− 1)− v.
Now, we turn to the traveling wave equation (2.16). Rewrite (2.16) as
φ′ = ψ, ψ′ = g(φ)− vψ. (2.29)
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The equilibria (or static solutions or stationary points) of this system are (φ0, 0) where φ0
solves g(φ) = 0. For g(φ) = φ(φ−1) we have the equilibria (0, 0) and (1, 0). To determinewhether these equilibria are stable or not, we compute the linearized vector field (Jacobi
derivative of the vector field on the r.h.s.) at (φ0, 0). It is
(0 1
g′(φ0) −v
). The eigenvalue
equation, λ(λ+ v)− g′(φ0) = 0, gives
λ =−1
2v ±
√v2
4+ g′(φ0). (2.30)
At φ0 = 0 we have g′(0) = −1, which gives two negative eigenvalues for v ≥ 2 and twocomplex eigenvalues with negative real parts for v < 2. Hence the equilibrium (0, 0) is astable equilibrium. (0,0) is a stable focus for v > 2 and stable spiral for v < 2. For φ0 = 1we find g′(1) = 1 and the point (1, 0) is a saddle point. For v > 2 there is a separatrix.This is a bit subtle, see Figure 5 above.
The upshot of this is that for v ≥ 2 the solutions is monodically decreasing and forv ≤ 2 , oscillating, see Figure 3 (where vn and vm denote the monotonic and oscillatingsolutions, respectively.)
2.3 Symmetries and solutions of PDEs
It turns out that special solutions we considered above come from specific symmetriesof the equation we consider. We explore the relation between symmetries of evolutionequations and solutions of such equations in more depth.
Transformations which map solutions of an equation into solutions are called sym-metries of this equation. More precisely, we say a bounded transformation T on Y is asymmetry of (2.1) (∂tu = F (u)) iff the fact that u is a solution implies that so is Tu.
We say a map F : X → Y, X ⊂ Y, is invariant under a bounded transformation T onY iff
F (Tu) = TF (u), (2.31)
We call such a T a symmetry of a map F . If F is invariant under T , then T is a symmetryof (2.1).
All symmetries of a given equation form a group. Namely, we have
Proposition 2.6. Symmetries of a map F and therefore of the corresponding dynamicalsystem form a group.
Proof. If T and T ′ are symmetries of a map F , then F (TT ′u) = TF (T ′u) = TT ′F (u) andtherefore so is TT ′.
The group of all symmetries of equation (2.1) (resp. the map F ) is called the symmetrygroup of (2.1) (resp. F ). We distinguish
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- the space - time symmetries, i.e. symmetries of the form
u(x, t) 7→ u(g(x, t)), (2.32)
where g is a transformation of the space time Rn+1, and- the gauge symmetries, i.e. symmetries of the form
u(x, t) 7→ gu(x, t), (2.33)
where g is a transformation of the target space of u(x, t).Here are examples of transformation which could be space - time symmetries:Translation transformation: Assume the additive group Rn acts on the space of u’s
as: for any h ∈ Rn,
Th : u(x, t) 7→ u(x+ h, t); (2.34)
Rotation and reflection transformations: for any R ∈ O(n),
TR : u(x, t) 7→ u(Rx, t) (2.35)
(including the reflections u(x, t)→ u(−x, t));Scaling transformation: for any λ > 0,
Tλ : u(x, t) 7→ λαu(λβx, λγt). (2.36)
Translations and rotations are called rigid motions. They are a part of Galilean group:
Examples of gauge symmetries will be special cases of the following general situation.Assume V is a vector space and G, a group acting on V , via a representation g 7→ Tg(group homomorphism into bounded operators). Then for ψ : Rn × R+ → V , we define
ψ(x, t) 7→ Tgψ(x, t), ∀g ∈ G, ∀x, t. (2.39)
The group G is called the gauge group and (2.39), the gauge transformation with the
corresponding gauge group G. Specifically, consider V = Cm, or = Rm. Then G = U(m)or O(m), the unitary or orthogonal (rotation) group. In particular, for the gauge groupU(1), we have
ψ(x, t) 7→ eiαψ(x, t), α ∈ R. (2.40)
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Remark. Transformations (2.34), (2.35), (2.36), (2.40) are representations of the fol-lowing groups: the group, T (n), of translations of Rn, identified with Rn, the group, O(n),of rotations of Rn, the group, R+, of dilatations of Rn, and the unitary group U(1). Allthese groups, except for O(n), are Abelian ones.
The largest group whose representation of Y consists of symmetries of (2.1) is calledthe symmetry group of (2.1).
Symmetries of some basic equations.
Theorem 2.7. (i) The NLS (or GPE) NLH (=RD) and NLW equations (see (9.12)and (2.14)) have translational and rotational symmetries. (ii) The NLS has the gaugesymmetry.
Exercise 2.8. Prove the theorem above.
We will learn in Section 7.4 below that continuous symmetries lead to conservationlaws. Presently, we will use symmetries to classify solutions in two different ways, namelyby invariance or time-covariance under certain subgroups of the symmetry group.
Remark. A genuine space-time symmetry are the Galilean symmetry of the nonlinearSchrodinger equation (NLS):
ψ(x, t) 7→ ei(12v·x− 1
4|v|2t)ψ(x− vt, t). (2.41)
and the scaling symmetries u(x, t) 7→ λu(λx, λ2t) and u(x, t) 7→ λβ−2αγ−1 u(λαx, λβt) of the
NLS and porous medium equations, respectively.
Group-invariant (symmetric) solutions. Consider solutions of the dynamical system(2.1), invariant under some subgroup of its symmetry group.
Static solutions. Solutions invariant under time translations are called static solutions.They are independent of time and form the most important class of special solutions. Theyare also called equilibria and sometimes stationary solutions. Thus u∗ is an equilibriumor static solution iff u∗ is independent of time and satisfy the equation F (u∗) = 0.
Examples: NLH, NLS, NLW.
Homogeneous solutions. Solutions invariant under spatial translations are independentof x and are called the homogeneous solutions.
Spherically symmetric solutions. Solutions invariant under spatial rotations are calledspherically symmetric solutions. They are independent of angular variables and dependonly on two variables, |x| and t.
Exercise 2.9. Find spherically symmetric solutions of the Laplace equation ∆u = 0 andof the Poisson equation −∆u = f on Rn/0 and Rn, respectively (see [6], Section 2.2.1).
Lectures on Applied PDEs 21
Equivariant solutions. We begin with auxiliary definitions. We say two solutions areequivalent if one can be obtained from the other by a gauge transformation. The groupof rigid motions, Grm, is defined as a semi-direct product of the groups of translationsand rotations. We denote by Tg, g ∈ Grm, the action of the group, Grm, of rigid motionson space of solutions. For the groups of translations, and rotations, is given (2.34) and(2.35), respectively.
We say a solution u is equivariant (under a subgroup, G, of the group of rigid motions)iff the action of G on u takes it into the (gauge-) equivalent function, i.e., for any g ∈ G,there is γ = γ(g) s.t.
Tgu = Γγu,
where Γγ is the action of for the gauge group, given in (2.40). Here are two importantexamples coming from the Gross - Pitaevski equation, mentioned above in (2.42), and theGinzburg-Landau equations studied in detail later.
Vortices. We consider the static Gross - Pitaevskii (or Ginzburg-Landau) equation, ap-pearing in the condensed matter physics (the theory of superfluidity and Bose - Einsteincondensation). It is given by
∆Ψ = κ2(|Ψ|2 − 1)Ψ, (2.42)
for Ψ : Rd → C. In the dimension d = 2 and for G, the group of rotations, O(2), theseequations have O(2)− equivariant solutions, which are labeled by integers n, which aregiven by
Ψ(n)(x) = f (n)(r)einθ, (2.43)
where (r, θ) are the polar coordinates of x ∈ R2 (see Fig. 6). These solutions are calledvortices. (n labels the equivalence classes of the homomorphisms of S1 into U(1).) Theysatisfy
Ψ(n)(R−1α x) = einαΨ(n)(x), (2.44)
where Rα is the contra-clockwise rotation by the angle α ∈ [0, 2π).
s1s1
s2 nt
Figure 6: Portraits of vortices
22 Lectures on Applied PDEs
2.4 Solitons
Symmetries allow us to introduce special classes of time-dependent solutions: stationary,rotating and traveling waves, which sometimes grouped as solitons. We begin with somedefinitions. The family Ts, s ∈ R of bounded operators is called a one-parameter groupiff
T0 = 1; Tt Ts = Tt+s. (2.45)
We define the generator, A, of the group Ts as Aϕ = ∂θTθϕ|θ=0, for all ϕ’s for which theabove derivative exists. The set of all such ϕ’s form the domain, D(A), of A. We have∂tTtu0 = ATtu0, t ≥ 0.
To cover a larger class of equations, we also consider a one-parameter groups of gen-eralized symmetries. We say thatt Ts is a one-parameter generalized symmetry groupof the equation (2.1) iff for each s, Ts is the symmetry in the sense of
F (Tsϕ) = b(s)TsF (ϕ), (2.46)
for some Ggauge−valued function b(s), where Ggauge is the gauge group of F . Since Tsis a one-parameter group, we must have that b(s) satisfies b(s+ t) = b(s)b(t).
Proposition 2.10. The NLS and porous medium maps F (u) := −∆u + κ|u|2u andF (u) := ∆(uγ) have the generalized scaling symmetries u(x) 7→ eθu(eθx) and u(x) 7→eνθu(eθx), with b(θ) := e2θ and b(θ) := e[(γ−1)ν+2]θ, respectively.
Exercise 2.11. Prove the proposition above.
Remark. To covert the scaling transformations u(x, t) 7→ λu(λx, t) and u(x, t) 7→λνu(λx, t) into groups w.r. to additions, rather than multiplication, we set λ := eθ.
Consider evolution equation (2.1). Let Tθ be a one-parameter generalized symmetrygroup of the equation (2.1), that is for each θ, Tθ is the symmetry in the sense of
F (Tθϕ) = b(θ)TθF (ϕ), (2.47)
for some T (Ggauge)−valued function b(θ), where Ggauge is the gauge group of F andT (Ggauge) is its representation on Y . Since Tθ is a one-parameter group, we musthave that b(θ) satisfies b(s+ t) = b(s)b(t), i.e. b(θ) is a one-parameter group in T (Ggauge).
We define the (T -) soliton as a solution to equation (2.1) of the form
usol = Tθϕ, (2.48)
where θ depends on t, θ = θ(t), while ϕ is t−independent.
Lectures on Applied PDEs 23
Theorem 2.12. Let Tθ be a one-parameter generalized symmetry group of equation(2.1) with a generator A. Then (2.1) has a (T -) soliton (2.48) iff there is a constant v,s.t. θ and ϕ satisfy the equations
b(θ)−1∂θ
∂t= v, (2.49)
F (ϕ)− vAϕ = 0. (2.50)
Proof. Substituting u = Tθϕ into (2.1), using (2.47) and multiplying the resulting equationby T−1
λ and dividing by b(θ), we obtain the equation for ϕ and θ:
b(θ)−1∂θ
∂tAϕ = F (ϕ), (2.51)
where A is the generator of the group Tθ. Since the r.h.s. of (2.51) is independent of t,we see that ∂θ
∂tb(θ)−1 must be independent of t as well, say = v, which is equivalent to
equation (2.49). Then (2.51) becomes (2.50).
Note that (2.50) is a PDE and is usually hard to solve, while (2.49) is an ODE for thesoliton parameter θ which can be easily solved as∫ θ
0
b−1(s)ds = vt+ θ0, (2.52)
for some constant θ0. (More generally, (2.49) is an equation on the gauge group Ggauge ofF .) In the simplest case of b(θ) = 1, the equation (2.52) becomes
θ = θ0 + vt, (2.53)
Eq (2.50) is a time-independent equation for ϕ and v. (v is like a nonlinear eigenvalue.)The function ϕ is called the solitary wave profile and v velocity of the soliton.
Considering U(m) gauge group and the groups of translations, rotations and dilata-tions, we arrive at the stationary, traveling wave, rotating and scaling soliton solutionssome of which we studied above. The corresponding generators are i (for U(1)), ∂xj and(x×∇)j and x · ∇+ α, respectively.
We describe this in more detail. To be specific we assume that the space, Y , ofsolutions is a space of functions on Rn.
Stationary waves. The notion of stationary waves is related to the U(1)−gauge symme-try. Let Y be a complex space and consider the one-parameter group of transformationsTαf := eiαf, α ∈ R, on Y . The generator of this transformation is Af = if , the solitarywaves in this case are of the form
u(x, t) = ei(α0+λt)ϕ(x), (2.54)
24 Lectures on Applied PDEs
where ϕ(y) and λ satisfy the equation
F (ϕ) + iλϕ = 0, (2.55)
the nonlinear eigenvalue problem. Such solutions are called the stationary or standingwaves or breathers.
The above can be extended in a straightforward way to Y beong a space of complexfunctions on Rn with values in Cm and the group U(m).
Traveling waves. The traveling waves arise from the translational symmetry w.r.to thegroup of translations,
(Thf)(x) := f(x+ h).
Exercise 2.13. Show that the generator Aj for translations along the coordinate xj isAj = ∂xj .
For the translation invariance, the equations (2.48), (??) and (2.50) become
u(x, t) = ϕ(x− vt), (2.56)
where ϕ(y) and v satisfy the equation
F (ϕ) + v · ∇ϕ = 0. (2.57)
They are called the traveling waves. The function ϕ(y) is called the traveling wave profile.
Rotating solitons. These solitons are associated with the group O(m) of rotations. IfY be a space of functions on Rn with values in Rm, then we can define an action of O(n)on Y by
(TRf)(x) := f(R−1x), R ∈ O(n),
or an action of O(m) on Y by
(TRf)(x) := Rf(x), R ∈ O(m).
Note that the generators of TRf in the first case is −i times the angular momenta
Lj := −i(x×∇)j (2.58)
of Quantum Mechanics. (To have a one parameter group, we can consider U rots : ψ(x)→
ψ(R−1s x), where Rs ≡ Rω
s is the contra-clockwise rotation around the axis along theunit vector ω by the angle s ∈ [0, 2π). The generator of this group is ω · L, whereL = x× (−i∇x).)
Scaling solitons (shrinkers and expanders). Consider the scaling transformation
(Tθf)(x) := eνθf(eθx),
for some ν corresponding to the group R.
Lectures on Applied PDEs 25
Exercise 2.14. Show that the generator of the scaling transformation above is A = x ·∇+ ν.
By Theorem 19 and the exercise above, (2.1) has a solitary wave solution of the formu = Tθϕ, where Tθ is given above and θ depends on t, while ϕ is a t−independent function,iff there is a constant v s.t. the wave profile ϕ solves the equation
F (ϕ)− v(y · ∇+ ν)ϕ = 0. (2.59)
In this case, the soliton parameter θ is given by (2.52). Note that we passed to the variabley = eθ(t)x (the moving (shrinking or expanding) coordinate frame).
Such solutions are called variously the self-similar solution or shrinker or expanders,depending on whether λ(t) grows or decreases. The function ϕ(y) is called the self-similarwave profile.
Consider the porous medium equation (see (2.23))
∂u
∂t= ∆(uγ), (2.60)
for u ≥ 0 and γ > 1 and on Rn. By Proposition 2.10, it has the generalized scalingsymmetry
u(x, t) 7→ eνθu(eθx, t), with b(θ) := eµθ,
where µ := (γ − 1)ν + 2. Hence, we expect to find scaling solitons for this equation. By(2.59), the profile satisfies the equation
∆(φγ)− v(y · ∇+ ν)ϕ = 0. (2.61)
This coincides with equation (2.24) provided α := −vν and β := −v.Next, equation (??) and Proposition 2.10 imply the equation for θ:
1
µ(1− e−µθ) = vt+ θ0, (2.62)
This, after solving for λ = eθ, gives eθ = (c − µvt)−1/µ, where c := 1 − µθ0. The latterequation makes sense only if µv < 0. Since µ > 0, we have to require that v < 0.
Hence, if (2.61) has a solution, then we have the scaling soliton of the form
usol(x, t) = (c− µvt)−ν/µφ((c− µvt)−1/µx), (2.63)
This coincides with self-similar solution (2.22), up to a linear transformation of t, providedα = ν/µ and β = 1/µ. Together with the previous relations α := −vν and β := −v, thisgives v = −1/µ.
26 Lectures on Applied PDEs
Now, α = ν/µ and β = 1/µ satisfy α+ 1 = αγ+ 2β. We know from Exercise 2.5, thatequation (2.61) has a one-parameter family of solutions, φ(y), with φ, ∂yφ→ 0 sufficientlyfast, as |y| → ∞. Moreover, this family is of the form
φ(y) = (b− γ − 1
2γβ|y|2)1/γ−1, (2.64)
where b > 0 (the Barenblatt-Kompaneetz-Zeldovich solutions).
As a second example, we consider the mean curvature flow (MCF) equation of a surfaceS. The mean curvature flow is the family of hypersurfaces S(t) in Rn+1 given by localparametrizations x(·, t) : U → Rn+1 (or immersions, i.e. dx(t) are one-to-one), where Uis an open set in Rn, which satisfy the evolution equation
∂x
∂t= −H(x)ν(x), (2.65)
where H(x) and ν(x) are mean curvature and the outward unit normal vector at x ∈ S(t),respectively. The terms used above are explained in Appendix G.2, for more details andextensions, see [29].
If S is given by a graph, S = graph f , of function f : U → R, where U is an openset in Rn+1 i.e. S := x = (u, f(u)) : u ∈ U, then the MCF equation reduces to (seeSubsection 8.1)
∂f
∂t= −
√|∇f |2 + 1H(f), (2.66)
where H(f) := −div
(∇f√|∇f |2+1
), the mean curvature of S.
The MCF is translationally (Th : f(u, t) → f(u + h, t)), rotationally (TR : f(u, t) →f(R−1u, t)), scaling (Tλ : f(u, t)→ λf(λ−1u, λ−2t)) and gauge (Ts : f(u, t)→ λf(u, t)+s)invariant. Consequently, we can consider translationally, rotationally, scaling and gaugeinvariant solutions.
(ii) Rotationally invariant (‘spherically symmetric’ (equivariant)) solutions:
a) Sphere: upper hemisphere can be given as the graph of the function f =√R2 − |x|2.
Using the formula H(f) := −div
(∇f√|∇f |2+1
), it is easy to compute H(f) = n
Rand
therefore we get R = − nR
which implies R =√R2
0 − 2nt. So this solution shrinksto a point.
b) Cylinder: it is a sphere in in the variables (x1, . . . , xn−1) H(x) = n−1R
and R = −n−1R
which implies R =√R2
0 − 2(n− 1)t.
Lectures on Applied PDEs 27
Exercise 2.15. Show that the MCF equation ( (2.66)) is invariant under the rescalingtransformation
Tλ : f(u, t)→ λf(λ−1u, λ−2t). (2.67)
Transformation (2.67) corresponds to surfaces re-scaled as S → λS. Indeed, droppingthe t−dependence, we have (u, λf(λ−1u)) = λ(λ−1u, f(λ−1u)). For the parametrizationψf (u) = (u, f(u)), this gives
ψTλf (u) = λψf (λ−1u).
We can define translationally, rotationally, scaling and gauge invariant solitons. Here,we consider scaling solitons. First, to covert the scaling transformations like (2.67) intogroups w.r. to additions, rather than multiplication, we set λ := eθ. (As above with(2.67), this transformation corresponds to surfaces re-scaled as S → λS.)
Exercise 2.16. Prove that the map F (f) := −√|∇f |2 + 1H(f) has the generalized scal-
ing symmetry f(x) 7→ eθf(e−θx), with b(θ) := e−2θ.
This leads to the scaling soliton
f(u, t) = λφ(λ−1u), λ depends on t, (2.68)
where φ and λ satisfy the equations√1 + |∇yφ|2H(φ) = a(y∂y − 1)φ, (2.69)
λλ = −a, a is time-independent. (2.70)
(In this case, the surface changes by rescaling S(t) = λ(t)S.) In the non-trivial case, a isa non-zero constant, which implies λ =
√λ0 − 2at. So
i) a > 0⇒ λ→ 0 as t→ T :=λ202a⇒ (2.68) (Sλ) is a shrinker.
ii) a < 0⇒ λ→∞ as t→∞⇒ (2.68) (Sλ) is an expander.
Solution to (2.69): Sphere φ(y) =√R2 − |y|2, with R =
√na.
Combined symmetries. We consider solitons generated by two groups, one a spacegroup, say the group of translations, Th, and another, a gauge group, say the U(1)−gaugegroup, Γα = eiα. (In other words, we allow the soliton to move inside the gauge-equivalentclass of solutions.) Hence we look solutions of the form
u = ΓαThϕ, (2.71)
where h = h(t) and α = α(t), while ϕ is t−independent. To be specific, we consider thenonlinear Schrodinger or Gross - Pitaevskii equation (9.12) and look for solution of the
28 Lectures on Applied PDEs
form (2.71). A simple computation gives α(t) = 12v · x − 1
4|v|2t and h(t) = x − vt − x0,
which shows that (9.12) has the solutions of the form
u(x, t) := ei(12v·x− 1
4|v|2t−λt)φ(x− vt− x0) (2.72)
where e−iλtφ(x) is a stationary solution to (9.12). This solution is obviously a travelingwave. We have already encountered it above, in (2.19).
Put differently, the solution (2.19) comes from applying the Galilean symmetry trans-formation (2.41) to the stationary solution e−iλtφ(x).
Moving reference frame. We transform the equation to the moving frame by settingu = Tλw. Then w satisfies the equation
∂w
∂t= F (w) + vAw. (2.73)
Now, if u = Tλϕ is a solitary wave for (2.1), then the solitary wave profile ϕ is a stationarysolution to the new equation (2.73). Thus one can apply general definitions and resultsfor the static solutions to solitary waves as well.
For example consider the scaling transformations. Passing to the variables y = λ(t)xis the same as using the frame of reference expanding or shrinking together with thetraveling wave. In this variable, the traveling wave is a stationary solution to the newequation
∂w
∂t= F (w) + v · (y · ∇+ α)w. (2.74)
Remark: Manifolds of static solutions and traveling waves. Let the dynamicalsystem (2.1) have the symmetry group G which is a Lie group and which acts on the spaceof solution via a representation g → Tg (i.e. a Lie group homomorphism T : G→ GL(X)).If (2.1) has a static solution u∗, then it has a manifold of static solutions
M = Tgu∗ : g ∈ G.
Indeed, if u∗ is a static solution to (2.1), then, due to (2.31) with T = Tg, so is Tgu∗.For example, since the Allen-Cahn equation (2.2) is translational invariant and has
the kink and anti-kink solutions, (2.4) and its negative, the functions χ±ε ( (x−x0)·e√2
) ≡±χ( (x−x0)·e√
2ε), are also solutions for any x0 ∈ R. These are kink and antikink solutions
centered at a. Thus the equation (2.2) has a family (manifold) of stationary solutions,
χ±ε ( (x−x0)·e√2
) ≡ ±χ( (x−x0)·e√2ε
), x0 ∈ Rn, e ∈ Sn−1.The same of course is true for traveling wave, as they can be reduced to static solution
when the equation is considered in the moving reference frame.
Lectures on Applied PDEs 29
2.5 Local gauge symmetry and vortex lattices
A more complicated example is presented by the Ginzburg-Landau equations of super-conductivity (and particle physics). These equations describe equilibrium configurationsof superconductors, and of the U(1) Higgs model from particle physics, and are given by
∆AΨ = κ2(|Ψ|2 − 1)Ψ, (2.75a)
curl∗ curlA = Im(Ψ∇AΨ). (2.75b)
for u := (Ψ, A) : R2 → C × R2, ∇A = ∇ − iA, and ∆A = ∇2A, the covariant derivative
and covariant Laplacian, respectively.
The Ginzburg-Landau equations lie in the foundation of the macroscopic theory ofsuperconductivity. Here Ψ(x) is called the order parameter and |ψ(x)|2 gives the localdensity of (Cooper pairs of) superconducting electrons, and the vector field A is themagnetic potential, so that B(x) := curlA(x) = the magnetic field. The vector quantityJ(x) := Im(Ψ∇AΨ) = the superconducting current. The parameter κ > 0 depends onthe material properties of the superconductor.
These equations form also a fundamental block of particle physics - the Abelian-Higgsmodel. Here, Ψ and A are the Higgs and U(1) gauge (electro-magnetic) fields, respectively.
Geometrically, one can think of A as a connection on the principal U(1)-bundle Rn ×U(1), n = 2, 3.
These equations are discussed in detail in Section 14. Here we discuss a key and veryinteresting class of solutions - vortex lattice solutions - the discovery of which by A.A.Abrikosov was recognized with a Nobel prize in physics.
To begin with we mention that the Ginzburg-Landau equations (2.75) admit, besidesthe translation and rotation symmetry, (2.34) - (2.35), also
Gauge symmetry: for any sufficiently regular function η : R2 → R,
Equivalence classes of solutions, gauge conditions.
Exercise 2.17. Prove that the above transformations are symmetries of the Ginzburg-Landau equations (2.75).
Thus the set of all solutions of the Ginzburg-Landau equations can be split into equiv-alence classes of solutions related by gauge transformations. A condition which pick asubclass of each equivalence class is called the gauge condition. An example of a gaugecondition is divA = 0. This can be also arranged: if divA 6= 0 we can always find a gaugeη s.t. div(A+∇η) = 0, namely, we take η solving the equation −∆η = divA.
One of the analytically interesting aspects of the Ginzburg-Landau theory is the factthat, because of the gauge transformations, the symmetry group is infinite-dimensional.
30 Lectures on Applied PDEs
Abrikosov lattices. If G is the group of lattice translations for a lattice L, then we callthe corresponding equivariant solution a lattice, or L-gauge-periodic, solution. Explicitly,
where gs : R2 → R is, in general, a multi-valued differentiable function, with differencesof values at the same point ∈ 2πZ, and satisfying
gs+t(x)− gs(x+ t)− gt(x) ∈ 2πZ. (2.78)
The latter condition on gs can be derived by computing Ψ(x+s+ t) in two different ways.Note that for Abrikosov lattices, all physical properties, the density of superconducting
pairs of electrons, ns := |Ψ|2, the magnetic field, B := curlA, and the current density,J := Im(Ψ∇AΨ), are doubly-periodic with respect to some lattice L.
One can also show the converse: a state (Ψ, A) ∈ H1loc(R2;C)×H1
loc(R2;R2) for whichthe physical properties above are doubly-periodic with respect to some lattice L (we callsuch a state the (generalized) Abrikosov (vortex) lattice) is a L-gauge-periodic state.
Lectures on Applied PDEs 31
3 Fourier transform and partial differential equations
In this section, we describe one of the most powerful tools in analysis – the Fouriertransform. This transform allows us to analyze a fine structure of functions and to solvedifferential equations. The Fourier transform takes functions of time to functions offrequencies, functions of coordinates to functions of momenta, and vice versa.
3.1 Definitions and properties
Initially, we define the Fourier transform on the Schwartz space S(Rn) = S:
S = f ∈ C∞(Rn) : 〈x〉N |∂αf(x)| is bounded ∀N and ∀α, (3.1)
where 〈x〉 = (1 + |x|2)1/2 and α = (α1, ..., αn), with αj non-negative integers, ∂α :=∏nj=1 ∂
αjxj and |α| = ∑n
i=1 αj. On S, we define the Fourier transform (FT) F : f 7→ f by
f(k) := (2π)−n/2∫f(x)e−ik·xdx. (3.2)
Define also the inverse Fourier transform (IFT) of f(k) as
f(x) := (2π)−n/2∫f(k)eix·kdk. (3.3)
Some key properties of the Fourier transform are collected in the following
Theorem 3.1. Assume f, g ∈ S(Rn). Then we have:
(a) The FT and IFT are linear maps;
(b) (−i∂)αf 7→ kαf , and xαf 7→ (−i∂)αf ;
(c) fg 7→ (2π)−n/2f ∗ g, and f ∗ g 7→ (2π)n/2f g;
(d) (f ) = f = (f ) ;
(e)∫f g =
∫fg.
Properties (a) - (e) hold (possibly, with signs changed in (b)) also whenˆ is replaced by .
Exercise 3.2. Prove this theorem (see Appendix C).
32 Lectures on Applied PDEs
(a) is obvious. We give a formal proof of the first relation in (b) and the secondrelation in (c). Integrating by parts, we compute
−i(∂xjf ) (k) = (2π)−n/2∫
(−i)∂xjf(x)e−ik·xdx
= (2π)−n/2∫f(x)i∂xje
−ik·xdx
= kj f(k).
This gives the first relation in (b).Now we prove the second relation in (c). Using e−ik·x = e−ik·(x−y)e−ik·y and changing
the variable of integration as x′ = x− y, we obtain
f ∗ g (k) := (2π)−n/2∫e−ik·x(
∫f(x− y)g(y)dy)dx
= (2π)−n/2∫
(
∫e−ik·(x−y)f(x− y)dx) e−ik·yg(y)dy
= (2π)n/2f(k)g(k).
Exercise 3.3. Derive the first relation in (c) from the second one and (d).
The two key functions appearing often in applications as well as in theoretical researchare the gaussian e−x·Ax/2 and the power |x|−α. Fortunately, their transforms can becomputed explicitly (see Appendix C):
F : e−x·Ax/2 7→ (detA)−1/2e−k·A−1k/2, (3.4)
F : |x|−α 7→cn,α|k|−n+α if α 6= n,cn,n ln |k| if α = n.
(3.5)
The coefficients cn,α are given for α = 2 by
cn,2 =
((2− n)σn−1)−1 for n 6= 2,−(σn−1)−1 = −(2π)−1 for n = 2,
(3.6)
where σn is the volume of the n–dimensional unit sphere Sn = x ∈ Rn+1 : |x| = 1.Though it is easy to compute the Fourier transform of |x|−α, it is not easy to justify it.Indeed, the function |x|−α is rather singular and definitely does not belong to S(Rn).
Remarks. 1) Clearly, the FT and IFT are defined on the space L1(Rn) (see AppendixA.3 for the definition of the Lp-spaces).
2) A special case of Property (e) is the Plancherel theorem:
Lectures on Applied PDEs 33
∫|f |2 =
∫|f |2. (3.7)
Using the L2-norm, this becomes ‖f‖L2 = ‖f‖L2 , i.e. the FT and its inverse are isometries.Using this one can extend the FT, by the continuity, as an isometry to the space L2(Rn).
3) Using the L2-inner product, Property (e) can be written as 〈f , g〉L2 = 〈f, g〉L2 , i.e.the FT is a unitary operator.
3.2 Application of Fourier transform to partial differential equa-tions
Our goal in this section is to apply the Fourier transform in order to solve elementary butvery basic partial differential equations (PDE’s).
The Poisson equation on Rn:−∆u = f, (3.8)
where u : Rn → R is an unknown function, f : Rn → R is a given function, and ∆ is theLaplace operator (the Laplacian):
∆u :=n∑j=1
∂2u
∂x2j
.
The Poisson equation first appeared in the problem of determining the electric potentialu(x), created by a given charge distribution ρ(x) = f(x)/(4π). Since then, it came up invarious fields of mathematics, physics, engineering, chemistry, biology and economics.
In order to solve the Poisson equation, we apply the Fourier transform to both sidesof (3.8) to obtain:
|k|2u(k) = f(k).
This equation can be easily solved: u = f/|k|2. We can now apply the inverse Fouriertransform to the last equality to get
u = g ∗ f, where φ(k) = |k|−2. (3.9)
But the inverse Fourier transform of φ(k) = |k|−2 is known:
φ(x) =
[(2− n)σn−1]−1|x|−n+2 if n 6= 2[2π]−1 ln |x| if n = 2,
where σn is the volume of the unit–sphere Sn = x ∈ Rn+1 : |x| = 1 in dimension n.
34 Lectures on Applied PDEs
Explicitly, (3.9) can be written as
u(x) = [(2− n)σn−1]−1
∫f(y)
|x− y|n−2dy,
for n 6= 2, and similarly for n = 2. In particular, for n = 3, we have the celebratedNewton formula
u(x) = − 1
4π
∫f(y)
|x− y|dy.
Of course, the functions appearing in the above derivation are not necessarily from theSchwartz space S and therefore these manipulations must be justified. We leave this asan exercise, while proceeding in a similar fashion with other equations.
The heat equation on Rn:
∂u
∂t= ∆u and u|t=0 = u0, (3.10)
where u : Rnx × R+
t → R is an unknown function, and u0 : Rn → R is a given initialcondition. Problem (3.10) is called an initial value problem. It first appeared in the theoryof heat diffusion. In that case, u0(x) is a given distribution of temperature in a body attime t = 0, and u(x, t) is the unknown temperature–distribution at time t. Presently, thisequation appears in various fields of science, including mathematical modeling of stockmarkets.
As before, we apply the Fourier transform to (3.10) and solve the resulting equation
∂u
∂t= −|k|2u and u|t=0 = u0
to get u = e−|k|2tu0. Applying the inverse Fourier transform, and using that (e−|k|
2t) =(4πt)−n/2e−|x|
2/(4t), we obtainu = g√
2t∗ u0, (3.11)
where gs(x) = s−ng(x/s) with g(x) = (2π)−n/2e−|x|2/2. In particular, u→ u0 as t→ 0, as
it should be.
The Schrodinger equation on Rn:
i∂ψ
∂t= −∆ψ and ψ|t=0 = ψ0. (3.12)
This is an initial value problem for the unknown function ψ : Rnx × R+
t 7→ C. Equation(3.12) describes the motion of a free quantum particle. Proceeding as with the heatequation, we obtain
ψ = g√2it∗ ψ0, (3.13)
Lectures on Applied PDEs 35
where, recall, gs(x) = s−nϕ(x/s) with g(x) = (2πt)−n/2e−|x|2/(2t). Observe that this
formula can be obtained from (3.11) by performing the substitution t→ t/i.
Exercise 3.4. Derive equation (3.13) using the Fourier transform.
The wave equation on Rn:
∂2u
∂t2= ∆u with u|t=0 = u0 and ∂tu|t=0 = u1. (3.14)
This is a second order equation in time and consequently, it has two initial conditionsu0 and u1. The wave equation (3.14) describes various wave phenomena: propagation oflight and sound, oscillations of strings, etc. Proceeding as with the heat equation, we find
u = ∂tWt ∗ u0 +Wt ∗ u1, (3.15)
where Wt(x) is the inverse Fourier transform of the function sin(|k|t)/|k|. The latter canbe computed explicitly for n = 1, 2, 3:
Wt(x) =
12χρ2≥0 for n = 1,
(2π)−1ρ−1χρ2≥0 for n = 2,(2π)−1δ(ρ2) for n = 3,
where ρ2 := t2 − |x|2, and χρ2≥0 stands for the characteristic function of the set (x, t) ∈R3+1 : ρ2 ≥ 0, i.e.
χρ2≥0 =
1 if ρ2 ≥ 0,0 otherwise,
and δ(x) is the Dirac δ–function, a generalized function, or distribution.Thus the dependence of W on x and t comes through the combination ρ2 = t2 − |x|2,
which is the Minkowski–distance in space–time, playing a crucial role in relativity. Observethat χρ≥0 = χ|x|≤t and δ(ρ2) = (2t)−1δ(t− |x|).Exercise 3.5. Prove (3.15), and find Wt(x) for n = 1.
We examine closer the special case when n = 3 and u0 = 0. Then we get
u = Wt ∗ u1 ≡1
4πt
∫δ(|x− y| − t)u1(y)dy
=1
4πt
∫S(x,t)
u1(y)dS(y)
=t
4π
∫S(0,1)
u1(x+ tz)dS(z),
where S(x, t) = y ∈ R3 : |y − x| = t is a sphere of radius t centered at x. We see thatonly the initial condition evaluated on the sphere S(x, t) matters in order to determinethe solution at time t and at position x. This is called the Huygens’ principle.
36 Lectures on Applied PDEs
Discussion. We can write the equations considered above as the linear initial valueproblem
∂tu = Au, u|t=0 = u0, (3.16)
where A = ∆ for the heat equation and A = i∆ for the Schrodinger equation. For thewave equation, ∂2v
∂t2= ∆v, we have (3.16) with u = (v, w) and
A =
(0 1∆ 0
). (3.17)
We have shown above that equation (3.16) has a unique solution for every reasonableu0. We denote this solution by u(t) = etAu0.
The family U(t) = etA has the following properties (see Appendix D):
(a) U(t) are bounded ∀t ≥ 0,
(b) U(0) = 1 and U(t+ s) = U(t)U(s),
(c) ∂tU(t) = AU(t) = U(t)A.
Exercise 3.6. Prove relations (a) - (c).
Remarks. 1) A family of operators, U(t), satisfying (a) - (c) is called the evolutionsemigroup or a propagator. Evolution semigroups are discussed in Appendix D.
2) For A = ∆, the family et∆ is also denoted as Pt = et∆. It, or sometimes its integralkernel, is called the heat kernel and it plays an important role in the theory of stochasticprocesses.
3) The semi-group Pt = et∆ has the following properties
(i) Pt is positivity improving, i.e. if f ≥ 0, then Ptf > 0,
(ii) Pt1 = 1.
Semi-groups with such properties are called the stochastic semigroups.4) U(t) = ei∆t has properties (a) - (c) for all t ∈ R and for each t ∈ R, it is an isometry,
‖U(t)u‖L2 = ‖u‖L2 . In fact, U(t) = ei∆t is unitary:
〈U(t)f, U(t)g〉L2 = 〈f, g〉L2 .
Exercise 3.7. Prove statements 3) and 4).
Exercise 3.8. Prove that the energy E(u) = 12
∫|∇u|2 is conserved (∂tE(ut) = 0 or
E(ut) = E(u0)) for the SE and WE and decreases (∂tE(ut) < 0) for the HE. Here ut isthe solution of the corresponding equation at time t.
Lectures on Applied PDEs 37
4 Local Existence for Key Evolution Equations
We address the problem of the short time existence of solutions for key evolution PDEs:the reaction-diffusion (nonlinear heat), Hartree and nonlinear Schrodinger equations. Inall these case, the equation of interest can be written as
∂tu = Au+ f(u), u|t=0 = u0. (4.1)
where A is a linear operator: A = ∆ for the reaction-diffusion (nonlinear heat) equationand A = i∆ for the Hartree and nonlinear Schrodinger equations, and f is a nonlinearity:f(u) is a real function on R satisfying f(0) = f ′(0) = 0 for the reaction-diffusion (nonlinearheat) equation and f(u) = (v ∗ |u|2)u, where v is a given function (potential), and f(u) =|u|2u for the Hartree and nonlinear Schrodinger equations, respectively. We will call fthe nonlinearity.
If f(u) = 0, then we arrive at the linear initial value problem
∂tu = Au, u|t=0 = u0, (4.2)
which we tackled in Subsection 3.2. We have shown there that this equation has a uniquesolution for every reasonable u0. This solution is of the form
u = g√2τ∗ u0, where gs(x) = s−ng(x/s), with g(x) = (2π)−n/2e−|x|
2/2,
and τ = t, if A = ∆, and τ = it, if A = i∆. We denote this solution by u(t) = etAu0.
4.1 Reduction to a fixed point problem
Duhamel principle. Consider the inhomogeneous linear initial value problem
∂tu = Au+ f, u|t=0 = u0, (4.3)
where A is either ∆ or i∆ and f = f(t) is a given function of x and t.
Proposition 4.1 (Duhamel principle). Provided the integral below exists, the solution,u(t), of (4.3), is given by
u(t) = etAu0 +
∫ t
0
e(t−s)Af(s) ds. (4.4)
In opposite direction, the family u(t) given by (4.4), which is differentiable in t and is inthe domain of the operator A, satisfies (4.3).
Exercise 4.2. Prove this proposition. (Hint: For the first part, verify directly that (4.4)satisfies (4.3). If v(t) := e−tAu(t) is well-defined, then one can prove the proposition byshowing that it satisfies the equation
∂tv = f, v|t=0 = u0
and then solving this equation by using the Fundamental Theorem of calculus.)
38 Lectures on Applied PDEs
Mild (weak) solutions. Consider the initial value problem (4.1). We apply (4.4) to(4.1) to obtain
u(t) = etAu0 +
∫ t
0
e(t−s)Af(u(s)) ds. (4.5)
If u(t) solves (4.1), then it also solves the equation (4.5). Conversely, if u(t) solves (4.5)and is differentiable in t and twice differentiable in x, then it solves the equation (4.1).
If u(t) solves (4.5), but we do not know whether it is differentiable or not, we call u(t)a mild (or weak) solution to (4.1).
Remark. There are several definitions of weak solutions depending on the methodsused. The above definition is adapted to the fixed point formulation of the local existenceproblem for evolutions PDEs. Another common definition appears in the variationalapproach (see Section 6 below) and states that u is a weak solution iff it satisfies theequation
−∫ ∫
u∂tgddxdt =
∫ ∫ (uA∗g + f(u)g
)ddxdt (4.6)
for all g’s of compact support and which are differentiable in t and are in the domain of(formally) adjoint operator A∗.
Fixed point problem. Eq (4.5) can be written as the equation u = H(u), where
H(u)(t) := etAu0 +
∫ t
0
e(t−s)Af(u)(s) ds, (4.7)
called the fixed point equation or the fixed point problem. A solution of such an equationis called a fixed point. In our next step we learn how to solve fixed point equations.
4.2 The contraction mapping principle
A key to dealing with a large class of equations is to reduce them to a fixed point problem,
H(u) = u, (4.8)
for some map H and then use one of several fixed point theorem stating existence anduniqueness of solutions of the latter problem. (The equation (4.8) is called the fixedpoint equation and its solution, a fixed point of the map H.) The most useful among thesetheorems is the Banach contraction mapping principle which we now formulate and prove.
Let X be a Banach space. Denote by d(u, v) = ‖u − v‖ the distance between thevectors u and v. We remark that actually all we need for the next theorem is that X isa complete metric space (i. e. it does not have to have a norm). Let B be a closed setin X. A map H : B → B is called a strict contraction if and only if there is a numberα ∈ (0, 1) s.t.
d(H(u), H(v)) ≤ α d(u, v), ∀u, v ∈ B.
Lectures on Applied PDEs 39
Theorem 4.3 (the contraction mapping principle). If H is a strict contraction in B,then H has a unique fixed point in B.
Proof. We use the method of successive approximations to solve the equation u = H(u).Pick some u0 ∈ B and define u1 = H(u0), . . . , un = H(un−1). Since H is a contraction,un ∈ B.
We claim that un is a Cauchy sequence in X. Indeed, let n ≥ m, then
d(un, um) ≤ αmd(un−m, u0).
Taking here n = m + 1, we find d(um+1, um) ≤ αmd(u1, u0). Next, by the triangleinequality (i.e. d(v, u) ≤ d(v, w) + d(w, u), ∀w ∈ X), we obtain
Applying d(um+1, um) ≤ αmd(u1, u0) to each term on the r.h.s. gives
d(uk, u0) ≤(αk−1 + αk−2 + . . .+ 1
)d(u1, u0)
≤ 1
1− αd(u1, u0).
(If B is a bounded set in X, then we do not need the step above, since in this case ‖uk‖are uniformly bounded.) The last two inequalities imply that
d(un, um) ≤ αm
1− αd(u1, u0)→ 0 as m,n→∞.
Thus un is a Cauchy sequence in X. Now since X is complete, un ∈ B and B isclosed, there is a u∗ ∈ B s.t. un → u∗. Since d(H(un), H(u∗)) ≤ αd(un, u∗) → 0, wehave also that H(un) → H(u∗). This and the equation un = H(un−1) imply that thelimit u∗ satisfies the equation u∗ = H(u∗). This demonstrates existence of a fixed pointin B, and we finish the proof by showing its uniqueness. Suppose that H(u∗) = u∗, andH(v∗) = v∗. Then we have d(v∗, u∗) = d(H(v∗), H(u∗)) ≤ αd(v∗, u∗). Since α ∈ (0, 1),this gives d(v∗, u∗) = 0 and so v∗ = u∗.
Remark. As was mentioned above, there are many fixed point theorems and the Banachcontraction mapping principle is one of these, albeit the most useful one.
In the next two subsections, we use the contraction mapping principle to prove localexistence of solutions for the reaction-diffusion (nonlinear heat), Hartree and nonlinearSchrodinger equations.
40 Lectures on Applied PDEs
4.3 Local existence for the nonlinear heat equation
We show local existence to the initial value problem for the nonlinear heat (or reaction-diffusion) equation:
∂u
∂t= ∆u+ f(u), u|t=0 = u0, (4.9)
on Rn. To fix ideas, for the nonlinearity f(u) we take f(u) = λ|u|p−1u, with 1 < p <∞,and bounded initial conditions u0 on Rn. The special case of equation (4.9) withoutnonlinearity first appeared in the theory of heat diffusion. In that case, u0(x) is agiven distribution of temperature in a body at time t = 0, and u(x, t) is the unknowntemperature–distribution at time t. Presently, this equation appears in various fields ofscience, including the theory of chemical reactions and mathematical modeling of stockmarkets. Similar equations appear in the motion by mean curvature flow, vortex dynamicsin superconductors, surface diffusion and chemotaxis.
For the next result, let L∞ be defined as (see Appendix A.3 for more details)
L∞(Ω) := f : Ω→ C | f is measurable, and ess sup |f | <∞, (4.10)
where, recall that ess sup |f | := infsup |g| : g = f a.e., with the norm defined as
‖f‖∞ := ess sup|f |. (4.11)
( L∞ is one of the standard Lp-spaces. Strictly speaking, elements of Lp(Ω) are equivalenceclasses of measurable functions: two functions define the same elements of Lp(Ω) if theydiffer only on a set of measure 0, for more details see Appendix A.3.) We often use theabbreviations Lp and ‖v‖p for Lp(Ω) and ‖v‖Lp .
Furthermore, for T > 0, we define the Banach space XT := C([0, T ], X), where X :=L∞(Rn).
Theorem 4.4. Consider initial value problem (4.9) for the nonlinear heat equation withf(u) = λ|u|p−1u, 1 < p <∞ and u0 ∈ L∞(Rn) and let
T := cp/(|λ|‖u0‖p−1∞ ), cp :=
1
p
(1− 1
p
)p−1.
Then (4.9) has a mild solution u ∈ C([0, T ], L∞), satisfying ‖u‖XT ≤ R, where R :=‖u0‖∞1− 1
p
> ‖u0‖∞, and unique in the ball u ∈ L∞(Rn) : ‖u‖XT ≤ R.
Proof. Using Duhamel’s principle, Eq (4.9) can be written as the fixed point equationu = H(u), where
H(u)(t) := et∆u0 +
∫ t
0
e(t−s)∆f(u)(s) ds (4.12)
Lectures on Applied PDEs 41
and we have written f(u)(s) for f(u(s)). The proof of existence and uniqueness will followif we can show that the map H has a unique fixed point in the ball
BR := u ∈ XT , ‖u‖XT ≤ R,for some R > 0. We prove this statement via the contraction mapping principle.
We begin by proving that there is R > 0 s.t. H is a well-defined map from BR to BR.First, we show the estimate ∥∥et∆u∥∥
X≤ ‖u‖X (4.13)
We have shown above that the operator et∆ has the integral kernel pt(x, y), t > 0, i.e.(et∆u)(x) =
∫pt(x, y)u(y) dy, with the following properties: pt(x, y) > 0 and
∫pt(x, y) dy =
1. Using these properties, we obtain the estimate (4.13).Next, the elementary bound ||u|p−1u| ≤ Rp, for |u| ≤ R, shows that, if t < T and
u ∈ BR, thensup‖w‖Y ≤R
‖f(w)‖X = sup|w|≤R
|f(w)| ≤ |λ|Rp (4.14)
(remember that X := L∞(Rn)), which, together with the estimate (4.13), gives∥∥∥∥∫ t
0
e(t−s)Af(u)(s) ds
∥∥∥∥XT
≤ supt≤T
∫ t
0
‖f(u)(s)‖X ds ≤ T |λ|Rp. (4.15)
Estimates (4.13) and (4.15) and the definition (4.12) of the map H imply that H : BR →BR, provided
‖u0‖X + T |λ|Rp ≤ R.
Now, we prove that H : BR → BR is a strict contraction. Recalling the definition fand using the elementary bound
||u1|p−1u1 − |u2|p−1u2| ≤ pRp−1|u1 − u2|,for |u|, |u1|, |u2| ≤ R, we obtain, for u1, u2 ∈ BR,
sup‖w1‖X ,‖w2‖X≤R
‖f(w1)− f(w2)‖X / ‖w1 − w2‖X ≤ |λ|pRp−1, (4.16)
which, together with the definitions of H and ‖u‖X and estimate (4.13), gives
‖H(u1)−H(u2)‖XT ≤ supt≤T
∫ t
0
‖f(u1)(s)− f(u2)(s)‖X ds
≤ T |λ|pRp−1‖u1 − u2‖XT .Therefore, if |λ|pRp−1T < 1, then H is a strict contraction in BR. We see that theinequalities
‖u0‖X + |λ|TRp ≤ R and pRp−1|λ|T < 1
are satisfied if we choose R so that(pRp−1
)−1= (R−‖u0‖X)R−p, i.e. R := ‖u0‖X
1− 1p
> ‖u0‖X ,
and T := 1p(1− 1
p)p−1/(|λ|‖u0‖∂−1
X ). This gives the existence and uniqueness of initial value
problem (4.9) for the stated R and T , as well as the estimate on the solution.
42 Lectures on Applied PDEs
4.4 Local existence for the Hartree equation
In this subsection we show local existence to the initial value problem for the Hartreeequation
i∂u
∂t= −∆u+ (v ∗ |u|2)u, u|t=0 = u0, (4.17)
on Rn, where, as usual v∗f is the convolution of two functions, v∗g(x) :=∫v(x−y)g(y)dy.
We assume that v is a ’nice’ function, i.e. sufficiently smooth and fast decaying at infinity.Equation (4.17) arises in the problem in quantum physics of many-body systems.
Due to the physical interpretation of (4.17), we consider mild solutions in the spaceL2(Rn). (See Appendix A.3 for a discussion of Lp− spaces.)
Theorem 4.5. Assume v ∈ L∞. Then, for any u0 ∈ L2(Rn), the Hartree equation (4.17)
has a unique mild solution u ∈ C([0, T ], L2), where T = 4(27‖u0‖2
2‖v‖∞)−1
, satisfying‖u‖C([0,T ],L2) ≤ 3
2‖u0‖2.
Proof. We proceed as in the proof of Theorem 4.4 for (4.9), but with Y = L2(Rn). Letwt := eit∆w. Passing to the Fourier transform and using the Plancherel theorem, weobtain ‖wt‖2 = ‖wt‖2. Next, since wt = e−i|k|
2tw, we have ‖wt‖2 = ‖w‖2 = ‖w‖2, whichimplies the estimate ∥∥eit∆w∥∥
2= ‖w‖2, (4.18)
uniformly in t. Next, using the elementary inequality ‖v ∗ g‖p ≤ ‖v‖p‖g‖1, ∀p ≥ 1, weobtain ∥∥(v ∗ |u|2)u
∥∥2≤ ‖v‖∞‖u‖3
2.
Moreover, using the triangle inequality |(v∗|u1|2)u1−(v∗|u2|2)u2| ≤ |(v∗(|u1|2−|u2|2))u1|+|(v ∗ |u2|2)(u1 − u2)| and again the above inequality, we find∥∥(v ∗ |u1|2)u1 − (v ∗ |u2|2)u2
∥∥2≤ 3‖v‖∞(max
i‖ui‖2)2‖u1 − u2‖2.
Now, proceeding exactly as in the proof of Theorem 4.4 for (4.9), we arrive at the state-ment of the theorem.
Exercise 4.6. (a) Give a complete proof of Theorem 4.5. (b) Show the local existenceproperty for the Hartree equation, (4.17), in the Sobolev spaces Hs(Rn), s ≥ 0 (seeAppendix A.6).
Exercise 4.7. Extend the above theorem from −∆ to an operator A =∑
ij ∂xiaij∂xj ,where aij is a smooth, symmetric real strictly positive definite matrix (in particular,∑
ij ξiaijξj ≥ δ∑
j |ξj|2 for some δ > 0).
Lectures on Applied PDEs 43
4.5 Local existence for the nonlinear Schrodinger (Gross-Pitaevskii)equation.
Consider the initial value problem for the nonlinear Schrodinger (or Gross-Pitaevskii)equation:
i∂ψ
∂t= −∆ψ + λ|ψ|p−1ψ, ψ|t=0 = ψ0, (4.19)
with an initial condition ψ0 ∈ Hs(Rn). We assume p > 1 and s > 0. Equation (4.19)arises in nonlinear optics, plasma physics, theory of water waves and in condensed matterphysics.
Proceeding as above, we find
Theorem 4.8. Let p be an odd integer. Then there are constants c, c′ > 0, s.t, for anyψ0 ∈ Hs(Rn), with s > n/2, the nonlinear Schrodinger equation, (4.19) has a uniquesolution ψ ∈ C([0, T ], Hs), where T = c/‖ψ0‖2
Hs, satisfying ‖ψ‖C([0,T ],Hs) ≤ c′‖ψ0‖Hs.
Exercise 4.9. Prove the above theorem. (Hint: Use Sobolev embedding theorems, e.g.Hs(Rn) ⊂ L∞(Rn) for s > n/2, so that products of Hs functions are again Hs functions,i.e. Hs is an algebra.)
Techniques developed above apply to the generalized nonlinear Schrodinger equation(gNLS) (cf. (4.19)),
i∂tψ = −∆xψ + g(|ψ|2)ψ, (4.20)
where g(u) is a real function. We assume that f(ψ) = g(|ψ|2)ψ satisfies the conditions
|f (k)(ψ)| . |ψ|p−k, ∀k = 0, ..., s+ 1, (4.21)
with p < 1 + 4d.
Discussion. 1) Notice the difference in behaviour between the heat kernel Pt := et∆ andthe propagator Ut := eit∆. The family et∆ is a one parameter semi-group, which is welldefined on the entire space, say Lp, only for t ≥ 0, while eit∆ is a one parameter group,defined for all t.
While∥∥et∆u∥∥
Lp< ‖u‖Lp , for u not identical 1, the characteristic property of the
propagator Ut is its unitarity. In particular it preserves the L2− inner product, 〈u, v〉 :=∫uvdx: 〈u, Utv〉 = 〈u, v〉.
2) We can also show that the solutions u of the above equations depend continuouslyon the initial condition u0.
44 Lectures on Applied PDEs
4.6 Classical solutions
One would like to show that mild (weak) solutions, i.e. solutions of Eq (4.5) in C([0, T ], X)are in fact classical ones of (4.1), i.e. the are once differentiable in t and twice in x inappropriate sense. This is done by ‘bootstrapping’ estimates. We summarize the resultsin
Proposition 4.10. The local mild solutions of the NLH, Hartree and NLS equations (see(4.1), with A and f described after this equation), i.e. solutions of Eq (4.5) in C([0, T ], X),where X is the corresponding Banach space, are in fact classical ones of (4.1).
Sketch of the proof of the NLH equation, (4.9). For the NLH equation, (4.9), X := L∞(Rn)and we can start, for instance, by showing that u ∈ C([0, T ],W 1,∞(Rn)), where, recall,
W s,p(Rn) := u ∈ Lp(Rn) : ∂αu ∀α, |α| ≤ s,the Sobolev space. Differentiating (4.5), where A = ∆, we find
∂u(t) = ∂et∆u0 +
∫ t
0
∂e(t−s)∆f(u(s)) ds, (4.22)
where ∂ ≡ ∂xj for some j. Using the explicit integral kernel representation (3.11) of
et∆/2u, we find (∂et∆/2f)(x) =∫∂xjpt(x, y)f(y)dy, from which we estimate
‖∂et∆/2‖L∞→L∞ . t−1/2, (4.23)
Using this estimate, one can show easily that ∂u ∈ C([0, T ], L∞(Rn)) and therefore u ∈C([0, T ],W 1,∞(Rn)). Now, we can prove that u(t) is twice differentiable in x and oncedifferentiable in t. For example for x, we have
∂2u(t) = ∂2et∆u0 +
∫ t
0
∂e(t−s)∆∂f(u(s)) ds, (4.24)
For the first term on the r.h.s. we use the estimate
‖∂2et∆/2‖L∞→L∞ . t−1, (4.25)
is obtained similarly to the one in (4.23) and for the term on the r.h.s., we estimate
which together with the relation (4.24) and the estimate (4.25) gives
‖∂2u‖C([0,T ],L∞) . t−1‖u0‖L∞+ ‖u‖p−1
C([0,T ],L∞)‖∂u(s)‖C([0,T ],W 1,∞), (4.34)
as desired.
Exercise 4.11. Prove estimates (4.23) and (4.25).
5 Global Existence
5.1 A priori estimates
Let X and Y be Banach spaces and U , an open set in X. Consider the abstract nonlinearevolution equation
∂tu = F (u), u|t=0 = u0, (5.1)
where F is a map defined from U to Y . We assume
(LE) (Local existence) There is a positive, monotonically decreasing function c on R+ :=(0,∞), s.t. for any u0 ∈ U , (5.1) has a unique solution in C([0, T ], U), with T =c(‖u0‖X).
(AE) (A priori estimate) There is a positive function e on R+ := (0,∞), s.t. e(s) ≥ s andany solution u = u(t) of (5.1) satisfies the estimate
‖u(t)‖X ≤ e(‖u0‖X).
The estimate above is called a priori estimate. It allows one to iterate the local existenceargument and to obtain the existence on the infinite time interval, i.e. the global existence:
46 Lectures on Applied PDEs
Proposition 5.1. Assume the (LE) and (AE). Then (5.1) has a global solution for everyinitial condition u(0) = u0 ∈ X.
Proof. Define T := c(e(‖u0‖X)). Since c(‖u0‖X) ≥ c(e(‖u0‖X)) = T , by Condition (LE),(5.1) has a unique solution, u(t) ≡ u1(t), on an interval [0, T ]. Now consider (5.1) with the
new initial condition u(1)0 = u(3
4T ). Since c(‖u(1)
0 ‖X) ≥ c(e(‖u0‖Y )) = T , it has a uniquesolution, u(t) ≡ u2(t), on the interval [3
4T, 7
4T ] and so on. This way we construct solutions
uj(t) on the intervals Ij, where Ij := [Tj, Tj+T ], with T1 = 0, Tj+1 = Tj+ 34T, j = 1, 2, . . . .
By the uniqueness uj(t) = uj+1(t) on Ij∩Ij+1. Since the intervals Ij cover R, this gives thesolution, u(t), on R defined as u(t) ≡ uj(t) on the interval Ij for every j = 1, 2, . . . .
Global existence of solutions of the Hartree equation. We demonstrate howProposition 5.1 works on the Hartree equation.
Proposition 5.2. Any solution to the Hartree equation (4.17) in C([0,∞), H2)∩C1([0,∞), L2)satisfies
‖ψ(t)‖L2 = ‖ψ0‖L2 . (5.2)
Exercise 5.3. Prove this proposition.
Now, we use this conservation law and Propositions 4.10 and 5.1 to obtain
Theorem 5.4 (GWP). Assume v ∈ L∞. Then, for any ψ0 ∈ L2(Rn), the Hartreeequation (4.17) has a unique mild solution ψ ∈ C([0,∞), L2).
5.2 Energy and Entropy
For most of evolution equations appearing in natural sciences and especially in Physicsthere are conserved quantities, like energy, and/or monotonic quantities, like the entropy.Such quantities allow one to control the solutions at large times and, often, to prove theglobal existence and stability.
In application to the equations we considered in the previous section, these are
1) Under the Hartree and nonlinear Schrodinger (Gross-Pitaevskii) evolutions, (a) theL2−norm of solutions,
N(ψ) =
∫|ψ|2 (5.3)
(the ‘charge’, or the number of particles), and (b) the energy,
E(ψ) =
∫ (1
2|∇ψ|2 +G(ψ)
), (5.4)
where G(ψ) := 14(v ∗ |ψ|2)|ψ|2 and G(ψ) := 1
pλ|ψ|p, respectively, are conserved,
E(ψ) = E(ψ0), N(ψ) = N(ψ0). (5.5)
For the generalized nonlinear Schrodinger evolution, G is an anti-derivative of g.
Lectures on Applied PDEs 47
2) Under the nonlinear heat (reaction-diffusion) evolution, (4.9), the energy
E(u) =
∫(1
2|∇u|2 +G(u)), (5.6)
where G(u) := 1p|u|p, decreases, as t increases: ∂tE(u) < 0.
3) For the linear heat equation, ∂tu = ∆u, the entropy S(u) =∫u log u is decreasing:
∂tS(u) = −4
∫|∇√u|2 < 0. (5.7)
4) the (reduced) Keller-Segel equation of chemotaxis or nonlinear Fokker-Planck equa-tion,
∂ρ
∂t= ∆ρ−∇ · (ρ∇c) ,
0 = ∆c+ ρ,(5.8)
where ρ and c represent the organism density and chemical concentration (for theKeller-Segel equation) or the mass density and gravitational potential (for the non-linear Fokker-Planck equation). In this case, the ‘free energy’, defined by
F(ρ) =
∫R2
[− 1
2ρ(−∆)−1ρ+ ρ ln ρ
]dx, (5.9)
is decreasing and the total mass,∫ρ, is conserved.
Exercise 5.5. Prove the above claims.
5.3 A priori estimates for concrete equations
A priori estimates for the Hartree equation. In the last subsection, we illustratedhow this work on the Hartree equation on L2, using only the conservation of the L2−norm.Using also energy, we can also show the global existence of the Hartree equation on H1.
Proposition 5.6. We have
‖ψ‖2H1 ≤ 2E(ψ) +
1
2‖v‖∞‖ψ‖4
2 + ‖ψ‖2L2 . (5.10)
Proof. We estimate the term∫G(ψ) in (5.4), where G(ψ) := 1
Using this result and the local existence of the Hartree equation on H1, one can showthe global existence of this equation on H1.
Similar proof could be also done for the generalized nonlinear Schrodinger equation(gNLS). It is slightly more complicated. Instead of the trivial estimate (5.11), it uses themore sophisticated Gagliardo-Nirenberg’s inequality( ∫
|ψ|p+1) 1p+1 ≤ C‖ψ‖aH1‖ψ‖1−a
L2 , (5.14)
where a = d2p−1p+1
. This proof can be found in the next subsection.
A priori estimates for the Keller-Segel equations. The proof of the global exis-tence for the Keller-Segel equations (5.8), which we reproduce here,
∂ρ
∂t= ∆ρ−∇ · (ρ∇c) ,
0 = ∆c+ ρ,(5.15)
relies on a priori estimates of its entropy. We start with computing the change in theentropy
∂t
∫ρ ln ρ = −4
∫|∇√ρ|2︸ ︷︷ ︸
entropy dissipation
+
∫ρ2︸︷︷︸
entropy production
. (5.16)
Depending on whether the entropy dissipation or the entropy production wins we expecteither dissipation of the solution or the collapse (blowup). The Gagliardo - Nirenberg-Sobolev inequality in dimension d = 2,
‖f‖24 ≤ cgn‖∇f‖2‖f‖2
shows that the dissipation wins if Mc2gn ≤ 4 where M :=
∫f . This gives ∂t
∫ρ ln ρ ≤ 0
and therefore∫ρ ln ρ ≤
∫ρ0 ln ρ0. Using this a priori estimate one can show the global
exitence for the Keller-Segel equations in dimension d = 2 and for Mc2gn ≤ 4.
To sharpen this result one uses that the free energy decreases together with the loga-rithmic Hardy-Littlewood-Sobolev inequality in dimension d = 2,∫
f ln f ≥ 1
(M/8π)
1
2
∫f(−∆)−1f − C(M),
Lectures on Applied PDEs 49
where M :=∫f and C(M) := M(1 + log π − logM), which gives the following lower
bound on the free energy,
F(ρ) ≥ (1
(M/8π)− 1)
1
2
∫ρ(−∆)−1ρ− C(M).
Combining this inequality together with the fact that the free energy is decreases, F(ρ) ≤F(ρ0), one finds the bound on the entropy
(1−M/8π)
∫ρ ln ρ ≤ F(ρ0)− 1
4πC(M),
which is used to prove the global existence for M ≤ 8π (see [3] for details).
Blowup for the Keller-Segel equations. We note that∫R2 Rdx = 8π. This mass
turns out to be the threshold for the blowup of solutions of (5.15). Indeed, we have thefollowing result:
Theorem 5.7 (See [2]). Take ρ0 ≥ 0. If the dimension d = 2 and the total mass satisfies
M :=
∫R2
ρ0 dx > 8π, (5.17)
then the solution to (5.15) breaks down in finite time.
Proof. The time derivative of the second moment of mass
W :=
∫R2
|x|2ρ(x) dx
is, using that ρ is a solution to (5.15),
∂tW =
∫R2
|x|2(∆ρ+∇
(ρ∇∆−1ρ
))dx.
Integrating by parts and using that∫R2 |x|2∆ρdx = 4
∫R2 ρdx = 4
∫R2 ρ0dx = 4M , by the
conservation of total mass, we obtain
∂tW = 2(dM − J). (5.18)
where J :=∫ρ (∇∆−1ρ) · x dx. Using that in two dimensions ∆−1ρ = 1
2π
∫R2 ln |x −
y| ρ(y) dy and symmetrizing the integral we compute
J =1
2π
∫R2×R2
x · x− y|x− y|2ρ(y)ρ(x) dy dx
=1
4π
∫R2×R2
x · x− y|x− y|2ρ(y)ρ(x) dy dx
+1
4π
∫R2×R2
y · y − x|x− y|2ρ(y)ρ(x) dy dx
50 Lectures on Applied PDEs
This gives
J =1
4π
∫R2×R2
ρ(y)ρ(x) dy dx =1
4πM2.
Substituting this into (5.18), we obtain that
∂tW = 4M(1− 1
8πM).
If M > 8π, then the right hand side is negative and constant and hence, W becomesnegative in finite time. When this happens we have a contradiction since W is by definitionalways positive (recall that if ρ0 ≥ 0 then ρ(t) ≥ 0 by the maximum principle). Thus, ifM > 8π, then the solution ρ exists only for a finite time (t < t∗, where t∗ is the point oftime when W vanishes).
Thus we conclude that
• (Blanchet, Dolbeault, Perthame) If the initial total mass satisfies M :=∫R2 ρ0 dx ≤
8π, then the solution to (5.15) exists globally;
• (Biler) If the initial total mass satisfies M :=∫R2 ρ0 dx > 8π, then the solution to
(5.15) blows up in finite time.
Global existence of solutions of the generalized NLS. We address the questionof global existence, i.e., existence for all t ≥ 0 of solutions of the generalized NLS (gNLS)given in (4.20), which we reproduce here,
i∂tψ = −∆xψ + g(|ψ|2)ψ. (5.19)
Here g(u) is a real function, s.t. f(ψ) = g(|ψ|2)ψ satisfies the conditions (4.21). If G ≥ 0(the non-focusing nonlinearity), then the conservation of energy, implies
‖∇xψ(t)‖L2 ≤ E(ψ) = E(ψ0).
Using, in addition, the conservation of charge or the number of particles, N(ψ) = N(ψ0),we find
‖ψ(t)‖H1 ≤ E(ψ0) +N(ψ0).
Hence ψ(t) is uniformly bounded in the Sobolev norm. Thus if we have the local existenceresult of the type of Theorem 4.8 in the Sobolev space H1(Rn), then we also have theglobal one.
In the more difficult case G ≤ 0 (the focusing nonlinearity), we have to impose addi-tional conditions on the growth of nonlinearity. We follow [35].
Theorem 5.8 (GWP). Assume (4.21) with 1 ≤ p < d+2d
and s = 1. Then (5.19) has aunique solution in H1(Rd) for all t ≥ 0.
Lectures on Applied PDEs 51
We will derive this theorem from the following
Proposition 5.9. Let p < 1+ 4d. Define the function J(h, n) := Cd,p(h+n+n
2c2−b ), where
b := d(p−1)2
< 2 and c := p+ 1− d(p−1)2
. Then
‖ψ‖2H1 ≤ J(H(ψ), N(ψ)). (5.20)
Proof. For f(ψ) = g(|ψ|2)ψ satisfying (4.21) and F (ψ) = G(|ψ|2), we have
|F (ψ)| ≤ c|ψ|p+1 (5.21)
We have by the Gagliardo-Nirenberg’s inequality
( ∫|ψ|p+1
) 1p+1 ≤ C‖ψ‖aH1‖ψ‖1−a
L2 , (5.22)
where a = d2p−1p+1
. Then ∫|F (ψ)| ≤ C‖ψ‖bH1‖ψ‖cL2 (5.23)
where b := d(p−1)2
and c := p+ 1− d(p−1)2
. This gives
E(ψ) ≥ 1
2
∫|∇ψ|2 − C‖ψ‖bH1‖ψ‖cL2
=1
2‖ψ‖2
H1 − C‖ψ‖bH1‖ψ‖cL2 − 1
2‖ψ‖2
L2 . (5.24)
Now, since p < 1 + 4d, we have that b < 2 and therefore
1
4‖ψ‖2
H1 − C‖ψ‖bH1‖ψ‖cL2 ≥ −C ′‖ψ‖2c2−bL2 . (5.25)
The last two inequalities give (5.20).
Proof of Theorem 5.8. Thus if ψ is a solution to (5.19) with the initial condition ψ0 wehave by the conservation of energy and number of particles that
‖ψ‖H1 ≤M0, where M0 := J(H(ψ0), N(ψ0)). (5.26)
Now the global existence follows from the local existence result of the type of Theorem4.8 and Propositions 4.10 and 5.1.
We summarize some results on the relation between the global existence and functionalinequalities.
52 Lectures on Applied PDEs
Relation between the global existence and functional inequalities
• the generalized nonlinear Schrodinger equations vs the Gagliardo-Nirenberg-Sobolevinequality,
‖f‖p+1 ≤ C‖ψ‖aH1‖ψ‖1−aL2 , (5.27)
where a = d2p−1p+1
;
• the fast diffusion equation, ∂tu = ∆um, 0 < m < 1, vs the Hardy-Littlewood-Sobolev inequality (p = 2d/(d+ 2))∫ ∫
f(x)f(y)
|x− y| dxdy ≤ ‖f‖2p;
• the (reduced) Keller-Segel or nonlinear Fokker-Planck equations, (5.15), vs the log-arithmic Hardy-Littlewood-Sobolev inequality,
− 2
M
∫ ∫f(x)f(y) log |x− y|dxdy ≤ −
∫f log f + C(M),
where M :=∫f (in dimension d = 2).
Lectures on Applied PDEs 53
6 Elements of Variational Calculus
In this section, we introduce an important structure into the manifold of partial differ-ential equations - the variational structure. It distinguishes a class of partial differentialequations for which we can provide a uniform treatment of existence and stability prob-lems.
6.1 Functionals
Functionals are maps which have R as the target space. More precisely, let X be a vectorspace, and M ⊂ X, a not necessarily open subset of X. Then a functional is a mapE : M → R. Usually, X is a space of functions. If X has a basis, then functionals on Xcan be represented as functions of an infinite number of coordinates along the basis. If Xis a finite–dimensional space (which we are not concerned with here), then a functionalon X is just a usual function of several variables. In the following list of examples, Ω isa domain in Rn and G : R→ R, u : Ω→ R, or u : Ω→ C:
1) E(u) =∫
ΩG(u(x))dnx,
2) E(u) = 12
∫Ω|∇u|2dnx,
3) E(u) =∫
Ω(1
2|∇u|2 +G(u))dnx.
We also have to specify the spaces on which the functionals are defined. The (notnecessarily linear or vector) spaces are chosen according to the specific functional and theproblem at hand. Of course, we always try to choose the simplest possible space for agiven problem.
For instance consider example 1). Assume that the domain Ω is bounded, and thatthe function G satisfies the estimate
|G(u)| ≤ C|u|p + C, (6.1)
for some constant C > 0. It is then natural to define E on the space Lp(Ω).In example 2), we define E on the Sobolev–space H1(Ω), defined in Section A.6. In
example 3), if Ω is bounded, and G satisfies (6.1), then we define E on H1(Ω) ∩ Lp(Ω).
Exercise 6.1. For which p and n, the functional E(u) =∫
Ω(1
2|∇u|2 +G(u))dnx satisfying
(6.1) and with Ω bounded is defined on H1(Ω)?
An important example of functionals are linear functionals on a vector space X. Thisis the special case of linear operators when the target space is Y = C. If X is a normedspace, then we consider bounded or continuous linear functionals, for which the followingnorm is finite:
‖l‖ = sup‖x‖=1
|l(x)|. (6.2)
54 Lectures on Applied PDEs
Functionals from classical field theory. A very important class of functionals arecoming from the classical mechanics and classical field theory. Let φ : Ω× [0, T ]→ R andφ := ∂φ
∂t. Functionals of the form
S(φ) =
∫ T
0
L(φ, φ
)dt, (6.3)
where L(φ, φ) are functionals of two fields, φ and φ, are called the action functionals.L(φ, φ) is called the Lagrange function or functional.
Here are two important examples of the action functionals:
S(ϕ) =
∫ T
0
[1
2|∂ϕ∂t|2 − V (ϕ)]dt, (6.4)
where ϕ : [0, T ]→ Rm, and V : Rm → R;
S(φ) =
∫ T
0
∫Ω
[1
2|∂φ∂t|2 − 1
2|∇φ|2 −G(φ)]dnxdt, (6.5)
where φ : Ω× [0, T ]→ R.We see that in the first example L(φ, φ) := 1
2|φ|2 − V (φ) and in the second one,
L(φ, φ) :=
∫Ω
[12|φ|2 − 1
2|∇φ|2 −G(φ)
]dnx,
Functionals from geometry. The first example is given by the length of a curve γgiven parametrically as γ : [0, T ]→ Rm:
4) L(γ) =∫ T
0|γ′(t)|dt, where γ : [0, T ]→ Rm.
The next example is the area A(f) of the hypersurface S in Rn+1 given as a graphof a function f : Ω → R, defined on an open subset Ω of the hyperplane x⊥n+1 in Rn+1:S =graphf := (x, f(x))|x ∈ Ω:
5) A(f) =∫
Ω
√1 + |∇f |2dnx (see Appendix G.1 to this section).
Functionals from image recognition. In conclusion, we give examples of functionalsfrom image recognition:
6) E(u) =∫
Ω(|u− f |2 + λ|∇u|2) dnx, where u, f : Ω→ R (f is fixed) and λ ≥ 0,
7) E(u, γ) =∫K\γ (|u− f |2 + λ|∇u|2) d2x + µL(γ), where u, f : K → R (f is fixed),
λ, µ ≥ 0, and γ is a closed curve in K, i.e. γ : [0, 1]→ K and γ(0) = γ(1).
Lectures on Applied PDEs 55
Here E(u) is an error functional measuring how much a smooth function u differs froma given function f , and E(u, γ) is the Mumford-Shah functional in image segmentation(f : K → R is a given image, u is a piecewise smooth approximation of f , with jumpsacross the curve γ are allowed, and γ is a segmentation of the image f , into two regions:interior of γ and exterior of γ). Here L(γ) is the length of a curve γ given parametricallyas γ : [0, 1]→ K ⊂ R2.
6.2 The Gateaux derivative for functionals
Consider a functional E : M → R. If M is an open subset of a vector space X, then theGateaux derivative dE(u), u ∈M , is a functional on X defined, for every ξ ∈ X, as
dE(u)ξ =∂
∂λE(uλ)
∣∣λ=0
, (6.6)
where uλ := u+ λξ if the latter derivative exists and is a bounded map in the sense thatsup‖ξ‖≤1 |dE(u)ξ| . 1.
It is shown in Theorem B.3(a) of Appendix B that dE(u) is also linear. Since dE(u)ξ ∈R, we conclude dE(u) is a bounded, linear functional on X.
We say that E is C1 in U ⊂ M if and only if ∀u ∈ U , dE(u) is a bounded linearfunctional.
Exercise 6.2. Show that the definition (6.6) is independent of the choice of the curve uλ(as long as uλ
∣∣λ=0
= u and ∂∂λuλ)∣∣λ=0
= ξ). (Hint: see Appendix B.)
Let us now consider a simple example. Let G be a real differentiable function on Rsatisfying the estimate
|G(u)|+ |G′(u)| ≤ C|u|2,where C is independent of u ∈ R, and let Ω ⊂ Rn . Then the functional
V (u) :=
∫Ω
G(u(x))dnx
is defined on L2(Ω). Using the definition, dV (u)ξ = ∂∂λV (u + λξ)|λ=0, we compute its
Gateaux derivative (d
∫Ω
G u dnx)ξ =
∫Ω
G′(u(x))ξ(x)dnx.
Thus the Gateaux derivative in this case is the linear functional standing on the r.h.s..
Exercise 6.3. Compute the Gateaux derivatives in examples 1)–3).
56 Lectures on Applied PDEs
6.3 Critical points and connection to PDEs
Let M be an open set in a real vector space. Given a C1–functional E : M → R, we saythat u∗ ∈M is a critical point (CP) of E if and only if dE(u∗) = 0.
Exercise 6.4. Find the equations for the critical points in examples 1)–3) given at thebeginning of this section.
The equation dE(u∗) = 0 (or, in detail, dE(u∗)ξ = 0, for every ξ ∈ X) for criticalpoints of E is sometimes called the Euler or Euler-Lagrange equation. Let us discuss verybriefly the Euler-Lagrange equations for the functional
E(u) =
∫Rn
(1
2|∇u|2 + uf
)dnx, (6.7)
which is a modification of the Dirichlet functional in example 2). We consider this func-tional on the Sobolev space H1(Rn). We compute
dE(u)ξ =∂
∂λE(uλ)|λ=0 =
∫Rn
(∇u · ∇ξ + fξ) dnx,
where uλ ∈ H1(Rn) s.t. u0 = u and ∂∂λuλ|λ=0 = ξ. Hence the Euler-Lagrange equation,
dE(u)ξ = 0, reads ∫Rn
(∇u · ∇ξ + fξ) dnx = 0, ∀ξ ∈ H1(Rn). (6.8)
If u is twice differentiable, then we can integrate by parts to find
dE(u)ξ =
∫Rn
(−∆u+ f)ξ.
Hence the Euler-Lagrange equation becomes∫Rn(−∆u + f)ξ = 0, ∀ξ ∈ H1(Rn) and
therefore∆u = f in Rn. (6.9)
This is the classical Poisson equation.The above discussion motivates the following definition:
Definition 6.5. We say that u satisfying (6.8) is a (variational) weak solutions to theequation (6.9) (see the remark around equation (4.6) in in Subsection 4.1). Thus criticalpoints of E are (variational) weak solutions to the corresponding differential equations.
Using the elliptic regularity, we can promote weak solutions to classical ones, seeSubsection 4.6.
If we consider the functional similar to the one above on a domain Ω ⊂ Rn, ratherthan the entire Rn, i.e.
E(u) =
∫Ω
(1
2|∇u|2 + uf
)dnx, (6.10)
Lectures on Applied PDEs 57
then we have to be more careful. We first consider this functional on the Sobolev spaceH1g (Ω) of functions in H1(Ω), which are equal to a given function g on ∂Ω. (Since the
boundary ∂Ω has n–dimensional Lebesgue-measure zero, we have to be careful about themeaning of “u = g on ∂Ω”.) We compute
dE(u)ξ =∂
∂λE(uλ)|λ=0 =
∫Ω
(∇u · ∇ξ + fξ) dnx,
where uλ ∈ H1g (Ω) s.t. u0 = u and ∂
∂λuλ|λ=0 = ξ. Integrating by parts, we find
dE(u)ξ =
∫Ω
(−∆u+ f)ξ +
∫∂Ω
∂u
∂νξ,
where ν is the outward unit normal vector to ∂Ω.Since our functions u are fixed (to g) on ∂Ω, we must take ξ vanishing on ∂Ω (so
that to have uλ = u + λξ fixed at the boundary). Hence the Euler-Lagrange equation,dE(u)ξ = 0, reads
∫Ω
(−∆u + f)ξ = 0, for every ξ ∈ H1(Ω) that vanishes on ∂Ω (i.e.ξ ∈ H1
0 (Ω)), and therefore we have in the weak sense discussed above
∆u = f in Ωu = g on ∂Ω.
(6.11)
This is the Dirichlet boundary value problem.If we consider the functional (6.7) on H1(Ω), then ξ is an arbitrary function in H1(Ω)
and varying its values in Ω and on ∂Ω independently, we see that the Euler-Lagrangeequation dE(u)ξ = 0, ∀ξ ∈ H1(Ω), becomes
∆u = f in Ω∂u∂ν
= 0 on ∂Ω.
This is the Neumann boundary value problem. Solutions of these two problems are ratherdifferent, so we see that the space on which a variational problem is considered plays animportant role.
Exercise 6.6. Compute the Gateaux derivative and find the Euler-Lagrange equation inexample 4).
As another example we consider the problem of minimal surfaces from differentialgeometry. Let A(f) be the area of the hypersurface S in Rn+1 given as a graph of afunction f : Ω→ R, defined on an open subset Ω of the hyperplane x⊥n+1 in Rn+1:
A(f) =
∫Ω
√1 + |∇f |2dnx (6.12)
(see Appendix G.1 to this section). We define A on C2(Ω). Critical points of the functionalA(f) are called minimal surfaces. We have
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Proposition 6.7. The equation for a minimal surface given locally as the graph of afunction f is given by
H(x′) = 0, (6.13)
where x′ = (x, f(x)) ∈ S =graphf and H(x′) is the mean curvature of S at x′ ∈ S givenexplicitly by
H(x′) := div
(∇f√
1 + |∇f |2
). (6.14)
Exercise 6.8. Show that the Euler-Lagrange equation for A(f) is is given by (6.13). (Inother words show that dA(f)ξ = −
∫ΩH(x)ξ(x)dnx.)
Hence the equation for a minimal surface states that the mean curvature on it vanishes.Sometimes this equation is taken for a definition of the minimal surface.
One can also define surface in Rn+1 locally via a parametrization φ : U → Rn, whereU is an open subset of Rn, as
S = Image(φ) ⊂ Rn+1.
This is a parametric surface. Then the area A(φ) of S is given by the functional A(φ) =∫U
∣∣∣ ∂φ∂x1 ∧ ∂φ∂x2
∣∣∣ d2x.
Finally, we consider the two important action functionals:
S(ϕ) =
∫ T
0
[1
2|∂ϕ∂t|2 − V (ϕ)]dt, (6.15)
where ϕ : [0, T ]→ Rm, and V : Rm → R; and
S(φ) =
∫ T
0
∫Ω
[1
2|∂φ∂t|2 − 1
2|∇φ|2 −G(φ)]dnxdt, (6.16)
where φ : Ω× [0, T ]→ R.The Euler-Lagrange equation in (6.15) and (6.16) are Newton’s equation and the
classical relativistic field theory. In the second case,
∂2φ
∂t2−∆φ−G′(φ) = 0.
Exercise 6.9. Prove the above statement.
These functionals are examples of the action functional
S(φ) =
∫ T
0
L(φ, φ
)dt, (6.17)
where L(φ, φ) are functionals of two fields, φ and φ, called the Lagrange function orfunctional.
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Exercise 6.10. Show that the Euler-Lagrange equation for the functional (6.3) is givenby
−∂t(dφL(φ, φ
)) + dφL
(φ, φ
)) = 0. (6.18)
Exercise 6.11. Verify that the Euler-Lagrange equations for the examples (6.15) and(6.16) above are of the form (6.18).
A key fact about Hamiltonian systems is that they have a conserved functional, calledthe energy. Indeed, define the energy functional by
E(φ) := dφL(φ, φ
)φ− L
(φ, φ
)). (6.19)
This functional is a constant of motion, i.e. the energy is conserved:
∂tE(φ) = 0. (6.20)
We call such systems conservative.
Exercise 6.12. Prove (6.20). Verify it for the examples (6.15) and (6.16) above.
∗ ∗ ∗ ∗ ∗ ∗ ∗∗
The following elementary but important result connects the main problem of varia-tional calculus to the problem of existence of solutions of differential equations
Theorem 6.13. Let F be a functional on an open subset M of a vector space X. Ifu0 ∈M is a minimizer of F , then u0 is a critical point of F .
Proof. Let ξ be an arbitrary vector from X, and λ sufficiently close to 0 so that thereis a curve uλ s.t. uλ=0 = u0 and duλ
dλ|λ=0 = ξ. Then the function f(λ) := F (uλ) has a
minimum at λ = 0, and therefore λ = 0 is a critical point of this function, f ′(0) = 0.This is equivalent to ∂
∂λF (uλ)|λ=0 = 0, which by the definition of the Gateaux derivative
implies that dF (u0)ξ = 0. This holds for every ξ ∈ X, and we conclude that dF (u0) = 0as a functional on X.
A powerful method of proving existence of solutions of a large class of static equationsis showing that these equations are Euler - Lagrange equations of certain functionals andthen proving existence of minimizers (or saddle points) by the direct methods of variationalcalculus. This will be done in Section 18 below.
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6.4 Constraints and Lagrange multipliers
The problem we address now is concerns with minimizing or maximizing a functional,E(u), under a constraint, J(u) = 0. Such problems appear naturally in applications:
Examples.(1) Minimize the energy for a fixed entropy, or total mass, or total number of particles.(2) Maximize output for a fixed cost.
Let U be an open set in a Banach space X and J : U → R, a functional on U . Weconsider minimization of a functional E(u), on a set, M , defined by side conditions orconstraints, i.e., M is of the form
M := u ∈ U | J(u) = 0, (6.21)
where J is a functional on U . Such a set can be thought of as an infinite dimensionalhypersurface in X. The restriction J(u) = 0 is called the constraint.
Problems of minimizing functionals on sets of the form (6.21) also come up in provingexistence of weak solutions of partial differential equations. This will be demonstratedlater.
Since M is not open in X, the definition of the Gateaux derivative is more subtle,as we cannot in general take pieces of straight lines uλ = u + λξ in the definition ofdF (u). For M not open in X, to define the Gateaux derivative, we take differentiablepaths λ 7→ uλ in M , s.t. u0 = u. Here λ ∈ [−ε, ε] for some ε = ε(u) sufficiently small.Then, for ξ ∈ X s.t. there is an ε > 0, and a differentiable path [−ε, ε] 3 λ 7→ uλ ∈M forwhich u0 = u, duλ
dλ|λ=0 = ξ, we define
dF (u)ξ :=d
dλF (uλ)|λ=0.
if this derivative exists and represents a bounded map.The set of all the ξ ∈ X s.t. there is an ε > 0, and a differentiable path [−ε, ε] 3 λ 7→
uλ ∈M for which u0 = u, duλdλ|λ=0 = ξ is called the tangent space to M at u, TuM . Thus,
dF (u) : TuM → R. By the definition, we have that dF (u) ∈ (TuM)′. Here X ′ is the spaceof linear functionals on X, called the dual space to X. The dual space to TuM is calledthe cotangent space to M at u and is denoted by T ∗uM := (TuM)′
Exercise 6.14. Show that TuX = X and therefore in particular, TuHs(Ω) = Hs(Ω).
As before the points u ∈M for which dF (u) = 0 are called the critical points of F .Now we consider critical points of a functional E : X → R on the set (6.21) where J is
another functional on X. The key result here goes back to Lagrange and it is called themethod of Lagrange multipliers:
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Theorem 6.15. Let E and J be C1 functionals, and let M be defined by (6.21). Thenu0 is a critical point of E on M iff dE(u0)− λdJ(u0) = 0 on X for some λ ∈ R.
λ is called the Lagrange multiplier. It is determined by the condition that J(u0) = 0.First we prove the following
Proposition 6.16. Let J be a C1 functional and let M is given by (6.21). Then TuM =Null dJ(u).
Proof. By the definition of TuM , for any ξ ∈ TuM , there is a differentiable path usin M (i.e. a differentiable path in X satisfying J(us) = 0) satisfying us=0 = u0 anddusds|s=0 = ξ. Differentiating J(us) = 0 in s at s = 0 gives dJ(u0)ξ = 0, which implies that
ξ ∈ Null dJ(u0). This shows that TuM ⊂ Null dJ(u). The proof that Null dJ(u) ⊂ TuMis given in Proposition 6.19 at the end of this subsection.
Proof of Theorem 6.15. The fact that u0 is a minimizer of E in M implies that s = 0 is aminimizer of E(us) in s, which means that dE(us)
ds|s=0 = 0. The latter equation gives
dE(u0)ξ = 0, ∀ξ ∈ Tu0M = Null J(u0), (6.22)
(in the last step we used Proposition 6.16). Now, let e ∈ X, e /∈ Null dJ(u0). We construct
a projection P : X → Null dJ(u0) as Pη := η − dJ(u0)ηdJ(u0)e
e ∈ Null dJ(u0) for all η ∈ X. Letξ := Pη. Then
0 = dE(u0)ξ = dE(u0)η − λdJ(u0)η (6.23)
with λ = dE(u0)edJ(u0)e
, ∀η ∈ X. Hence dE(u0) − λdJ(u0) = 0 on X follows. Clearly, we canreverse the steps to prove the opposite implication.
Eq (6.22) shows that dE(u0) is perpendicular to the subspace dJ(u0)⊥ := Null J(u0)and therefore (as we have shown) dE(u0) is parallel to dJ(u0).
Example. Below Ω is a domain in Rn. Consider the Dirichlet functional
E(u) =1
2
∫Ω
|∇u|2dnx
on the M = u ∈ H1g (Ω) : J(u) = 1, where J(u) = 1
p
∫Ω|u|pdnx. To find the equation
of the critical points on the space M , we use the theorem. Since dE(u) = −∆u, anddJ(u) = |u|p−2u, the theorem implies that u satisfies
∆u+ λ|u|p−2u = 0 in Ω,
u = g on ∂Ω.
Exercise 6.17. Find the equations for critical points of the following functionals:
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(1) 12
∫Ω|∇u|2 on the space M = u ∈ H1
0 (Ω) :∫
Ω|u|2 = 1;
(2) E(u) =∫
Ω
(12|∇u|2 +G(u)
)dnx, on the space M = u ∈ H1(Ω)|
∫uf = a for some
a ∈ R and f ∈ C(Ω);
(3) the mean energy E(f) =∫hfdnxdnp of a classical particle distribution f : Rn →
R+, where h(x, p) is a classical hamiltonian of the system and for entropy, S(f) =∫f log fdnxdnp;
(4) E(u) = 〈u,Au〉, where A is a self-adjoint operator on a complex Hilbert space Hand u ∈M := u ∈ D(A) | ‖u‖ = 1;
(5) (Isoperimetric problem) the area of a closed n−dimensional surface (hypersurface)in Rn+1 for a fixed enclosed volume (see the next paragraph and Exercise 6.18).
There is a quantum analogue of the problem 3), with the classical particle distributionf replaced by the density operator ρ, the mean energy given by E(ρ) = Tr(Hρ), where His a quantum hamiltonian, and the Boltzmann entropy S(f) is given by the Boltzmannentropy S(ρ) := −Tr(ρ log ρ).
Isoperimetric problem. The celebrated isoperimetric problem is differential geometryrequires to minimize the surface area of a closed surface for a given enclosed volume.
To formulate the problem formally, let S ⊂ Rn+1 be a closed hypersurface (i.e. ann−dimensional surface). Denote its area and enclosed volume by A(S) and V (S), respec-tively. The problem is to minimize A(S) for a fixed V (S) = c.
First one would like to find critical points for this problem, i.e. to find critical pointsof A(S), given V (S) = c. This is what we do now.
We can find critical points for this problem as follows. Take a piece, S ′, of S whichcan be written as the graph, graphf , of a function f : Ω→ R, defined on an open subsetΩ of the hyperplane Rn := (x1, . . . , xn, 0) : (x1, . . . , xn) ∈ Rn in Rn+1 and satisfyingf ≥ 0 and f = 0 on ∂Ω. One can also find such a S ′ by translating and rotating S.
Then the area of S ′ is given by (see Appendix G.1)
A(f) =
∫Ω
√1 + |∇f |2dnx.
The enclosed volume between Ω and the graphf is equal to
V (f) =
∫Ω
fdnx.
One can show that the problem of finding critical points of A(f) on the set M = f ∈H1
0 (Ω)|∫
Ωf dnx = c is locally equivalent to an original problem.
Lectures on Applied PDEs 63
Exercise 6.18. Show that critical points of A(f) on the set M = f ∈ H10 (Ω)|
∫Ωf dnx =
c satisfy the Euler- Lagrange equation
div(∇f√
1 + |∇f |2) = h,
where h is the Lagrange multiplier.
A solution to this differential equation gives a surface of a constant mean curvature.
∗ ∗ ∗
Finally, we prove
Proposition 6.19. If M = u ∈ X : J(u) = 0, where J is a C1 functional on X, thenNull dJ(u) ⊂ TuM for u ∈M .
Proof. Let u ∈ M and ξ ∈ Null dJ(u) and find uλ such that uλ=0 = u, ξ = ∂λ|λ=0uλ andJ(uλ) = 0. Let η /∈ Null dJ(u) and define the path uλ = u+ λξ + λaη, where a solves theequation f(a, λ) = 0, where
f(a, λ) :=1
λJ(u+ λξ + λaη).
We write J(u+ v) = J(u) + dJ(u)v+Ru(v), where Ru(v) is defined by this equation, i.e.Ru(v) := J(u + v) − J(u) − dJ(u)v. Now, we take v = λξ + λaη in this equation anduse J(u) = 0 and dJ(u)ξ = 0, to obtain f(a, λ) = adJ(u)η + 1
Hence ∂af(a, λ) = dJ(u)η+o(λ) and, in particular, ∂af(0, 0) = dJ(u)η 6= 0. Therefore, bythe implicit function theorem, the equation f(a, λ) = 0 has a unique solution for a = a(λ)for λ sufficiently small and this solution satisfies a(λ) = oλ(1).
By the definition of f(a, λ) the family uλ = u + λξ + λaη, with a = a(λ) = oλ(1)solving f(a, λ) = 0, has the properties mentioned above. Hence ξ ∈ TuM .
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6.5 Minimization problem and spectrum.
We would like to explain relation between the minimization (or in general critical point)theory and the spectral theory of operators. Consider an operator A acting on a Banachspace X with a domain D(A) (i.e., A : D(A) → X). Recall that the spectrum, σ(A), ofan operator A is the set in C defined by
σ(A) := z ∈ C : A− z1l is not invertible . (6.24)
Clearly, eigenvalues of A belong to σ(A) (in fact, if λ is an eigenvalue, then Auλ = λuλfor some nonzero uλ ∈ X (uλ is called an eigenvector), so (A−λ)uλ = 0, and A−λ is notinvertible). In general, the spectrum can also contain continuous pieces and it can takevery peculiar forms.
Now, let A be a self-adjoint operator on a complex Hilbert space H and define M :=u ∈ D(A) | ‖u‖ = 1. Consider the functional E(u) = 〈u,Au〉, where u ∈ M . InQuantum Mechanics, if A is a quantum hamiltonian, then E is the expectation of theenergy in the state u.
There is a deep relation between existence of minimizers (or saddle points) for thefunctional 1
2〈u,Au〉 on the set M (we can assume here that the operator A is bounded
below, i.e., 〈u,Au〉 ≥ −C‖u‖2 for some C < ∞) and spectral properties of the operatorA. In particular, we have
Theorem 6.20. (i) infu∈M〈u,Au〉 = infλ∈σ(A) λ;
(ii) 〈u,Au〉 has a minimizer on M if and only if infλ∈σ(A) λ is an eigenvalue of A;
A proof of this theorem can be found in [12] where one can also find a theorem on arelation between saddle points of the functional 〈u,Au〉 and eigenvalues of the operatorA greater than the smallest eigenvalue infλ∈σ(A) λ. Here we mention only that the Euler-Lagrange equation for the functional 〈u,Au〉 on the space M is
Au = λu
where λ is the Lagrange multiplier. This is the eigenvalue equation for A.In the case when A is a Schrodinger operator, the number λ0 := infλ∈σ(A) λ is called the
ground state energy of A and the theorem above expresses the variational characterizationof eigenvalues used frequently in Quantum Mechanics.
The above motivates the following definitions. We define the discrete spectrum of anoperator A as
σd = λ ∈ C | λ is an isolated eigenvalue of A with finite multiplicity.
The part of the spectrum which complements the discrete spectrum is called the essentialspectrum of an operator A:
σess(A) := σ(A) \ σd(A).
Lectures on Applied PDEs 65
(Some authors use the term ”continuous spectrum” rather than ”essential spectrum”.)Hence we have σ(A) = σd(A) ∪ σess(A).
Examples of operators and their spectra are given in Appendices A.7 and E.If A is a quantum hamiltonian (the Schrodinger operator), then the eigenfunctions
corresponding to discrete eigenvalues are called the bound states. They describe themotion of quantum system localized essentially to a bounded domain of the physicalspace. The essential spectrum is related to the scattering states, describing the systembroken into freely moving fragments.
A more difficult and very useful result is the following
Theorem 6.21. If infu∈M〈u,Au〉 < infλ∈σess(A) λ, then infu∈M〈u,Au〉 is an eigenvalueof A.
(spectrum of ∆ on bounded domains)
6.6 Hessians and local minimizers.
Like in a finite dimensional case, to determine whether a critical point is a minimizer,or maximizer, or a saddle point, we use the hessian operator. However, in the finitedimensional context, this operator acts on an infinite dimensional space and handling itis a more delicate mater.
We define the hessian of E at u as the linear operator defined by the quadratic formgiven by the relation
〈ξ,Hess E(u)η〉 = d2E(u∗)(ξ, η), (6.25)
where d2E(u)(ξ, η) is the hessian bilinear form defined as d2E(u)(ξ, η) := d(dE(u)ξ)η =∂t∂s
∣∣s=t=0
E(ust), where ust := u+ sξ + tη.With the definitions below of the gradient in (7.1) and the Gateaux derivative (B.1)
of maps, we can give a more succinct definition of the hessian of E at u (compared to(6.25)) as the linear operator given by
Hess E(u) := d gradg E(u). (6.26)
Similarly to the finite dimensional case, we have the following statement
Theorem 6.22. If u∗ is a minimizer of E, then Hess E(u∗) ≥ 0. If Hess E(u∗) ≥ θ, forsome θ > 0, then u∗ is a minimizer of E.
6.7 Convexity and uniqueness
Recall that a set A is said to be convex iff for every u, v ∈ A, we have λu+ (1− λ)v ∈ Aand a functional F : A→ R is convex iff
F (λx+ (1− λ)y) ≤ λF (x) + (1− λ)F (y)
66 Lectures on Applied PDEs
and f is said to be strictly convex iff
F (λx+ (1− λ)y) < λF (x) + (1− λ)F (y).
We have the following standard result
Proposition 6.23. Let A be a convex set. Then
• F : A→ R be a convex C1–function iff F (x)− F (y) ≥ dF (y)(x− y)
and similarly, for the strictly convex functionals.
Proof. We give a proof in one direction. If F is convex, then so is f(s) := F (u+sv), ∀v ∈X. Hence the function f ′(s) := ∂sF (u + sv) is monotonically non-decreasing. Let v :=
u′−u. Then F (u′)−F (u) = F (u+ (u′−u))−F (u) =∫ 1
0ds∂sF (u+ sv) =
∫ 1
0ds[∂sf(s)−
∂sf(0)] + ∂sf(0). By the monotonicity of ∂sf(s), we have that F (u′) ≥ F (u) + ∂sf(0).Now, by the definition of dF (u), we have ∂sf(0) = dF (u)(u′ − u). The last two relationsgive F (u′) ≥ F (u) + dF (u)(u′ − u).
Theorem 6.24. Let A be a convex set and F : A → R be A convex C1–function. Wehave
(i) If a is a critical point of F , then a is a minimizer.
(ii) The set of all minimizers of F is a convex set.
(iii) If F is strictly convex, then it has at most one minimizer.
Proof. (i) and (iii) follow from Proposition 6.23 with y = a and the relation dF (a) = 0and (ii), from the definition of the convexity.
Corollary 6.25. Weak solutions of the Laplace and Poisson equations with the Dirichletboundary conditions are unique.
For more details see [6, 17].
6.8 Dual space
The set of all bounded linear functionals on X is called the dual space of X (or simplythe dual (or adjoint space) of X, and it is denoted as X ′. Thus dE(u) ∈ X ′. It is a vectorspace with the norm (6.2). In fact, since C is complete, one can show that X ′ is alwaysa Banach space, whether X is complete or not.
If X is a space of functions, then X ′ can be identified with either a space of functionsor a space of distributions or a space of measures. Here are some examples of dual spaces:
1) (Lp)′ = Lq, where 1/p+ 1/q = 1, if 1 ≤ p <∞ (space of functions),
2) (L∞)′ is a space of measures which is much larger than L1,
3) (Hs)′ = H−s (space of distributions if s > 0).
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6.9 Functionals on complex spaces
We consider functionals on open subsets, M , of complex vector spaces, Z. For suchfunctionals, we define the complex Gateaux derivatives as
where ψ = ψ1 + iψ2, and similarly the complex gradients ∂ψE(ψ) and ∂ψE(ψ). One way tocompute dψE(ψ) and dψE(ψ) is to treat ψ and ψ as independent functions and computethe corresponding objects as partial Gateaux derivatives.
To connect this to the real Banach theory considered above, we note that Z can bewritten as Z = V + iV , for some real vector space. For example Z = H1(Rd,C) =H1(Rd,R) + iH1(Rd,R). We associate with such a Z, a real space Z := V ⊕ V . There isone - to - one correspondence between Z and Z:
φ⇐⇒ ~φ := (φ1, φ2), φ1 := Reφ, φ2 := Imφ.
Consider the map compl : Z → Z, given by ~φ := (φ1, φ2) =⇒ φ = φ1+iφ2, and its inverse vect :
φ = φ1 + iφ2 =⇒ ~φ := (φ1, φ2). Using these maps identify a functional E(φ) on Z witha functional E real(vect(φ)) = E(φ) on Z and we can define the variational (or Gateauxor Frechet) differentiability and derivative, ∂~φE(φ), and partial derivatives, ∂φ1E(φ) and∂φ2E(φ), with respect the real, φ1, and imaginary, φ2, parts of the field φ for E(φ), by
computing them for E real(~φ). After that we introduce the derivatives with respect φ andφ as follows
Here are some examples (below Ω ⊂ Rd, d = 2, 3):6) The Ginzburg-Landau energy functional (ψ : Ω→ C, a : Ω→ Rd)
EΩ(ψ, a) :=1
2
∫Ω
|∇aψ|2 +
κ
2(|ψ|2 − 1)2 + | curl a|2
, (6.30)
where ∇a = ∇− ia, the covariant derivative.
7) The action functional for nonlinear Schrodinger equation (ψ : Ω× [0, T ]→ C)
S(ψ) =1
2
∫ T
0
∫Ω
(−Im(ψψ) + |∇ψ|2 +G(|ψ|2))ddxdt. (6.31)
Exercise 6.26. Compute the complex and real Gateaux derivatives and the equationsfor the critical pints in examples 6) – 7) above.
For complex vector spaces, we say that u0 ∈ M is a critical point (CP) of E if andonly if dE(u0) = 0.
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6.10 Harmonic maps
In the simplest context the harmonic map or the stationary sigma model is described amap, φ, from a d−dimensional euclidean space, Rd to the space Rn+1, satisfying |φ| = 1,i.e. φ maps Rd to Sn embedded in Rn+1 in the standard way, and with the energy givenby
E(φ) =1
2
∫Rd
∑i
∂iφ · ∂iφddx. (6.32)
(a · b denotes the dot product in Rn+1.)Critical points of (6.32) satisfy the Euler-Lagrange (stationary) equations
φ ∧∆φ = 0. (6.33)
To derive this equation, we observe that the variations of φ are along maps Rd → thetangent space TφS
3 = φ⊥ and that the orthogonal projection, P⊥φ , onto TφS3 = φ⊥ is
given by P⊥φ ξ = φ ∧ ξ, which leads to (6.33). (To obtain this expression we use that any
ξ ∈ φ⊥ can be written as ξ = P⊥φ η, where η is an arbitrary vector.)
6.10.1 Symmetries
Gauge transformation: for any sufficiently regular function R ∈ O(n+ 1),
T gaugeρ : φ(x) 7→ R−1φ(x); (6.34)
Translation transformation: for any h ∈ Rd,
T translh : φ(x) 7→ φ(x+ h); (6.35)
Rotation and reflection transformation: for any R ∈ O(d) (including the reflectionsf(x)→ f(−x))
T rotR : φ(x) 7→ φ(Rx). (6.36)
Scaling symmetry:
φ(x)→ φ(λx).
6.10.2 Topological invariants
For a map, φ, to have finite energy, it should converge to a constant at infinity. In thiscase, φ can be extended to a continuous map from Sd to Sn taking the point at infinityto the limit of φ(x) at the spatial infinity. (In other words, we pass to the one-pointcompactification of Rd.) Then one can define the degree, deg φ, as the homotopy class ofφ as a map from Sd to Sn, i.e. a member of the homotopy group πd(S
n).
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In the particle physics one takes d = 2, 3 and n = 2, and in the condensed matterphysics one takes d = n = 2, i.e. φ : Rd → S2, d = 2, 3. In both cases, the degree of φ isan integer (the degree for maps from Sd to S2), i.e. πd(S
2) = Z, for d = 2, 3.Let d = n = 2. Then the degree is given by
degφ =1
8π
∫R2
φ · (∂1φ ∧ ∂2φ)d2x (=1
4π
∫S2
φ∗(dS)). (6.37)
By Kronecker’s integral formula that deg φ is an integer and is equal to the Brouwerdegree of m. In the present case, degm is called the skyrmion number.
6.10.3 Complex representation
For computations as well as for theoretical studies it is convenient to pass to the projective,or complex, representation of the harmonic map functional and equation. To this end, weperform the stereographic projection, φ→ w, of S2 to the complex plane C, as
φ =( 2 Rew
1 + |w|2 ,−2Imw
1 + |w|2 ,1− |w|21 + |w|2
). (6.38)
Here we identified R2 with the complex plane C, by z := x1 + ix2. In the new variablesthe energy is given by
E(w) :=
∫ |∂w|2 + |∂w|2(1 + |w|2)2
dz, (6.39)
where we used the complex derivatives ∂ := ∂1 − i∂2 and ∂ := ∂1 + i∂2, while the degreeis given by
deg φ ≡ degw =1
4π
∫∂w∂w − ∂w∂w
(1 + |w|2)2dz. (6.40)
The Euler - Lagrange equation, dwE(w) = 0, where dw := 12(dw1 + idw2), for w = w1 + iw2,
is given by
∆w =2w
1 + |w|2 (|∂w|2 + |∂w|2). (6.41)
6.10.4 Bogomolny bounds
Let d = n = 2. A key point here is that one has the Bogomolnyi-type inequality
E(φ) ≥ 4π|degφ|.
70 Lectures on Applied PDEs
Indeed, we have the Bogomolnyi-type identity
E(φ) = ±4πdegφ+1
4
∫R2
∑i
|∂iφ± εijφ ∧ ∂jφ|2d2x,
where εij is the Levi-Cevita antisymmetric symbol with ε12 = −ε21 = 1, ε11 = ε22 =0, which implies the inequality above. Moreover, the last relation yields that in everyhomotopy class solutions of the self-dual/ anti-dual equations,
∂iφ± εijφ ∧ ∂jφ = 0 (6.42)
are the minimizers of E(φ). (φ∧ in the second term gives a complex structure, see Murray.)In the complex representation, the Bogomolnyi identity becomes
E(w) = ±2π deg φ+
∫ |∂#w|2(1 + |w|2)2
dz, (6.43)
where ∂# stands for either ∂ or ∂. Eqs (6.43) implies that E(w) is minimized by wsatisfying either ∂w = 0 or ∂w = 0 (the Cauchy - Riemann equations), i.e. w is eithera holomorphic or anti-holomorphic function. They are mapped into each other by thecomplex conjugation.
6.10.5 Solutions
Ground states vs excitations: Usually, the ground states are stationary solutions corre-sponding to energy minimizers. In translationally invariant problems like ours they haveinfinite energy and are either homogeneous (x−independent) or lattice gauge-periodicsolutions, while finite energy localized solutions are interpreted as excitations.
We have the following solutions:
(a) Ferromagnetic states: φfm = constant.
(b) Skyrmions: These are solutions with F (m) <∞ and deg =1. They can be classifiedby T rot
ρ φ = T gaugeRρ
φ (equivariance) for every ρ ∈ O(2) and some group homomor-phisms Rρ : O(2)→ O(2). The existence is given in Theorem 6.27 below.
(c) Skyrmion lattices: These are solutions periodic w.r.to a lattice L. They can beclassified by T trans
s φ = T gaugeRs
φ, for every s ∈ L and for some group homomorphismsRs : L → O(2).
Topological invariants for skyrmions and skyrmion lattices. Denote by ΩL ' T2
a fundamental cell of L. One has the following results
(a) Skyrmions: the topological invariant is given by degree (6.37);
Lectures on Applied PDEs 71
(b) Skyrmion lattices: the topological invariant is given by
c1(φ) =1
8π
∫T2
φ · (∂1φ ∧ ∂2φ)d2x.
Here c1(φ) is the first Chern number of the map φ : T2 → S2.
Self-dual and anti-self-dual solutions. For any degφ = k, minimizers of satisfy self-dual/ anti-dual equations (6.42), which are the first order equations. These equationshave explicit solutions (harmonic or anti-harmonic maps), φstatk :
Theorem 6.27. For any degφ = k, minimizers of E(φ) are given by
φstatk (ρ, α) = (Uk(ρ), kα),
where (ρ, α) are the polar coordinates in R2 and (θ, ϕ) are the spherical coordinates in S2
andUk(ρ) = 2arctan ρk.
Proof. The solutions above can be found by using the projective representation. As wefound above, in this representation the minimizers of the energy are either holomorphicor anti-holomorphic functions. They are mapped into each other by the complex con-jugation. We consider holomorphic solutions. One can show that they are of the formw(x) = Q(z)
P (z), where P (z) and Q(z) are polynomials ?? with no common factors. The
degree of denominator = deg φ (the harmonic maps of degree n.).
72 Lectures on Applied PDEs
7 Gradient and hamiltonian systems
In this section we isolate two key classes of evolution equations - gradient and hamilto-nian systems. An example of the gradient flow is the nonlinear heat (reaction-diffusion)evolution, (4.9). The examples of the hamiltonian systems are the Hartree and nonlinearSchrodinger equations, (4.17) and (4.19). Former relies on the inner product on thespace of solutions and the latter, on the symplectic form.
In what follows we use definitions of functional, the Gateaux derivative and criticalpoint from Section 6.
7.1 Gradient systems
To define the gradient evolution equations (or systems), we first define the gradient of afunctional. Let X be a real space with a real inner product, which we denote g, so thatg(v, w) is a bilinear functional satisfying all the conditions of the inner product. Let Mbe an open subset of X and E : M → R and differentiable functional on M . We definethe gradient of E at u w.r. to the inner product g as an element gradg E(u) ∈ X definedby the equation
g(gradg E(u), v) = dE(u)v. (7.1)
(By the Riesz representation theorem gradg E(u) is well defined, provided X is a Hilbertspace; we will not go into further details on this and refer to [16].) Thus the gradientdefines the map, or the vector field, gradg E : u ∈ X → gradg E(u) ∈ X. Hence we candefine the evolution equation
∂tu = − gradg E(u). (7.2)
Such equations are called the gradient systems or gradient flows.
Energy dissipation. For a gradient system the energy, i.e. the functional E(u) de-creases (we say that a gradient system is dissipative):
where X is the corresponding Hilbert space. Indeed, using the relation (following fromthe Leibniz rule)
E(ut) = E(u0) +
∫ t
0
d
dsE(us) ds, (7.3)
one shows that E(ut) < E(u0).We consider a few examples.
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The nonlinear heat equation. The nonlinear heat (or reaction-diffusion) equation,
∂tu = ∆u+ g(u),
is a gradient system on the phase space X = H2(Rn) with the functional
E(u) =
∫(1
2|∇u|2 +G(u))ddx (7.4)
where G(u) is the anti-derivative of g(u), (one of the most widely used functionals) andL2−metric. Indeed, we compute
grad E(u) = −∆u+G′(u).
In particular, the Allen - Cahn initial value problem (2.2) is a gradient system,
∂tu = − grad E(u), (7.5)
on the phase space X, where the energy E is defined as
E(u) :=
∫Ω
1
2(ε2|∇u|2 +G(u)) dx, (7.6)
with G′ = g, and where the gradient is defined with respect to the real L2 inner product
〈u, v〉 :=
∫Rduv dx
on the tangent spaces of X. Notice that E is well defined on the space X. (Recallthat grad E(u) ∈ L2 is defined by the relation 〈v, grad E(u)〉 = ∂E(u)v, where ∂E(u) isthe Frechet derivative of E . Clearly, grad E(u) is the L2 function corresponding to thefunctional ∂E(u) in the L2 pairing.)
To have finite energy, one imposes the boundary conditions
u(x)→ ±1 as x→ ∂Ω.
In this section we consider Ω = Rd, though the results can be extended to (and eveneasier for) other domains.
The Cahn-Hilliard equation. The Cahn-Hilliard equation is given by
ut = −∆(ε2∆u− g(u)), (7.7)
It is a gradient system with the energy functional (7.6) and the metric
〈u, v〉H−10
:=
∫Ω
u∆−1vdx,
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that isdu
dt= −gradH−1
0E(u).
If X is a real inner product space with the inner product g(ξ, η), then there exists avector, gradgE(u), such that
〈gradgE(u), ξ〉 = dE(u)ξ. (7.8)
This vector is called the gradient of E(u) at u w.r.t. the metric g). Specifically, if X is anL2−space, then the L2– gradient, grad E(u), of the functional E(u) is defined by
dE(u)ξ =
∫grad E(u)ξdx. (7.9)
For example the L2– gradient of the functional∫
ΩGu dnx is the L2(Ω)–function G′(u(x)).
Exercise 7.1. Compute the L2– gradients in examples 1)–3) at the beginning of thissupplement.
Similarly, we define partial L2−gradients gradu E(u, v) by fixing v and differentiatingw.r.to u.
Gradient formulation of the Keller-Segel equations
7.2 Hamiltonian Equations
Let Z be a real vector space and assume there is a skew-symmetric, non-degenerate twoform, ω(v, w), on Z, called the symplectic form or symplectic structure. For a functionH : Z → R, we define the vector field XH : Z → Z by the equation
ω(XH(u), ξ) = dH(u)ξ, (7.10)
∀ξ ∈ Z. We define the hamiltonian system or hamiltonian flow as the equation
∂tu = VH(u). (7.11)
This equation is called the Hamiltonian equation, Z, the phase space and the functionH, the Hamiltonian.
The pair (Z, ω), a vector space Z and symplectic form ω is called a symplectic space. AHamiltonian system is a pair: a symplectic space, (Z, Ω), and a Hamiltonian function,H : Z → R.
In what follows we assume that Z is a real inner-product space with an inner product〈·, ·〉 and denote the gradient corresponding to this inner product by ∂ (or sometimes by∇) and similarly for partial gradients. Recall that for a bounded operator A, the adjointis defined by 〈A∗u, v〉 = 〈u,Av〉.
Lectures on Applied PDEs 75
Proposition 7.2. Suppose there is a linear, invertible, bounded operator J : Z∗ → Zsuch that J∗ = −J (J is called a symplectic operator), then the bi-linear form
ωJ(v, w) = 〈v, J−1w〉 (7.12)
is a symplectic form on Z and the hamiltonian vector field corresponding to a hamiltonianH : Z → R is given by
XH(u) = −J∂H(u). (7.13)
Consequently the hamiltonian equation in this case is ∂tu = −J−1∂H(u).
Proof. Comparing the two definitions 〈∂H(u), ξ〉 = dH(u)ξ and dH(u)ξ = ω(XH(u), ξ) =〈XH(u), J−1ξ〉 = 〈(J−1)∗XH(u), ξ〉, we conclude that 〈∂H(u), ξ) = 〈(J−1)∗XH(u), ξ〉.Since this is true for all ξ ∈ Z, we conclude that ∂H(u) = (J−1)∗XH(u) and thereforeXH(u) == J∗∂H(u) = −J∂H(u), which gives (7.13).
Examples:1) The hamiltonian system of classical mechanics: the phase space and symplectic
form are given byZ = R3
x × R3k, ω(z, z′) := x · k′ − x′ · k, (7.14)
where z := (x, k), and the hamiltonian H : Z → R. We can write the symplectic form in(7.14) as ω(z, z′) = z · J−1z′, where the symplectic operator J is given by
J =
(0 1−1 0
), (7.15)
yielding the hamiltonian vector field XH(z) = (∂kH,−∂xH) and the equations
x = ∂pH and p = −∂xH. (7.16)
(These can also be computed directly using (7.10).) For a classical particle of mass m in apotential V (x), we have and H(x, k) = 1
2m|k|2 +V (x), so that XH(z) = ( 1
mk,−∇V (x)).
2) The hamiltonian system of classical field theory: the phase space and symplecticform are given by
Z = H1(Rd,Rm)×H1(Rd,Rm), ω(v, v′) :=
∫Rd
(ξη′ − ξ′ · η) , (7.17)
where v := (ξ, η). We can write the symplectic form in (7.17) as ω(v, v′) = 〈v, ·J−1v′〉,where the symplectic operator J is given by (7.15) (but defined on a different space). Givena hamiltonian H(φ, π), this yields the hamiltonian vector field XH(φ, π) = (∂πH,−∂φH),which leads to the equations
∂tφ = ∂πH and ∂tπ = −∂φH. (7.18)
76 Lectures on Applied PDEs
For the Klein-Gordon CFT,
H(φ, π) =
∫Rd
1
2|π|2 +
1
2|∇φ|2 + f(φ)
dx, (7.19)
the hamiltonian vector field is XH(φ, π) = (π,−∆φ−f ′(φ)) and the corresponding hamil-tonian equation, which, after elimination of π, becomes the nonlinear wave (or Klein -Gordon) equation
φ− f ′(φ) = 0, := ∂2t −∆x. (7.20)
( = ∂µ∂µ in the Minkowski metric.)
3) More generally, let V be a Banach space and V ′ denotes its dual, i.e. the space ofbounded linear functionals. It is also a Banach space in the norm ‖α‖ := sup‖v‖=1 |α(v)|.We consider Z = Ω× V ′, where Ω ⊂ V and define a symplectic form on Z by ω(v, v′) :=η′(ξ) − η(ξ′), where v := (ξ, η). If V is reflexive, i.e. V ′′ = V , and there is a linearinvertible, bounded operator J : Z ′ → Z, which is anti-self-dual (i.e. J ′ = −J , where thedual J ′ is defined by J ′α(v) = α(Jv), or 〈J ′u, v〉 = 〈u, Jv〉), then again we can define asymplectic form on Z by (7.12), where 〈η, ξ〉 is understood as a coupling between V andV ′ (i.e. linear functionals η(ξ) on V are written as η(ξ) = 〈η, ξ〉). Then the symplecticform ω(v, v′) := η′(ξ)− η(ξ′) is given by (7.12) with J given by (7.15).
For more details see Appendix F.
7.3 Energy-momentum tensor (to-do)
7.4 Symmetries and conservation laws (to-do)
Lectures on Applied PDEs 77
8 Mean curvature flow
8.1 Definition and general properties
The mean curvature flow (starting with a hypersurface S0 in Rn+1) is the family of hy-persurfaces S(t) given by local parametrizations x(·, t) : U → Rn+1 (or immersions, i.e.dx(t) are one-to-one), where U is an open set in Rn, which satisfy the evolution equation
∂x∂t
= −H(x)ν(x)x|t=0 = x0
(8.1)
where x0 parametrizes (or is an immersion of) S0, H(x) and ν(x) are mean curvatureand the outward unit normal vector at x ∈ S(t), respectively. The terms used above areexplained in Appendix G.2, for more details and extensions, see [29].
In this lecture we describe some general properties of the mean curvature flow, (8.1).We begin with writing out (8.1) for various explicit representations for surfaces St.
Mean curvature flow (MCF) for level sets and graphs. We rewrite out (8.1) forthe level set and graph representation of S. Below, all differential operations, e.g. ∇,∆,are defined in the corresponding Euclidian space (either Rn+1 or Rn).
1) Level set representation S = ϕ(x, t) = 0. Then, by Proposition G.3 of AppendixG.2, we have
ν(x) =∇ϕ(x)
|∇ϕ(x)| , H(x) = div
( ∇ϕ|∇ϕ|
)(x). (8.2)
We compute 0 = dϕdt
= ∇xϕ · ∂x∂t + ∂ϕ∂t
and therefore ∂ϕ∂t
= ∇ϕ · ∇ϕ|∇ϕ| div(∇ϕ|∇ϕ|
), which
gives∂ϕ
∂t= |∇ϕ| div
( ∇ϕ|∇ϕ|
). (8.3)
2) Graph representation: S = graph of f . In this case S is the zero level set of thefunction ϕ(x) = xn+1 − f(u), where u = (x1, . . . , xn) and x = (u, xn+1), and using(8.3) with this function, we obtain
∂f
∂t=√|∇f |2 + 1 div
(∇f√|∇f |2 + 1
). (8.4)
Denote by Hess f the standard euclidean hessian, Hess f :=(
∂2f∂ui∂uj
). Then we can
rewrite (8.4) as∂f
∂t= ∆f − ∇f Hess f∇f
|∇f |2 + 1. (8.5)
78 Lectures on Applied PDEs
Exercise 8.1. Using (8.2), find the graph representation of the mean curvature,
H(x) = div
(∇f√
1 + |∇f |2
). (8.6)
Different form of the mean curvature flow. Multiplying the equation (8.1) in Rn+1
by ν(x), we obtain the equation
ν(x) · ∂x∂t
= −H(x). (8.7)
In opposite direction we have
Proposition 8.2. If x satisfies (8.7), then there is a (time-dependent) reparametrizationϕ of S, s.t. x ϕ satisfies (8.1).
Proof. Denote (∂x∂t
)T := ∂x∂t− (ν · ∂x
∂t)ν (the projection of ∂x
∂tonto TxS) and let ϕ satisfy the
ODE ϕ = −(dx)−1(∂x∂tϕ)T (parametrized by u ∈ U). Then ∂
∂t(x ϕ) = ( ∂
∂tx) ϕ+ dx ϕ.
Substituting ϕ = −(dx)−1(∂x∂t ϕ)T into this, we obtain ∂
∂t(x ϕ) = (ν · ( ∂
∂tx) ϕ)ν =
−H.
Thus the MCF in the form (8.7) is invariant under reparametrization, while the form(8.1) is obtained by fixing a specific parametrization (fixing the gauge).
Now we describe some general properties of MCF. We summarize these properties asfollows
• (8.1) is invariant under rigid motions of the surface, i.e. ψ 7→ Rψ + a, whereR ∈ O(n + 1), a ∈ Rn+1 and ψ = ψ(u, t) is a parametrization of St, is a symmetryof (8.1).
• (8.1) is invariant under the scaling, for any λ > 0,
S(t)→ λS(λ−2t) ⇔ x→ λx, t→ λ−2t. (8.8)
• (8.1) is area shrinking. Actually, ddtV (ψ(t)) = −
∫StH2dσ ≤ 0.
• Static solutions of (8.1) are minimal surfaces, i.e. critical points of the area func-tional, in particular they satisfy the equation H = 0.
The first property comes from the fact that the mean curvature is invariant under trans-lations and rotations which is obvious. The remaining properties are proven below (thelast property was already proven as an example in Section 6.3, see Proposition 6.7).
Proposition 8.3. (8.1) is invariant under the scaling x 7→ λx and t 7→ λ−2t for anyλ > 0.
Lectures on Applied PDEs 79
Proof. To prove the invariance under scaling, we first observe the scaling property of themean curvature, (8.2),
H(λx) = λ−1H(x). (8.9)
To show this, we use the level set representation S = ϕ(x, t) = 0 of S. Then λS =ϕ(λ−1x, t) = 0. Hence, denoting ϕλ(x, t) := ϕ(λ−1x, t) and using (8.2), we compute
H(ϕλ)(x, t) = div
( ∇ϕ(λ−1x, t)
|∇ϕ(λ−1x, t)|
)(8.10)
= λ−1div
( ∇ϕ|∇ϕ|
)(λ−1x, t) (8.11)
= H(ϕ)(λ−1x, t), (8.12)
as claimedNow, let τ := λ−2t, and xλ(t) := λx(τ). Then
∂txλ = λλ−2(∂τx)(τ) = −λ−1H(x(τ))ν(x(τ)).
On the other hand, νλ ≡ ν(xλ(τ)) = ν ≡ ν(x(τ)) and Hλ ≡ H(xλ(τ)) = λ−1H(x(τ)) ≡λ−1H (by (8.9)), which gives
∂txλ = −Hλνλ. (8.13)
For S = graph f , f(u) → λf(λ−1u) ⇔ ψ(u) → λψ (λ−1u) , where ψ(u) := (u, f(u)).Indeed,
ψ(u) := (u, f(u))→(u, λf(λ−1u)
)= λ
(λ−1u, f(λ−1u)
)=: λψ
(λ−1u
).
Exercise 8.4. Check (8.9) using the graph representation of the mean curvature, (8.6).
Volume functional. Recall that for a surface S given locally by an immersion ψ : U →Rn+1, the surface volume functional can be written as
V (ψ) =
∫U
√gdnu,
where g := det(gij) for a metric gij :=⟨∂ψ∂ui, ∂ψ∂uj
⟩. Recall, from Lemma G.1 that for a
surface S given by a graph of a function f : U → R, the surface area is given by theformula
V (f) ≡∫S
1 =
∫U
√1 + |∇f |2dnu. (8.14)
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Proposition 8.5. • If a closed surface S evolves according to (8.1), then ∂tV (S) =−∫SH2 < 0 and consequently, its area, V (S), decreases.
• Critical points of V (S) are static solution to the MCF (8.1).
Proof. We prove the proposition for a surface S given by a graph of a function f : U → R,the general case can be derived from this result. In this case, the surface area is given bythe formula (8.14). Let ξ ∈ H1
0 (U). We compute, using that ξ|∂U = 0,
dV (f)ξ = ∂s|s=0V (f + sξ) =
∫U
∇f√1 + |∇f |2
· ∇ξdnu,
which, after integrating by parts and using the equation H(x) = div
(∇f√
1+|∇f |2
)(see
(8.6)), gives
dV (f)ξ =
∫U
Hξdnu. (8.15)
Hence f is a critical point iff∫UHξdnu = 0 for all ξ ∈ H1
0 (U) which gives H = 0.Using (8.15) we find
dV (f)ξ =
∫U
Hξdnu =
∫U
√1 + |∇f |2Hξ dnu√
1 + |∇f |2.
This proves the second statement. To prove the first one, we compute ∂tV (f) = dV (f)∂tf ,which due to the previous relation gives
∂tV (f) =
∫U
H∂tfdnu = −
∫U
√1 + |∇f |2H2dnu = −
∫S
H2,
as stated in the first statement.
For an immersion ψ : U → Rn+1, we can rewrite (8.15) as
dV (ψ)ξ =
∫U
Hξdnu. (8.16)
8.2 Special solutions
Static solutions. Static solutions satisfy the equation H(x) = 0 on S, which, as shownabove, is the equation for critical points of the volume functional V (ψ) (the Euler - La-grange equation for V (ψ)). Thus, static solutions of the MCF (8.1) are minimal surfaces.
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Spherically and axi - symmetric (equivariant) solutions:
a) Sphere. The n−dimensional sphere of the radius R(t) centred at the origin can begiven either locally by the immersion x(t) = R(t)(u,
√1− |u|2), u ∈ Rn, |u| < 1, or
as the level set ϕ(x, t) = 0, where ϕ(x, t) := |x|2 − R(t)2, or as graph f , wheref(u, t) =
√R(t)2 − |u|2. (SR = RSn, where Sn is the unit n-sphere.) Then we have
H(x) = div
( ∇ϕ|∇ϕ|
)= div(x) =
n
R
and therefore we get R = − nR
which implies R =√R2
0 − 2nt. So this solutionshrinks to a point.
b) Cylinder. Define the n−dimensional cylinder of the radius R(t) with the axis alongthe xn+1−axis locally by the immersion x(t) = (u,
√R2(t)− |u|2, x′′), u ∈ Rn, |u| <
R(t), where x = (x′, x′′) ∈ Rn+1 × R1. Then H(x) = n−1R
and R = −n−1R
which
implies R =√R2
0 − 2(n− 1)t. (In the implicit function representation the cyliner
is given by ϕ = 0, where ϕ := r −R, with r =(∑n−1
i=1 x2i
) 12 .)
Exercise 8.6. Show that the above expressions give solutions to the mean curvature flowequation, (8.1).
Motion of torus (H. M. Soner and P. E. Souganidis). (to be done)
8.3 Self-similar surfaces
Recalling (??) - (2.50), or (8.8), we consider solutions of the MCF of the form S(t) ≡Sλ(t) := λ(t)S (standing waves), or x(u, t) = λ(t)y(u), where λ(t) > 0. Plugging thisinto (8.1) and using H(λy) = λ−1H(y), gives λy = −λ−1H(y)ν(y), or λλy = −H(y)ν(y).Multiplying this by ν(y), we obtain (cf. (2.50))
H(y) = a〈ν, y〉, and λλ = −a. (8.17)
Since H(y) is independent of t, then so should be λλ = −a. Solving the last equation, wefind λ =
√λ2
0 − 2at.
i) a > 0⇒ λ→ 0 as t→ T :=λ202a⇒ Sλ is a shrinker.
ii) a < 0⇒ λ→∞ as t→∞⇒ Sλ is an expander.
Recall, that static solutions of the MCF (8.1) satisfy the equation H(x) = 0 on S andtherefore are minimal surfaces. Hence the minimal surfaces are special cases of self-similarones corresponding to a = 0. In fact, a = 0 separates two types of evolution: contracting
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a > 0 ( λ decreasing) and expanding a < 0 ( λ increasing). (Remember that a = −λ∂tλ isthe negative of the speed of scaling λ.) (We see that scaling solitons generalize the notionof the minimal surface.)
The equation (8.17) has the solutions: a is time-independent and x is one of thefollowing
a) Sphere x = Rx, where R =√
na.
b) Cylinder x = (Rx′, x′′), where x = (x′, x′′) ∈ Rn+1 × R1, where R =√
na.
As stated in the following theorem, these solutions are robust.
Theorem 8.7 (Huisken). Let S satisfy H = ax · ν and H ≥ 0. We have(i) If n ≥ 2, and S is compact, then S is a sphere of radius
√na
.
(ii) If n = 2 and S is a surface of revolution, then S is the cylinder of radius√
n−1a
.
Now we consider self-similar surfaces in the graph representation. Let
f(u, t) = λχ(λ−1u), λ depends on t. (8.18)
Substituting this into (8.4) and setting y = λ−1u and a = −λλ, we find√1 + |∇yχ|2H(χ) = a(y∂y − 1)χ. (8.19)
Translation solitons. These are solutions of the MCF of the form S(t) ≡ S + h(t)(traveling waves), or x(u, t) = y(u) + h(t), where h(t) ∈ Rn+1. Plugging this into (8.1)and using H(y + h) = H(y), gives h = −H(y)ν(y), or
H(y) = v · ν(y), and h = v. (8.20)
Since H(y) is independent of t, then so should be h = v. (more to come)
Breathers. MCF periodic in t.
Self-similar surfaces and rescaled MCF. Minimal surfaces are static solutions of theMCF (8.1). Are more general the self-similar surfaces static solutions of some equation?The answer is yes, the self-similar surfaces are static solutions of the rescaled MCF,
∂τϕ = −(H(ϕ)− aϕ · ν(ϕ))ν(ϕ), (8.21)
where a = −λλ, which is obtained by rescaling the surface ψ and time t as
ϕ(u, τ) := λ−1(t)ψ(u, t), τ =
∫ t
0
λ−2(s)ds, (8.22)
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and then reparametrizing the obtained surface Sresc(τ) := λ(t)S(t) as in the proof ofProposition 8.2. Indeed, since λH = Hλ (or λH(ψ) = H(λ−1ψ)), we find
∂τϕ = λ2∂tϕ = λ2(−λλ−2ψ + λ−1∂tψ)
= −λλϕ− λH(ψ)ν(ψ) = aϕ−H(ϕ)ν(ϕ).
Then the mean curvature flow equation (8.1) implies
∂τϕ = −H(ϕ)ν(ϕ) + aϕ, (8.23)
which after the reparametrization gives (8.21). This is another analogy with minimalsurfaces.
For static solutions, a =const. Then solving the equation λλ = −a, we obtain theparabolic scaling:
λ =√
2a(T − t) and τ(t) = − 1
2aln(T − t), (8.24)
where T := λ20/2T , which was already discussed in connection with the scaling solitons.
Now, by Proposition 8.5, we know that minimal surfaces are critical points of thevolume functional V (ψ). Are self-similar surfaces critical points of some modification ofthe volume functional? (Recall that, because of reparametrization (see Proposition 8.2),it suffices to look only at normal variations, ψs, of the immersion ψ, i.e. generated byvector fields η, directed along the normal ν: η = fν.) The answer to this question is yesand is given in the following
Proposition 8.8. Let ρa(x) = e−a2|x|2 and Va(ϕ) :=
∫Sρa. For a surface S given locally
by an immersion ϕ and normal variations, η = fν, we have
dVa(ϕ)η =
∫U
(H − aϕ · ν)ν · η ρadnu, (8.25)
Proof. The definition of Va(ϕ) gives Va(ϕ) =∫Uρa(ϕ)
√g(ϕ), where ρa(ϕ) = e−
a2|ϕ|2 and
g(ϕ) := det(gij). We have dρa(ϕ)η = −aρa(ϕ)η. Using this formula and the equationd√gη = H
√gν · η (see [29]) and the fact that we are dealing with normal variations,
η = fν, we obtain (8.25).
The equation (8.21) and Proposition 8.8 imply
Corollary 1. Assume a in (8.21) is constant (i.e. the rescaling (8.22) is parabolic,(8.24)). Then the modified area functional Va(ϕ) is momotonically decaying under therescaled flow (8.21), more precisely,
∂τVa(ϕ) = −∫Sλρa|H − aν · ϕ|2. (8.26)
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The relation (8.26) is the Huisken monotonicity formula (earlier results of this typewere obtained by Giga and Kohn and by Struwe).
Most interesting minimal surfaces are not just critical points of the volume functionalV (ψ) but are minimizers for it. What about self-similar surfaces? We see that
(i) inf Va(ϕ) = 0 and, for compact minimal surfaces, Va(ϕ) is minimized by any sequenceshrinking to a point.
(ii) Va(ϕ) is unbounded from above. This is clear for a < 0. To see this for a < 0, weconstruct a sequence of surfaces lying inside a fixed ball in Rn+1 and folding tighterand tighter.
Thus self-similar surfaces are neither minimizers nor maximizers of Va(ϕ). We conjecturethat they are saddle points satisfying min-max principle: supV infϕ:V (ϕ)=V Va(ϕ). One cantry use this principle (say in the form of the mountain pass lemma) to find solutions of(8.17).
For more on the self-similar surfaces see Appendix H.
8.4 Volume preserving mean curvature flow
The volume preserving mean curvature flow (VPF) is a family St; t ≥ 0 of smoothclosed hypersurfaces in Rn+1, given say by immersions ψ, satisfying the following evolutionequation:
∂tψN = H −H, (8.27)
where ∂tψN denotes the normal velocity of St at time t, ∂tψ
N = 〈∂tψ, ν〉, where ν isthe unit normal vector field on St, H = H(t) stands for the mean curvature of St andH = H(t) is the average of the mean curvature on St, i.e.,
H :=
∫StHdσ∫
Stdσ
, t ≥ 0. (8.28)
As an initial condition, we consider a simple hypersurface S0 in Rn+1, with no boundaryin Ω (e.g. either entirely Ω or ∂S0 ⊂ ∂Ω) given by an immersion x0.
Like the MCF, the VPF is invariant under rigid motions (translations and rotations)and appropriate scaling. Moreover, it shrinks the area, A(ψ) = V (ψ), of the surfaces,but, unlike the MCF, the VPF preserves the enclosed volume, Vencl(ψ). As the result,it has stationary solutions - the Euclidean spheres (for closed surfaces) and cylinders forsurfaces with flat boundaries. We summarize these properties as follows
• (8.27) is invariant under rigid motions of the surface, i.e. ψ 7→ Rψ + a, whereR ∈ O(n + 1), a ∈ Rn+1 and ψ = ψ(u, t) is a parametrization of St, is a symmetryof (8.27).
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• (8.27) is invariant under the scaling x 7→ λx and t 7→ λ−2t for any λ > 0.
• (8.27) is volume preserving.
• (8.27) is area shrinking. Actually, ddtV (ψ(t)) = −
∫St
(H −H)2dσ ≤ 0.
• Static solutions of (8.27) are surfaces of constant mean curvature (CMC), H = h.
The first two statements are proven as for the MCF. The third and fourth ones followfrom d
dtVencl(ψ(t)) =
∫St〈∂tψ, ν〉dσ =
∫St
(H −H)dσ = 0 and (cf. Proposition 8.3 and itsproof)
d
dtV (ψ(t)) =
∫St
H〈∂tψ, ν〉dσ =
∫St
(HH −H2)dσ = −∫St
(H −H)2dσ ≤ 0.
The fifth one is obvious.The third and forth properties suggests that, as t → ∞, St converges to a closed
surfaces with the smallest surface area for a given enclosed volume. The limiting surfacemust be a static solution to the VPF and therefore, by the fifth property, it is a surfaceof constant mean curvature (CMC). This leads to the isoperimetric problem:
• minimize the area V (ψ) given the enclosed volume Vencl(ψ).
Namely, we have
Proposition 8.9. (i) Minimizers of the area V (ψ) for a given enclosed volume Vencl(ψ)are critical points of the area functional V (ψ) on space of immersions with the givenenclosed volume Vencl(ψ).
(ii) The (Euler-Lagrange) equation for these critical points is exactly the CMC equationH = h.
(iii) These critical points are critical points of the functional
Vh(ψ) := V (ψ)− hVencl(ψ),
where h is determined by c = Vencl(ψ) and vice versa.
Proof. The first and third statements are standard results (the third statement is followsfrom the Lagrange multiplier theorem). The second statement follows from the relationsdV (ψ)ξ =
∫UHν · ξ √gdnu and dVencl(ψ)ξ =
∫St〈ξ, ν〉dσ and the definition of Vh(ψ).
Thus finding stationary solutions to the VPF is the same as finding closed CMCsurfaces. This leads to the following problems: (a) Find CMC surfaces and (b) determinetheir stability w.r. to the VPF.
However, we expect that the VPF converges to a minimal CMC surface (i.e. onesolving the isoperimetric problem), not just to a CMC surface. For such we have anadditional characterization, which is a standard fact from the variational calculus (seee.g. [?]):
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Proposition 8.10. If ϕ minimizes the area V (ψ) for a given enclosed volume Vencl(ψ),then HessNV (ϕ) ≥ 0 on TϕXc.
Remark 8.11. The property of minimal CMC surfaces isolated in this proposition playsan important role in their analysis and deserves the name. We say that a CMC surfaceθ is stable iff HessNV (θ) ≥ 0 on TθXc. (In dynamical systems and partial differentialequations, this notion is called (weak) linear or energetic stability,see Sections 9.1 and9.2.)
8.4.1 The gradient property of the VPF
Furthermore, we have the following important property
• (8.27) is a gradient flow for the area functional on closed surfaces with given enclosedvolume.
Proof. To prove the last statement we use the formula (8.16), which we rewrite in thepresent notation as
dV (ψ)ξ =
∫ψ
Hν · ξ =
∫U
Hν · ξ √gdnu, (8.29)
and the fact that ξ is a vector field for deformations with a fixed enclosed volume, i.e.it is a tangent vector field to the manifold Xc := ψ : U → Rn+1, Vencl(ψ) = c (we donot specify the topology here, we could take the topology of differentiable or Sobolevfunctions), and therefore it satisfies
∫ψξ · ν = 0. Indeed, let ψs be a family of constant
enclosed volume surfaces deforming ψ and let ξ be the corresponding vector field at s = 0,i.e. ξ = ∂sψs
∣∣s=0
. Since Vencl(ψs) = c, we have dVencl(ψ)ξ = ∂sVencl(ψs)∣∣s=0
= 0. Hencethe result follows from the formula
dVencl(ψ)ξ =
∫ψ
ξ · ν. (8.30)
This formula can be proven by either considering an infinitesimal change in the enclosedvolume under the variation of S or by using that, by the divergence theorem, Vencl(ψ) =
1n+1
∫Ω
div x = 1n+1
∫Sx · ν, where Ω is domain enclosed by the surface S, described by
the immersion ψ, and then differentiating the latter integral, see Appendix ?? (to beadded). Conversely, if f satisfies
∫Sf = 0, then there is a volume preserving normal
variation with the vector field fν. This implies that
TθXc = ξ : S → Rn+1,
∫S
ξ · ν = 0.
Now, for an arbitrary normal vector field η = fν, the vector field ξ := (f − f)ν,where f := 1
|S|
∫Sf , satisfies
∫Sξ · ν = 0 and therefore we have dV (ψ)ξ =
∫ψHν · ξ =∫
ψH(f − f) =
∫ψ(H −H)f . This shows that the L2−gradient of V (ψ) on closed surfaces
with given enclosed volume is H −H.
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8.5 Surface diffusion flow (not finished)
We consider the surface diffusion flow (SDF). A family of immersions ψ(t, ·) : Σ →Rn+1, t ≥ 0, of a compact, connected, orientable manifold, Σ, into Rn+1 is called thesurface diffusion flow (SDF), iff it satisfies the equation
∂tψN = ∆H(ψ), (8.31)
where ∂tψN is the normal component of the velocity vector ∂tψ, defined as ∂tψ
N =(∂tψ, ν(ψ)
)Rn+1 , with ν(ψ), the outward unit normal to the surface St := ψ(t,Σ), ∆ is
the surface Laplace-Beltrami operator on St and H(ψ) is the mean curvature of St.Like the MCF and VPF, the SDF is invariant under rigid motions (translations and
rotations) and appropriate scaling and shrinks the area, A(ψ) = V (ψ), of the surfaces.Moreover, like VPF, it preserves the enclosed volume, Vencl(ψ), and has a rich class ofstationary solutions. We summarize these properties as follows
• (8.31) is invariant under rigid motions of the surface, i.e. ψ 7→ Rψ + a, whereR ∈ O(n + 1), a ∈ Rn+1 and ψ = ψ(u, t) is a parametrization of St, is a symmetryof (8.31).
• (8.31) is invariant under the scaling x 7→ λx and t 7→ λ−4t (check) for any λ > 0.
• (8.31) is volume preserving.
• (8.31) is area shrinking. Actually, ddtV (ψ(t)) = −
∫St|∇H|2dσ ≤ 0.
• Static solutions of (8.31) are surfaces of constant mean curvature (CMC), H = h.
The first two statements are proven as for the MCF. The third and fourth ones follow fromddtVencl(ψ(t)) =
∫St〈∂tψ, ν〉dσ =
∫St
∆Hdσ = 0 and (cf. Proposition 8.3 and its proof)
d
dtV (ψ(t)) =
∫St
H〈∂tψ, ν〉dσ =
∫St
H∆Hdσ = −∫St
|∇H|2dσ ≤ 0.
The fifth one is obvious.As a consequence, the enclosed volume of St is constant in time under (8.31), while
its area is decreasing.For more on the surface diffusion flow see Appendix 9.3.
8.6 Droplets on a surface (to-be-done)
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9 Linear stability
9.1 Set-up and definitions
In this section we study the linear stability of stationary (and traveling wave) solutions.The general theory is discussed in the next section. At the moment we give an ad hocdefinition of the linear stability, which we will explain below. Consider the abstractdynamical system, (2.1), which we reproduce here:
∂u
∂t= F (u). (9.1)
Here F : U → Y , with U ⊂ X and X and Y Banach spaces. Let dF (u) denote theGateaux derivative of the map F (u) at u defined as
dF (u)ξ :=∂
∂λ
∣∣∣∣λ=0
F (u+ λξ), (9.2)
if the derivative on the r.h.s. exists and is a bounded and linear map map from X toY . (In fact, the boundedness and continuity in u is enough, see Appendix B for moredetails.)
We say that a stationary solution u∗ of (9.1) is linearly stable iff
σ(dF (u∗)) ⊂ Re z < 0 (9.3)
and linearly unstable iff
σ(dF (u∗) ∩ Re z > 0 6= ∅. (9.4)
Note that this definition says nothing about hamiltonian equations for which we havealways σ(dF (u∗)) ⊂ Re z = 0 .
To exhibit the origin of this definition, we write the equation (9.1) in the canonicalform as follows. First, we expand F (u) around u∗ and use that F (u∗) = 0 to obtain
F (u∗ + ξ) = dF (u∗)ξ + f(ξ), (9.5)
where the term f(ξ) is defined by this expansion. According to (B.3) of Theorem B.3of Appendix B, it is of a higher order, f(ξ) = o(‖ξ‖). Substituting u = u∗ + ξ and thisexpansion into (9.1), we find
∂ξ
∂t= Lξ + f(ξ), (9.6)
where L := dF (u∗). This is the abstract dynamical system in the canonical form.
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If the initial condition, u0, for (9.1) is close to u∗, then the initial condition, ξ0 = u0−u∗,for (9.6) is close to 0. If u∗ is stable, then ξ remains small for all times. This suggeststhat f(ξ) which is of a higher order in ξ does not have significant effect on the behaviourof the solution and therefore can be omitted on the first step. This leads to the linearequation
∂ξ
∂t= Lξ. (9.7)
In a very large variety of situations (e.g. when L∗ is self-adjoint), the spectrum of thelinear operator L∗ determines the long time behaviour of the latter equation (see AppendixD), which leads to the definition above. (In all situations, the spectrum of the linerizedmap is the first step in finding asymptotic behavior for the evolution equation in questionfor initial conditions close to the static solution.)
Example 9.1. Let X = R2 and let L be a real 2 × 2 matrix with eigenvalues λ1, λ2.According to the nature of the spectrum of L∗, we have the qualitative pictures for the flowφt(u) = eLtu shown in Figure 7.
λ1
, λ2
< 0 λ1
,λ2
>0 λ1
< 0, λ2
>0
unstableasymptotically stable unstable
Reλ1
= Reλ2
< 0
asymptotically stable
Re λ1
= Reλ2
= 0
Lyapunov stable unstable
Reλ1=Reλ2
>0
Figure 7: Flow for Eqn. (9.7) and X = R2
Remark. If the spectrum of dF (u∗) is real, then the stability condition says that
dF (u∗) ≤ −θ, for some θ > 0. (9.8)
The exercise below shows that in this case the solutions to (9.7) tends to 0 exponentiallyfast
‖ξ‖ ≤ e−θt‖ξ0‖. (9.9)
90 Lectures on Applied PDEs
Exercise 9.1. Prove (9.9). Hint: let L := dF (u∗) and define the Lyapunov functionalΛ(ξ) := 1
2‖ξ‖2. Using the Leibnitz rule, (9.7) and (9.8) show the differential inequality
∂tΛ(ξ) ≤ −2θΛ(ξ). Solve this inequality to obtain (9.9).
As an example consider the Allen - Cahn equation (see (2.2))
∂u
∂t= ∆u− g(u), (9.10)
where g : R → R is the derivative, g = G′, of a double-well potential: G(u) ≥ 0 and hastwo non-degenerate global minima, say at ±1, with the minimum value 0:
(For g(u) = u3 − u, we take G(u) = 12(u2 − 1)2. G(u) is also called a bistable potential,
see Figure 8.)This equation has three homogeneous static solutions, u∗ = ±1, 0.
G
1−1
u
Figure 8: Double well potential
Exercise 9.2. (i) Consider (9.10) on R. Prove that u∗ = ±1 are linearly stable staticsolutions and u∗ = 0 is not. Hint: Show that the linearized operator around u∗ is
L := ∆− g′(u∗).
Assuming that L is self-adjoint, show that that
σ(L) = (−∞, G′′(u∗)].
Then observe the sign of G′′(u∗).(ii) Consider (9.10) on [0, 2π] with the Dirichlet boundary conditions, u(0) = u(2π) =
0. More precisely, we consider (9.10) on H20 ([0, 2π]) (where the subindex 0 signifies the
the Dirichlet boundary conditions). (By the Sobolev embedding theorem, H2([0, 2π]) ⊂C([0, 2π]) and therefore the boundary conditions u(0) = u(2π) = 0 make sense. As usual,we say that ∆ is defined on L2([0, 2π]) with the domain H2
0 ([0, 2π]).) Investigate whetherthe static solutions u∗ = 0,±1 are linearly stable or not.
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Now, instead of the Allen - Cahn equation (9.10), we consider the Gross-Pitaevskii ornonlinear Schrodinger equation
i∂u
∂t= ∆u− f(|u|2)u, (9.12)
with u : Ω → C and f(|u|2) = G′(|u|2), where G(|u|2) is a double well potential, sayG(|u|2) = 1
2(|u|2 − 1)2 or more generally as shown on Figure 8. Then, as follows from
general results in the next subsection and can be verified explicitly in this example, thelinearized operator around u∗ = ±1 has always zero eigenvalue. This is related to thegauge symmetry of this equation.
9.2 Systems with symmetry and zero modes
Often, the presence of symmetries implies that 0 is an eigenvalue of the linearized operatorand therefore condition (9.3) is never satisfied (see Proposition 9.3 below). This leads tomodification of the notion of linearized stability.
Recall from Subsection 2.3, that we say that dynamical system (9.1) has a symme-try group G if G acts on the space of solutions, X, via a representation (i.e a grouphomomorphism) T : G→ GL(X), T : g → T (g), and each T (g) satisfies
F (T (g)u) = T (g)F (u). (9.13)
(See Subsections 2.3 and 6.10.1 for examples.) We assume that G is a (matrix) Lie group(see [15] for definitions related to Lie groups and Lie algebras).
In the case of symmetry, condition (9.3) never holds: the Gateaux derivative dF (u∗)has always zero eigenvectors. Indeed, let g be its Lie algebra, i.e. the tangent space of Gat g = 1, or the space of all matrices a, such that eat is in G for all t ∈ R. Then thereis a unique representation (i.e. a Lie algebra homomorphism) τ of g acting on the samespace such that T (ea) = eτ(a), ∀a ∈ g. (τ(g) is the Lie algebra of T (G).) We have
Proposition 9.3. τ(g)u∗ ⊂ Null(dF (u∗)). Specifically, for any A ∈ τ(g), the vector Au∗is an eigenvector of dF (u∗) with the eigenvalue 0,
dF (u∗)Au∗ = 0. (9.14)
Proof. If u∗ is a static solution to (9.1), then, due to (16.2), so is Tgu∗, for any g ∈ G. Inparticular, F (esAu∗) = 0, for any A ∈ τ(g) and s ∈ R. Differentiate this equation withrespect to s at s = 0 and use that ∂sF (esAu∗)
∣∣s=0
= Au∗ to obtain
0 = ∂sF (esAu∗)∣∣s=0
= F (u∗)Au∗.
Hence (9.14) follows.
92 Lectures on Applied PDEs
Specific examples are given in the next two subsections. A symmetry Ts = esA is saidto be unbroken if Tsu∗ = u∗, or Au∗ = 0, and broken otherwise.
Thus in the case of symmetries, if Au∗ 6= 0 for some A ∈ τ(g), then the assumption(9.8) fails. We replace it by the following weaker assumption
• Assume for simplicity that X is a Hilbert space and dF (u∗) is self adjoint. We sayu∗ is linearly orbitally stable iff
σ(dF (u∗)|(τ(g)u∗)⊥) ⊂ Re z < 0 . (9.15)
For a given u∗, consider the maximal subgroup, G∗, of G, which is not broken by u∗,i.e. T (g)u∗ = u∗,∀g ∈ G∗. In what follows, we use the quotient group G/G∗ instead ofG.
9.3 Linear stability of spheres, cylinders and planes
We consider the surface diffusion flow (SDF). A family of immersions ψ(t, ·) : Σ →Rn+1, t ≥ 0, of a compact, connected, orientable manifold, Σ, into Rn+1 is called thesurface diffusion flow (SDF), iff it satisfies the equation
∂tψN = ∆H(ψ), (9.16)
where ∂tψN is the normal component of the velocity vector ∂tψ, defined as ∂tψ
N =(∂tψ, ν(ψ)
)Rn+1 , with ν(ψ), the outward unit normal to the surface St := ψ(t,Σ), ∆ is
the surface Laplace-Beltrami operator on St and H(ψ) is the mean curvature of St.Recall that, like the MCF and VPF, the SDF, (9.16), is invariant under rigid motions
(translations and rotations) and appropriate scaling and has a large class of static solutions- the constant mean curvature immersions:
H = h. (9.17)
(See Subsection 8.5 for more details.)Let F (ψ) := ∆H(ψ) and let ψh be a surface of constant mean curvature h. By above,
the symmetry algebra τ(g) of (9.16) is generated by translations, rotation and scaling.Then by Proposition 9.3, τ(g)ψh ⊂ Null(dF (ψh)).
In general, τ(g)ψh is a n + 1 + (n + 1)(n + 2)/2 + 1-dimensional vector space. Forspecial symmetric surfaces this dimension is reduced correspondingly. For instance, forn-spheres, which are invariant under rotations, this space is of the dimension n + 1 + 1,for n-cylinders, which are invariant under rotations which their axes invariant and undertranslations along the axes, the dimension is n+ 1 + 1.
For spheres and cylinders, we can compute the nullspace expliocitly but this is a verylaborious operation (see [29] for some computations). Say, for n-sphere, SnR, of radius R,
Lectures on Applied PDEs 93
differentiating in the direction normal to the surface (to exclude reparametrizations, seeAppendix H.1), we find
LsphR = ∆Sn(
1
R2∆Sn +
n
R2), (9.18)
on L2(Sn), where ∆Sk is the Laplace-Beltrami operator on the standard k−sphere Sk.It is a standard fact that the operator −∆Sn is a self-adjoint operator on L2(Sn). Its
spectrum is well known (see [30]): it consists of the eigenvalues l(l+ n− 1), l = 0, 1, . . . ,
of the multiplicities m` =
(n+ ln
)−(n+ l − 2
n
). Moreover, the eigenfunctions cor-
responding to the eigenvalue l(l+n− 1) are the restrictions to the sphere Sn of harmonicpolynomials on Rn+1 of degree l and denoted by Ylm (the spherical harmonics),
In particular, the first eigenvalue 0 has the only eigenfunction ω0 = 1 and the secondeigenvalue n has the eigenfunctions ω1, · · · , ωn+1, where ωj = xj/|x|, j = 1, . . . , n+ 1.
Consequently, the operator LsphR = 1
R2 ∆Sn(∆Sn + n) is self-adjoint and its spectrumconsists of the eigenvalues 1
R2 l(l+n−1)(l(l+n−1)−n), l = 0, 1, . . . , of the multiplicities
m`. In particular, the first n + 2 eigenvectors of LsphR (those with l = 0, 1) correspond to
the non-positive eigenvalues,
LsphR ωj = 0, j = 0, . . . , n+ 1, (9.20)
and are due to the scaling (l = 0) and translation (l = 1) symmetries.
For the n−cylinder CnR = Sn−kR × Rk of radius R =√
n−ka
in Rn+1, we have
LcylR = −∆y −
1
R2∆Sn−k − 2
n− kR2
, (9.21)
acting on L2(Cn). The situation here is a bit more subtle due to the presence of theoperator −∆y with purely continuous spectrum [0,∞).
9.4 Linearized stability of kinks
Consider the linearized stability of the kink solutions of the Allen - Cahn equation (see(2.2))
∂u
∂t= ∆u− g(u), (9.22)
94 Lectures on Applied PDEs
where g(u) = u3 − u, or, more generally, g : R → R is the derivative, g = G′, of adouble-well potential: G(u) ≥ 0 and has two non-degenerate global minima, say at ±1,with the minimum value 0:
For g(u) = u3 − u, we take G(u) = 12(u2 − 1)2. G(u) is also called a bistable potential,
see Figure 18.
G
1−1
u
Figure 9: Double well potential
To keep the notation simple, assume we are in the dimension 1, i.e. u : R× R+ → R.Recall from Section 2.1 that (9.22) has the kink static solution by χ(x), see Figure 19 (forg(u) = u3 − u, we have ). (9.22) is translationally symmetric. Hence we expect that the
ïG
ba
a
b
x
u
Figure 10: Hill-valley-hill structure for −G, and the kink for u.
definition (9.3) - (9.4) carries no information in this case and we use the definition (9.15).We have
Theorem 9.4. The kink solutions of (9.22) are linearly orbitally stable w.r.to H1(R)−perturbationsin the sense of definition (9.15).
Proof. If we denote the negative of the r.h.s. of (9.22) by F (u),
F (u) := ∆u− g(u), (9.24)
then the linearization L := −dF (χ) of F at χ isis computed explicitly as
L := −∆ + g′(χ). (9.25)
The statement of the theorem follows from the theorem below.
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Theorem 9.5. (a) The operator L is self-adjoint.(b) 0 is a simple eigenvalue of L, with the eigenfunction χ′.(c) the spectrum of L lies in [0,∞) and in (0,ming′(1), g′(−1)) consists of isolated
eigenvalues of finite multiplicities, which might accumulate only at ming′(1), g′(−1).
Proof. It is straightforward to show that the operator L is symmetric. The self-adjointnessis a stronger properties and follows from observing that L is of a Schrodinger type andstandard results about Schodinger operators (see e.g. [12]).
To prove the second statement, we observe that, by Proposition 9.3 and the transla-tional invariance of (2.2), 0 is an eigenvalue of L with eigenfunction ζ := ∂xχ,
Lχ′ = 0. (9.26)
For an exercise we prove this directly. Since χ solution to F (u) = 0 (see Section 2.1 and(2.5)), where the map F is defined in (17.26), hence, for any a ∈ R, χa is also a solutionto F (χa) = 0, where χa = u(x+ a). Differentiate this equation with respect to a:
∂aF (χa)|a=0 = ∂F (χa)∂aχ|a=0 = Lζ ,
where, recall L := dF (χ). Hence we have Lζ = 0 as claimed.Now, since χ′ is positive, we have by the Perron - Frobenius theory (see Appendix
E.3) that 0 is the smallest eigenvalue of L and it is simple. This proves also a part of (c).To complete the proof of (c), given an operator L on L2(Rn), we define the number
µ(L) := limR→∞
inf〈ξ, Lξ〉 : ξ ∈ C∞0 (Rn/BR),
where BR is the ball of the radius R > 0 in Rn. It is a result in the spectral theory(see e.g. [12]) that the spectrum of a Schrodinger type operator L in (−∞, µ) consists ofisolated eigenvalues of finite multiplicities, which might accumulate only at µ(L).
We show now that for (9.25),
µ(L) ≥ ming′(1), g′(−1), (9.27)
which would prove the first statement. To prove this statement, we will use that |χ(x)∓1| ≤ |x|θ, for some θ > 0, as x → ±∞. (In fact, it is exponentially small.) We will notprove the latter statement, but just mention that for g(u) = u3 − u, we have the explicitsolution
χ(x) = tahn
(x√2
), (9.28)
which shows that χ(x)→ ±1, as x→ ±∞, exponentially.Now, we compute, differentiating by parts
〈ξ, Lξ〉 =
∫(|ξ′|2 + g′(χ)|ξ|2) ≥
∫g′(χ)|ξ|2.
96 Lectures on Applied PDEs
Since ξ ∈ C∞0 (Rn/BR), we can write∫g′(χ)|ξ|2 =
∫ −R−∞
g′(χ)|ξ|2 +
∫ ∞R
g′(χ)|ξ|2 =
∫ −R−∞
g′(−1)|ξ|2 +
∫ ∞R
g′(1)|ξ|2
+
∫ −R−∞
(g′(χ)− g′(−1))|ξ|2 +
∫ ∞R
(g′(χ)− g′(1))|ξ|2.
Since |χ(x)∓ 1| ≤ |x|θ and g′(±1) > 0 (see (9.23)), we have∫g′(χ)|ξ|2 ≥ g′(−1)
∫ −R−∞|ξ|2 + g′(1)
∫ ∞R
|ξ|2
−cR−θ(∫ −R−∞|ξ|2 +
∫ ∞R
|ξ|2),
which, due to ξ ∈ C∞0 (Rn/BR), gives∫g′(χ)|ξ|2 ≥ min(g′(±1))
∫∞−∞ |ξ|2 − cR−θ
∫∞−∞ |ξ|2),
which implies (9.27). This proves the property above.
Lectures on Applied PDEs 97
10 The implicit function theorem and applications
10.1 The implicit function theorem
The implicit function theorem is a key theorem in analysis. Consider three Banach spacesV,X and Y and a map F : M × U → Z, where M and U are open sets in V and X,respectively. We wish to solve the equation
F (µ, u) = 0 (10.1)
for u, i.e. we want to find a function u = g(µ) s.t. F (µ, g(µ)) = 0. For a map F (µ, u) de-pending on two arguments, µ and u, we introduce the notion of partial Gateaux derivative,say duF (µ, u), by fixing u and taking the Gateaux derivative in y.
Theorem 10.1 (The implicit function theorem). Assume
(1) F : M × U → Y is C1 in y and C in µ;
(2) F (a, b) = 0, where a ∈M and b ∈ U ;
(3) duF (a, b) has a bounded inverse.
Then there is a neighborhood M ′ ⊂ M of a ∈ M and a map g : M ′ → X such thatF (µ, g(µ)) = 0, ∀µ ∈ U ′, and g(a) = b.
First we explain the idea of the proof. Without loss of generality, we can take (a, b) =(0, 0). We want to solve F (µ, u) = 0 for y near (µ, u) = (0, 0). Expand F in u around 0(see Theorem B.3): F (µ, u) = F (µ, 0) + duF (µ, 0)u+R(µ, u), with R(µ, u) = o(||u||). Sowe can rewrite equation (10.1) as the following equation for u:
F (µ, 0) + duF (µ, 0)u+R(µ, u) = 0.
We will show in a moment that, since the operator L0 = duF (0, 0) is invertible, then sois Lµ = duF (µ, 0) for sufficiently small ‖µ‖. Hence the equation above can be rewrittenas the fixed point equation,
u = −L−1µ (F (µ, 0) +R(µ, u)) , (10.2)
or u = Hµ(u), where Hµ(u) is the map given by
Hµ(u) := −L−1µ (F (µ, 0) +R(µ, u)) . (10.3)
If we neglect the remainder term R(µ, u), then equation (10.2) yields for each given x thecorresponding u = G(µ) = −L−1
µ F (µ, 0). In the general case, the remainder is not zero,but o(‖u‖) for u small, and we can use the fixed point argument to show existence of asolution to (10.2).
98 Lectures on Applied PDEs
We proceed to the proof. Our goal is to show that the map Hµ is well defined andhas a fixed point u = u(µ). First we show that there is a δ1 such that Lµ is invertiblefor ‖µ‖ ≤ δ1. Write Lµ = L0 + Vµ, where Vµ := Lµ − L0. Since Lµ is continuous in x bythe conditions of the theorem we can choose δ1 such that ‖Vµ‖ ≤ (2‖L−1
0 ‖)−1 if ‖µ‖ ≤ δ1.Hence Lµ is invertible if ‖µ‖ ≤ δ1 by Theorem A.11 in Section A.7.
Denote by BY (y, r) the open ball in Y of radius r centered at z ∈ Z.
Claim 10.1. ∃ε > 0 and δ > 0 such that ∀µ ∈ BV (0, δ)
(i) Hµ : BX(0, ε)→ BX(0, ε),
(ii) ‖dHµ(u)‖ ≤ 1/2 ∀u ∈ BX(0, ε).
Given (i) and (ii), we see that for any µ ∈ BV (0, δ), Hµ is a contraction map andtherefore it has a unique fixed point in BX(0, ε). Call this fixed point u = g(µ). It solvesthe equation u = Hµ(u) which is equivalent to the equation F (µ, u) = 0. Thus, thetheorem follows with M ′ = BV (0, δ).
It remains to prove the claim. (i) First, we pick δ > 0 so that for m = 2‖L−10 ‖ and
given ε > 0 and for any µ ∈ BV (0, δ), we have
(a) ‖L−1µ ‖ ≤ m;
(b) ‖F (µ, 0)‖ ≤ ε/2m.
This is possible by the continuity of duF (µ, 0)−1 (due to the continuity of duF (µ, 0), seeabove) and of F (µ, 0) in µ and the relation F (0, 0) = 0. Next, we pick ε > 0 so that, forany µ ∈ BV (0, δ) and any u ∈ BX(0, ε),
(c) ‖R(µ, u)‖ = o(‖u‖) ≤ ε/2m;
(d) ‖duF (µ, u)− duF (µ, 0)‖ ≤ 1/2m.
Again this is possible by a property of R(µ, u) and the continuity of duF (µ, u).Now, inequalities (a)-(c) and the definition of Hµ imply
‖Hµ(u)‖ ≤ m (ε/2m + ε/2m) = ε
∀µ ∈ BV (0, δ) and u ∈ BX(0, ε). So (i) follows.(ii) Using the definition, R(µ, u) := F (µ, u)− F (µ, 0)− duF (µ, 0)u, of the remainder
R(µ, u) and definition (10.3) of the map Hµ, we compute
duHµ(u) = −duF (µ, 0)−1duR(µ, u)
= −duF (µ, 0)−1 [duF (µ, u)− duF (µ, 0)] .
Applying inequalities (a) and (d), we conclude that ‖duHµ(u)‖ ≤ 1/2 for any µ ∈ BV (0, δ)and any u ∈ BX(0, ε). This proves (ii) and therefore completes the proof of the claim andwith it, the theorem.
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Remark. To prove that some linear operator is invertible is usually a difficult task.It is significantly simplified if we know that the operator in question is self-adjoint (seeAppendix A.8 and [12] for the definition). Then it amounts to showing that 0 is not inthe spectrum of the operator and there are many techniques to find spectra of self-adjointoperators (see Appendix E and [12]).
10.2 Existence of surfaces with prescribed mean curvature.
Colloquially, given a function h on a surface S, we would like to find S whose meancurvature is h.
We formulate this precisely. Assume S is a hypersurface in Rn+1, given as a graph ofa function ψ : Ω ⊂ Rn → R, S = graph ψ (see Fig. 11). We assume Ω is a boundeddomain with smooth boundary.
,,
,,
xn+1
S = graph
1
Figure 11: Graph of ψ
We can lift any function h(x) on Ω to a function, h(x′), on S as h(x′) ≡ h(x), x′ =(x, ψ(x)) ∈ S. Now, given a function h(x′) on Ω, is there a surface S = graph ψ whichhas mean curvature h(x)? This problem can be reformulated as finding a solution ψ tothe equation H(ψ) = h. The mean curvature of S at a point x′ = (x, ψ(x)) ∈ S is givenby (see Appendix G.2)
H(x′) ≡ H(ψ)(x) = div
(∇ψ(x)√
1 + |∇ψ(x)|2
), (10.4)
100 Lectures on Applied PDEs
for x ∈ Ω. Hence the equation H(ψ) = h can be given explicitly as
div
(∇ψ(x)√
1 + |∇ψ(x)|2
)= h(x), (10.5)
on Ω. We sketch a proof that
• for any h ∈ Hr−2(Ω), r > n/2 + 1, sufficiently small and satisfying∫
Ωh = 0, there
is a unique ψ ∈ Hr(Ω), ψ∣∣∂Ω
= 0 solving equation (10.5).
To solve the equation H(ψ) = h, we note H(0) = 0 and apply the implicit (or inverse,see the next subsection) function theorem to the equation F (h, ψ) := H(ψ) − h in aneighbourhood of (0, 0). So we need to show that for some spaces V,X and Y we have
1) F : M × U → Y , where M and U are a neighborhoods of 0 ∈ V and 0 ∈ X,
2) F is C1 in u and C in h,
3) dF (0, 0) has a bounded inverse.
For the spaces V,X and Y we take the Sobolev spaces X = Hr0(Ω), V = Y = Hr−2
0 (Ω),with r > n/2 + 1, where Hs
0(Ω) := u ∈ Hs(Ω) : u∣∣∂Ω
= 0 (which is well defined forr > n/2 + 1).
To show 1), we first observe that by the Sobolev embedding theorem (see TheoremA.7), ∇jψ are continuous and bounded functions. (Here we use the condition r > n/2+1.)Furthermore, differentiating through we have
H(ψ) =∆ψ
(1 + |∇ψ|2)1/2− ∇ψ · ∇
2ψ∇ψ(1 + |∇ψ|2)3/2
.
Assuming for simplicity that r = 2, we see that, since ∇jψ are continuous and boundedfunctions and ∆ψ,∇2ψ ∈ L2, the function H(ψ) is in L2, which shows 1).
In order to check 2), i.e. F ∈ C1, we remember from Exercise B.1 #1 that
dH(ψ)ξ = div
( ∇ξ(1 + |∇ψ|2)1/2
− (∇ψ · ∇ξ)∇ψ(1 + |∇ψ|2)3/2
).
Exercise 10.2. Let n = 1 and r = 2. Show dH(ψ) : Hr(Ω) → Hr−2(Ω) is bounded andcontinuous in ψ (i.e. ‖dH(ψ′)− dH(ψ)‖ → 0, as ‖ψ′ − ψ‖ → 0).
Finally, we have to verify that 3) is satisfied, i.e. that dF (0, 0) = dH(0) has a boundedinverse. Now dH(0) = ∆ on X = Hr
0(Ω), with r > n/2+1. For Ω = Rn, we have discussedthe existence of ∆−1 in Appendix C. For a general bounded domain, Ω, analysis is moresubtle and involves some non-trivial spectral theory (see Appendix E and e.g. [12, 16]).The upshot of it is that for Ω bounded, the operator ∆ on X = Hr
0(Ω) has purely discrete
Lectures on Applied PDEs 101
and strictly positive spectrum and hence it is invertible as a map from X = Hr0(Ω) to
Y = Hr−2(Ω), with r > n/2 + 1, and its inverse is bounded as ∆−1 : Hr−2(Ω)→ Hr0(Ω).
Modulo the proof of the fact above (we might do this in a later section), wehave thus shown that the conditions of the implicit function theorem are satisfied, andtherefore, for any sufficiently small h ∈ Hr−2(Ω), r > n/2 + 1, the equation F (ψ) = h hasa unique solution ψ ∈ Hr
0(Ω). In other words, there exists a surface S = graph ψ withprescribed small mean curvature h.
If h is constant, then the corresponding surface is called the constant mean curvaturesurface. For h = 0, this is a minimal surface.
Exercise 10.3. Let Ω be an SnR-sphere of radius R. Prove existence of surfaces withprescribed mean curvature close to n
R. Hint: Observe that H(ψR) = n
R, where ψR(x) :=√
R2 − |x|2, a spherical cap over Ω = SnR. Apply the implicit function theorem in aneighbourhood of ( n
R, ψR).
10.3 Appendix: Existence of breathers
Consider the discrete nonlinear Schrodinger equation (DNLS)
i∂ψ
∂t= −ε∆ψ − |ψ|2ψ
in l2(Zd). Here Zd = α = (α1, . . . , αd) |αj integers, l2(Zd) = ψ : Zd → C | ∑x∈Zd |ψ(α)|2 <∞ and ∆ is the discrete Laplacian
(∆f)(α) =∑|α−β|=1
f(β)− 2df(α). (10.6)
Remark. The discrete Laplacian can be also defined as follows. Let E := ej = (0, ..., 1, ..., 0)|j =1...d be the basis of the elementary cell at the origin in Zd. Define ∆ := − div ·∇ where
(∇f)(α) =∑e∈E
(f(α + e)− f(α))e
and div (the divergence) is the adjoint negative of operator, div = −∇∗, i.e., 〈∇f,~g〉 =−〈f, div ~g〉. Hence
Exercise 1. 1) Derive (10.6) from the formula ∆ := −div · ∇ and, separately, from theresult of Example 10.1 (applying the latter in each coordinate). 2) Show ∆ is a boundedoperator on l2(Zd).
Breathers are time-periodic solutions to DLNS of the form
ψ(α, t) = eiλtφ(α)
This implies that φ(α) satisfies
− ε∆φ− |φ|2φ+ λφ = 0 (10.7)
Theorem 10.4. If ε is sufficiently small and λ > 0, then (10.7) has a solution in l2(Zd).
Proof. We look for real solutions φ to (10.7). Define F (ε, φ) := −ε∆φ+ λφ− |φ|2φ. Notethat (10.7) ⇐⇒ F (ε, φ) = 0. Therefore, we have to solve F (ε, φ) = 0 for φ, providedε sufficiently small. We want to use IFT. Note that 1) F (ε, φ) : R × l2(Zd) → l2(Zd) iscontinuous in ε and φ (since l2 ⊂ lp ∀p ≥ 2); 2) F (0, φ0) = 0, where φ0 =
√λ δα0 for any
α0; 3) F (ε, φ) is C1 in φ: dφF (ε, φ) = −ε∆ +λ+ 3φ2 is a bounded operator on l2(Zd), andis continuous in ε and φ; 4) dφF (0, φ0) = λ(1 − 3δα0) is invertible if λ > 0. By 1) − 4),IFT is applicable and hence (10.7) has a unique solution for ε sufficiently small.
Remark. One can extend the above theorem to large ε: let F (ε, φε) = 0 for ε small;dφF (ε, φε) = −ε∆ + λ− 3φ2
ε is invertible if
λ /∈ Ran 3φε and ε <minα |λ− 3φ2
ε |‖∆‖ . (10.8)
Exercise 10.5. Show that if (10.8) holds, then dφF (ε, φε) = −ε∆ + λ− 3φ2ε is invertible.
Hint: Write−ε∆ + λ− 3φ2
ε =(λ− 3φ2
ε
) (1l− ε(λ− 3φ2
ε)−1∆
).
If λ /∈ Ran 3φε, then the first factor on the r.h.s (the multiplication operator λ− 3φ2ε) is
invertible. Show that the second factor is invertible as well.
10.4 Appendix: Inverse function theorem
The implicit and inverse function theorems easily follow from one to the other. For theinverse function theorem, we have to solve the equation
F (u) = h
for u. Recall that a map G : X → Y is called the inverse of the map F : Y → X if andonly if G F = 1lX and F G = 1lY . Here, 1lZ denotes the identity on the space Z. Wewrite G = F−1.
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Recall that a linear map L is called invertible if and only if it has a bounded inverse.If the map F is linear, F (y) = Ly, where L is a linear operator on Y , then the problemwe want to solve reads Ly = x, which is reduced to inverting the operator L. Thus theinverse function theorem is a generalization to the nonlinear setting of the key problemof linear analysis of inverting an operator.
Applying the implicit function theorem to the equation F (h, u) = 0, where F (h, u) :=F (u) − h, we obtain the following theorem generalizing the corresponding theorem inmultivariable calculus:
Theorem 10.6 (The inverse function theorem). Let U be an open neighborhood of 0 ∈ X,and let F : U → Y be a C1 map s.t. dF (0) : Y → X has a bounded inverse (i.e.dF (0) : Y → X is bijective). Then there is a neighborhood W of F (0) in Y and a uniquemap G : V → X s.t. F (G(u)) = u, for all u ∈ W .
G
F
. .0 F(0)V
Y
U
X
Figure 12: Maps F and G
Note that the problem of proving of the existence of surfaces with prescribed meancurvature fits nicely in the framework of the implicit function theorem. In this case wehave to show that F = H satisfies:
1) H : U → Hr−2⊥ (Ω), where U is a neighborhood of 0 in Hr
0(Ω), r > n/2 + 1,
2) H is C1,
3) dH(0) has a bounded inverse.
104 Lectures on Applied PDEs
11 The bifurcation theory
11.1 Set-up
Let X and Y be Banach spaces and I and U be an open interval and an open set in Rand X, respectively. Consider a C1–map F : I × U → Y , where. We would like to find afunction u = u(µ), implicitly defined by the equation
F (µ, u) = 0. (11.1)
By the implicit function theorem, we know that if F (µ0, u0) = 0 and duF (µ0, u0) has abounded inverse, then equation (11.1) has a unique solution in a neighborhood of (µ0, u0).Now, we are interested in a situation when duF (µ0, u0) does not have a bounded inverse.What happens then?
Eq (11.1) could have multiple wild solutions in a neighbourhood of the point (µ0, u0).We consider the simplest situation of two curves or a curve and a surface intersectingintersecting at (µ0, u0). Furthermore, we assume that one of the curves is known and call‘trivial’ and denote it by (µ, u(µ)), µ ∈ I (see Fig.13).
bifurcation point trivial branch
nontrivial branch
µ
u
Figure 13: Bifurcation
We give some definitions. The “curve” (µ, u(µ)), µ ∈ I, is called a branch of solutionsiff for every µ ∈ I, u(µ) is a solution to (11.1),
F (µ, u(µ)) = 0, ∀µ ∈ I. (11.2)
Thus, we assume (11.1) has a branch of solutions (µ, u(µ)) for all µ ∈ I, called the trivialbranch.
A point (µ0, u0) at which a new branch of solutions emerges from (µ, u(µ)) is called abifurcation point.
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At a bifurcation point, the operator duF (µ, u0) is not invertible, since otherwise, theimplicit function theorem would apply and give a unique solution in a neighbourhoodof (µ0, u0), which must be a piece of the trivial branch (µ, u(µ)). Hence we have thefollowing
Proposition 11.1. If (µ0, u0) is a bifurcation point, then duF (µ0, u0) does not have abounded inverse.
An important example of bifurcation is given by the linear equation F (µ, u) = 0,where F : R×X → Y is a linear (in u) map,
F (µ, u) = µLu− u,
where L is a linear operator from X to Y . Then (µ, 0) is the trivial branch of solutions.If µ−1
0 is an eigenvalue of L, then in a neighbourhood of (µ0, 0), the equation F (µ, u) =µLu − u = 0 has another branch of solution: (µ0, au0), for any a ∈ C, where u0 isan eigenfunction of L corresponding to the eigenvalue µ−1
0 . Since dF (µ, 0) = µL − 1l,duF (µ0, 0) is not invertible, which verifies the conclusion above. The conclusion above istrue for every eigenvalue of L (see 14). (If 1/µ is an eigenvalue of L, then we call µ acharacteristic value of L.)
µ
Ru1
Ru2
(λ−1,0 ) (λ−1
21,0 ) . . . . . . .
. . . . . . .
. . .
u
Figure 14: Bifurcation:linear case
In a sense, we would like to address a nonlinear version of this example and constructsnonlinear manifolds tangent to the spaces above at the bifurcation points.
11.2 Key result
We now give a sufficient condition for a bifurcation to take place. To minimize technical-ities, we formulate and prove the next result for the special case of when duF (µ0, u0) is aself–adjoint operator (see a remark below for a discussion and Appendix A.8 and [12] forthe definition).
For an eigenvalue λ of an operator A, we define the (geometric) multiplicity of λ asdim Null(A − λ1l). We say an eigenvalue λ of A is isolated iff there is a neighbourhood,U , of λ in C s.t. U ∩ σ(A) = λ.
106 Lectures on Applied PDEs
Theorem 11.2 (Krasnoselski). Assume that X is dense in a Hilbert space Y and considera map F : R×X → Y . Assume that
(i) F is C1 in u and C in µ, with duF (µ, u) being C1 in µ;
(ii) F has a trivial branch of solutions (µ, u(µ)) : µ ∈ I;
(iii) duF (µ0, u0) is a self–adjoint operator and has the isolated eigenvalue 0 of odd mul-tiplicity;
(iv) there exists a v0 ∈ Null duF (µ0, u0) such that
〈v0, ∂µduF (µ0, u0)v0〉 6= 0.
Then (µ0, u0) is a bifurcation point of (11.1).
We conduct the proof in two steps. One the first step, under very general conditions,we reduce the problem of solving the equation (11.1) to the problem of solving an equationin a few dimensions (the reduced or effective, or bifurcation equation). On the secondstep, we solve the latter equation.
On the first step, we use the powerful technique, called the Lyapunov-Schmidt reduc-tion. We will use this technique repeatedly below.
Proof of the Krasnoselski theorem. Let L(µ) := duF (µ, u0), and denote by P the orthog-onal projection onto the subspace NullL(µ0) and let P⊥ := 1l − P , the orthogonal pro-jection onto the orthogonal complement (NullL(µ0))⊥ of the subspace NullL(µ0). (Forthe definition and properties of (orthogonal) projections, see Appendix A.8.) Note thatL(µ0)P = PL(µ0) = 0.
By changing the unknown variables if necessary, we assume in what follows that u(µ) =0. We project u ∈ X and the equation F (µ, u) = 0 onto the subspaces Ran P and RanP⊥:u = v + w, where v := Pu and w :=P⊥u, and
PF (µ, v + w) = 0, (11.3)
P⊥F (µ, v + w) = 0. (11.4)
We have thus two equations, (11.3) and (11.4), for two variables v and w. Observe thatsince dim Ran P <∞, variable v is finite–dimensional. This step is called the Lyapunov-Schmidt decomposition.
We now show equation (11.4) has a unique solution w = w(µ, v). Define
F1(µ, v, w) :=P⊥F (µ, v + w) : R× PX ×P⊥X →P⊥Y.
Then the problem (11.4) can be reformulated as solving the equation F1(µ, v, w) = 0 forw. Now observe that
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(α) F1 is C1,
(β) F1(µ, 0, 0) = 0 for any µ,
(γ) dwF1(µ0, 0, 0) is invertible.
Indeed, (α) follows from the condition that F is C1, (β) results from the relation F1(µ, 0, 0) =P⊥F (µ, 0) = 0, and (γ) is due to the relation
dwF1(µ0, 0, 0) =P⊥duF (µ0, 0)P⊥
plus the fact that, by (iii) and the spectral theory (see Proposition 11.19 below), the r.h.s.is invertible as an operator fromP⊥X toP⊥Y .
Hence we can apply the implicit function theorem to the equation F1(µ, v, w) = 0,which shows thus that for any (µ, v) sufficiently close to (µ0, 0), the equation F1(µ, v, w) =0, and therefore (11.4), has a unique solution. We denote this solution by w = w(µ, v).
We substitute the solution, w = w(µ, v), of (11.4) into (11.3) to obtain the equation
f(v, µ) = 0, (11.5)
with f(v, µ) := PF (µ, v+w(µ, v)). Eq (11.5) is called the bifurcation equation. It is a sys-tem ofm algebraic equations form+1 variables µ and v, wherem :== dim Null duF (µ0, 0),and is expected to describe the bifurcating branches.
Furthermore, below, we show that
w = o(‖v‖) +O(|µ− µ0|‖v‖). (11.6)
This completes the first step of the proof. We summarize the result of this step in thefollowing
Theorem 11.3 (Reduction to effective equation). Assume conditions (i) - (iii) of The-orem 11.2 hold. Then (11.1) has a solution, u, iff (11.5) has a solution, v. Moreover,these solutions are related as
v = Pu and u = v + w(µ, v). (11.7)
The process of reducing Eq (11.1) to Eq (11.5) (in much fewer variables) is called theLyapunov-Schmidt reduction. Eqs (11.7) and (11.6) give that the solution of the originalproblem has the form
u = v + w(µ, v), with w(µ, v) = o(‖v‖) +O(|µ− µ0|‖v‖). (11.8)
Now, we proceed to the second step: solving equation (11.5). Recall the notationL(µ) := duF (µ, 0). Assume for simplicity that dim NullL(µ0) = 1, i.e. that the eigenvalue0 of L(µ0) is simple and show that equation (11.5) has a unique solution for µ as a function
108 Lectures on Applied PDEs
of v ∈ R. Let v0 the zero eigenvector of the operator L(µ0), normalized as 〈v0, v0〉 = 1.Then Pu = 〈v0, u〉v0. Since v = sv0 for some s ∈ R, equation (11.5) is equivalent to theequation f(s, µ) = 0, where
f(s, µ) :=
1s〈v0, F (µ, sv0 + w(µ, sv0))〉 for s 6= 0,
〈v0, L(µ)v0〉 for s = 0.(11.9)
Let u1 := s−1u. Next, using the fact that F (µ, 0) = 0 for any µ, we expand the mapF (µ, u) around u = 0 as
F (µ, u) = L(µ)u+R(µ, u), (11.10)
with R(µ, u) = o(‖u‖) and, recall, L(µ) := duF (µ, 0). Using this expansion, we rewritef(s, µ) as
Since s−1‖R(µ, su1)‖ → 0 and, by (11.6), u1 − v0 = s−1w → 0, as s → 0, we have thatf(s, µ) → 〈v0, L(µ)v0〉, as s → 0. We compute the µ−derivative of f . Hence f(s, µ) iscontinuous at s = 0. Next, we have
∂f
∂µ(s, µ) = 〈v0, ∂µL(µ)u1〉+ 〈v0, L(µ)∂µu1〉+ 〈v0, s
−1(∂µR(µ, su1) + sduR(µ, su1))∂µu1〉.
Write u1 := s−1u = v0 + w1, where w1 := s−1w. Below, we show that
w, ∂µw = o(‖v‖) +O(|µ− µ0|‖v‖). (11.12)
This implies ‖w1‖, ‖∂µw1‖ → 0, as s → 0 and |µ − µ0| → 0. by the last relation ands−1‖∂µR(µ, su1)‖, ‖duR(µ, su1)‖ → 0, as s→ 0, we have that
∂f
∂µ(0, µ) = 〈v0, ∂µduF (µ, 0)v0〉, (11.13)
which shows that f(s, µ) is differentiable in µ also at s = 0. Moreover, by condition (v)of the Krasnoselski theorem, ∂f
(11.9) has a unique solution µ = µ(s), for µ as a function of s, if s is in a neighbourhoodof s = 0. This completes the second step.
We have shown above that the solution of the original problem has the form (11.8),where v satisfies the equation (11.5). In the case when 0 is a simple eigenvalue of L(µ0),we have shown that the equation (11.5) can be solved for µ as a function of v, µ = µ(v).This give the bifurcation branch of solutions
(u = v + w(µ, v), µ = µ(v)), (11.14)
parametrized by v ∈ NullL(µ).
Lectures on Applied PDEs 109
Proof of the estimate (11.12). To show (11.12), we use the expansion (11.10), to rewritethe equation (11.4), as
L⊥(µ)w +P⊥L(µ)v +P⊥R(µ, u) = 0, (11.15)
where L⊥(µ) := P⊥µ L(µ)P⊥µ . Since the operator L⊥(µ) : P⊥µ Y → P⊥µ Z is invertible, wederive
w = −L⊥(µ)−1P⊥µ (R(µ, u) + L(µ)v) .
The relations R(µ, u) = o(‖u‖) and L(µ)v = (L(µ) − L(µ0))v = O(|µ − µ0| ‖v‖) implyw = o(‖u‖)+O(|µ−µ0| ‖v‖). Since u = v+w, this shows (11.12) for w. To prove (11.12)for ∂µw, we differentiate (11.15) w.r.to µ and then proceed with the resulting equation aswe did with (11.15).
The proof of the Krasnoselski’s theorem (see (11.14)) implies the following
Corollary 11.4. Assume the operator duF (µ0, 0) has a simple, isolated eigenvalue at 0with an eigenfunction ϕ. Then the bifurcating at µ0 branch of solutions of (11.1) is ofthe form
(u = sϕ+O(s2), µ = µ0 +O(s2)), (11.16)
parametrized by s := 〈ϕ, u〉.
Corollary 11.5. Let F be a C1 in u and C in λ map satisfying duF (λ, u0) = L − λ1l,where L is a linear, self–adjoint operator. If λ0 is an isolated eigenvalue of L of oddmultiplicity, then (λ0, u0) is a bifurcation point.
Proof. Clearly the first two conditions of the Krasnoselski’s theorem are satisfied. Tocheck the third condition, we compute ∂λduF(λ, 0) = −1l, and 〈v0, ∂λduF(λ, u0)v0〉 =−〈v0, v0〉 = −‖v0‖2 6= 0 ∀v0 ∈ Null (L − λ1l). Hence the third condition is satisfied aswell.
Example 11.1. Let X = Y = R, and F (µ, u) = µu − u3. Clearly we have F (µ, 0) = 0∀µ ∈ R, so (µ, 0) is the trivial branch. We calculate the derivative duF (µ, u) = µ − 3u2,so duF (µ, 0) = µ = 0 has the solution µ0 = 0, hence (0, 0) is a candidate for a bifurcationpoint. Next, ∂µduF (0, 0) = 1, so the condition (ii) is satisfied as well. Therefore (0, 0) isa bifurcation point. On the other hand, we can solve the equation F (µ, u) = 0 explicitly,confirming this conclusion.
Now we give a cautionary example
Example 11.2. For F : R×R2 → R2 given by F (µ, u1, u2) = (u1, u2)−µ(u1−u32, u2+u3
and therefore duF (µ, 0) = (1 − µ)1l. Thus duF (1, 0) is not invertible. However, (1, 0) isnot a bifurcation point! Indeed, look at the two components of the equation F (µ, u) = 0.Multiplying the first one by −u2, the second one by u1, we obtain
−(1− µ)u1u2 − µu42 = 0
(1− µ)u1u2 − µu41 = 0.
Adding the above two equations yields −µ(u41 + u4
2) = 0, so u1 = u2 = 0 (for µ 6= 0),which shows that F (µ, u1, u2) = 0 has only the trivial solution (u1, u2) = (0, 0), ∀µ ∈ R(if µ = 0, then this follows directly from the definition of F ). (1, 0) is therefore not abifurcation point.
11.3 Applications
The nonlinear eigenvalue problem. Consider the nonlinear eigenvalue problem
Lu+ f(u) = λu, with f ∈ C1 and f(0) = f ′(0) = 0. (11.17)
Then the corollary implies that if λ0 is an eigenvalue of a self–adjoint operator L of oddmultiplicity, then equation (11.17) has a nontrivial branch of solutions near the bifurcationpoint (λ0, 0).
We specify this example further by considering the following nonlinear eigenvalueproblem
(−∆ + V )u+ f(u) = λu, (11.18)
where u ∈ H2(Rn,R), ∆ is the Laplacian on Rn and the function f : Rn → R satisfiesf(0) = 0, f ′(0) = 0 and |f(u)| ≤ c|u|p with n
2− n
2p< 2.
Exercise 11.6. Find the bifurcation points for the equation
∆u+ u+ u5 = 0, (11.19)
(a) for u ∈ H2([−a, a]), with Dirichlet boundary conditions (i.e. u = 0 on the boundary),and (b) for u ∈ H2
sym([−a, a]n), n ≥ 2. Here H2sym([−a, a]n) denotes the subspace of the
Sobolev space H2([−a, a]n) consisting of functions symmetric with respect to permuta-tions, i.e., ψ(x1, . . . , xn) = ψ(xπ(1), . . . , xπ(n)) for all permutations π. Hint: Here a is abifurcation parameter. You can break the solution in the following steps:
- Rescale (11.31) by passing from u(x) to v(x) = u(ax) to find an equivalent equationfor v depending on a but defined on the fixed domain [−1, 1]n;
- find the trivial branch of solutions;- find the candidate for the bifurcation point;- for this candidate check the conditions of the bifurcation (Krasnoselski) theorem.
Lectures on Applied PDEs 111
Lamellar phase. Problem (open? references?): Show the existence of periodic staticsolutions (lamellar phases) of the Allen-Cahn equation, (2.2) (see Section 2.1). Staticsolutions of the Allen-Cahn equation satisfy the static Allen-Cahn equation (see (2.3)),
− ε2∆u+ g(u) = 0 (11.20)
where g(u) = G′(u), with G(u) a function of the form shown in Fig 2, or Fig 18 below.
G
1−1
u
Figure 15: Double well potential.
(Recall that in lamellar phase, the layers of +1 and −1 phases (substances) coexist ina periodic array.)
There are three ways to proving existence of such solutions. A variational approachwill be discussed in Section 19. Another way is to construct an approximate solutionsby gluing together kinks and antikinks (see Fig 16) and then using either the implicitfunction theorem or its generalization due to the Lyapunov-Schmidt decomposition, toprove the existence of the true solutions nearby.
C
CCCCC
C
CCCCC
or
\\
+1
-1
+1
-1 R
Figure 16: Periodic solution built out of kinks and antikinks.
The third approach which we discuss here is to use the bifurcation theory. The last twoapproaches give the existence of the periodic solutions in two opposite regimes (the longperiod large solutions and short period small solutions), while the variational approachdoes not give much specific information about the solutions.
We discuss briefly the bifurcation approach. This equation has three homogeneous(x−independent) solutions u = 0, u = a and u = b, which give three solution branches
112 Lectures on Applied PDEs
(ε, 0), (ε, a) and (ε, b), ∀ε ≥ 0. Consider bifurcation from one of these branches. If wedefine F (ε, u) := −ε2∆u+ g(u), then
dF (ε, u) = −ε2∆ + g′(u). (11.21)
We are interested in u = 0, a, b. Since G has a strict local maximum at u = 0 and strictlocal minima at u = a and u = b, we have
Hence the linearized operator is strictly positive for v = a, b and the only branch whichmight lead to bifurcations is (ε, 0).
If we consider the operator dF (ε, 0) on the entire space L2(R), then σ(dF (ε, 0)) =[g′(0),∞) (see Exercise 11.18 above) and the bifurcation problem becomes intractable: 0is a part of the continuous spectrum of dF (ε, 0).
Since we are looking for the bifurcation of periodic solutions, we consider the operatordF (ε, 0) on an interval, say [0, c] ⊂ R, with periodic boundary conditions. The spectrumof dF (ε, 0) in this case is
σ(dF (ε, 0)) = ε22π
ck2 + g′(0), k = 0, 1, . . . .
Exercise 11.7. Find all bifurcation points (proving existence of bifurcating periodicsolutions) of the static Allen-Cahn equation
−∆u+ (u2 − τ)u = 0 (11.22)
on R, satisfying u(−x) = −u(x). (Hint: Since we are looking for periodic solutions, weconsider equation (11.22) for odd functions on an interval, say [−a/2, a/2] ⊂ R, withperiodic boundary conditions. Consider a as the bifurcation parameter. You can breakthe solution in the following steps:
rescale (11.22) by passing from u(x) to v(x) = u(ax) to find an equivalent equationfor v depending on a but defined on the fixed domain [−1/2, 1/2]; write this equation asF (a, v) = 0 (not to confuse with (11.21));
find the trivial branch of solutions;
find the candidates for the bifurcation points;
for these candidates check the conditions of the bifurcation (Krasnoselski) theorem.)
Exercise 11.8. Prove existence of bifurcating periodic solutions of the static Allen-Cahnequation (11.22) on R2, satisfying u(−x, y) = −u(x, y) and u(x,−y) = −u(x, y). (Hint:Here we are dealing with double periodic solutions, hence consider equation (11.22) on arectangle, say [0, a]× [0, b] ⊂ R2, with periodic boundary conditions.)
Lectures on Applied PDEs 113
The FitzHugh-Nagumo model. Consider the system of equations describing con-duction of electric pulses along nerve exon and known as the FitzHugh-Nagumo model:
∂u
∂t= ∆u− (u− a)(u− 1)u− v, (11.23)
∂v
∂t= ε(u− v). (11.24)
Exercise 11.9. (open? references?) Prove existence of bifurcating static periodic solu-tions of the FitzHugh-Nagumo equations, (11.23) – (11.24), on R.
Exercise 11.10. (open? references?) Consider bifurcation of traveling wave periodicsolutions (trains) of the FitzHugh-Nagumo equations, (11.23) – (11.24), on R.
Bifurcation of instantons. Let g(u) be given by g(u) = G′(u), where G(u) is a func-tion of the form shown in Fig 17.
u0
u1
G
−G
Figure 17: Function G and solution u1
We consider the equation−∆u+ g(u) = 0, (11.25)
on the strip [−a/2, a/2]× (−∞,∞), with periodic boundary condition in x and L2 in y:
u(−a/2, y) = u(a/2, y) ∀y, (11.26)
u(x, ·) ∈ L2(R) ∀x.
This equation describes tunnelling of a string (or line vortex) placed at the local minimumof the potential G(u) (see Fig 17) through the barrier. The parameter a is inverse tem-perature. Solutions of this equation are called instantons (for the physical background,see [25]).
Equation (11.25) with boundary conditions (11.26) has the following branches of so-lutions which are independent of x: (a, u0), ∀a, and (a, u1), ∀a, where u0 is a constant
114 Lectures on Applied PDEs
function equal to the local minimizer of G(u) (see Fig 17) and u1 satisfies 0 ≤ u1(y) ≤α, u1 → 0 as y → ±∞ and u1(0) = α (where α is the first positive zero of G(u), see Fig17). To prove the existence of the solution u1, we observe that it satisfies the equation
− ∂2u
∂y2+ g(u) = 0 on (−∞,∞). (11.27)
The last equation is just Newton’s equation if we interpret y as being the time-variable,and −G as a potential (whose derivative is a force). Then the existence of u1 is obviousfrom the picture above.
The problem here is to find solutions bifurcating from the branches above as a varies.
Exercise 11.11. (a) Check whether any solutions bifurcate from (a, u0).
(b) Find the bifurcation points from the branch (a, u1). (Hint: Let F (a, u) := −∆u +g(u). We can write La := dF (a, u1) as
La = −∂2x + `,
where ` is the operator acting on the variable y and given by
` := −∂2y + g′(u1(y)).
Prove the following statements
– the eigenvalues, νi, of ` are of the form
νi = λi + µi, (11.28)
where λi and µi are the eigenvalues of the operators −∂2x and ` acting on
L2per([−a/2, a/2]) and L2(R), respectively;
– the operator ` has the eigenvalue zero with the eigenfunction φ1(y) := ∂yu1(y);
– φ1(y) has exactly one zero and hence by the Sturm theory the operator ` hasexactly one negative eigenvalue, say, λ0, and this eigenvalue is simple (i.e. ofthe multiplicity 1 or non-degenerate).
Find the eigenvalues of −∂2x on L2
per([−a/2, a/2]) and use (11.28) to find the eigen-values of La. Use these facts to solve the problem.
11.4 Change of stability at a bifurcation
In this subsection we study the linearized stability of the trivial and bifurcating solutions.We consider dynamical system (9.1) with F (u) replaced by F (µ, u). Static solutions for(9.1) satisfy the static equation
F (µ, u) = 0. (11.29)
Lectures on Applied PDEs 115
We assume that the conditions of the bifurcation Theorem 11.2 (Krasnoselski’s Theorem),are satisfied. In particular (11.29) has a family of (trivial, static) solutions (µ, u(µ))and duF (µ∗, u∗), u∗ ≡ u(µ∗), is self-adjoint and has the isolated eigenvalue 0 of oddmultiplicity. According to Theorem 11.2, µ∗ is a bifurcation point of this equation.
We assume, for simplicity, that duF (µ, u) is self-adjoint and the multiplicity of theeigenvalue 0 of duF (µ∗, u∗) is one.
Theorem 11.12 (Change of stability at a bifurcation point). At the bifurcation pointµ∗, the stability property of the trivial solution u(µ) changes to the opposite: if u(µ) isstable/unstable for µ < µ∗, then it is unstable/stable for µ > µ∗.
Proof. Denote Lµ := duF (µ, u(µ)). Since Lµ∗ has has the isolated eigenvalue 0 of mul-tiplicity one and since Lµ is differentiable in µ, by the perturbation theory, for |µ − µ∗|sufficiently small, we have (see [12])
1) Lµ has an isolated eigenvalue, νµ, of the multiplicity one, s.t. νµ=µ∗ = 0;2) νµ, as well as the corresponding eigenfunction, is differentiable in µ and satisfies
νµ=0 = 0;3) if νµ∗ = 0 the largest spectral point of Lµ∗ , then νµ the largest spectral point of Lµ.Let φµ be the eigenfunction of Lµ corresponding to the eigenvalue νµ.
Lemma 11.13.∂µνµ = 〈φµ, ∂µLµφµ〉.
Proof. We compute ∂µνµ by differentiating the eigenvalue equation
Lµφµ = νµφµ,
w.r. to µ, multiplying the result, (∂µLµ)φµ + Lµ∂µφµ = (∂µνµ)φµ + νµ∂µφµ, by φµ andusing that φµ is normalized, to obtain
〈φµ, (∂µLµ)φµ〉+ 〈φµ, (Lµ − νµ)∂µφµ〉 = ∂µνµ.
Now, since Lµ is self-adjoint, we have that
〈φµ, (Lµ − νµ)∂µφµ〉 = 〈(Lµ − νµ)φµ, ∂µφµ〉 = 0
and similarly for the other term. The last two equations give the desired relation.
For µ = µ∗, we have φ∗ ≡ φµ∗ ∈ NullLµ∗ and therefore the r.h.s. is non zero by thecondition (iii) of Theorem 11.2:
∂µνµ∗ = 〈φ∗, ∂µLµ∗φ∗〉 6= 0.
Hence the eigenvalue νµ changes the sign at µ = µ∗.
116 Lectures on Applied PDEs
Now, assume the trivial solution u(µ) is stable for µ < µ∗. Then, for µ < µ∗, thespectrum of Lµ is negative and therefore νµ < 0. Since νµ=µ∗ = 0, νµ < 0 is the largesteigenvalue of Lµ. As νµ changes the sign at µ = µ∗, it becomes positive for µ > µ∗ andtherefore u(µ) looses its stability at µ = µ∗ and becomes unstable for µ > µ∗.
We can assume the trivial solution u(µ) is stable for µ > µ∗. Then the same argumentshows u(µ) looses its stability at µ = µ∗ and becomes unstable for µ < µ∗.
Assume ∂µνµ∗ > 0. Does the equation F (µ, u) = 0 have stable solutions for µ > µ∗?The answer is yes: it is the bifurcating solution, u(µ). One can show that the spectrumof duF (µ, u(µ)) for µ > µ∗, but close to µ∗ is strictly negative.
Exercise 11.14. Find values of λ for which the trivial solutions of the nonlinear eigenvalueproblem (11.18) are linearly stable and for which they are linearly unstable.
Exercise 11.15. Find values of a for which the static solution, u = 0 to the time-dependent equation
∂tut = ∆ut + ut + u5t , (11.30)
for ut ∈ H2sym([−a, a]3), is linearly stable and for which it is linearly unstable. See Exercise
11.6 for the definition of ut ∈ H2sym([−a, a]3). (Note that u = 0 is the trivial solution of
the bifurcation problem in in Exercise 11.6.)Find the bifurcation points for the equation
∆u+ u+ u5 = 0, (11.31)
for u ∈ H2sym([−a, a]3), with Dirichlet boundary conditions (i.e. u = 0 on the boundary).
Here H2sym([−a, a]n) denotes the subspace of the Sobolev space H2([−a, a]n) consisting
of functions symmetric with respect to permutations, i.e., ψ(x1, . . . , xn) = ψ(xπ(1), . . . ,xπ(n)) for all permutations π.
Exercise 11.16. Find values of the bifurcation parameter for which the trivial solutionof the static Allen-Cahn equation (11.22) in one and two dimensions (see Exercises 11.7and 11.8) are linearly stable and for which they are linearly unstable.
Exercise 11.17. Find values of a for which the trivial solutions of problem (11.25) –(11.26) (see Exercise 11.11) are linearly stable and for which they are linearly unstable.
11.5 Appendix: Connection to the spectral theory
The non-invertibility of duF (µ0, 0) is equivalent to the statement that
0 ∈ σ(duF (µ0, 0)).
Here σ(A) denotes the spectrum of a linear operator A, i.e.
σ(A) := z ∈ C : A− z1l is not invertible.
Lectures on Applied PDEs 117
Clearly, eigenvalues of A belong to σ(A). The (geometric) multiplicity of an eigenvalue λof A is defined as the number of linearly independent eigenvectors with eigenvalue λ, i.e.
• the (geometric) multiplicity of an eigenvalue λ of A is dim Null(A− λ1l).
We say an eigenvalue λ of A is isolated iff there is a neighbourhood, U , of λ in C s.t.U ∩ σ(A) = λ.
In general, the spectrum can also contain continuous pieces and it can take verypeculiar forms. For details on spectral theory see Appendix E and [12]. This motivatesthe following definitions. The set of all isolated eigenvalues of finite multiplicities is calledthe discrete spectrum and denotes, σd(A). The rest of the spectrum is called the essentialspectrum,
σess(A) := σ(A)/σd(A) (11.32)
The spectral analysis simplifies considerably if an operator A acts on a Hilbert spaceand is self–adjoint (see Appendix A.8 and [12] for the definition). For instance, thespectrum of a self–adjoint operator is real.
Self-adjoint operators are symmetric operators, i.e. operators obeying
〈Au, v〉 = 〈u,Av〉,
which in addition satisfy certain domain condition. (The latter condition is triviallysatisfied for bounded operators.) To show that a given operator is self-adjoint usuallyrequires some work. However, in the situations we will be dealing with, all symmetricoperators are self-adjoint. Thus in these situations to show that that a given operator isself–adjoint it suffices to show that it is symmetric.
Exercise 11.18. Show that(a) the multiplication operator Mf : u(x) → f(x)u(x) is symmetric provided the
function f is real and bounded and its spectrum is σ(Mf ) = Ran f ;(b) the differentiation operator −i ∂
∂xjis symmetric and its spectrum is σ(−i ∂
∂xj) = R;
(c) the identity operator has the spectrum consisting of one point σ(1l) = 1;(d) the Laplacian ∆ :=
∑n1
∂2
∂x2jon L2(Ω) with the domain D(∆) = H2(Ω) has the
spectrum [0,∞);(e) the Laplacian ∆ on [−a, a]n with (i) Dirichlet boundary conditions (i.e. u = 0
on the boundary) or (ii) the periodic boundary conditions has only eigenvalues of finitemultiplicities; find these eigenvalues.
Hint: in cases (b) and (d) use the Fourier transform to reduce these problems to onesof the type of (a).
We have the following result (see [12])
118 Lectures on Applied PDEs
Proposition 11.19. Let A be a linear, self–adjoint operator with the eigenvalue 0. ThenA∣∣NullA
is invertible iff 0 is an isolated eigenvalue of A.
How to determine that a given eigenvalue is isolated? There are two notable cases:(a) elliptic differential operators on bounded and Schrodinger operators with confining
potentials have purely discrete spectrum and(b) Schrodinger operators with potentials vanishing at infinity have the essential spec-
trum filling in the semi-axis [0,∞) and therefore their negative spectrum if exists isdiscrete.
In the latter case, one uses the variational calculus to find negative spectrum ofSchrodinger operators, see Section 6.5 and [12], for more details.
Lectures on Applied PDEs 119
12 Bifurcation of surfaces of constant mean curva-
ture
12.1 Surfaces of constant mean curvature
Let S be a (n + 1)−dimensional surface in Rn+2 (a hypersurface). Denote by H(x) itsmean curvature at a point x ∈ S. If H(x) satisfies H(x) =const, then we say S is aconstant mean curvature (CMC) surface. One of the sources of CMC surfaces is theisoperimetric problem (IP):
• minimize surface area for a fixed enclosed volume.
Solutions of this problem are CMC surfaces. More generally, they are critical points ofthe surface area for a fixed enclosed volume.
There is a vast number of constant mean curvature surfaces, e.g. spheres, cylinders,rotationally symmetric surfaces in R3, known as Delaunay surfaces, and various surfacesobtained by perturbing several known surfaces, glued together ([?, ?]), see Fig. 12.1above.
Our goal is to find CMC surfaces satisfying certain conditions. If S is described locallyby a map (an immersion) θ : U → Rn+2 (where U , an open set in Rn+1), then we writeH(x) = H(θ)(x). We try to solve the equation
H(θ) = h (12.1)
for θ, for some constant h.Obvious solutions of the equation (12.1) are spheres of radius n+1
hand cylinders of
radius nh.
However, it is probably only spheres which solve the isoperimetric minimization prob-lem.
We would like to find periodic CMC surfaces. For examples of double periodic CMCsurfaces, see Fig 12.1. Fig 12.1 shows some periodic CMC surfaces ocuring in nature. Wewill be more modest and look for CMC surfaces periodic in one direction, say along thexn+2−axis. Hence, we consider solutions to (12.1) periodic along the xn+2−axis of perioda. In this case,
120 Lectures on Applied PDEs
4 ANTONIO ROS
Figure 3. (a) Doubly periodic surface with constant meancurvature and genus 2 (modulo translations). These surfaceswere first constructed by Lawson [11] as conjugate surfacesof minimal surfaces in the 3-sphere. (b) Periodic patternformed by a thin layer of liquid on a planar surface, [12]:this self-assembled wetting phenomena may be modeled bystable doubly periodic constant mean curvature surfaces.
convex curve in the (volume, area)-plane that projects injectively on, boththe volume and the area axes.
Among self-assembled materials, mesoscopic wetting phenomena are par-ticularly related to doubly periodic, stable, constant mean curvature sur-faces, see [12]: under suitable conditions, a thin layer of liquid wetting anhydrophobic planar surface produces a pattern as in Figure 3b, exhibit-ing a periodic array of dry spots. As doubly periodic constant mean cur-vature surfaces always have a horizontal mirror symmetry (assuming theprescribed translations are horizontal), it follows that the correspondingperiodic isoperimetric problems in R3 and in the halfspace x3 ! 0 areequivalent. As another consequence of Theorem 2 below, we conclude thatthe assumptions
(i) the pattern is doubly periodic,
(ii) the volume fraction of liquid is given, and
(iii) the energy of the system is just the area (per unit cell) of the liquidsurface,
predict the right experimental wetting phases: after reflection in the hori-zontal plane we obtain either round spheres, or horizontal right cylinders,or constant mean curvature doubly periodic surfaces with genus 2 (modulotranslations), like in Figure 3.
R38 M W Matsen
1.0
0.8
1.2
0.6
0.4
0.2
0.0
HaN1/2
H aN1/2 =0.970!H aN1/2 =0.003
H aN1/2 =0.704!H aN1/2 =0.121
H aN1/2 =0.636!H aN1/2 =0.146
H aN1/2 =0.747!H aN1/2 =0.311
PL
D
G
C
Figure 13. Interfacial curvature, H , distributions for the C, G, PL and D microstructures. Thegreen patches on the schematic diagrams to the left indicate elementary interfacial units. Thedetailed shapes of these units are then displayed on the right by means of a SCFT calculation alongthe L/C phase boundary at !N = 20 and f = 0.3378. The curvature distribution over each unitis specified by the colour scale with the average and standard deviation quoted to the far right.Adapted from [69].
calculations [69] show that minority-type homopolymer reduces the packing frustration in Drelative to G, whereas majority-type homopolymer does not. With PL, the converse is true. Infact, calculations [37,73] predict that sufficient minority homopolymer can stabilise D over G,and that adequate majority homopolymer causes PL to replace G.
The presence of the close-packed spherical (Scp) phase in figure 2(a) seems to contradictthe idea that packing frustration favours the bcc arrangement. The explanation is that, athigh asymmetries, a significant fraction of minority blocks are dislodged from their domains
• Every round cylinder along the xn+2−axis and every sphere are solutions of (12.1).
To see whether spheres or cylinders are minimizers of the IP, one can compare thesurface area of the cylinder of the length a and sphere of the same volume, to see whichone is smaller. For a = 1, the surface area of a cylinder of length a and volume V isAcyl = 2
√πV and the surface area of a sphere with volume V is Asph = (36πV 2)1/3.
Acyl ≤ Asph is equivalent to V ≤ 814π
. Since Vcyl = 2πRa, this gives aR ≤ 818π2 . In
particular, one expects that cylinders of a fixed radius R are stable when a is small andas a increases cylinders become unstable.
We are interested in rotationally symmetric, periodic static solutions. Our goal isto show their existence and their (linear) stability and how the latter depends on theparameter a.
The notion of linear stability is defined in Subsections 9.1 and 9.2 In the presentcontext, it should be applied to the volume preserving mean curvature flow. Therefore we
Lectures on Applied PDEs 121
should consider the variations (perturbations) which preserve the enclosed volume. Suchvariations, ξ, satisfy
∫Sξ · ν = 0. Hence, we say that a CMC surface θ is stable (in the
sense of geometric analysis) iff dH(θ) ≥ 0 on the subspace
Vθ = ξ : S → Rn+1 :
∫S
ξ · ν = 0 (12.2)
and unstable, otherwise.
12.2 Bifurcation of new surfaces
We consider solutions of Eq. (12.1) which are surfaces of revolution around the xn+2−axis,periodic in the xn+2−direction of period a. Such surfaces are obtained by rotating graphsof functions ρ(xn+2) around the xn+2−axis. (They are special cases of graphs, θρ : (ω, x) 7→(ρ(ω, x)ω, x), over the unit round cylinder Cn+1
a = Sn×[0, a] = (ω, x) : ω ∈ Sn, x ∈ [0, a],with the graph functions ρ.) Let x = xn+2. Eq. (12.1) has the cylindrical branch(a, ρcyl ≡ n
h),∀a > 0, of solutions (a cylinder is periodic along its axis with an arbitrary
period). This leads us to the following
Theorem 12.1. • At a = 2πk nh
, a new branch of even/odd periodic solutions of Eq.(12.1) bifurcates from the cylindrical branch (a, ρcyl).
• The bifurcating even solutions are periodic surfaces of revolution of period a givenby the functions ρs(x) = n
h+ s cos(πx) +O(s2) (and similarly for the odd solutions).
• For 0 < a < 2π nh
, the cylindrical static solution, ρcyl, of Eq. (12.1) is unique in asmall even/odd neighbourhood of ρcyl and is stable. At a = 2π n
h, it looses its stability
and is unstable for a > 2π nh
.
• The bifurcating branch is stable for 2π nh< a < 4π n
h, at a = 4π n
h, it looses its stability
and a new stable branch of solutions bifurcates at this point and so forth.
In the next subsection we prove existence of bifurcating solutions. After that we provetheir stability. Before proceeding to the proofs, we give the expression of the meancurvature H, of surfaces of revolution.
First, we write out the expression for the mean curvature for the level set representa-tion of S. Below, all differential operations, e.g. ∇,∆, are defined in the correspondingEuclidian space (Rn+2). First, we recall that, if a surface S given in the level set repre-sentation, S = x′ ∈ Rn+2 : ϕ(x′) = 0, then we have (see Proposition G.3)
ν(x′) =∇ϕ|∇ϕ| , H(x′) = div ν(x′) = div
( ∇ϕ|∇ϕ|
). (12.3)
Using this expression, we can obtain an expression for the mean curvature of surfaces ofrevolution. Let x′ = (x⊥, x) ∈ Rn+2, with x⊥ = (x1, . . . , xn+1) ∈ Rn+1. Then a surface
122 Lectures on Applied PDEs
of revolution can be written as S = x′ ∈ Rn+2 : ϕ(x′) := |x⊥| − ρ(x) = 0, where
r⊥ ≡ |x⊥| = (∑n+1
i=1 x2i )
12 . We use this and (12.3) and the notation x⊥ := x⊥/|x⊥| to
compute ν(ρ) = (x⊥,∂xρ)√1+(∂xρ)2
, which gives the following expression for the mean curvature
H(x′), of S:
H(ρ) ≡ H(x′) =−1√
1 + (∂xρ)2
( ∂2xρ
1 + (∂xρ)2− n
ρ
). (12.4)
We define the map F (ρ, h, a) := H(θρ)−h. Then the equation (12.1) can be rewrittenas
F (ρ, h, a) = 0. (12.5)
Note that the round cylinder of the radius R is given by the equation ρ = R.
12.3 Proof of the existence part
We consider a solution to (12.1) which are surfaces of revolution with the graph functionρ = ρ(x). Then the mean curvature is given by the expression (12.4) and the equation(12.1) can rewritten as F (ρ, h, a) = 0, with
F (ρ, h, a) :=−1√
1 + (∂xρ)2
( ∂2xρ
1 + (∂xρ)2− n
ρ
)− h, (12.6)
where ρ is periodic of a period a. Note the following properties of H(ρ):
• H(ρ = R) = n/R;
• H(ρλ)(x) = λ−1H(ρ)(x/λ), where ρλ(x) := λρ(x/λ);
• dρH(ρ)(ρ− x · ∇ρ) = −H(ρ)− x · ∇H(ρ).
Proof. The first and second properties are obvious from the expression (12.6). They canbe also deduced from the general facts that H(θcyl) = n
Rand (see (8.9) and e.g. [29])
H(λθ) = λ−1H(θ). (12.7)
Because of θ(ω, x) = (ρ(ω, x)ω, x), the rescaling θ → λθ induces the rescaling ρ→ λρ(x/λ)of ρ.
Differentiating the equation in the second property w.r.to λ at λ = 1 and using that∂ρλ∂λ
∣∣λ=1
= ρ− x · ∇ρ, we arrive at the equation in the third property.
These properties imply the following properties of F :
• cylinder of radius R = R(h) := nh
(ρcyl) solves F (ρcyl, h, a) = 0, ∀a, h.
Lectures on Applied PDEs 123
• F (ρλ, λ−1h, λa)(x) = λ−1F (ρ, h, a)(λ−1x), where ρλ(x) := λρ(x/λ).
• dρF (ρ∗, h, a)(ρ∗ − x · ∇ρ∗) = −h, for any (ρ∗, h, a) solving F (ρ, h, a) = 0.
The third property with ρ∗ = ρcyl (so that ρ∗ = R and∇ρ∗ = 0) implies that dρF (ρcyl, h, a)
has the eigenvalue − hR
= −h2
nwith the eigenfunction equal identically to 1.
Using the second property we rescale the equation F (ρ, h, a) = 0, to eliminate one ofthe parameters, e.g. by taking λ = h, or λ = a−1, in the equation F (ρλ, λ
−1h, λa)(x) =λ−1F (ρ, h, a)(λ−1x). (This might depend on which parameter physically we want to keepfixed and which to vary.) This gives the new equation F (ρ′, a′)(x) = 0, 0 ≤ x ≤ ha, whereF (ρ′, a′)(x) := h−1F (ρ′, 1, a′)(x), ρ′(x) := hρ(x/h) and a′ := ha, or
F (ρ′, h′)(x) = 0, ρ′(x+ 1) = ρ′(x)
where F (ρ′, h′)(x) := (h′)−1F (ρ′, h′, 1)(x), ρ′(x) = a−1ρ(ax) and h′ := ah. We choose thesecond rescaling and drop the primes. Consequently, we have the equation
F (ρ, h) = 0, where F (ρ, h) := H(ρ)− h, (12.8)
H(ρ) :=1√
1 + (∂xρ)2
(− ∂2
xρ
1 + (∂xρ)2+n
ρ
), (12.9)
on R with ρ periodic of the period 1:
ρ(x+ 1) = ρ(x).
The latter equation has the family of cylindrical (homogeneous) solutions:
ρh =n
h,
which in the old variables become ρcyl = nah
.To prove the bifurcation of the new solutions, we use the bifurcation theory, specifically,
the Krasnoselski theorem. We check the conditions of this theorem (below R(h) := nh):
(a) F is C1;
(b) F (ρh, h) = 0, ∀h, where ρh = R(h);
(c) dρF (ρh, h) has the eigenvalue 0 for h = π√nk and this eigenvalue is of the multi-
plicity 2
(d) ∂h(dρF (ρ, h)∣∣ρ=R(h)
) = −h−2 nR(h)
= −h−1 6= 0.
Properties (a), (b) and (d) are straightforward. So, we check (c). We linearize F (ρ, h)at ρh ≡ R ≡ R(h) := n
hand let Lh := −dρF (ρh, h). Then by direct computation
Lh = −∂2x −
n
R2,
acting on the space L2per(R) of periodic, locally L2−functions of the period 1.
124 Lectures on Applied PDEs
Proposition 12.2. The operator Lh is self-adjoint and its spectrum, σ(Lh), is purelydiscrete with the eigenvalues − n
R2 , of the multiplicity 1, and (2πk)2 − nR2 , k = 1, 2, · · · ,
of the multiplicity 2, with the eigenfunctions 1, cos(2πkx) and sin(2πkx).
The self-adjointness and the form of the spectrum are obvious. We mention that thefact that Lh has the eigenvalue − n
R2 is not accidental. As shown above, it is related tothe fact that ρh ≡ R breaks the scaling covariance of the equation.
Recall R ≡ R(h) := nh. If h satisfies (2πk)2− n
R2 = 0, k = 1, 2, · · · , where R = nh, then
the operator Lh has a zero eigenvalue. These values, h = 2π√nk, are the candidates for the
bifurcation points. This proves the property (c). To get eigenvalues of odd multiplicitiesas required by the conditions of the Krasnoselski theorem, we restrict to either odd oreven functions, ρ(x) to get the multiplicity one.
Hence the Krasnoselski theorem is applicable and implies that the equation (12.8) hasnon-trivial solution branches of even/odd solutions bifurcating at from the trivial branch(R(h), h), ∀h at h = 2π
√nk, k = 1, 2, . . . .
By Corollary 11.4 (of the proof of the Krasnoselski theorem), the first non-trivial evensolution branch is of the form
ρ =n
h+ s cos(πx) + ws, with ws = O(s2), (12.10)
and h depending on s.
12.4 Proof of the linearized stability
In this section we show that (a) for 0 < a < 2π nh, the cylindrical static solution, ρh = n
h,
of Eq. (12.1) is stable, (b) at a = 2π nh, it looses its stability and (c) for a > 2π n
h, but
sufficiently close to 2π nh, the new solution that bifurcated at a = 2π n
his stable.
Again we pass to the rescaled problem (12.8). The notion of linear stability is definedin at the beginning of Subsubsection 12.1. It translates in the present context as follows.
Let ρ∗ be a solution of F (ρ, h) = 0 and dρF (ρ∗, h) denote the Gateaux derivative ofthe map F (ρ, h) at ρ = ρ∗. Recall that ρ describe surfaces of a fixed enclosed volume,say V (ρ) = c > 0. Hence we have to define dρF (ρ∗, h) on the tangent space Tρ∗ρ ∈H2
per(R,R) : V (ρ) = c, where H2per(R,R) is the Sobolev space of order 2 of real periodic
functions on R of the period 1.For surfaces of revolution we have by integrating over radial slices, V (ρ) =
∫ 1
02πρ2(x)dx.
Since by Proposition 6.16, Tρ∗V (ρ) = c = Null dV (ρ∗), we have that
V∗ = ξ ∈ H2per(R,R) :
∫ 1
0
ρ∗ξdx = 0. (12.11)
This subspace descends from (12.2).We say that a solution ρ∗ of (12.8), i.e. a solution of F (ρ, h, a) = 0, is linearly stable
iff the spectrum of dρF (ρ∗, h) on V∗ lies in the half-plane Re z > 0 and we say that ρ∗
Lectures on Applied PDEs 125
is linearly unstable iff σ(dρF (ρ∗, h))∣∣V∗∩ Re z < 0 6= ∅. (Since in our case, dρF (ρ∗, h)
is self-adjoint, the above relations can be replaced by duF (ρ∗, h))∣∣V∗≥ 0 and the smallest
eigenvalue of duF (ρ∗, h)∣∣V∗
is ≤ 0, respectively.)
Now, we apply this to ρ∗ = ρh. In this case space (12.11) becomes
Vh = ξ ∈ H2per(R,R) :
∫ 1
0
ξdx = 0. (12.12)
The operator duF (ρh, h))∣∣Vh
=: Lh is self-adjoint. Hence its spectrum is real and therefore
the static solution ρh is linearly stable iff σ(Lh∣∣Vh
) ⊂ λ ∈ R : λ > 0, and unstable iff
σ(Lh∣∣Vh
) ∩ λ ∈ R : λ < 0 6= ∅.Proposition 12.2 implies that on the subspace Vh, the eigenvalues of Lh on even/odd
functions are (2πk)2 − nR2 , k = 1, 2, · · · , of the multiplicity 1, with the eigenfunctions
cos(2πkx)/ sin(2πkx).If 2π
√n > h, then σ(Lh) on Vh is positive and for h = 2π
√n, the lowest eigenvalue
vanishes. This suggests that a new stable stationary solution bifurcates at this pointfrom the old one. Moreover, for 2π
√n < h, the operator Lh has a negative eigenvalue
2π2− nR(h)2
= 2π2− h2
n) and therefore, while the old solution, the cylinder ρh, is stable for
2π√n > h, it is unstable for 2π
√n < h. We expect that the new bifurcating solution is
stable, at least for h not too large, more precisely, for 2π√n < h < 2π 3
2
√n.
To sum up, the cylinder ρh = R(h) is stable for h < π√n, and unstable for h > π
√n.
Non-axi-symmetric surfaces. Let n = 1 (two dimensional surfaces in 3 dimensionalambient space). We consider solutions to (12.1) which are graphs over the standardcylinder given by functions ρ depending also on the angle θ: ρ = ρ(x, θ). Then the meancurvature is given by1
H(ρ)(x, θ) =1√
1 + (∂xρ)2 + (∂θρ/ρ)2
( −∂2xρ− ρ−2∂2
θρ
1 + (∂xρ)2 + (∂θρ/ρ)2+
1
ρ
). (12.13)
We assume as before that ρ is periodic in x of a period a. Then equation (12.1) canrewritten as
F (ρ, h, a) := H(ρ)− h = 0, ρ is periodic in x of a period a, (12.14)
where H(ρ) is the mean curvature given by the expression (12.13).
where x⊥ := (x1, x2), x⊥ := x⊥/|x⊥| and x⊥ := (x2,−x1), x⊥ := x⊥/|x⊥|, which gives expression (12.13)for the mean curvature.
126 Lectures on Applied PDEs
Exercise 12.3. (a) Show that the cylinder are still solutions to (12.14) for every a;(b) Find the bifurcation points from the cylindrical branch;(c) Find the values of the parameter a for which the cylindrical solution is linearly
stable/unstable.(Hint: As in the axi-symmetric case, consider ρ’s even and odd in x separately. In
addition, one might want to separate the cases of ρ’s satisfying ρ(x,−θ) = ρ(x, θ) andρ(x,−θ) = −ρ(x, θ).)
Remark. Consider Eq. (12.8) on periodic ρ’s of the period 1. If ρ∗ is a non-constantsolution to (12.8), then dρF (ρ∗, h)∂xρ∗ = 0, i.e. dρF (ρ∗, h) has the eigenvalue 0, with theeigenfunction ∂xρ∗.
This property is related to the fact that ρ∗ breaks the translational invariance of theequation: if ρy(x) := ρ∗(x+ y), then F (ρy, h)(x) = F (ρ, h)(x+ y). Therefore, if ρ∗ solvesF (ρ, h) = 0, then ρλ satisfies F (ρy, h)(x) = 0. Differentiating the latter equation w.r.to y
at y = 0, we find dρF (ρ∗, h)∂ρy∂y
∣∣y=0
= 0. Since ∂ρy∂y
∣∣y=0
= ∂xρ∗, the result follows.
Due to the third property of F (dρF (ρcyl, h, a)(ρ∗ − x · ∇ρ∗) = −h, for any (ρ∗, h, a)solving F (ρ, h, a) = 0), to study the stability of ρ∗, we have to consider dρF (ρ∗, h) on thesubspace
V∗∗ = ξ ∈ H2Neum([0, 1],R) :
∫ 1
0
ξ(1− x∂xρ∗) = 0,
∫ 1
0
ξ∂xρ∗ = 0. (12.15)
Lectures on Applied PDEs 127
13 Turing instability and pattern formation (needs
editing)
We describe the Turing model for pattern formation designed originally to describe for-mation of animal coat patterns. In this model, the dyes colouring animal coats evolveaccording to a system of reaction diffusion equations. At some values of diffusion con-stants, the homogeneous (x-independent) solutions describing a uniform coating becomeunstable and new, stable solutions describing various patterns bifurcate.
13.1 Turing instability
(needs editing) Let Ω ⊆ Rn be a bounded open set with smooth boundary. We considerthe following reaction-diffusion system for u : Ω× (0,∞)→ Rn with Neumann boundaryconditions:
where f is a continuously differentiable function f : Rn → Rn, n ≥ 2 and δ be a positivedefinite n× n matrix (the diffusion tensor).
Homogeneous (x-independent) solutions satisfy the associated reaction equation
∂u(t)
∂t= f(u(t)). (13.2)
Notice that static solutions to (13.2) are also static, homogeneous solutions of (13.1) and,if a static, homogeneous solution to (13.2) is stable, then it is also a stable solution of(13.1) under homogeneous perturbations.
Turing proposed that pattern formation in biological systems could arise througha process where a stable static solution of (13.2) became unstable in the presence ofdiffusion, i.e., the static solution is not stable as a solution of (13.1). This is called Turingor diffusion-driven instability.
Let u ≡ u∗ be a static homogeneous solution to (13.1), i.e. u∗ solves f(u∗) = 0.u = u∗ is also a static solution of (13.2). If u(t) = u∗ is a stable equilibrium (staticsolution) of (13.2), then u(x, t) ≡ u∗ is also a stable solution of (13.1) under homogeneousperturbations.
The first problem then is: if u = u∗ is a stable solution of (13.2), under what conditionson f , a variation of δ induces instability of u = u∗? Here is a result on this.
Theorem 13.1. Suppose n = 2 and δ is a diagonal matrix δ = diag (d1, d2). Let f =(f1, f2) and fij = ∂fi
∂xj(u∗).
1. If f11 ≤ 0, then u∗ is a linearly stable solution of (13.1).
128 Lectures on Applied PDEs
2. If f11 > 0, then there exists ε such that u = 0 is linearly stable for d2 < βd1
and linearly unstable for d2/d1 > β, provided σ(−∆) ∩ (αδ, βδ) 6= ∅, for some0 < αδ < βδ. (β is given explicitly in (13.9).)
First we observe that −∆ is a self-adjoint operator on L2(Ω,C) (with the appropriateboundary conditions) and, since, Ω is bounded, it has a purely discrete spectrum withnon-negative eigenvalues, which we denote by 0 ≤ λ1 ≤ λ2 ≤ . . . , marching off to ∞,λn →∞, as n→∞. We define the n× n matrix
Lδ(λ) = −δλ+ f ′(u∗). (13.3)
We derive Theorem 13.1 from the following more general result
Theorem 13.2. Suppose n is even (n = 2) and u(t) = u∗ is a stable equilibrium (staticsolution) of (13.2). Then the homogeneous static solution u∗ of (13.1) is a linearly stable,if and only if detLδ(λ) > 0 for all λ ∈ σ(−∆).
Proof of Theorem 13.2. For the proof of the theorem, we will require a number of lemmas.We begin investigating the linearization of the r.h.s. of (13.1) around u∗, which is
Lδ = δ∆ + f ′(u∗). (13.4)
We prove the following result on the spectrum of Ld.
Lemma 13.3.σ(Lδ) =
⋃λ∈σ(−∆)
σ(Lδ(λ)). (13.5)
Proof. Let φλ λ∈σ(−∆) be a orthonormal basis of eigenfunctions of −∆. Then for anyu : Ω→ Rn, there exist vectors cλ ∈ Rn such that
u =∑λ
cλφλ.
We then have(Lδ − z)u =
∑λ
(Lδ(λ)− z) cλφλ.
Now, since Ω ⊆ Rm is a bounded open set with smooth boundary, Lδ has purely pointspectrum. Suppose that z ∈ σ(Ld). Then there is non-zero u such that (Lδ−z)u = 0. Foreach µ ∈ σ(−∆), we multiply by φµ and integrate, to obtain, using the orthonormality ofthe basis,
0 =∑λ
(Lδ(λ)− z) cλ
∫Ω
φµφλ = (Lδ(µ)− z) cµ,
which implies that either cµ = 0 or z is an eigenvalue of Lδ(λ). Since u is non-zero, notall cµ are zero, and therefore z ∈ σ(Ld(µ)) for some µ.
Lectures on Applied PDEs 129
Conversely, suppose that z ∈ σ(Lδ(λ)) for some λ. Fix non-zero c ∈ Rn such thatLδ(λ)c = 0. Then if we set u = cφλ, (Lδ − z)u = 0. Therefore z ∈ σ(Ld), and thatcompletes the proof.
Lemma 13.4. Suppose that λ ≥ 0. Then TrLδ(λ) < 0.
Proof. We then have
TrLδ(λ) = −λTr δ + Tr f ′(0).
Since f ′(u∗) is negative definite, we know that Tr f ′(u∗) < 0. Since δ > 0 and thereforeTr δ > 0, this implies the result.
For n = 2, the equation TrLδ(λ) < 0 implies
Corollary 13.5. One eigenvalue of Lδ(λ) is always negative. For n = 2, the othereigenvalue is negative iff detLδ(λ) > 0.
This corollary implies Theorem 13.2.
13.2 Appendix: Derivation of Theorem 13.1 from Theorem 13.2
Suppose n = 2 and δ is a diagonal matrix δ = diag (d1, d2). We have
detLδ(λ) = d1d2λ2 − (d1f22 + d2f11)λ+ det f ′(u∗) =: hδ(λ). (13.6)
This means that detLd(λ) > 0 if and only if hδ(λ) > 0. Hence Lδ(λ) is negative definiteif and only if hδ(λ) > 0.
We we begin with studying the function hδ(λ) and prove the following result.
Lemma 13.6.
1. If f11 ≤ 0, then for all δ > 0, hδ(λ) > 0 for all λ ≥ 0.
2. If f11 > 0, then there exists β > 1 with the following properties:
(a) If d2/d1 < β, hδ(λ) > 0 for all λ ≥ 0.
(b) If d2/d1 = β, there exists λ0 > 0, such that hδ(λ) = d1d2(λ− λ0)2.
(c) If d2/d1 > β, there exist λ2 > λ1 > 0, such that hδ(λ) = d1d2(λ− λ1)(λ− λ2).
Proof. It is clear from the definition of hδ(λ), (13.6), that hδ(λ) has a unique globalminimum point, λmin, which can be easily calculated to be
λmin =f11
2d1
+f22
2d2
. (13.7)
130 Lectures on Applied PDEs
We let hmin be the global minimum value. A number of properties of hδ(λ) can also bedetermined using its discriminant, D, which is
D = (d1f22 + d2f11)2 − 4d1d2 det f ′(0). (13.8)
Now suppose f11 ≤ 0. Since f11 + f22 < 0, we have f22 < −f11. If f11 = 0, thenλmin < 0 for all δ > 0. Now hδ(0) = det f ′(u∗) > 0, so it follows that for all λ > 0 > λmin,hδ(λ) ≥ hδ(0) > 0. So suppose f11 < 0. Fix d1 and set ε = −f22
f11. Then if d2/d1 > ε, then
again λmin < 0 and the same argument gives hδ(λ) > 0 for all λ ≥ 0. Suppose then thatd2/d1 < ε. We will show that hmin > 0, or equivalently that D < 0. Since d2/d1 < ε,d2
2f211 < d2
1f222. Also, since det f ′(0) > 0, we have f12f21 < f11f22. Therefore,
D = d21f
222 + d2
2f211 − 2d1d2f11f22 + 4d1d2f12f21
< 2d21f
222 + 2d1d2f11f22
< 2d21f
222 − 2d2
1f222 = 0.
That establishes (1).Now suppose f11 > 0. Then f22 < −f11 < 0. Set ε = −f22
f11, so ε > 1. For d2/d1 ≤ ε,
one can check that λmin ≤ 0, and therefore the same argument as before shows thathδ(λ) > 0 for all λ ≥ 0.
For d2/d1 > ε, on the other hand, λmin > 0. As can be seen from above, the discrim-inant Disc is a quadratic polynomial in d2 and D → ∞ as d2 → ∞. Now for d2 = ε, asimple calculation shows that
D = −4d1d2 det f ′(0) < 0.
This means there exist a β > ε such that D < 0 for ε ≤ d2/d1 < β, D = 0 at d2/d1 = β,and D > 0 for d2/d1 > β. One can calculate that
β =1
f 211
[2√−f12f21 Tr f ′(u∗) + Tr f ′(u∗)− f12f21
]. (13.9)
This then implies (2), except for the assertion that λ0, λ1, λ2 > 0, but this follows fromthe fact that λmin > 0 and hδ(0) > 0.
We now turn to the proof of Theorem 13.1.Suppose first that f11 ≤ 0. Then we have seen that for all δ, hδ(λ) > 0 for all λ ≥ 0.
This means that for all λ ≥ 0, Lδ(λ) is negative definite. Now if λ ∈ σ(−∆), then λ ≥ 0,so this means that the eigenvalues of Lδ lie in the plane Re z < 0 , and therefore u∗ islinearly stable.
Now suppose that f11 > 0 . Then if d2/d1 < β, an argument similar to the one above,shows that u = 0 is stable. If d2/d1 > β, on the other hand, then there are λ ≥ 0, forwhich hδ(λ) < 0. If there is λ ∈ σ(−∆) such that hδ(λ) < 0, then Lδ(λ) has an eigenvaluewith positive real part, and therefore so does Lδ, which means that u∗ is not linearlystable.
Lectures on Applied PDEs 131
13.3 Pattern formation
(needs editing) Recall that for n is even (n = 2), the homogeneous static solution u∗ of(13.1) is a linearly stable, if and only if detLδ(λ) > 0 for all λ ∈ σ(−∆).
Theorem 13.7. Suppose n = 2 and δ is a diagonal matrix and u(t) = u∗ is a stableequilibrium (static solution) of (13.2). Let λ1 be the smallest eigenvalue of −∆ and δ∗ bethe smallest δ at which detLδ(λ1) vanishes. Then a non-homogeneous static solution to(13.2) bifurcates at from the homogeneous one. At the bifurcation point, the solution u∗looses its stability, while the new non-homogeneous static solution is stable (the exchangeof the stability). The bifurcating solution is of the form ???.
Proof. Let δ = diag (d1, d2) and d1 be fixed for simplicity’s sake and let α := d2/d1. Wewrite δ(α) := diag(d1, αd1) and define F (α, u) = δ(α)∆u+f(u). Then the static equationfor (13.1) can be written as
F (α, u) = 0. (13.10)
Recall the definition (13.9) of β. We have
1. (13.10) has a trivial branch: F (α, 0) = 0 for all α > 0.2. There exists α ≥ β, such that Lδ(α) = duF (α, 0) has a 0 eigenvalue for some δ. The
latter is s.t. 0 is an eigenvalue of Lδ(α)(λ) for some λ ∈ σ(−∆).
3. The multiplicity of the zero eigenvalue of Lδ = the multiplicity of λ as an eigenvalueof −∆.
4. Fix any non-zero u in the nullspace of Lδ and non-zero v in the nullspace of L∗.Then 〈v, dα,uF (α, 0)u〉 6= 0.
We prove these properties. We know from the proof of Lemma 13.6 that for α =d2/d1 > β, λmin > 0 and hmin < 0. We also know that λmin → ∞ as d2 → ∞, sotherefore there is a smallest α such that detLδ(α)(λ) = hδ(α)(λ) = 0 for some λ ∈ σ(−∆).Since hδ(α)(λ) has two roots, it is possible for there to be two such λ, but in the genericcase, there will only be one and we assume that from now on.
This means that 0 is an eigenvalue of Lδ(α)(λ), which, since n = 2, has at most oneeigenvalue in Re z ≥ 0, so it has multiplicity 1.
Now 0 is an eigenvalue of Lδ(α) if and only if it is an eigenvalue of Lδ(α)(λ) for some λ ∈σ(−∆). To show that the multiplicity of the 0 eigenvalue of Lδ is equal to the multiplicityof λ as an eigenvalue of −∆, let φ1, . . . , φn be the n eigenfunctions corresponding to λ.Then the nullspace of Lδ(α) is generated by cφ1, . . . , cφn , where c is any non-zero vectorin the nullspace of Lδ(α)(λ).
First we show that if c ∈ R2 is such that Lδ(λ)c = 0, then c2 6= 0. Suppose, onthe contrary, that c2 = 0. Then we have f21c1 = 0, so f21 = 0. This implies that0 < det f ′(u∗) = f11f22, and since f11 > 0, this implies f22 > 0. But that would meanTr f ′(u∗) = f11 + f22 > 0 and that contradiction proves c2 6= 0.
A similar proof shows that if c ∈ R2 is such that Lδ(λ)∗c = 0, then d2 6= 0.
132 Lectures on Applied PDEs
Now, a simple calculation gives dd2,uF (δ, u) =
(0 00 d2∆
), so dd2,uF (d1, αd1, u) =(
0 00 αd1∆
). Next, u = a1cφ1 + · · ·+ ancφn and v = b1dφ1 + · · ·+ bndφn. So
which can at least be made to not be zero. The four properties above imply the theorem.
Literature. [36] found conditions on f ′(0) and d for u = 0 to be unstable or a linearlystable. [27] have Ω ⊂ R. They find conditions for when Turing instability can and cannotoccur, see also [37]. For a textbook exposition see [22].
Lectures on Applied PDEs 133
14 The Ginzburg-Landau Equations
14.1 Equations and results
In this section we consider the Ginzburg-Landau equations of superconductivity (and par-ticle physics):
∆AΨ = κ2(|Ψ|2 − 1)Ψ, (14.1a)
curl∗ curlA = Im(Ψ∇AΨ). (14.1b)
for (Ψ, A) : R2 → C × R2, ∇A = ∇ − iA, and ∆A = ∇2A, the covariant derivative and
covariant Laplacian, respectively. The equations are discussed in detail in Section 2.3.
Recall that these equations describe equilibrium configurations of superconductors,and of the U(1) Higgs model from particle physics. In the superconductivity, the complex-valued function Ψ(x) is called the order parameter, with |Ψ(x)|2 giving the local densityof (Cooper pairs of) superconducting electrons, and A is a vector field called the magneticpotential, so that B(x) := curlA(x) = the magnetic field. The vector quantity J(Ψ, A) :=Im(Ψ∇AΨ) = the superconducting current.
In particle physics, Ψ and A are the Higgs and U(1) gauge (electro-magnetic) fields,respectively.
Geometrically, one can think of A as a connection on the principal U(1)-bundle Rn ×U(1), n = 2, 3.
We prove existence of the Abrikosov lattices defined in Section 2.3.
In the idealized situation of a superconductor occupying all space and homogeneousin one direction, we are led to a problem on R2 and so may consider ψ : R2 → C andA : R2 → R2. In this case, curlA := ∂1A2 − ∂2A1 is a scalar, and for a scalar function,B(x) ∈ R, curlB = (∂2B,−∂1B) is a vector.
After Abrikosov, we look for solutions (Ψ, A) defined on all of R2, whose all physicalproperties, namely the density of superconducting pairs of electrons, ns := |Ψ|2, themagnetic field, B := curlA, and the current density, J := Im(Ψ∇AΨ), are doubly-periodicwith respect to some lattice L. (For us, a lattice is the set
L = m1ω1 +m2ω2 : m1,m2 ∈ Z ⊂ R2
for some basis vectors ω1, ω2 in R2; note that L forms a group under addition.) We callsuch states (L−)Abrikosov lattice states.
We formulate a simplified version of the main result of this section. The general resultcan be found in [34] (see also [13]).
Theorem 14.1. For every lattice L satisfying 0 < κ2−b 1, where b = 2π|Ω| , the equations
(14.1) have an L−Abrikosov lattice solution in a neighbourhood of the branch of normalsolutions.
134 Lectures on Applied PDEs
The full proof of this theorem is given in Appendix 14.4 (see also Subsection 19.3 fora different, variational proof). In the next subsection we describe some main propertiesof equations (14.1) and introduce some general notions. Then, in Section 14.3, we presentthe key step in the proof. In Appendix ?? we describe the relation between equivariantsolutions and sections of and connection on line bundles.
14.2 Properties of the GLE
Symmetries. As we already mentioned in Section 2.3, the Ginzburg-Landau equa-tions (14.1) admit several symmetries, that is, transformations which map solutions tosolutions.
Gauge symmetry: for any sufficiently regular function χ : R2 → R,
T gaugeχ : (Ψ(x), A(x)) 7→ (eiχ(x)Ψ(x), A(x) +∇χ(x)); (14.2)
Rotation and reflection symmetry: for any R ∈ O(2) (including the reflectionsf(x)→ f(−x))
T rotR : (Ψ(x), A(x)) 7→ (Ψ(Rx), R−1A(Rx)). (14.4)
Exercise 14.2. Prove that the above transformations are symmetries of the Ginzburg-Landau equations (14.1), i.e. if (Ψ, A) is a solution to (14.1), then so is (Ψ, A), for T beany of the above transformations.
Thus the set of all solutions of the Ginzburg-Landau equations can be split into equiv-alence classes of solutions related by gauge transformations. A condition which pick asubclass of each equivalence class is called the gauge condition. An example of a gaugecondition is divA = 0. This can be also arranged: if divA 6= 0 we can always find a gaugeη s.t. div(A+∇η) = 0, namely, we take η solving the equation −∆η = divA.
One of the analytically interesting aspects of the Ginzburg-Landau theory is the factthat, because of the gauge transformations, the symmetry group is infinite-dimensional.
Energy. The Ginzburg-Landau equations (14.1) are the Euler-Lagrange equations forcritical points of the Ginzburg-Landau energy functional (written here for a domain Q ∈R2)
EQ(Ψ, A) :=
∫Q
|∇AΨ|2 + (curlA)2 +
κ2
2(|Ψ|2 − 1)2
. (14.5)
Lectures on Applied PDEs 135
Mathematically, this is a natural generalization of the basic Dirichlet functional∫Q|∇Ψ|2.
First, one passes from the standard gradient ∇ to the covariant one ∇A. Then one addsthe energy
∫Q
(curlA)2 of the magnetic potential/connection A. Finally, one adds the
self-interaction term,∫Qκ2
2(|Ψ|2 − 1)2, from physics.
L-equivariant states. The following proposition will play a key role in the proof ofTheorem 14.1:
Proposition 14.3. A pair u = (Ψ, A) is an Abrikosov (vortex) lattice iff there are, ingeneral, multi-valued differentiable functions, ηs : R2 → R, s ∈ L, s.t.
ηs u. (In other words, u = (Ψ, A) is mapped by latticetranslations into a gauge equivalent pair.)
Proof. If state (Ψ, A) satisfies (14.6), then all associated physical quantities are L−periodic,i.e. (Ψ, A) is an Abrikosov lattice. In the opposite direction, if (Ψ, A) is an Abrikosovlattice, then curlA(x) is periodic w.r.to L, and therefore A(x + s) = A(x) + ∇ηs(x),for some functions ηs(x). Next, we write Ψ(x) = |Ψ(x)|eiφ(x). Since |Ψ(x)| and J(x) =|Ψ(x)|2(∇φ(x)−A(x)) are periodic w.r.to L, we have that ∇φ(x+ s) = ∇φ(x) +∇ηs(x),which implies that φ(x+ s) = φ(x) + ηs(x), where ηs(x) = ηs(x) + cs, for some constantscs.
We call a pair satisfying (14.6) the lattice, or η-gauge-periodic state. In terminology ofSection 2.3 it is an equivariant state w.r. to the group of lattice translations for a latticeL.
Since T transs is a commutative group, we see that the family of functions ηs has the
important cocycle property
ηs+t(x)− ηs(x+ t)− ηt(x) ∈ 2πZ. (14.7)
This can be seen by evaluating the effect of translation by s+ t in two different ways. Wecall ηs(x) ≡ η(s, x) the gauge exponent. (In algebraic geometry it is called the automorphyexponent.)
Automorphy factors. We denote by Ω and |Ω| the fundamental lattice cell and itsarea, respectively. We list some important properties of ηs:
• If (Ψ, A) satisfies (14.6) with ηs(x), then T gaugeχ (Ψ, A) satisfies (14.6) with ηs(x)→
is independent of x and of the choice of the basis ν1, ν2 and is an integer.
• Every exponential ηs satisfying the cocycle condition (14.7) is equivalent to theexponent
ηbs :=b
2s ∧ x+ cs, (14.10)
for b and cs satisfying b|Ω| = 2πc(gs) and
cs+t − cs − ct −1
2bs ∧ t ∈ 2πZ. (14.11)
• The condition (14.7) implies the magnetic flux quantization:
1
2π
∫Ω
curlA = deg Ψ = c(η). (14.12)
Indeed, the first and second statements are straightforward.For the fourth property, see e.g. [?, ?, ?, ?], though in these papers it is formulated
differently. In the present formulation this property was shown by A. Weil and generalizedin [?].
The third and fifth statements are proven below.
Proposition 14.4. • (characteristic class) The quantity (14.9) is independent of xand of the choice of the basis ν1, ν2 and is an integer. Thus (14.9) is the topologicalinvariant classifying Abrikosov lattice states (called the Chern number).
• (magnetic flux quantization) The quantity 12π
∫Ω
curlA (magnetic flux) is an integerand is equal to c(η) (magnetic flux quantization),
1
2π
∫Ω
curlA = c(η), (14.13)
where A is the corresponding connection and Ω is a fundamental lattice cell.
Proof. By the relation (14.7), ην2(x + ν1) + ην1(x) − ην1+ν2(x) ∈ 2πZ and ην1(x + ν2) +ην2(x) − ην1+ν2(x) ∈ 2πZ. Subtracting the second relation from the first shows that c(η)is independent of x and is an integer.
Lectures on Applied PDEs 137
To prove the second statement, we note that, by Stokes’ theorem, the magnetic fluxthrough a lattice cell Ω is
∫Ω
curlA =∫∂ΩA. Now, using the definition of the gauge
transformation, (14.2), and the condition (14.6), we obtain∫∂Ω
Remarks. 1) Relation (14.7) is well known in algebraic geometry and number theorywhere eiηs(x) is called the automorphy factor (see e.g. [11]).
2) In algebraic geometry and number theory, the automorphy factors eiη′s(x) and eiηs(x)
satisfyingη′s(x) = ηs(x) + χ(x+ s)− χ(x),
for some χ(x), are said to be equivalent. A function Ψ satisfying T transs Ψ = eiηsΨ is called
eiηs−theta function.3) The special form (14.10) is related to a general construction of line bundles over
the complex torus using symplectic form ω(z, w) to construct automorphy factors, e.g.ηs(z) = bω(z, s) + cs, where b|C/L| = 2πn and cs satisfies cs+t − cs − ct − b
2ω(s, t) ∈ 2πZ.
The Chern number, c(η), is expressed in terms of ω as
c(η) = bω(ν1, ν2), (14.16)
where ν1, ν2 is a basis of L.4) The exponentials ηs satisfying the cocycle condition (14.7) are classified by the
irreducible representation of the group of lattice translations. This follows from the factthat cs’s satisfying (14.11) are classified by the irreducible representation of the group oflattice translations.
Flux quantization. The important property (14.12) of lattice states tells that themagnetic flux through a lattice cell is quantized:∫
Ω
curlA = 2πn (14.17)
138 Lectures on Applied PDEs
for some integer n. We give another proof of this statement. If |Ψ| > 0 on the boundaryof the cell, we can write Ψ = |Ψ|eiθ and 0 ≤ θ < 2π. The periodicity of |Ψ| and J ensurethe periodicity of ∇θ − A and therefore by Green’s theorem,
∫Ω
curlA =∮∂ΩA =
∮∂Ω∇θ
and this function is equal to 2πn since Ψ is single-valued.Equation (14.17) then implies the relation between the average magnetic flux, b, per
lattice cell, b = 1|Ω|
∫Ω
curlA, and the area of a fundamental cell
b =2πn
|Ω| . (14.18)
We note that due to the reflection symmetry of the problem we can assume that b ≥ 0.From now on we assume that
(Ψ, A) satisfy (14.6) with ηs given by (14.10) - (14.11) and (14.18). (14.19)
Normal solutions and bifurcation. The GLE have two basic homogeneous (x−independent)solutions. First we mention the homogeneous solution, describing the perfect supercon-ductor solution, namely uS := (ΨS ≡ 1, AS ≡ 0). Without this solution there would beno superconductors.
More important for our analysis are the normal (or non-superconducting) solutions,
uN ≡ ub := (ΨN ≡ 0, AN ≡ Ab),
where Ab is a vector potential with constant magnetic field curlAb =: b = constant.Below we use the bifurcation theory developed above to show existence of the Abrikosov
lattice solutions. The branch of ‘trivial’ solutions they bifurcate from is exactly the nor-mal branch displayed above. b is the bifurcation parameter. Interestingly, it does notenter the GLE explicitly, but only through the space of solutions.
As we remember from the bifurcation theorem, Theorem 11.2, the key step in thebifurcation analysis is solving the linearized problem, namely finding the values of thebifurcation parameter, b (the candidates for the bifurcation points), for which this problemhas a non-trivial solution. In the next subsection we solve the linearized problem, thenonlinear analysis is carried out in Appendix 14.4.
Lattices. By identifying R2 with C, any lattice L can be given a basis ν1, ν2 suchthat the complex number τ = ν2
ν1satisfies Imτ > 0. τ will be called the shape parameter
of the lattice. By rotating L, if necessary, we can bring it to the form
Lω = r(Z + τZ), where ω = (τ, r), with τ ∈ C, Imτ > 0, and r > 0. (14.20)
By the quantization condition (14.18),
r :=
√2πn
Imτb. (14.21)
Lectures on Applied PDEs 139
14.3 The linear problem
To find candidates for bifurcation points we have solve the linear problem:
−∆Abψ = κ2ψ, (14.22)
for ψ satisfying the gauge - periodic boundary condition (see (14.19))
ψ(x+ s) = ei(b2x∧s+cs)ψ(x), ∀s ∈ Lω. (14.23)
This quasiperiodic boundary condition is consistent with the fact that ψ is a single valuedfunction if and only if the magnetic flux, b|Ω|, through the fundamental cell Ω is quantized:b|Ω| = 2πn, for some integer n.
We consider this eigenvalue problem above on the Sobolev space of order two, H2(Ω,C),whose elements satisfy the quasiperiodic boundary condition (14.23). We identify R2 withC in the usual way as x = (x1, x2)↔ z = x1 + ix2. The key result here is
Theorem 14.5. Let b be determined by the quantization condition b = 2πn/|Ω|. Thesmallest |Ω| for which the problem (14.22) - (14.23) has a non-trivial solution is given byb = κ2. In this case the space solutions is of the dimension n.
Remark. The value b = κ2 is called the second critical magnetic field and is denotedHc2, so that Hc2 = κ2.
Proof of Theorem 14.5. We consider the operator −∆Ab on the space L2(Ω,C), with thedomain consisting of functions from H2(Ω,C) satisfying the boundary conditions (14.23).By a standard result, it is self-adjoint. Spectral information about −∆Ab can be obtainedby introducing the complexified covariant derivatives (harmonic oscillator annihilationand creation operators), ∂A and ∂∗A = −∂A, with
∂A :=1
2((∇A)1 + i(∇Ab)2). (14.24)
Remembering the definition of ∇A, we compute
∂A = ∂ + iAc, (14.25)
where ∂ := 12(∂x1 + i∂x2) and Ac := A1 − iA2. One can verify that these operators satisfy
the following relations:
[∂A, ∂∗A] =
1
4curlA; (14.26)
−∆A − curlA = 4∂∗A∂A. (14.27)
(As for the harmonic oscillator (see [12]), this, together with the relation curlAb = b,gives σ(−∆Ab) = (2k + 1)b : k = 0, 1, 2, . . . .)
140 Lectures on Applied PDEs
Exercise 14.6. Prove the relations (14.26) and (14.27).
The second property and the relation curlAb = b imply
Lemma 14.7.Null(−∆Ab − b) = Null ∂Ab . (14.28)
Proof. if ∂Abφ = 0, then by the second property, (−∆Ab−b)φ = 0. If (−∆Ab−b)φ = 0, onthe other hand (−∆Ab − b)φ = 0, then by the second property, ∂∗
Ab∂Abφ = 0. Multiplying
this by φ, we have 0 = 〈φ, ∂∗Ab∂Abφ〉 = 〈∂Abφ, ∂Abφ〉 = ‖∂Abφ‖2, which implies ∂Abφ =
0.
To find Null ∂Ab , we solve for h satisfying h−1∂Abh = ∂. A simple calculation givesh = e−
b2
(ix1x2−x22). (This solution is highly non-unique.) Hence we have
eb2
(ix1x2−x22)∂Abe− b
2(ix1x2−x22) = ∂.
This immediately proves that Ψ ∈ Null ∂Ab if and only if ξ(x) = eb2
(ix1x2−x22)ψ(x) satisfies∂ξ = 0.
For z′ = 1r(x1 + ix2), where r is given in (14.21), and z = x1 + ix2, we define θ(z′) by
the relation
ψ0 (z) = e−b4
(|z|2−z2)θ(z′), z′ :=1
rz, (14.29)
By the above, the function θ is entire and, due to the periodicity conditions on φ, satisfies
θ(z + 1) = θ(z), (14.30a)
θ(z + τ) = e−2πin(z+ 12τ)θ(z). (14.30b)
(θ is the theta function, see [21]).To complete the proof, we now need to show that the space of the analytic functions
which satisfy these relations form a vector space of dimension n. By the first relation, θhas the absolutely convergent Fourier expansion
θ(z) =∞∑
k=−∞
cke2πkiz. (14.31)
The second relation, on the other hand, leads to a relation for the coefficients of theexpansion. Namely, we have
ck+n = einπτe2kiπτck
and that means such functions are determined solely by the values of c0, . . . , cn−1 andtherefore form an n-dimensional vector space. This completes the proof of Theorem 14.5.
Lectures on Applied PDEs 141
Corollary 14.8. Let b = κ2. Then the solutions of the linear problem (14.22) - (14.23)are of the form(14.29) - (14.30).
Abrikosov considered the case n = 1. In this case, the space (14.28) is one-dimensionaland spanned by the function
ψ := ei2x2(x1+ix2)
∞∑k=−∞
ckeik√
2πImτ(x1+ix2), (14.32)
ck = ceikπτk−1∏m=1
ei2mπτ . (14.33)
This is the leading approximation to the Abrikosov lattice solution. The normalizationcoefficient c cannot be found from the linear theory and is obtained by taking into accountnonlinear terms by perturbation theory.
The rest of the proof of Theorem 14.1 is given in Appendix 14.4.
14.4 Appendix A: Proof of Theorem 14.1 (to finish)
14.4.1 Fixing the gauge
The gauge symmetry allows one to fix solutions to be of a desired form. Let Ab(x) = b2Jx,
where J is the symplectic matrix
J =
(0 −11 0
). (14.34)
We will use the following proposition
Proposition 14.9. Let (Ψ, A′) be an L-lattice state, i.e. it satisfies (2.77) and let b bethe average magnetic flux per cell. Then there is a L-lattice state (Ψ, A), that is gauge-equivalent to a translation of (Ψ, A), such that
Ψ(x+ s) = ei(b2x·Js+cs)Ψ(x) and A(x+ s) = A(x) +
b
2Js, ∀s ∈ L, (14.35)
where cs satisfy (14.11). Moreover, we have for α := A− Ab,
(i) α(x+ s) = α(x) for all s ∈ L;
(ii) α has mean zero:∫
Ωα = 0;
(iii) α is divergence-free: divα = 0.
A proof of this proposition is given in an appendix at the end of this section.From now on we assume that (Ψ, A) satisfy (14.35).
142 Lectures on Applied PDEs
14.4.2 Rescaling
Suppose, that we have a Lω-lattice state (Ψ, A), where ω = (τ, r). We now define therescaled fields (ψ, a) to be
(ψ(x), a(x)) := (rΨ(rx), rA(rx)).
Let Lτ be the lattice spanned by 1 and τ , i. e. Lτ := Z + τZ, with Ωτ being a primitivecell of that lattice. We note that |Ωτ | = 2πn. We summarize the effects of the rescalingabove:
(A) (ψ, a) is a Lτ -lattice state.
(B) Ψ and A solve the Ginzburg-Landau equations if and only if ψ and a solve
(−∆a − λ)ψ = −κ2|ψ|2ψ, (14.36a)
curl∗ curl a = Im(ψ∇aψ) (14.36b)
for λ = κ2r2. The latter equations are valid on Ωτ with the boundary conditionsgiven in the next statement.
(C) If (Ψ, A) is of the form described in Proposition 14.9, then (ψ, a) satisfies
ψ(x+ t) = eikn2x·Jtψ(x), (14.37)
where k = k(τ) := r2b/n = 2πImτ
, and, with a = an + α, an(x) := kn2Jx,
α(x+ t) = α(x), ∀t ∈ Lτ , and
∫Ωτα = 0, divα = 0. (14.38)
Our problem then is, for each n = 1, 2, . . ., find (ψ, a), solving the rescaled Ginzburg-Landau equations (14.36) and satisfying (i).
14.4.3 Reformulation of the problem
In this section we reduce two equations (14.36) for ψ and a to a single equation for ψ.
We introduce the spaces Ln(τ) := L2(Ωτ ,C) and ~L (τ) := a ∈ L2(Ωτ ,R2) | 〈a〉 =0, div a = 0, in the distributional sense and the Sobolev spaces of order two: Hn(τ) and~H (τ), whose elements, ψ and α, satisfy the quasiperiodic boundary condition (14.37)
and (14.38), respectively. Define the operators
Ln := −∆an and M := curl∗ curl, (14.39)
on the spaces Ln(τ) and ~L (τ), with the domains being . Their properties that will beused below are summarized in the following propositions:
Lectures on Applied PDEs 143
Proposition 14.10. Ln is a self-adjoint operator on Hn(τ) with spectrum σ(Ln) = (2m+1)kn : m = 0, 1, 2, . . . and dimC Null(Ln−kn) = n, where recall k = k(τ) = 2π
Imτ.
Proposition 14.11. M is a strictly positive operator on ~H (τ) with discrete spectrum.
The proofs of these results are standard and, for the convenience of the reader, aregiven below.
Proof of Proposition 14.11. The fact that M is positive follows immediately from its def-inition. We note that its being strictly positive is the result of restricting its domain toelements having mean zero.
Proof of Proposition 14.10. First, we note that Ln is clearly a positive self-adjoint oper-ator. To see that it has discrete spectrum, we first note that the inclusion H2 → L2 iscompact for bounded domains in R2 with Lipschitz boundary (which certainly includeslattice cells). Then for any z in the resolvent set of Ln, (Ln− z)−1 : L2 → H2 is boundedand therefore (Ln− z)−1 : L2 → L2 is compact. In fact, the operator Ln is unitary equiv-alent (by rescaling) to k times the operator L, whose the spectrum was found explicitlyin the previous section. This completes the proof of Proposition 14.10.
Our goal now is to solve the equation (14.36b) for α and, substituting the solutioninto the equation (14.36a), find an equation containing only ψ. First we reformulate theequations (14.36), by substituting a = an + α, to obtain
In Appendix 14.4.7 at the end of this section, we solve the second equation, (14.40b), forα in terms of ψ. We write the result as α = α(ψ), where
α(ψ) = (M + |ψ|2)−1Im(ψ∇anψ). (14.41)
We collect the elementary properties of the map α in the following proposition, where we
identify Hn(τ) with a real Banach space using ψ ↔ −→ψ := (Reψ, Imψ).
Proposition 14.12. The unique solution, α(ψ), of (14.40b) maps Hn(τ) to ~H (τ) andhas the following properties:
(a) α(·) is analytic as a map between real Banach spaces.
(b) α(0) = 0.
(c) For any δ ∈ R, α(eiδψ) = α(ψ).
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Proof. The only statement that does not follow immediately from the definition of α is(a). It is clear that Im(ψ∇anψ) is real-analytic as it is a polynomial in ψ and ∇ψ, andtheir complex conjugates. We also note that (M − z)−1 is complex-analytic in z on theresolvent set of M , and therefore, (M + |ψ|2)−1 is analytic. (a) now follows.
Now we substitute the expression (14.41) for α into (14.40a) to get a single equation
For a map F (λ, ψ), we denote by ∂ψF (λ, φ) its Gateaux derivative in ψ at φ. Thefollowing proposition lists some properties of F .
Proposition 14.13.
(a) F is analytic as a map between real Banach spaces,
(b) for all λ, F (λ, 0) = 0,
(c) for all λ, ∂ψF (λ, 0) = Ln − λ,
(d) for all δ ∈ R, F (λ, eiδψ) = eiδF (λ, ψ).
(e) for all ψ, 〈ψ, F (λ, ψ)〉 ∈ R.
Proof. The first property follows from the definition of F and the corresponding ana-lyticity of a(ψ). (b) through (d) are straightforward calculations. For (e), we calculatethat
〈ψ, F (λ, ψ)〉 = 〈ψ, (Ln − λ)ψ〉+ 2i
∫Ωτψα(ψ) · ∇ψ
+ 2
∫Ωτ
(α(ψ) · an)|ψ|2 +
∫Ωτ|α(ψ)|2|ψ|2 + κ2
∫Ωτ|ψ|4.
The final three terms are clearly real and so is the first because Ln−λ is self-adjoint. Forthe second term we calculate the complex conjugate and see that
2i
∫Ωτψα(ψ) · ∇ψ = −2i
∫Ωτψα(ψ) · ∇ψ = 2i
∫Ωτ
(∇ψ · α(ψ))ψ,
where we have integrated by parts and used the fact that the boundary terms vanish dueto the periodicity of the integrand and that divα(ψ) = 0. Thus this term is also real and(e) is established.
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14.4.4 Reduction to a finite-dimensional problem
In this section we reduce the problem of solving the equation F (λ, ψ) = 0 to a finitedimensional problem. We address the latter in the next section. We use the standardmethod of Lyapunov-Schmidt reduction. Let X := Hn(τ) and Y := Ln(τ) and letK = Null(Ln − kn). We let P be the Riesz projection onto K, that is,
P := − 1
2πi
∮γ
(Ln − z)−1 dz, (14.44)
where γ ⊆ C is a contour around n that contains no other points of the spectrum ofLn. This is possible since n is an isolated eigenvalue of Ln. P is a bounded, orthogonalprojection, and if we let Z := NullP , then Y = K ⊕ Z. We also let Q := I − P , and soQ is a projection onto Z.
The equation F (λ, ψ) = 0 is therefore equivalent to the pair of equations
PF (λ, Pψ +Qψ) = 0, (14.45)
QF (λ, Pψ +Qψ) = 0. (14.46)
We will now solve (14.46) for w = Qψ in terms of λ and v = Pψ. To do this, weintroduce the map G : R × K × Z → Z to be G(λ, v, w) := QF (λ, v + w). Applyingthe Implicit Function Theorem to G, we obtain a function w : R×K → Z, defined on aneighbourhood of (n, 0), such that w = w(λ, v) is a unique solution to G(λ, v, w) = 0, for(λ, v) in that neighbourhood. This solution has the following properties
w(λ, v) real-analytic in (λ, v); (14.47)
‖w‖ = O(‖v‖3) and ‖∂λw‖ = O(‖v‖3), (14.48)
where the norms are in the space Hn(τ). The last property follows from (14.41) and(21.31) together with (14.46). (Indeed, by (14.41), we have ‖α(ψ)‖H2 . ‖ψ‖2
H2 andtherefore combining (21.31) with (14.46) and using (as above) that the product of Hn(τ)functions is again a Hn(τ) function (and the norms are bounded correspondingly), oneconcludes that (14.48) holds.)
We substitute the solution w = w(λ, v) into (14.45) and see that the latter equationin a neighbourhood of (n, 0) is equivalent to the equation (the bifurcation equation)
γ(λ, v) := PF (λ, v + w(λ, v)) = 0. (14.49)
Note that γ : R ×K → C. We have shown that in a neighbourhood of (n, 0) in R ×X,(λ, ψ) solves F (λ, ψ) = 0 if and only if (λ, v), with v = Pψ, solves (14.49). Moreover, thesolution ψ of F (λ, ψ) = 0 can be reconstructed from the solution v of (14.49) accordingto the formula
ψ = v + w(λ, v). (14.50)
Finally we note that w and γ inherit the symmetry of the original equation:
146 Lectures on Applied PDEs
Lemma 14.14. For every δ ∈ R, w(λ, eiδv) = eiδw(λ, v) and γ(λ, eiδv) = eiδγ(λ, v).
Proof. We first check that w(λ, eiδv) = eiδw(λ, v). We note that by definition of w,
G(λ, eiδv, w(λ, eiδv)) = 0,
but by the symmetry of F , we also have G(λ, eiδv, eiδw(λ, v)) = eiδG(λ, v, w(λ, v)) = 0.The uniqueness of w then implies that w(λ, eiδv) = eiδw(λ, v). We can now verify that
γ(λ, eiδv) = PF (λ, eiδv + w(λ, eiδv))
= eiδPF (λ, v + w(λ, v))〉 = eiδγ(λ, v).
Solving the bifurcation equation (14.49) is a subtle problem, unless n = 1. In thelatter case, this is done in the next section.
14.4.5 Proof of Theorem 14.1
In this section we look at the case n = 1, and look for solutions near the trivial solution.Recall that by Theorem 14.5, the nullspace of the operator Ln−kn, where k = 2π
Imτ, acting
on Hn(τ) is a one-dimensional complex subspace for n = 1. We denote a0 = an=1 anddrop the (super)index n = 1 from the notation Ln. We begin with the following resultwhich gives the existence and uniqueness of the Abrikosov lattices.
Theorem 14.15. For every τ there exist ε > 0 and a branch, (λs, ψs, αs), s ∈ [0,√ε),
of nontrivial solutions of the rescaled Ginzburg-Landau equations (14.36), unique modulothe global gauge symmetry (apart from the trivial solution (1, 0, a0)) in a sufficiently small
neighbourhood of (1, 0, a0) in R×H (τ)× ~H (τ), and such thatλs = k + gλ(s
[0, ε) → H (τ), and gα : [0, ε) → ~H (τ) are real-analytic functions such that gλ(0) = 0,gψ(0) = 0, gα(0) = 0.
Proof. The proof of this theorem is a slight modification of a standard result from bifur-cation theory. Our goal is to solve the equation (14.49) for λ. Since the projection P ,defined there, is rank one and self-adjoint, we have
Pψ =1
‖ψ0‖2〈ψ0, ψ〉ψ0, with ψ0 ∈ Null ∂ψF (λ0, 0). (14.52)
Lectures on Applied PDEs 147
We can therefore view the function γ in the bifurcation equation (14.49) as a map γ :R× C→ C, where
γ(λ, s) = 〈ψ0, F (λ, sψ0 + w(λ, sψ0)〉. (14.53)
We now show that γ(λ, s) ∈ R. Since the projection Q is self-adjoint, Qw(λ, v) = w(λ, v),w(λ, v) solves QF (λ, v + w) = 0 and v = sψ0, we have
〈ψ0, F (λ, sψ0 + Φ(λ, sψ0))〉 = s−1〈sψ0 + w(λ, sψ0), F (λ, sψ0 + w(λ, sψ0))〉,
and this is real by property (e) of Proposition 14.13. Thus, since by Lemma 14.14,γ(λ, s) = ei arg sγ(λ, |s|), it therefore suffices to solve the equation
γ0(λ, s) = 0 (14.54)
for the restriction γ0 : R × R → R of the function γ to R × R, i.e. for real s. Since by(14.48), w(λ, sψ0) = O(s2) and therefore (14.54) has the trivial branch of solutions s ≡ 0for all λ. Hence we factorize γ0(λ, s) as γ0(λ, s) = sγ1(λ, s), i.e. we define the function
γ1(λ, s) := s−1γ0(λ, s), if s > 0, and = 0 if s = 0, (14.55)
and solve the equation γ1(λ, s) = 0 for λ. The definition of the function γ1(λ, s) and(14.47) imply that it has the following properties: γ1(λ, s) is real-analytic, γ1(λ,−s) =γ1(λ, s), γ1(1, 0) = 0 and, by (21.31) and (14.48), ∂λγ1(1, 0) = −‖ψ0‖2 6= 0. Hence by astandard application of the Implicit Function Theorem, there is ε > 0 and a real-analyticfunction φλ : (−√ε,√ε) → R such that φλ(0) = 1 and λ = φλ(|s|) solves the equationγ(λ, s) = 0 with |s| < √ε.
We also note that because of the symmetry, φλ(−|s|) = φλ(|s|), φλ is an even real-analytic function, and therefore must in fact be a function solely of s2. We therefore setφλ(s) = φλ(
√s) for s ∈ [0, ε), and so φλ is real-analytic.
We now define gψ : [0, ε)→H (τ) to be
gψ(s) =
1√sw(φλ(s),
√sψ0) s 6= 0,
0 s = 0,(14.56)
It is easily check that gψ is real-analytic and satisfies sgψ(s2) = w(φλ(s), sψ0) for anys ∈ [0,
√ε).
Now, we know that there is a neighbourhood of (1, 0) in R×H (τ) such that in thisneighbourhood F (λ, ψ) = 0 if and only if γ(λ, s) = 0 where Pψ = sψ0. By taking a smallerneighbourhood if necessary, we have proven that F (λ, ψ) = 0 in this neighbourhood if and
148 Lectures on Applied PDEs
only if either s = 0 or λ = φλ(s2). If s = 0, we have ψ = sψ0 + w(φλ(s), sψ0) = 0, which
gives the trivial solution. In the other case, ψ = sψ0 + w(φλ(s), sψ0) = sψ0 + sgψ(s2).If we now also define gλ(s) = 1 − φλ(s), then the above gives us a neighbourhood of
(1, 0) in R ×H (τ) such that the only non-trivial solutions of the equation (14.42) aregiven by the first two equations in (14.51). We now define the function ga(s) = α(ψs),where, recall, α(ψ) is defined in (14.41) and ψs = sψ0 + sgψ(s2) was found above. Thisfunction is real-analytic and satisfies ga(−s) = α(−ψs) = ga(s), and therefore is really afunction of s2, ga(s
2). Define as = a0 + ga(s2). Then (λs, ψs, αs), s ∈ [0,
√ε), solves the
rescaled Ginzburg-Landau equations (14.36).
For ψ0 a non-zero element in the nullspace Null(L− k), we define the function of τ as
β(τ) :=〈|ψ0|4〉〈|ψ0|2〉2
, (14.57)
which we call this function the Abrikosov function.Simple computations give the following expression for the derivative g′λ(0) at 0 of the
function gλ(s2) defined in (14.51)
g′λ(0) =2π
Imτ
[(κ2 − 1
2
)β(τ) +
1
2
]〈|ψ0|2〉. (14.58)
Note that the definition λ = κ2r2 (n = 1), the first equation (14.51) and the relation(14.58) imply that for (κ2− 1
2)β(τ) + 1
2≥ 0, the bifurcated solution exists for b ≤ κ2, and
for (κ2 − 12)β(τ) + 1
2< 0, it exists for b > κ2. Thus Theorem 14.15, after rescaling to the
original variables, implies Theorem 14.1.
Remark. The proof of Theorem 14.15 gives in fact the following abstract result.
Theorem 14.16. Let X and Y be complex Hilbert spaces, with X a dense subset of Y ,and consider a map F : R×X → Y that is analytic as a map between real Banach spaces.Suppose that for some λ0 ∈ R, the following conditions are satisfied:
(1) F (λ, 0) = 0 for all λ ∈ R,
(2) ∂ψF (λ0, 0) is self-adjoint and has an isolated eigenvalue at 0 of (geometric) multiplic-ity 1,
Then (λ0, 0) is a bifurcation point of the equation F (λ, ψ) = 0, in the sense that there isa family of non-trivial solutions, (λs, ψs), for s ∈ [0,
√ε), unique modulo the global gauge
symmetry (apart from the trivial solution (1, 0)) in a neighbourhood of (λ0, 0) in R×X.Moreover, this family has the form
λ = φλ(s2),
ψ = sψ0 + sφψ(s2).
Here ψ0 ∈ Null ∂ψF (λ0, 0), and φλ : [0, ε) → R and φψ : [0, ε) → X are unique real-analytic functions, such that φλ(0) = λ0, φψ(0) = 0.
14.4.6 Appendix: Proof of Proposition 14.9
Proof of Proposition 14.9. We begin by defining the function B : R→ R to be
B(ζ) =1
r
∫ r
0
curlA′(ξ, ζ) dξ.
It is clear that b = 1rτ2
∫ rτ20
B(ζ) dζ. A calculation shows that B(ζ + rτ2) = B(ζ).
We now define P = (P1, P2) : R2 → R2 to be
P (x) = (bx2 −∫ x2
0
B(ζ) dζ,
∫ x1
τ1τ2x2
curlA′(ξ, x2) dξ +τ ∧ xτ2
B(x2)).
A calculation shows that P is doubly-periodic with respect to L.We now define η′ : R2 → R to be
η′(x) =b
2x1x2 −
∫ x1
0
A′1(ξ, 0) dξ −∫ x2
0
A′2(x1, ζ)− P2(x1, ζ) dζ.
η′ satisfies ∇η′ = −A′ + A0 + P and let η′′ be a doubly-periodic solution of the equation∆η′′ = − divP . Also let C = (C1, C2) be given by
C = − 1
|Ω|
∫Ω
(P +∇η′′) dx,
where Ω is any fundamental cell, and set η′′′ = C1x1 + C2x2 and define η = η′ + η′′ + η′′′.We claim that the pair (eηΨ′, A′+∇η) satisfies (14.35) and (i) - (iii) of the proposition.
Let α := A − Ab = A′ + ∇η − Ab. We first note that α = P + ∇η′′ + C and bythe above, it is periodic. We also calculate that divα = divP + ∆η′′ = 0. Finally∫
Ωα =
∫Ω
(P +∇η − C) = 0.We note that, since α(x) and L−periodic, Ab(x) satisfies Ab(x + s) = Ab(x) +
b2
(−s2
s1
)and
(−s2
s1
)= ∇(s ∧ x), we have that A = Ab + α satisfies
A(x+ s) = A(x) +b
2∇(s ∧ x), ∀s ∈ L.
150 Lectures on Applied PDEs
Next, if (Ψ′, A′) satisfies condition (2.77) with the exponent g′s(x), then (eηΨ′, A′+∇η)satisfies condition (2.77) with the exponent gs(x) := g′s(x) + η(x + s) − η(x). We haveshown above that for our choice of η, ∇gs(x) = b
2∇(t ∧ x). Therefore gs(x) = b
2s ∧ x+ cs
for some constant cs. To establish (14.35), we need to have it so that cs = 0 for s = r, rτ .First let l be such that r ∧ l = − cr
band rτ ∧ l = − crτ
b. This l exists as it is the solution
to the matrix equation (0 r−rτ2 rτ1
)(l1l2
)=
(− cr
b
− crτb
),
and the determinant of the matrix is just r2τ2, which is non-zero because (r, 0) and rτform a basis of the lattice. Let ζ(x) = b
2l ∧ x. A straightforward calculation then shows
that Ψ(x) := eiζ(x)Ψ′(x + l) satisfies (14.35) and that α(x) := α′(x + l) + ∇ζ(x) stillsatisfies (i) through (iii). This proves the proposition.
14.4.7 Appendix: Solving the equation (14.40b)
(under construction)
14.5 The Ginzburg-Landau equations on a complex torus
We begin with rewriting the GLEs (14.1) treating A as a (differential) one-form,2,
−∆AΨ + κ2(|Ψ|2 − 1)Ψ = 0, (14.59a)
d∗dA = Im(Ψ∇AΨ). (14.59b)
Recall the notations Ψ : R2 → C, ∇A = ∇ + iA, and −∆A := ∇∗A∇A , and ∇∗A for theadjoint taken w.r. to the inner product functions. Furthermore, d and d∗ are the exteriorderivative and its adjoint (defined as usual on differential forms, see a discussion below),which replace curl and curl∗.
The GLEs present a gauge theory in the sense that they are invariant under the gaugetransformations
Ψ(x)→ g(x)Ψ(x), A(x)→ A(x)− g−1(x)dg(x), (14.60)
for g(x) ∈ C1(Rd, U(1)), so that, as before, ∇g∗(x)A(x) = g(x)−1∇A(x)g(x), where g∗A :=A− dgg−1, while dA is invariant.
Now, we consider the Ginzburg-Landau equations for L−equivariant pairs, (Ψ(x), A(x)),i.e. pairs satisfying
2For an elementary introduction to differential forms, see T. Tao, Differential forms andintegration, http://www.math.ucla.edu/ tao/preprints/forms.pdf, Donu Arapura, Introduction todifferential forms https://www.math.purdue.edu/ dvb/preprints/diffforms.pdf, Differential formshttp://math.colorado.edu/ jnc/lecture1.pdf and many other sources online and offline.
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where the function ηs(x) satisfies the cocycle condition
ηs+t(x)− ηs(x+ t)− ηt(x) ∈ 2π. (14.62)
As we show below, L−equivariant functions and vector fields are in one-to-one correspon-dence with sections and covariant derivatives (or connections) of the line bundle, L, overa complex torus.
We summarize this correspondence as
• L−equivariant functions on R2 ⇐⇒ sections on L
• L−equivariant vector fields (1-forms) on R2 ⇐⇒ connections on L
This allows to reformulate the Ginzburg-Landau equations as equations on a linebundle over a complex torus, which is one of the simplest Riemann surfaces. Then weextend the latter to an arbitrary Riemann surface and explore the consequences of this.
First, for a lattice L, we define the action of L on the space R2 by
s : x→ x+ s.
Next, given a function ηs(x) satisfying the cocycle condition (I.8), we define the action ofa lattice L on the space R2 × C by
s : (x,Ψ)→ (x+ s, eiηs(x)Ψ). (14.63)
Now, we can define the spaces T := R2/L and L := R2 × C/L of the equivalence classes,[x] and [(x,Ψ)], of elements x ∈ R2 and (x,Ψ) ∈ R2 × C under the action s : x → x + sand (I.9) of the group L, e.g.
One can show that(i) the spaces L := R2 × C/L and T := R2/L are manifolds (i.e. locally, the look like
R2 × C and R2, respectively);(ii) L is not of the form L = T × V , for some vector space V , i.e. it is not a trivial
line bundle.(iii) T := R2/L is a Riemann surface. It is homeomorphic to the standard torus:
[λν1 + µν2]→ (e2πiλ, e2πiµ), (14.64)
where (ν1, ν2) is a basis in L and λ, µ ∈ R. It is called the (complex, flat) torus.With the map p : L→ T, defined as p : [(x,Ψ)]→ [x], L is a line bundle over T in the
sense of the definition below.
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By a (complex) line bundle (over a manifold X) one means a triple (L,X, p), where Land X are manifolds and p is a map, p : L→ X, with the property that X can be coveredby neighbourhoods U s.t. p−1(U) are homeomorphic to U × C (i.e. locally, L looks likeU × C).3
Here we mention a couple of definitions:
• A line bundle (L,X, p) is said to be trivial if it is of the form L = X × V , for somevector space V and with p : (x, v)→ x.
• Subspaces Lx := p−1(x) are called the fibers over x. (Lx are vector spaces iso-morphic to C.) L can be presented as a disjoint union of its fibers, L = ∪x∈XLx.
• A line bundle (L,X, p) is said to be U(1)-, or unitary, bundle if transition functionsare required to have values in U(1).
We mention also that all operations with vector spaces pass to the line bundles: fortwo line bundles, L and L′, one can define direct sum L ⊕ L′ and the tensor productL ⊗ L′. For instance, writing L = ∪x∈XLx, etc, we have L ⊗ L′ = ∪x∈X(Lx ⊗ L′x) , withthe natural projection p : L⊗ L′ → X given by p(x, v ⊗ v′) = x.
We can think of line bundles as manifolds with the additional structure of a localsplitting intothe product of the base and a fiber.
In our case if we restrict to a fundamental cell Ω, then L ≈ Ω× C.To sum up, we constructed the line bundle
L→ T, where L := R2 × C/L and T := R2/L, (14.65)
with the base manifold T := R2/L and the projection p : [(x,Ψ)]→ [x].
Next, we need some more definitions:
• The sections of L are maps σ : T→ L, satisfying p σ = 1, i.e. they map points ofT to points on the fibers above them. (Sections are locally usual functions, but witha global twist (in our case provided by factoring by L), they generalize the notionof a function.)
3For a quick introduction to the line bundles, one can read pages 2-7 of Michael Murray, Line Bundles,http://www.maths.adelaide.edu.au/michael.murray/line bundles.pdf.
The definition above shows that a line bundle is determined by its transition functions ϕ−1V ϕU :
(U ∩ V ) × C → (U ∩ V ) × C, where ϕU : U × C → p−1(U) are homeomorphisms alluded to above.The transition functions give also a convenient description of line bundles. For instance, a line bundle(L,X, p) is said to be C∞/holomorphic, iff the transition functions C∞/holomorphic and L⊗L′ is a linebundle with the transition functions ϕ−1
V ϕU ⊗ (ϕ′V )−1 ϕ′U .Transition functions act as bundle maps, i.e. fiberwise: ϕ−1
V ϕU : (x,w)) → (x, a(x)w). We assumethat, for each x ∈ U ∩ V , the transformations a(x) : Lx → Lx are from the unitary group U(1) acting onLx.The corresponding line bundle is called the U(1) or unitary bundle.
Lectures on Applied PDEs 153
• A covariant derivative ∇ (or connection) on L is a map from sections to one-formson T with values in L, which (i) is linear and (ii) satisfies the Leibniz rule ∇(fσ) =f∇σ + df ⊗ σ. (Covariant derivatives are locally usual derivatives, they extend thenotion of a derivative to a manifold, but also the physics notion of the covariantderivative.)
• A curvature of connection a on L is the two-form Fa := da on T with values in L.
• Gauge transformations: for any g(x) ∈ C1(T, U(1)),
ψ(x)→ g(x)ψ(x), ∇ → g(x)−1∇g(x). (14.66)
∗ ∗ ∗ If sections of L are transformed as
ψ(x)→ g(x)ψ(x), (14.67)
for g(x) ∈ C1(T, U(1)), then the connection transforms as a(x) → a(x) − g−1(x)dg(x),so that, as before, ∇g∗(x)a(x) = g(x)−1∇a(x)g(x), where g∗a := a − dgg−1, while da isinvariant. ∗ ∗ ∗
In coordinates x, we can write the covariant derivative (connection) ∇A = ∇+ iA as∇A =
∑i∇idx
i, where A =∑
iAidxi is the connection one-form and ∇i := ∂i + iAi, the
components of ∇A.
Chern number. Line bundles have a topological invariant - the degree, which is definedas the Chern number, c(e) = c(η), of the automorphy exponents ηs(x), defined in (14.9),which we reproduce here
c(η) :=1
2πi
[ην2(x+ ν1) + ην1(x)− ην1(x+ ν2)− ην2(x)
], (14.68)
where ν1, ν2 is a basis of L. (The degree of line bundles can be also defined throughdevisors or zero of sections; the expression (14.68) is convenient for our purposes.) Formore discussion, see Subsection 14.2.
Proposition 14.17. Given an automorphy factor ρ and the associated line bundle L (see(I.11)), there is one-to-one correspondence between L−equivariant functions on R2 andsections of L and L−equivariant one-form on R2 and connections on L.
In particular, given an L−equivariant function, Ψ, on R2 and and an L−equivariantone-form, A, on R2, the corresponding section and connection on L are defined as
ψ([x]) = [(x,Ψ(x))], ∇aψ([x]) = [∇AΨ(x)],
where ∇A : Ψ→ dΨ− iAΨ.
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Before proceeding to the proof of this statement, we give some more definitions. Thefundamental cell of L ⊂ C is a subset, Ω, of C having the properties
(i) Ω ∩ sΩ = ∀s ∈ L/0 and(ii) ∪s∈LΩ = C.
(The second property says that Ω tessellates C.)Using the restriction π
∣∣Ω
of the map π : x → [x] to Ω, one can identify T with afundamental cell Ω of L:
T ≈ Ω.
By (i), π∣∣Ω
is one-to-one (if x ∈ Ω, then sx /∈ Ω). By (ii), for every [x] ∈ T, there iss = s[x] ∈ L s.t. x ∈ sΩ and therefore s−1x ∈ Ω. Since s−1x ∈ [x], this gives the inversemap T→ Ω ([x]→ s−1
[x]x).
(π : C→ Ω is a covering map of Ω and C is its universal cover.)
Proof of Proposition I.1. With an L−equivariant function Ψ on R2 we associate the sec-tion ψ of L as
ψ([x]) = [(x,Ψ(x))].
To check that this is consistent, let x′ be another representative of [x], then there is s inL s.t. x′ = x+ s. Since Ψ(x+ s) = eiηs(x)Ψ(x), the pair (x′,Ψ(x′)) belongs to [(x,Ψ(x))].
Conversely, given a section ψ in a line bundle L → T, we associate with it theL−equivariant function Ψ on R2, constructed as follows. First, using the constructionabove, we identify T with a fundamental cell Ω of L. For x ∈ Ω, we define Ψ = Ψ(x)uniquely by the relation
ψ([x]) = [(x,Ψ)]
(there is only one Ψ in [(x,Ψ)] associated with x ∈ Ω). Now, we define Ψ(x) = Ψ. Thisdefines Ψ(x) on Ω. For x ∈ Ω + s, we define
Ψ(x+ s) = eiηs(x)Ψ(x),
where ηs(x) is an automorphy exponent, constructed below. Since sΩ tessellate C, thefunction Ψ(x) is defined on C. It is straightforward to show that, due to the cocyclecondition (14.7), Ψ(x) is L−equivarinant.L−equivariant one-forms, A, are in one-to-one correspondence with covariant deriva-
tives (or connections) on L, constructed as follows. With L-equivariant one-form A, weassociate the covariant derivative on complex functions on R2, as before, by ∇A : Ψ →dΨ − iAΨ, but now dΨ − iAΨ is a one-form. Now, we define a([x]) = [A(x)] and, forψ ↔ Ψ(x), we set4
daψ([x]) = [dAΨ(x)],
4Recall that the equivalence class [dAΨ] is defined by the relation dAΨ ∼ dA′Ψ′ ⇔ ∃χ : R2 → G :(Ψ′, A′) = T gaugeχ (Ψ, A). (We will use the notation dA[Ψ] := [dAΨ] = [(d− iA)Ψ].)
daψ([x]) = [dAΨ(x)],
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and use the equivariance of A(x) to check that the r.h.s. is independent of the represen-tative x of [x] we have chosen.
Remembering Proposition I.1, we can transfer the GLE to sections and connections,ψ and a, on the line bundle L over a torus T:
∆aψ = κ2(|ψ|2 − 1)ψ, (14.69a)
d∗da = Im(ψdaψ). (14.69b)
Here ∇a is a covariant derivative on L, −∆a = d∗ada, where d∗a is adjoint to da in theinner product (I.16), and d and d∗ are the exterior derivative and its adjoint (defined asusual on differential forms). Since da maps sections to one-forms, d∗a maps one-forms intosections, so that ∆a maps sections to sections.
Note that the equations (14.69) are still invariant under the gauge transformations,
Since the extended Ginzburg-Landau equations (14.69) are invariant under the gaugetransformations (14.70), this correspondence allows us to move freely between equivariantfunctions and vector fields (more conveniently, one-forms) and sections and connectionson line bundles. In particular, we can consider the extended Ginzburg-Landau equations(14.69) as either for equivariant functions and vector fields on H or for sections andconnections on L.
and use the equivariance of A(x) to check that the r.h.s. is independent of the representative x of [x] wehave chosen.
Note that dA : X → (dA)X , where X is a vector field on the base manifold R2/L, and
(dA)X [Ψ]([x]) := [(dX(x) − iA(X)(x))Ψ(x)].
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15 The Yang-Mills and Yang-Mills-Higgs equations
Let X be an open domain in Rd. Recall the Ginzburg-Landau equations (GLE) on X ina geometrical form, (14.59),
−∆AΨ + κ2(|Ψ|2 − 1)Ψ = 0, (15.1a)
d∗dA = iIm(ΨdAΨ). (15.1b)
Here Ψ : X → C, A is a (differential) one-form on X, dA = d+A is a covariant derivativemapping functions from X to C into C valued one-forms, and −∆A := d∗AdA , where d∗Ais the adjoint w.r. to the inner product functions on X and d is the exterior derivativeand d∗ its adjoint taken w.r. to the standard inner product on one-forms were discussedin Subsection 14.5. (We changed the notation here by absorbing i into A, so dA = d+ iAis replaced by dA = d+ A and i appears on the r.h.s. of (15.1b).)
Recall that these equations are invariant under the gauge transformations
Ψ(x)→ g(x)Ψ(x), A(x)→ A(x)− g−1(x)dg(x), (15.2)
for g(x) ∈ C1(Xd,U(1)), so that, as before, dg∗(x)A(x) = g(x)−1dA(x)g(x), where g∗A :=A− dgg−1. The group U(1), or C1(X,U(1)), is called the gauge group.
A natural generalization of the GLE to non-abelian gauge groups is given by the Yang-Mills-Higgs (YMH) equations. The static Yang-Mills-Higgs (YMH) equations on X withthe gauge Lie group, G acting on a vector space V , read
d∗AdAΨ− λ(1− |Ψ|2)Ψ = 0, (15.3a)
d∗AdAA = J(Ψ, A). (15.3b)
Here Ψ : X → V , A is a one-form (connection, or gauge field) on X with values in theLie algebra, g, of G and dA = d + A and d∗A are a covariant derivative and its adjoint(dA maps functions (sections) from X to V into g−valued one-forms and ∇∗A goes in theopposite direction), and dA is the generalization of d and its adjoint to Lie algebra-valuedforms and d∗A is its adjoint in the appropriate metric. dA given by
dAB := dB +1
2[A ∧B],
with [A∧B] defined for g−valued p− and q−forms A =∑
aAa⊗ τa and B =
∑aB
a⊗ τa,where Aa and Ba are standard p− and q−forms and τa is a basis in g, as
[A ∧B] :=∑ab
(Aa ∧Bb)⊗ [τa, τb].
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Finally, J(Ψ, A) is the YMH current defined for V = Cm and G, a matrix group U(r)acting on Cm, as5
J(Ψ, A) := 2Im(Ψ⊗ dAΨ), (15.4)
where (ξ ⊗ η)ij = ξjηi for two one-forms ξ =∑
i ξidxi and η =
∑i ηidx
i, so that [(Ψ ⊗∇AΨ)]ab = Ψb(∇AΨ)a ((dAΨ)a =
∑(∂xi + AcibΨc)dx
i, see below).6
For G = U(r), connections, A are in u(r), i.e. they are anti-self-adjoint, A∗ = −A.
For the time-independent YMH equations, one takes the Euclidian metric in (15.3) (i.e.adjoints are taken in the Euclidian metric), for the time-dependent the YMH equationstime-dependent, in the Minkowski one (see below).
Physical background. In particle physics, Ψ is called the Higgs field and A, the G-YM, or gauge, field. The standard model of particle physics involves the Higgs field andthe U(2)- and SU(3)-YM gauge fields describing the electro-weak and strong interactionsrespectively (in addition to matter (quark and lepton) fields).
Curvature. Similarly to the U(1)-case, the curvature is defined as FA := dAA.
Gauge invariance. The YMH equations are invariant under the gauge transformations
for g(x) ∈ C1(Rd, G). This implies dρg(x)A(x) = g(x)−1dAg(x), for all g(x) ∈ C1(Rd+1, G),where
ρg(x)A(x) = g(x)A(x)g−1(x)− (∇g(x))g−1(x).
∗ ∗ ∗ The physical quantities are invariant under the gauge transformations (in thephysics language, independent of the choice of the gauge). Hence we consider gauge-equivalent classes of solutions. ∗ ∗ ∗
5The YMH current, J(Ψ, A), defined by 〈α, J(Ψ, A)〉 = δA(‖dAψ‖2)α, i.e. J(Ψ, A) := ∇L2
A (‖dAψ‖2).To show that it is of the form (15.4), we let As = A+α and compute δA(‖dAψ‖2)α = ∂s|s=0
∫‖dAsψ‖2V =∫
[〈iαijψj , (dAψ)i〉V + 〈(dAψ)i, iαijψj〉V ]. Next, using that α∗ = α, we find δA(‖dAψ‖2)α =−i∫αij [〈ψi, (∇Aψ)j〉V − 〈(dAψ)i, ψj〉V ]. We can rewrite this as δA(‖dAψ‖2)α = −i
∫αij [(ψ⊗∇Aψ)ij −
(ψ ⊗ dAψ)∗ij ].6We could have also taken Ψ : X → g. In this case, A acts on Ψ as Ψ → [A,Ψ], so that dAΨ =
dΨ− [A,Ψ] and the YMH current given by J(Ψ, A) := [dAΨ,Ψ]. More generally, one can replace Cm bya vector space V on which G acts via a representation ρ.
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Energy.
E(Ψ, A) =1
2‖∇AΨ‖2 +
λ
4‖|Ψ|2 − 1‖2 +
1
2‖FA‖2. (15.6)
Here |Ψ|2 = (Ψ,Ψ), where (Ψ,Ψ′) := Tr(Ψ∗Ψ′).In coordinates x on Rd and with the summation over repeated indices is understood,
we can write the gauge field, A, as A = Aidxi,
[A ∧B] := [Ai, Bj]dxi ∧ dxj. (15.7)
and the covariant derivative ∇A as ∇A = ∇idxi, where ∇i := ∂i +Ai, the components of
∇A. Then the curvature of ∇A is given by FA = Fijdxi ∧ dxj, with Fij = ∂iAj − ∂jAi +
[Ai, Aj]. (Note that Fij = [∇i,∇j], or FA =[∇i,∇j
]dxi ∧ dxj, or FA = ∇2
A ≡ ∇A ∧ ∇A
(as differential forms).)The above derivatives are dAΨ = ∇iΨdx
In the coordinate form, the conserved energy for the model is given by
E(Ψ, A) :=
∫〈∇kΨ,∇kΨ〉V +
1
2λ(1− |Ψ|2)2 +
1
2TrFijF
ij, (15.9)
where the lower case roman indices run over the spatial components 1, 2, . . . , d.
Topological invariants. (to do)
Monopoles. (to do)
The time-dependent Yang-Mills-Higgs equations. For the time-dependent Yang-Mills and Yang-Mills-Higgs theories, we take X = Rd+1 with the Minkowski metric.The YMH equations, (15.3), preserve their form but the underlying metric now is theMinkowski one. These equations are the Euler-Lagrange equations for the Lagrangian
LYMH(Ψ, A) :=1
2‖∇AΨ‖2 − λ
4‖|Ψ|2 − 1‖2 +
1
2‖FA‖2, (15.10)
respectively. The Lagrangians above are invariant under the Poincare group and thegauge transformations defined as above.
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The Yang-Mills equations. The second equation in (15.3) is the (static) YMH equa-tion with a current J . In the vacuum it becomes
d∗AdAA = 0 (15.11)
Note that, if, as usual, J(0, A) = 0, then (0, A) is a solution of the YMH equation (15.3)iff A solves the YM equation. Equation (15.11) generalizes the Maxwell equations in thevacuum,
d∗dA = 0, (15.12)
to general gauge groups. Unlike the Maxwell equations, this is a non-linear equation forthe connection A. (If the matter is present then the corresponding equations becomed∗AdAA = 0 and d∗dA = J .)
Gauge invariance. The YM equations are invariant under the gauge transformation
for g(x) ∈ C1(Rd, G). Under the gauge transformation, (15.13), the curvature, FA, trans-forms as
Fg∗(x)A(x) = g(x)−1FA(x)g(x).
Energy.
E(A) =1
2‖FA‖2 ≡ 1
2
∫X
(FA, FA) = −1
2
∫X
Tr(FA ∧ ∗FA). (15.14)
Recall that in coordinates x on Rd, writing the gauge field, A, as A = Aidxi and
dAB = ∇iBjdxi ∧ dxj, d∗AB = −∇iBi and d∗AFA = −∇iFijdx
j, where
∇iBj = ∂iBj − ∂jBi + [Ai, Bj],
∇iFij := ∂iFij + [Ai, Fij],
(the summation over repeated indices is understood) and the YM equations become
∇iFij = 0, (15.15)
where Fij are the components of the curvature of ∇A is given by FA = Fijdxi ∧ dxj, with
Fij = ∂iAj − ∂jAi + [Ai, Aj].
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The dimensional reduction to the Yang-Mills equations ([14], Section II.6).Following [14], we construct the map of the Yang-Mills equation in d dimensions to theYang-Mills-Higgs equations in d− 1 dimensions.
To show that the (static) Yang-Mills equation in the dimension d is equivalent to the(static) Yang-Mills-Higgs equations in the dimension d− 1, we assume A is independentof xd and rename Ψ ≡ Ad. Since FA = Fijdx
i ∧ dxj, with Fij = ∂iAj − ∂jAi + [Ai, Aj],the jd−component of the curvature FA becomes Fjd = ∂jAd + [Aj, Ad] = ∇jAd = ∇jΨ.
Now, let A := (A1, . . . , Ad−1) and FA :=∑d−1
1 Fijdxi∧dxj. Hence the YM energy (15.14)becomes
E(A) =1
2(‖FA‖2 + ‖∇AΨ‖2), (15.16)
which the energy functional for the YMH theory in d − 1 dimensions and with no self-interaction. (to be continued)
Instantons. (to do)
Homogeneous solutions. The simplest solutions of the YM equations, (15.11), areflat connections, i.e. A satisfying FA = 0 (zero curvature or magnetic field). For d = 2and g having a nontrivial center, this can be extended to constant curvature connections(ccc’s). A connection A is said to be a constant (or central or projectively flat) curvatureconnection (CC) if its curvature is of the form
FA = −ibJ ⊗ ω, J ∈ Z(g), (15.17)
for some b ∈ R, where ω is the volume form on X and Z(g) is the centre of the algebrag. We have
Proposition 15.1. Every constant curvature connection, a, on E satisfies the YM equa-tion (15.11).
Proof. Let A be a CC connection and let ∗ be the Hodge star operator (the properties ∗are listed in (I.17)7). Using the relation d∗A = (−1)k ∗ dA∗ on k−forms, we find d∗AdAA =− ∗ dA ∗ dAA. Since A is a CC connection, we have ∗dAA = bJ , for some constant b.Hence d∗AdAA = − ∗ dAbJ = 0.
Stability and bifurcation in Yang-Mills theory (R. Jackie and P. Rossi, PhysRev D21, No 2, 15 Jan 1980)
Non-Abelian Vortices on Riemann Surfaces (Lett Math Phys (2008) 84:139?148,2008)
7For d = 2 and a conformally flat metric ds2 = λ2dx1 ∧ dx2, ∗dx1 = dx2, ∗dx2 = −dx1, ∗1 =λdx1 ∧ dx2, ∗dx1 ∧ dx2 = 1/λ.
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16 Nonlinear stability
16.1 Stability: generalities
So far we studied mainly existence of static solutions and considered their linearized stabil-ity (see Subsection 9). (Remember that at the same time this gives also the existence andthe linearized stability of of stationary or standing waves and traveling wave solutions.)
Now, we address the full nonlinear stability, which is the next key step. The questionhere is, starting with initial conditions close to a static solution, how the solutions of thedynamical equation in question behave? This leads to the question of stability of staticsolutions of general dynamical systems,
∂u
∂t= F (u). (16.1)
Assume F : U → Y , where U is an open set in X and X and Y are Hilbert spaces, X ⊂ Y ,densely. Let (16.1) have a static solution, u∗, i.e., u∗ is time independent. (Such solutionsare also called equilibria and sometimes stationary solutions.) Thus u∗ is an equilibriumor static solution iff u∗ is independent of time and satisfy the equation F (u∗) = 0.
We would like to understand the behavior of solutions to equation (16.1) for initialconditions u0 near an equilibrium one u∗. There are the following general scenarios
• The solutions stay in a neighborhood of u∗ (Lyapunov stability);
• The solutions converge to u∗ as t→ +∞ (asymptotic stability);
• The solutions move away from u∗ as t→∞ (instability).
More precisely, we say that a static solution of (16.1) is Lyapunov stable if for anyneighborhood, U of u∗ have another neighborhood, V ⊂ U , of u∗ such that if an initialcondition, u0, is in V , then the solution, u, stays in U . Otherwise, u∗ is said to be unstable.
We say a static solution, u∗, is asymptotically stable if ∃δ > 0 such that for any initialcondition u0 satisfying ‖u0 − u∗‖ < δ, we have limt→∞ ‖φt(u0)− u∗‖ = 0.
For systems with symmetry, the notion of stability/instability should be modified.This is done below.
A weaker notion of stability - the linear stability - introduced in Subsection 9 is usuallyis the first step in proving asymptotic stability and often gives the necessary condition.
Stability in the presence of symmetry. Recall from Subsection 2.3, that we saythat the dynamical system (16.1) has a symmetry group G if a representation T : g → Tgof G on the space of solutions satisfies
F (Tgu) = TgF (u). (16.2)
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(See Subsection 2.3 for examples.) In this case, if the dynamical system (16.1) with thesymmetry group G has a static solution u∗, then it has the manifold of static solutions
M∗ = Tgu∗ : g ∈ G.
Indeed, if u∗ is a static solution to (16.1), then, due to (16.2), so is Tgu∗. Now, we haveto address the stability w.r.to this manifold. Clearly, we have to modify the definitionsatthe beginning of this section and the approach.
We say that a static solution u∗ (or more precisely, the manifold of static solutionsM∗)is orbitally stable if any solution to the equation (16.1), starting in a small neighborhoodof u∗ stays in a small neighborhood of the manifold M∗. In other words the solution utsticks very close to a possibly moving static solution Tg(t)u∗.
Note that the tangent space, Tu∗M∗ ⊂ X, toM∗ at u∗ is the vector space Gu∗, whereG was defined above.
As an example we consider Eq. (2.2). This equation is translation invariant and has thekink solution χ which breaks the translational symmetry. Hence if u is a solution to (2.2)then so is ua and therefore the functions χa(x) := χ(x+a) ∀a ∈ R are also static solutionsto (2.2). Thus we have an entire manifold of stationary solutions Mkink = χa : a ∈ R.(In the multidimensional case, we have also rotations.)
The energy method is particularly useful in proving the Lyapunov and orbital stability.We demonstrate this in Subsection 17 after developing some variational calculus.
16.2 Asymptotic stability of kinks
16.2.1 Problem and results
We are interested in the asymptotic stability of the families of kink solutions of the Allen-Cahn equation (2.2) on R, discussed in Subsection 9.4. We recall this equation here:
∂u∂t
= ∆u− g(u),u|t=0 = u0(x),
(16.3)
where u : R → R and g : R → R is C1, and is the derivative of a double-well potential(i.e. g(u) = G′(u), where G(u) ≥ 0 and has two non-degenerate global minima withthe minimum value 0 and a positive maximum at u = 0, see Figure 18, specifically,g(u) = u3 − u). (We set (say by rescaling) the parameter ε to 1.)
Recall that the equation (16.3) (i) is translationally invariant and (ii) has a (smooth)solution χ(z) satisfying χ(z)→ ±1, as z → ±∞ and χ(0) = 0, called the kink, see Figure19. These properties imply that (16.3) has a family of stationary solutions, χ(x−a), a ∈ R.(Recall that χ(−x+ a) are also solutions and they are called anti-kinks.)
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G
1−1
u
Figure 18: Double well potential
ïG
ba
a
b
x
u
Figure 19: Hill-valley-hill structure for −G, and the kink for u.
The family above of the kink solutions forms a one-dimensional manifold
Mkink := χ(x− a) | a ∈ R. (16.4)
Our goal is to prove asymptotic stability of this manifold. Let Uδ denote the δ-neighbourhoodof Mkink in the H1-metric. Let ρ > 0 denote the second eigenvalue of the linearized mapL := dF (χ), where F (u) denotes the negative of the r.h.s. of (16.3). We have
Theorem 16.1 (Asymptotic stability of kinks). Assume g is as above and let in addition,
|g′′(u)| . 1 + |u|k, k > 2. (16.5)
Then there is δ > 0 such that(i) (16.3) with u|t=0 = u0 ∈ Uδ has a unique global solution, u(t),(ii) as t→∞, this solution approaches the kink manifold Mkink as exponentially fast:
then the solution u to (16.3) with u|t=0 = u0 satisfies
‖u(t)− χa(t)‖H1 . e−ρ4t (16.6)
for some differentiable a(t) and for all times t ≥ 0 (in particular, Mkink is asymptoticallystable) and
(iii)a(t)→ a∞, for some real number a∞, exponentially fast, as t→∞.
Recall that (16.3) is a gradient system with energy functional given by (see (7.4) andalso (17.25) below)
E(u) =
∫(1
2|∇u|2 +G(u)) . (16.7)
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However, E(χ) =∞ and, and though formally E(u) is decreasing under the evolution, byitself, it is not very useful.
To prove the theorem we use, instead the energy dissipation property, the differentialinequality for the Lyapunov functional Λ(ξ) := 1
2〈Lξ, ξ〉, where, recall, L is the hessian of
the energy functions E(u) at χ: L := dF = E ′′(χ), which is closely related to the energyE(χ), via the Taylor expansion.
A key role here is played by the following
Theorem 16.2. Let ρ be the second lowest eigenvalue of L. Then
〈ξ, Lξ〉 ≥ 1
2ρ‖ξ‖2
H1 , ∀ξ⊥χ′. (16.8)
Proof. Theorem 9.5 and a spectral theorem (see e. g. [12]) imply that
〈Lξ, ξ〉 ≥ ρ‖ξ‖2L2 ∀ξ⊥χ′. (16.9)
We combine this estimate together with δ > 0 times the elementary estimate
〈Lξ, ξ〉 ≥ ‖∇ξ‖2L2 − C‖ξ‖2
L2 ,
for C = −min−1≤u≤1 g′(u), to obtain
〈Lξ, ξ〉 ≥ δ‖∇ξ‖2L2 + (ρ− δC)‖ξ‖2
L2 ,
and optimizing over δ (say taking δ = ρ/2C), completes the proof.
16.2.2 Orthogonal Decomposition
Throughout, we will use the notation fa(x) = f(x − a). Similarly to the general caseabove (see also (17.16) below), we introduce a neighbourhood of Mkink:
The following statement and its proof are specialization of Proposition 17.4 below and itsproof to the present situation.
Proposition 16.3. There is δ > 0 s.t. for any u ∈ Uδ there is a = a(u) s.t. u−χa ⊥ χ′a.Consequently, u can be decomposed as follows
u = χa + ξa, with
∫ξ(x)χ′ (x) dx = 0. (16.11)
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Proof. We define the function G(a, u) =∫
(u− χa)χ′adx and rewrite the statement of theproposition as solving the equation
G(a, χa) = 0 (16.12)
for a as a function of u. To do this, we use the Implicit Function Theorem. Clearly, G isC1 in u because is it a linear functional in u, and C1 in a since χa is C1 in a. Furthermore,G(a, χa) = 0. Finally, we show that ∂aG(a, u)
∣∣u=χa
6= 0:
∂aG(a, u) = −∫∂aχaχ
′adx+
∫(u− χa)∂aχadx. (16.13)
Hence, using that ∂aχ = −χ′, we have, at u = χa,
∂aG(a, u) =
∫(χ′a)
2dx. (16.14)
Therefore, the conditions of the Implicit Function Theorem are satisfied and it impliesthat (16.12) has a unique solution for a as a function of u ∈ Uδ, which gives the statementof the proposition.
16.2.3 Evolution of Fluctuation ξ
We insert the decomposition (16.11) (u = χa+ξa) into (16.3) project the resulting equationonto the tangent space Rχa of the kink manifold Mkink and on the orthogonal subspaceto get equations for a and ξ.
We have shown in Theorem 9.5 that since the kink χ breaks the translational symme-try, χ′ is a zero eigenfunction of the linearized map L := dF (χ), where F (u) denotes thenegative of the r.h.s. of (16.3),
Lχ′ = 0. (16.15)
The linearization L := dF (χ) is computed explicitly as
L := −∆ + g′(χ). (16.16)
Proposition 16.4. Assume that for every t, a solution u of Eq (16.3) is in Uδ, withδ > 0 given in Proposition 16.3. Then a(t) and ξ(x, t) given in that proposition satisfythe following equations
− ∂tξ = Lξ +N(ξ) + aχ′ + a∂xξ, (16.17)
where N(ξ) := g(χ+ ξ)− g(χ)− g′(χ)ξ, and L := dF (χ), given explicitly by (16.16), and
a(t)
[∫(χ′)2dx+
∫∂xξχ
′dx
]=
∫N(ξ)χ′dx. (16.18)
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Proof. For any t decompose a solution u of (16.3) according to Proposition 16.3, u(x, t) =χa(t) + ξa(t)(x, t). Substituting the decomposition u(x, t) = χa(t) + ξa(t)(x, t) into (16.3),we see that our equation becomes ∂xχaa + ∂tξa + ∂xξaa = −F (χa + ξa), where, recall, Fis the negative of the r.h.s. of (16.3) by F (u):
F (u) := −∆u+ g(u). (16.19)
By changing the variables x→ x+ a in the last equation, we obtain
−∂tξ − ∂xξa− ∂xχaa = F (χ+ ξ). (16.20)
We will need the Taylor expansion of F ,
F (χ+ ξ) = F (χ) + dF (χ)ξ +N(ξ), (16.21)
where N(ξ) := g(χ+ ξ)− g(χ)− g′(χ)ξ, and the observation that, by the definition of F ,F (χ) = 0 (see the sentence after (9.26)), to obtain
F (χ+ ξ) = Lξ +N(ξ), (16.22)
where L := dF (χ). Equations (16.20) and (16.22) give (16.17).
Equation (16.17) contains two unknowns: a(t) and ξ(x, t). To derive a separate equa-tion for a(t) we project equation (16.17) onto Rχ′ by multiplying by χ′ and integratingover x, to get
a(t)
∫(χ′)2dx+ a(t)
∫∂xξχ
′dxn =
∫∂tξχ
′dx+
∫Lξχ′dx+
∫N(ξ)χ′dx. (16.23)
For the first term on the r.h.s. we have∫∂tξχ
′dx = −∫ξ∂t (χ′) dx = 0. (16.24)
For the second term on the r.h.s., we use (9.26) and the self-adjointness of L to obtain∫Lξχ′dx = 0. The last relation together with (16.24) implies (16.18).
16.2.4 Bound on a(t)
Proposition 16.5. We assume that ‖ξ‖L2 . 1. Then, if u satisfies (16.3) and a(t) isdefined by (16.11) then a(t) satisfies
|a(t)| . ‖ξ‖2L2 . (16.25)
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Proof. We use equation (16.18) and estimate the right-hand side of this equation. Wehave, by the assumption (16.5) and a Sobolev embedding theorem,∣∣∣∣∫ N(ξ)χ′dx
∣∣∣∣ ≤ ‖N(ξ)‖L1‖χ′‖L∞
. ‖ξ‖2L2 + ‖ξ‖kLk
. ‖ξ‖2L2 + ‖ξ‖kH1 . (16.26)
We estimates the second term on the left-hand side of (16.18). Using integration byparts, we have
|∫∂xξχ
′dx| =
∫|ξχ′′dx| (16.27)
≤ ‖ξ‖L2‖χ′′‖L2 . (16.28)
Combining this estimate with (16.18) and (16.26) gives (16.25).
16.2.5 Upper Bound on ‖ξ‖H1
Proposition 16.6. If u solves (16.3), and ξ is the fluctuation defined by (16.11), then
Using that L is self-adjoint (see Theorem 9.5) and using (16.15), we have that 〈χ′, Lξ〉 = 0.Therefore, using (16.30), we obtain
∂tΛ(ξ) + ‖Lξ‖2L2 = −〈Lξ,N(ξ)〉. (16.31)
Now we prove the following estimate of the terms on the right-hand side:
|〈Lξ,N(ξ)〉| ≤ C(‖ξ‖4H1 + ‖ξ‖2k
H1) +1
2‖Lξ‖2
L2 . (16.32)
We start by using the bound on the nonlinearity, (16.5) to get
|〈Lξ,N(ξ)〉| ≤ ‖N(ξ)‖L2‖Lξ‖L2 (16.33)
. (‖ξ‖2L4 + ‖ξ‖kL2k)‖Lξ‖L2 (16.34)
≤ 1
4δ(‖ξ‖4
L4 + ‖ξ‖2kL2k) + δ‖Lξ‖2
L2 . (16.35)
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We use the Sobolev embedding inequality to find that we have (16.32).Finally, we prove the estimate
‖Lξ‖2L2 ≥ 1
2ρ〈Lξ, ξ〉. (16.36)
To this end, we use the Cauchy-Schwarz inequality and the inequality (16.8) to find
1
2ρ‖ξ‖2〈Lξ, ξ〉 ≤ 〈Lξ, ξ〉2 ≤ ‖Lξ‖2‖ξ‖2, (16.37)
which implies (16.36).Writing ‖Lξ‖2
L2 = 12‖Lξ‖2
L2 + 12‖Lξ‖2
L2 and using (16.36) gives ‖Lξ‖2L2 ≥ ρΛ(ξ) +
12‖Lξ‖2
L2 . Substituting this and the inequality (16.32) into (16.31), we obtain
∂tΛ(ξ) +1
2‖Lξ‖2
L2 + ρΛ(ξ) . C(‖ξ‖4H1 + ‖ξ‖2k
H1). (16.38)
Due to the inequality (16.8), this implies (16.29).
Proposition 16.7. If u solves (16.3) and u ∈ Uδ, where δ > 0 given in Proposition 16.3,so that (16.11) hold and if ξ is the fluctuation defined by (16.11), then
‖ξ‖H1 . e−ρ4t‖ξ0‖H1 , (16.39)
where ρ is the constant from Theorem 16.2.
Proof. We set X(t) := eρ2tΛ(ξ(t)) and use (16.29) to find X ≤ Ce−αρtX2, assuming for
simplicity that X . 1 and consequently dropping the term Xk. Therefore,
X(t) ≤ X0 + C
∫ t
0
e−αρtX2(s)ds. (16.40)
Setting M = sups∈[0,t] Λ(ξ(s)), we have
M . e−ρ2tM0 + CM2. (16.41)
Therefore, M ≤ e−ρ2tM0 provided t is sufficiently large. Since M ≥ 1
2〈Lξ, ξ〉 & ‖ξ‖2
H1 , wehave (16.39).
16.2.6 Global existence and symptotic stability
Now, we prove (i) of Theorem 16.1, the global existence of solutions. By Theorem 4.8and remarks after it, we have the local existence for (16.3). Indeed, by writing u = χ+ ξand plugging this into (16.3), we arrive at the equation for ξ of the NLS type discussedin the remarks after Theorem 4.8.
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Now, assume u0 ∈ Uδ/2, where δ > 0 given in Proposition 16.3. Then by the localwell-posedness, there is T > 0, s.t. the solution u(t) satisfies u(t) ∈ Uδ for 0 ≤ t ≤ T .Then Proposition 16.3 and therefore Proposition 16.7 hold for this u(t) for 0 ≤ t ≤ T .This gives the bound (16.39), which shows that, in fact, u(t) ∈ Uδ/2. Continuing in thisfashion, we find that the solution u(t) exists for all t > 0, is in Uδ and bound (16.39)holds.
The bound (16.39) and the definition of ξ in Proposition 16.3 imply (16.6).
16.2.7 Asymptotic behaviour of a(t)
In the last subsection we proved that the solution u = χa + ξa approaches the manifold ofkinks Mk; however, this does not imply the solution approaches a particular kink on themanifold. We prove in this subsection that the limit does exist. More precisely, we provethat a(t)→ a∞ as t→∞.
As above, we assume u0 ∈ Uδ/2, where δ > 0 given in Proposition 16.3. Then, as wasshown above, the solution u(t) exists for all t > 0 and satisfies u(t) ∈ Uδ and the bound(16.39), which, together with the definition of ξ in Proposition 16.3, implies (16.6).
In this case, Proposition 16.5 is applicable and gives the estimate (16.25) on a, whichtogether with (16.39) implies that a(t) satisfies
|a(t)| . e−ρ2t‖ξ0‖2
H1 . (16.42)
This shows that the function a(t) = a(0) +∫ t
0a(s) ds converges as t→∞ and
|a(t)− a(∞)| ≤∫ ∞t
|a(s)| ds . e−ρ2t‖ξ0‖2
H1 . (16.43)
We now complete the proof. The triangle inequality and the mean value theoremimply
‖u(t)− χa∞‖H1 .∥∥u(t)− χa(t)
∥∥H1 + |a(t)− a∞|,
which together with (16.6) and (16.43) completes the proof of (ii) of Theorem 16.1.
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17 Energy and orbital stability
17.1 Energy argument
The energy conservation or dissipation restricts severely possible dynamics of our system.We use this in investigating stability of static solutions, i.e. in understanding the behaviorof solutions to equation
∂u
∂t= F (u) (17.1)
near an equilibrium point u∗. Here F : U → Y , where U is an open set in X and X andY are Hilbert spaces.
We assume that (17.1) is a such that there exist a functional E(u) on U (called energy,or entropy or Lyapunov functional), defined on an open set U in a Hilbert space X, withthe following properties:
(i) E(u) is non-increasing under the evolution equation (17.1) (a stronger statement:∂tE(u(t)) ≤ 0, for any solution u(t) of (17.1))
(ii) Static solutions to (17.1) are critical points of E(u).
These conditions are satisfied for the hamiltonian (conservative) and gradient (dissipative)equations, two key classes of evolution PDE appearing in applications and in mathematics.These classes are discussed in Section 7.
On a basis of energy considerations one expects the following behavior
1. if u∗ is a strict local minimizer then u∗ is a stable solution,
2. if u∗ is either a saddle point or a maximizer then u∗ is an unstable solution.
We will turn Possibility #1 into a rigorous statement below. The Possibility #2 is notalways true. Sometimes conservation laws present in the equations can lead to stabilityin the cases of the second type. We will study this case below as well.
An important role below is played by the hessian operator
Hess E(u) := d grad E(u),
introduced in (6.26) above. With Theorem 6.22 in mind, we say that E is coercive at u∗if the following inequality satisfies
Hess E(u∗) ≥ θ, for some θ > 0. (17.2)
In what follows we use the notation E ′′ ≡ Hess E . We make the following assumptions:
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• (a) E(u) is C2, (b) E is coercive at u∗.
Since E is coercive at u∗, we can define the family (scale) of the Hilbert spaces Ys := Ls/2∗ Y ,
where L∗ := E ′′(u∗). We denote the norm in Ys by ‖ · ‖s. Clearly, E ′′(u∗) : Ys → Ys−2.We make a technical assumption that E ′′(u) maps Y1 into Y−1 and is continuous in u.
Theorem 17.1. Under the assumptions (i), (ii), (a) and (b), the static solution u∗ ofthe evolution equation (17.1) is Lyapunov stable.
Note that Theorem 17.1 and its proof (with some reinterpretations) remain valid ifsolutions considered have infinite energy, as is the case in Subsection 17.3. The sameholds for an extension of this theorem given in the next subsection.
The proof of Theorem 17.1 is based on the following important energy estimate
Theorem 17.2. Under the assumptions above, there is δ > 0 s.t., for any u satisfying‖u− u∗‖1 ≤ δ, we have the estimate
θ‖u− u∗‖2 ≤ ‖u− u∗‖21 ≤ 3(E(u)− E(u∗)). (17.3)
Proof. Using that E ∈ C2 and dE(u∗) = 0 and writing u = u∗+ ξ, we expand E(u) aroundu∗ to the third order as
E(u) = E(u∗) +1
2〈ξ, E ′′(u∗)ξ〉+R(ξ) , (17.4)
where R(ξ) is the remainder term defined by this expression. By the assumption that Eis C2 and the technical assumption, it satisfies the estimate R(ξ) = o(‖ξ‖2
1). The vectorξ is called a fluctuation of u around u∗. Hence, since 〈ξ, E ′′(u∗)ξ〉 = ‖ξ‖2
1, we have, for‖ξ‖1 = ‖u− u∗‖1 ≤ δ,
E(u)− E(u∗) ≥1
2‖ξ‖2
1 − ε‖ξ‖21 ≥
1
3‖ξ‖2
1, (17.5)
which gives (17.3).
Proof of Theorem 17.1. We write the solution ut of (17.1) as ut = u∗ + ξt. Let ‖ξ0‖ =‖u0− u∗‖ ≤ δ/2, where u0 is the initial condition, u|t=0 = u0, and δ > 0 is the same as inTheorem 17.2. The there is T > 0 s.t. ‖ξt‖ ≤ δ, for 0 ≤ t ≤ T . Then by Theorem 17.2,‖ut − u∗‖2
1 ≤ 3(E(ut) − E(u∗)), which, together with the assumption (i), that E(ut) is anon-increasing function of time, E(ut) ≤ E(u0), gives
‖ut − u∗‖21 ≤ 3(E(u0)− E(u∗)), (17.6)
for 0 ≤ t ≤ T .
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Next, by equation (17.4) and estimate the estimate R(ξ) = O(‖ξ‖21), we have
|E(u0)− E(u∗)| ≤1
2〈ξ0, E ′′(u∗)ξ0〉+ ε‖ξ‖2
1 ≤3
4〈ξ0, E ′′(u∗)ξ0〉 ≤
3
4‖ξ0‖2
1 , (17.7)
where ξ0 := u0−u∗ and we used again that 〈ξ, E ′′(u∗)ξ〉 = ‖ξ‖21, for ‖ξ‖1 sufficiently small.
Combining the last two estimates and remembering the definition ξ0 := u0 − u∗, we find
‖ut − u∗‖21 ≤
9
4‖u0 − u∗‖2
1, (17.8)
for 0 ≤ t ≤ T .Estimate (17.8) implies the Lyapunov stability of u∗. Indeed, for any ε > 0, we take
δ := 13ε. Then ‖ut − u∗‖1 ≤ ε for ‖u0 − u∗‖1 ≤ δ, which is the precise statement of the
Lyapunov stability.
2nd proof of Theorem 17.1: Lyapunov functional. We introduce the functional (Lyapunovfunctional) Λ(ξ) := 1
2〈ξ, E ′′(u∗)ξ〉. Using that, by the assumption, ‖ξ‖2 ≤ 1
θΛ(ξ) and
R(ξ) ≥ C‖ξ‖3, for some 0 < C <∞, we have
E(u)− E(u∗) ≥ Λ(ξ)− µΛ(ξ)3/2 , (17.9)
where µ := C/θ3/2. Denote λ = Λ(ξ)1/2 and α = E(u) − E(u∗) and define the functionF (λ) by
F (λ) = α− λ2 + µλ3 .
We have shown that λ satisfies F (λ) ≥ 0. Consider the graph of F (x), see Figure 20.For sufficiently small α (54αµ2 < 1), the function F (λ) positive in two disjoint intervals
A
x
F
Figure 20: Observe the first crossing of the graph of F and the x-axis is for sufficientlysmall α at λ∗ ≈
√2α, (
√2α ≤ λ∗ ≤ 2
√α when 52C2α < 1) in the graph the y-axis is in
units of α.
[0, λ∗) and (λ∗∗,∞), where λ∗∗ > λ∗ > 0 are the zeros of F (λ), λ > 0. For α sufficiently
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small, λ∗ ≈√α and λ∗∗ ≈ 1/µ. Then if F (λ) ≥ 0 and λ < λ∗∗, we must have λ ≤ λ∗.
Taking δ := 1θλ2∗∗, we conclude that if ‖ξ‖2 ≤ 1
θΛ(ξ) < δ, then
‖ξ‖2 ≤ 1
θΛ(ξ) ≤ 1
θλ2∗ ≈
1
θα. (17.10)
By (17.9), the function λt := Λ(ξt)1/2 satisfies F (λt) ≥ 0, where the function F is
defined there. By the (17.10) and the continuity of Λ(ξt), if ‖ξ0‖2 ≤ 1θΛ(ξ0) < δ, then
‖ξt‖2 ≤ 1θΛ(ξt) ≤ 1
θλ∗(t)
2 ≈ 1θ(E(u) − E(u∗)). Here ξ0 is the initial condition, ξ|t=0 = ξ0.
(If λt starts at t = 0 at λ0 ≤ λ∗(0), then since f(λt) > 0, λt must stay in the interval[0, λ∗(t)].)
Next, by the assumption (i), E(ut) is a non-increasing function of time: E(ut) ≤ E(u0),where u0 is the initial condition, u|t=0 = u0. The last two results give
‖ξ‖ ≤ 4
δ(E(u0)− E(u∗)), (17.11)
which shows that by making u0 to be close to u∗, we can make ut to be arbitrary close tou∗. Hence the static solution u∗ is Lyapunov stable. (needs cleaning)
17.2 Systems with symmetry
For systems with symmetry, the notion of stability/instability should be modified. Recallfrom Subsection 2.3, that we say that the dynamical system (17.1) has a symmetry groupG if a representation T : g → Tg of G on the space of solutions satisfies
F (Tgu) = TgF (u). (17.12)
As was discussed in Subsection 16.1, in this case, if the dynamical system (17.1) with thesymmetry group G has a static solution u∗, then it has the manifold of static solutions
M∗ = Tgu∗ : g ∈ G.
Indeed, if u∗ is a static solution to (17.1), then, due to (17.12), so is Tgu∗. Now, we haveto address the stability w.r.to this manifold. Clearly, we have to modify the definitionsat the beginning of this section and the approach.
We say that a static solution u∗ (or more precisely, the manifold of static solutionsM∗)is orbitally stable if any solution to the equation (17.1), starting in a small neighborhoodof u∗ stays in a small neighborhood of the manifold M∗. In other words the solution utsticks very close to a possibly moving static solution Tg(t)u∗.
As far as the energy is concerned, the symmetry requirement means that
E(Tgu) = E(u). (17.13)
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This implies that, if u∗ is a critical point of E(u), then so is Tgu∗. (This can be provedindependently by differentiating the equality E(Tg(u∗ + sξ)) = E(u∗ + sξ) w.r. to s.)
Note that the tangent space, Tu∗M∗ ⊂ X, to M∗ at u∗ is the vector space τ(g)u∗,where τ(g) is the representation of the Lie algebra g of G acting on X. (This can be seenby differentiating Tg(s)u∗, where Tg(s) is any one-parameter subgroup of G, w.r. to s.)
As we know from Subsection 9.2, in the case of symmetry, the inequality (17.2) neverholds: the hessian E ′′(u∗) has always zero eigenvectors (see Proposition 9.3). Indeed, wehave Thus in the case of symmetries, if Au∗ 6= 0 for some A ∈ τ(g), then the assumption(b) fails. We replace it by the following weaker assumption
(b’) E is coercive at u∗ in the sense that
〈ξ, E ′′(u∗)ξ〉 ≥ θ‖ξ‖2, for some θ > 0 and ∀ξ ⊥ Tu∗M∗. (17.14)
If Au∗ 6= 0, or what is the same esAu∗ 6= u∗, for some A ∈ τ(g), then we say that u∗breaks the symmetry corresponding to the one-parameter subgroup esA. For a given u∗,consider the maximal subgroup, G∗, of G, which is broken by u∗, i.e. gu∗ 6= u∗,∀g ∈ G∗.In what follows, we use this subgroup instead of G.
Now, we have the following strengthening of Theorem 17.1:
Theorem 17.3. Under the assumptions (i), (ii), (a) and (b’), the static solution u∗ ofthe evolution equation (17.1) is orbitally stable.
Proof. In the case of symmetry the argument above should be modified. In a neighborhoodof the manifold M∗ we decompose u as
u = Tgu∗ + ξ , 〈ζa, ξ〉 = 0 , (17.15)
for some symmetry transformation g. (This is a nonlinear, orthogonal decompositionwith respect to the manifold M∗, see Figure 21.) Note that a in (17.15) is determined
u
xi
M
chi
zeta
Figure 21: Manifold decomposition
by u uniquely. This follows from the implicit function theorem applied to the functionF (g, u) := 〈u− Tgu∗, Au∗〉.
We introduce a neighbourhood of M∗:
Uδ = u ∈ X : ∃g s.t. ‖u− Tgu∗‖X ≤ δ. (17.16)
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Proposition 17.4. There is δ > 0 s.t. any u ∈ Uδ can be decomposed as follows
u = Tgu∗ + ξ, with g = g(u) and ξ ⊥ TTgu∗M∗. (17.17)
The latter condition can be written as
〈ξ, Au∗〉 = 0, ∀A ∈ g. (17.18)
Proof. Assume for simplicity that G is a one-parameter (one-dimensional) group, para-metrized by a ∈ R and write Ta for Tg(a). Let A be the corresponding generator. We definethe function G(a, u) = 〈u− Tau∗, ATau∗〉 and rewrite the statement of the proposition assolving the equation
G(a, u) = 0 (17.19)
for a as a function of u. To do this, we use the Implicit Function Theorem. Clearly,G is C1 in u because is it a linear functional in u, and C1 in a since Tau∗ is C1 in a.Furthermore, G(a, Tau∗) = 0. Finally, we show that ∂aG(a, u)
Hence, using that ∂aTau∗ = ATau∗, we have, at u = Tau∗,
∂aG(a, u)|u=Tau∗ = −‖ATau∗‖2 = −‖Au∗‖2. (17.21)
By the definition of G∗, we have Au∗ 6= 0. Therefore, the conditions of the ImplicitFunction Theorem are satisfied and it implies that (17.19) has a unique solution for a asa function of u ∈ Uδ, which gives the statement of the proposition.
The importance of this proposition lies in the fact that for ξ ⊥ Tu∗M∗, we have theestimate (17.14).
Proceeding as in the proof of Theorem 17.1 and using (17.14) instead of (17.2) andusing that E(Tgu∗) = E(u∗), we find as in (17.3),
‖u− Tgu∗‖ ≤2
θ(E(u)− E(u∗)), (17.22)
where g = g(u) is the same as in (17.17). However, as little contemplation shows,‖u− Tgu∗‖ = dist(u,M∗), which gives the estimate
dist(u,M∗) ≤2
θ(E(u)− E(u∗)). (17.23)
Continuing as in the proof of Theorem 17.1, we obtain the conclusion of our theorem.
We apply the analysis above on the problem of stability of kinks in the Allen - Cahnequation. This will demonstrate the concepts introduced above.
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17.3 Orbital stability of kink solutions of the Allen - Cahn equa-tion
Consider the question of stability of the kink solutions of the Allen - Cahn equation (see(2.2))
∂u
∂t= ∆u− g(u), (17.24)
where g(u) = u3− u, or, more generally, g : R→ R is the derivative, g = G′, of a double-well potential: G(u) ≥ 0 and has two non-degenerate global minima with the minimumvalue 0. For g(u) = u3 − u, we take G(u) = 1
2(u2 − 1)2. G(u) is also called a bistable
potential, see Figure 18.To keep the notation simple, assume we are in the dimension 1, i.e. u : R× R+ → R.
Recall from Section 2.1 that (17.24) has the kink static solution by χ(x), see Figure 19;and Eq. (17.24) is translation invariant and therefore the functions χa(x) := χ(x + a)∀a ∈ R are also static solutions to (17.24). Thus we have an entire manifold of stationarysolutions Mkink = χa : a ∈ R. (In the multidimensional case, we have also rotations.)
We now use the results above to derive orbital stability of the kinks. We have
Theorem 17.5. The kink solutions of (17.24) are orbitally stable w.r.to H1(R)−perturbations.
Proof. It is not difficult to see that (17.24) is an L2−gradient system (in the sense of thedefinition given in Subsection 7) with the energy functional (cf. (??) or (18.4))
E(u) =
∫(1
2|∇u|2 +G(u)) , (17.25)
defined on χ + H1(R). Consequently, we have the properties (i) and (ii) of Subsection17.1. (Note however that E(χ) =∞.)
Because of the translational symmetry mentioned above, we use Theorem 17.3. Ac-cording to this theorem, to prove the theorem, it suffices to show that the energy (17.25)satisfies Conditions (a) and (b’) on the space H1(R). Verifying Conditions (a) is straight-forward. Conditions (b’) requires more work.
Let L be the hessian of the energy functions E(u) at χ: L := E ′′(χ). If we denote thenegative of the r.h.s. of (17.24) by F (u),
F (u) := −∆u+ g(u). (17.26)
then the hessian L := E ′′(χ) of the energy functions E(u) at χ is given by L := dF (χ)and is computed explicitly as
L := −∆ + g′(χ). (17.27)
The key here is Theorem 16.2. This theorem implies Condition (b’) for the energy (17.25).Since satisfies Condition (a) (E(u) is is C3), the statement of the theorem follows.
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18 Minimization: direct methods
18.1 General result
The problem we address is the following: given a functional E on a space M , find afunction u0 ∈M (if such exists) that minimizes E :
E(u0) = infu∈ME(u).
Such a function u0 is called a minimizer for E . Thus to begin with, we want to assumethat E is bounded below, i.e.
E0 := infu∈ME(u) > −∞.
We also assume that M is a closed subset of a Banach space, X. Let us first analyze thefinite dimensional situation: X = RN . How would we minimize a functional on M? Wedo this in three steps (that will be suitable to be generalized to the infinite dimensionalcase):
• Step 1. Since E is bounded on M from below, E0 := infu∈M E(u) > −∞, we canpick a sequence un ⊂M s.t. E(un)→ E0, as n→∞. Such a sequence is called aminimizing sequence. Clearly we can take the sequence un s.t. every element unsatisfies
E(un) ≤ E0 + 1 (18.1)
(just throw out those un for which E(un) > E0 + 1).
• Step 2. We hope that either such a sequence converges or at least contains aconvergent subsequence. The limit of such a subsequence clearly is a candidatefor a minimizer if the latter exists. How do we show that un has a convergentsubsequence? Assume that
E(u)→∞ as ‖u‖X →∞. (18.2)
A functional E on M satisfying (18.2) is called coercive. Due to (18.1) and (18.2), wehave that ‖un‖X ≤ C, ∀n, and for some C > 0. Hence by the Bolzano–Weierstrasstheorem, un has a convergent subsequence, which for notational convenience wedenote again by un.
• Step 3. Let u0 := limn→∞ un. If E is continuous, then
limn→∞
E(un) = E(u0).
Since on the other hand we have limn→∞ E(un) = E0, we conclude that E(u0) = E0,i.e. u0 is a minimizer of E .
178 Lectures on Applied PDEs
Let us look closer at the last two steps. Recall that a set K s.t. every infinite sequenceof elements of K contains a convergent subsequence is called (sequentially) compact. TheBolzano–Weierstrass theorem states that a closed ball in RN is compact. This propertydoes not hold in general in the infinite dimensional case. For instance, closed balls in L2(Ω)are not compact. As a concrete example, take for instance L2(Rn) 3 un(x) := u(x − n),for some u ∈ L2(Rn), ‖u‖2 = 1. Clearly, this sequence does not have a convergentsubsequence.
We have however, the following weaker result (this result follows from the Banach-Alaoglu theorem): If X is a reflexive, separable Banach space, then every uniformlybounded sequence un in X (i.e. ‖un‖X ≤ C) has a weakly convergent subsequenceunk. (In the previous example, un(x) := u(x − n) converges to 0 weakly. In thefinite dimensional spaces, weak convergence is equivalent to strong convergence, and theBanach-Alaoglu theorem reduces to the Bolzano-Weierstrass theorem.) 8
Thus we have to assume that M is weakly closed: We say that M is weakly closed inX if and only if un
w−→u0 in X and un ∈M , imply u0 ∈M .
Next, the continuity w.r.to the weak convergence is a property hard to come by forfunctionals on infinite dimensional spaces (e.g. u→ ‖u‖2
X is not weakly continuous in X).But there is a weaker property which often holds: the weak lower semicontinuity, which isdefined as follows. E is called weakly lower semicontinuous (w.l.s.c.) if and only if un
w−→u0
in M implies lim infn→∞ E(un) ≥ E(u0) (recall that lim inf un := limk→∞(infn≥k un)) (e.g.u→ ‖u‖2
X is w.l.s.c. in X).
We now check that in fact w.l.s.c. suffices to carry out step 3 above. If unw−→u0,
then lim infn→∞ E(un) ≥ E(u0). On the other hand, by the definition of un, we havelim infn→∞ E(un) = E0 = infu∈M E(u). Therefore E(u0) = E0, and hence u0 is a minimizerof E .
Thus, we have essentially proven the following
Theorem 2 (Key Theorem). Assume that
(α) M is weakly closed in X,
(β) E is w.l.s.c.,
(γ) E is coercive.
Then E is bounded below and attains its minimum in M (i.e. there is a minimizer of Eon M).
8Here we recall some definitions. A Banach space is said to be reflexive, iff its second dual (the dualof the dual) is isometrically isomorphic to the space itself. Every Hilbert space is reflexive, see e.g., [20],Section 6.3, Theorem 2 and its Corollary. Furthermore, we say a sequence un in X is weakly convergent
iff ∃u0 s.t. `(unk)→ `(u0), ∀` ∈ X ′. The weak convergence is denoted by
w−→, as in unw−→u0.
Lectures on Applied PDEs 179
Proof. Let E0 := infu∈M E(u), and let un be a minimizing sequence for E , i.e.
limn→∞
E(un) = E0. (18.3)
Clearly, we can assume that E(un) ≤ E0 + 1 (we get rid of those un’s in the minimizingsequence for which E(un) > E0 + 1). Then by the coercivity of E , there is a constantK s.t. ‖un‖ ≤ K, ∀n. Hence, by the Banach-Alaoglu theorem, un contains a weaklyconvergent subsequence, un′, un′ w−→u0 ∈ X. The element u0 is a candidate for aminimizer. Since M is weakly closed, u0 ∈ M . W.l.s.c. of E gives lim infn′→∞ E(un′) ≥E(u0). This together with equation (18.3) implies that E0 ≥ E(u0). On the other hand,E0 = infu∈M E(u) ≤ E(u0), and therefore E(u0) = E0. This shows that u0 is a minimizerand that infu∈M E(u) > −∞.
18.2 Applications
This is a simple but powerful result. It says that in order to show that E has a minimizer,we have to check three conditions (α) - (γ). We begin with the discussion of theseconditions. We begin with (β).
Lemma 1. Let X be a Hilbert space. Then the functional u→ ‖u‖2X is w.l.s.c. on X.
Proof. Dropping the subindex X, we have ‖un‖2 = 〈un, un〉 = 〈un − u+ u, un − u+ u〉 =〈u, u〉+ 2 Re〈un−u, u〉+ 〈un−u, un−u〉 ≥ ‖u‖2 + 2 Re〈un−u, u〉. Since 〈un−u, u〉 → 0,this gives lim inf ‖un‖2 = ‖u‖2.
In particular, this lemma implies
Corollary 3. The functional E(u) := 12
∫Ω|∇u|2 is w.l.s.c. on H1(Ω).9
Lemma 2. Let G(x, u) ≥ 0, be l.s.c. in u and s.t.∫
ΩG(x, u) is well defined for u ∈
H1(Ω). Then the functional E(u) :=∫
ΩG(x, u) is w.l.s.c. on H1(Ω).
9 We show this by the direct computation:
1
2
∫Ω
|∇un|2 =1
2
∫Ω
|∇(u+ un − u)|2
=1
2
∫Ω
|∇u|2 + Re
∫Ω
∇u · ∇(un − u) +1
2
∫Ω
|∇(un − u)|2
≥ 1
2
∫Ω
|∇u|2 + Re
∫Ω
∇u · ∇(un − u).
If unw−→u in H1(Ω), then
∫Ω∇u · ∇(un − u)→ 0, and therefore
lim infn→∞
1
2
∫Ω
|∇un|2 ≥1
2
∫Ω
|∇u|2.
180 Lectures on Applied PDEs
Proof. Indeed, let unw−→u in H1(Ω) and let Q be a bounded subset of Ω (Q = Ω if Ω is
bounded). Then by the Rellich-Kondrashov theorem, un → u in L2(Ω) (up to picking asubsequence) and therefore un → u a.e. in Q. Since Q is arbitrary bounded subset of Ω, wehave that un → u a.e.. Therefore, by Fatou’s lemma10, we get lim infn→∞
∫ΩG(x, un(x)) ≥∫
Ωlim infn→∞G(x, un(x)) ≥
∫ΩG(x, u(x)), and E is w.l.s.c. on H1(Ω).
Lemmas 1 and 2 imply
Corollary 4. If G(x, u) ≥ 0, l.s.c. in u and s.t.∫
ΩG(x, u) is well defined for u ∈ H1(Ω),
then the functional
E(u) =
∫Ω
(1
2|∇u(x)|2 +G(x, u(x)))dx. (18.4)
is w.l.s.c. on H1(Ω).
Now, we discuss (γ), the coercivity.(c) The functional E(u) =
∫Ω
(12|∇u|2 + c|u|2), c > 0, is obviously coercive. Indeed,
if c > 0, then E(u) ≥ δ‖u‖2(1), where δ = min(1/2, c) > 0, and, recall, ‖u‖(1) = ‖u‖H1(Ω),
and therefore E is coercive on H1(Ω), i.e. E(u)→∞ whenever ‖u‖(1) →∞.A more delicate situation is with the Dirichlet functional. We have(d) The functional E(u) := 1
2
∫Ω|∇u|2 is is coercive onH1(Ω), provided Ω is bounded in
one direction. Indeed, by the Poincare inequality (see (18.12) below), we have∫
Ω|u|2 ≤
D2∫
Ω|∇u|2, for any u ∈ H1
0 (Ω), where D is the smallest diameter of Ω. So we getE(u) ≥ 1
4min(1, D−2)||u||2(1) for every u ∈ H1
0 (Ω). Therefore E is coercive on H10 (Ω).
Collecting above statements we obtain
Proposition 1. Let G(x, u) ≥ c|u|2, for some c ≥ 0. If c > 0, then the functional (18.4)is coercive on H1(Ω); if c = 0 and Ω is bounded in one direction, then (18.4) is coerciveon H1
0 (Ω).
Now we give examples of weakly closed sets.
(f) M = X or M = g + X (for a fixed g), where X is a reflexive Banach space (inparticular, a Hilbert space). Hence H1
g (Ω) is weakly closed in H1(Ω), if g has anextension, g, from ∂Ω to Ω. Indeed, in the latter case can write H1
g = g + H10 (Ω).
Since H10 (Ω) is a Hilbert space, the result follows.
(g) A convex, closed subset of a reflexive Banach space (by Mazur’s theorem).
(h) M0 = u ∈ H10 (Ω) :
∫Ω|u|pdnx = 1, where Ω is a bounded, smooth domain in Rn
and p < 2nn−2
if n > 2 and p <∞ if n ≤ 2.
Proposition 2. The set M0 defined above is weakly closed in H10 (Ω).
10For any sequesnce fn of non-negative and measurable functions, lim infn→∞∫
Ωfn(x) ≥∫
Ωlim infn→∞ fn(x).
Lectures on Applied PDEs 181
Proof. By the Rellich-Kondrashov theorem, H10 (Ω) is compactly embedded into
Lp(Ω), for p as in the proposition, and Ω bounded. This means that any weaklyconvergent sequence un
w−→u0 in H10 (Ω) contains a subsequence un′ s.t. un′ → u0
in Lp(Ω). Hence ‖u0‖p = limn→∞ ‖un‖p = 1, and therefore u0 ∈M0.
Let us draw some conclusions from what we have just shown:
Theorem 5. Assume G(x, u) ≥ 0, l.s.c. in u, is integrable over Ω and satisfies G(x, u) ≥c|u|2. Then the functional (18.4) has a minimizer in H1
0 (Ω) and in M0, if c ≥ 0 and Ω isbounded in one direction, and in H1(Ω), if c > 0.
Exercise 2. (Nonlinear Dirichlet problem) Use results above to show that the Dirichletproblem
∆u− |u|2u = 0 in Ω,u = g on ∂Ω,
(18.5)
has a (weak) solution if the domain Ω is bounded in one direction and g ∈ L2(∂Ω,R).
Thie result above deals with the functional (18.4) with G(x, u) ≥ 0. What happenswhen G(x, u) ≤ 0? To see possible pitfalls, consider the functional
F (u) =
∫Ω
(1
2|∇u|2 − 1
p|u|p − λ
2|u|2)dnx. (18.6)
However, for p > 2, this functional is not bounded from below. Indeed, take uµ = µuwith a fixed function u, and some µ > 0. Then
F (uµ) = µ2 1
2
∫Ω
(|∇u|2 − λ|u|2
)dnx− µp
p
∫Ω
|u|p → −∞, as µ→∞.
Consequently, this functional does not have a minimizer. Taking u(µ) := µαu(µx) with afixed function u and some µ, we find
F (u(µ)) = µ2+2α−n1
2
∫Ω
|∇u|2 − µαp−n1
p
∫Ω
|u|p − µ2α−nλ
2
∫Ω
|u|2.
Take now α so that 2 + 2α − n > 0, and 2 + 2α − n > αp − n, i.e. 2p−2
> α > n2− 1.
Since 2p−2
> n−22
(because p2< 2
n−2+ 1 = n
n−2), this is possible. Then we get F (u(µ))→∞
as µ → ∞, which shows that F is not bounded from above, and consequently, it has nomaximizer either.
We discuss below how to go around this problem.
182 Lectures on Applied PDEs
The nonlinear eigenvalue problem. Now we show how to use the Key Theorem inorder to prove existence of solutions of differential equations. Let Ω be a smooth boundeddomain in Rn. For λ ∈ R and p > 2, we consider the problem
−∆u− |u|p−2u = λu in Ω,u = 0 on ∂Ω.
(18.7)
We want to prove existence of solutions of this boundary value problem. Denote λ1 thelowest eigenvalue of −∆ on Ω with Dirichlet boundary conditions. We have the following
Theorem 6. Let 2 < p < 2nn−2
if n > 2 and 2 < p < ∞ if n ≤ 2. Then for any λ < λ1,there is a positive (weak) solution to the problem (18.7).
Exercise 3. Prove the theorem above. (Hint: Minimize the functional
E(u) =1
2
∫Ω
(|∇u|2 − λ|u|2
)dnx, (18.8)
subject to the constraint J(u) = 1, where
J(u) :=1
p
∫Ω
|u|pdnx. (18.9)
Show that the solution to the corresponding Euler-Lagrange equation can be rescaled tosatisfy (18.7). Specifically, show that the Euler-Lagrange equation for the problem (18.8)- (18.9) is
−∆v − λv − µ|v|p−2v = 0, (18.10)
and1
p
∫Ω
|v|pdnx = 1, (18.11)
for some µ ∈ R. But now we have the undesirable coefficient µ. (The parameter µ is afunction of λ determined through equation (18.11)). To get rid of this coefficient, we firstshow that µ > 0. Indeed, multiplying (18.10) by v, integrating the result over Ω, andthen integrating by parts, we obtain
µ
∫Ω
|v|pdnx =
∫Ω
(|∇v|2 − λ|v|2
)dnx.
The r.h.s. is∫
Ω(v(−∆)v−λ|v2|) ≥ (λ1−λ)
∫Ω|v|2, where, recall, λ1 is the lowest eigenvalue
of the negative Laplacian, −∆, with Dirichlet boundary conditions. Since λ < λ1, ther.h.s is positive and so µ > 0. Now we rescale v as
u(x) := µ1p−2v(x),
then clearly u satisfies (18.7).)
Lectures on Applied PDEs 183
Discussion. Differential equation (18.7) is the Euler–Lagrange equation for the func-tional (18.6), which as was shown above, for p > 2, is unbounded below and above and,consequently, does not have a minimizer or maximizer. To get out of this dilemma, weconsider instead the constraint problem (18.8) - (18.9).
Discussion. One can show that µ ↓ 0 as λ ↑ λ1. This shows that a branch of non-trivialsolutions of (18.7) bifurcates from the trivial solution u ≡ 0 at λ = λ1.
We give the following general result about w.l.s.c..
Theorem 7. If E is convex, then E is w.l.s.c..
Proof. By Proposition 6.23, we have E(un) ≥ E(u) + dE(u)(un− u). By the weak conver-gence we have dE(u)(un− u)→ 0, as n→∞, which, together with the previous relation,implies that lim infn→∞ E(un) ≥ E(u).
This result implies, in particular, that if C ≥ V (x) ≥ 0, then the functional∫
ΩV (x)|u|2
is w.l.s.c. in L2(Ω).
Exercise 4. 1) Let V (x) ≥ 0. Show that if unw−→u in L2(Ω), then
lim infn→∞
∫Ω
V (x)|un|2 ≥∫
Ω
V (x)|u|2.
2) Let Ω be a bounded domain in Rn, and let f ∈ L2(Ω). Show that the functional12
∫Ω|∇u|2dnx−
∫Ωfu is coercive and w.l.s.c. on H1
g (Ω).3) Let Ω be a bounded domain in Rn, and let g(u) = (gij(u)) be a family of m × m
positive definite matrices satisfying g(u) ≥ δ1l, for some δ > 0. Show that the functional
E(u) =1
2
∫Ω
∑i,j
gij(u)∇ui · ∇ujdnx
is coercive and w.l.s.c. on H1g (Ω,Rm). Here, u = (u1, . . . , um) : Ω→ Rm.
We give more examples of weakly closed sets.
(h) Let Ω ⊂ Rn be bounded and
M := u ∈ L1(Ω) |∫
Ω
|∇u|2 <∞ and
∫Ω
u = c
for some c ∈ R.
Proposition 3. M ⊂ H1(Ω) and is weakly closed there.
184 Lectures on Applied PDEs
Proof. By passing from u(x) to u(x) − c we reduce our problem to the case ofc = 0. By the Poincare inequality, Theorem 9,
∫Ω|u|2 ≤ C
∫Ω|∇u|2, we have that
M ⊂ H1(Ω).
Now, if un → u weakly in H1(Ω), then by the Kondrashov-Rellich theorem (see e.g.[20] and Section A.6), un → u strongly in L2(Ω) and therefore in L1(Ω) (rememberthat Ω is a bounded domain). Hence, if un ∈ M , i.e.,
∫Ωun = 0, then
∫u = 0, i.e.,
u ∈M . Thus M is weakly closed in H1(Ω).
(i) Let Ω ⊂ Rn be a bounded domain, and let M ⊂ H1(Ω) be given by M = u ∈H1(Ω) : u(x0) = 0 for some x0 ∈ Ω (say x0 = 0).
Proposition 4. M is weakly closed in H1(Ω) for n = 1 and is not weakly closedfor n > 1.
Proof. Let first n = 1. Since Ω is bounded, we have by the Rellich–Kondrashovtheorem that if un
w−→u0 in H1(Ω), then there is a subsequence un′ s.t. un′ → u0
in C(Ω). If un ∈M , then un(0) = 0, and therefore u0(0) = limn′→0 un′(0) = 0.
Now let n ≥ 2. Assume for simplicity that Ω is a ball and take ϕn ∈ M , ϕn(x) =f(n|x|) with f(r) = r for r ≤ 1 and f(r) = 1 for f ≥ 1. Then (see in the exercisebelow) ϕn → 1 weakly in H1(Ω) as n→∞.
Exercise 5. Show the statement above.
The key theorem implies that the following functionals have minimizers on the specifiedsets:
1.∫
Ω(1
2|∇u|2 + G(x, u)) on M , if G(x, u) ≥ 0, where M is either H1
g with Ω boundedin one direction or
u ∈ H10 :
∫Ω
|u|pdnx = 1
with Ω bounded and 2 ≤ p < 2n(n−2)+
, m+ =
m if m > 00 if m ≤ 0
.
2. 12
∫Ω
∑ij gij(u)∇ui ·∇uj on H1
g (Ω,Rm), provided Ω is bounded in one direction, andg(u) = (gij(u)) ≥ δ1l, for some δ > 0.
The Euler-Lagrange equations for the variational problems in the examples 1 and 2above are
−∆u+G′(x, u) + λ|u|p−2u = 0 and∑ij
(g′ij(u)∇ui · ∇uj + div(aij(u)∇u)
)= 0,
Lectures on Applied PDEs 185
respectively, where λ is a Langrange multiplier, G′(x, u) = ∂uG(x, u) and g′ij(u) =∂ugij(u). Existence of minimizers yields existence of weak solutions of the above equationsin a bounded Ω ⊂ Rn.??
We have the following special cases for example 1:
i) If G has a minimum at u0 which is independent of x, then u0(x) ≡ u0 is a minimizer:E(u0) = 0,
ii) G(x, u) = V (x)|u|2 for some V (x) ≥ 0, i.e. G is quadratic (remember that inthis case the equation for the critical points of E is linear !). We can write E(u) onH2
0 (Ω) as E(u) =∫
Ωu(−1
2∆+V (x))u, i.e. E is the quadratic form of the Schrodinger
operator −12∆ + V (x).
Exercise 6. Prove existence of (weak) solutions of the following boundary value problems(below, Ω is a bounded domain in Rn):
(a) the nonlinear eigenvalue problem:
∆u+ a(x)|u|p−1u = µu in Ω,
u = 0 on ∂Ω,
where a(x) is a smooth and positive function on Ω, n ≥ 3, 2 < p < 2nn−2
and µ ≥ 0;
(b) the nonlinear Dirichlet problem:
∇(|∇u|2∇u
)= f in Ω,
u = 0 on ∂Ω,
for any f ∈ L4/3(Ω). (Hint: reduce this to a minimization problem on the Sobolevspace
W 4,10 (Ω) = u ∈ L4(Ω) :
∫Ω
|∇u|4 <∞ and u|∂Ω = 0).
How to gain smoothness: elliptic regularity. Assume we show that the followingequation has a (weak) solution in H1(Ω):
∆u = a(x)u4 in Ω,
and u = 0 on ∂Ω (Dirichlet boundary conditions). Here, a is smooth and Ω ⊂ R3 isbounded. This is not so good since u4 does not below to a good space and ∆u ∈ H−1(Ω).But it turns out that in fact u is smooth!
We can show this in the following way, which we just sketch. By the Sobolev embeddingtheorem (i.e. H1(Ω) ⊂ Lα(Ω) with α < 2n
n−2= 6 for n = 3), we have that u ∈ Lα(Ω) with
α < 6. Hence u4 ∈ Lβ(Ω) with β < 3/2, so ∆u ∈ Lβ(Ω) since a is smooth. Then by theSobolev embedding theorem, u ∈ Lα(Ω) for any α < ∞. Iterating this we obtain that uis bounded and differentiable.
186 Lectures on Applied PDEs
Poincare inequality
Theorem 8 (Poincare inequality). Let Ω have a diameter d <∞ in some direction (i.e.it is possible to place Ω between two parallel hyperplanes at a distance d from each other).Then for any u ∈ H1
0 (Ω), we have∫Ω
|u|2 ≤ (2d)2
∫Ω
|∇u|2. (18.12)
Proof. We can assume that this hyperplanes are x1 = 0 and x1 = d. Assume u isreal and estimate
‖u‖22 =
∫Ω
1 · |u|2 = −∫
Ω
x1∂
∂x1
|u|2 = −2Re
∫Ω
x1u∗∂u
∂x1
≤ 2d
∫Ω
|u|∣∣∣∣ ∂u∂x1
∣∣∣∣ .Applying now the Schwartz inequality to the integral on the r.h.s., we obtain
‖u‖22 ≤ 2d‖u‖2
∥∥∥∥ ∂u∂x1
∥∥∥∥2
≤ 2d‖u‖2 ‖∇u‖2,
where ‖∇u‖22 =
∫Ω|∇u|2 =
∫Ω
∑n1 | ∂u∂xn |
2. The latter inequality implies ‖u‖2 ≤ 2d‖∇u‖2.
We mention without proof, the following variant of the result above.
Theorem 9. Let Ω be bounded and u := 1|Ω|
∫Ωu. Then∫
Ω
|u− u|2 ≤ C
∫Ω
|∇u|2.
18.3 Existence of ground state of nonlinear Schrodinger equa-tion (without and with potential)
*(under construction)* (see [35], Introduction and Section II, and also [?], Sections 1( in particular, Theorem 1 i) - ii)) and 3 and Appendix)
19 Interfaces, vortices, vortex lattices and harmonic
maps
19.1 Allen-Cahn energy functional and interfaces
The difference in free energy of two phases of the same substance or of two substances ina domain Ω ⊂ Rd can be often described by the following simple functional:
E(u) =
∫Ω
(1
2|∇u|2 + λG(u)
)ddx, (19.1)
Lectures on Applied PDEs 187
where u : Ω → R, λ > 0, and G(u) ≥ 0, has two strict, non-degenerate minima at u = 1and u = −1, and G(u)→∞ as |u| → ∞. In other words, G is of the form of a double-wellpotential:
The typical and most important example is G(u) = 14(|u|2 − 1)2, see Fig 18. This
G
1−1
u
Figure 22: Double well potential.
functional is called the Allen - Cahn (or Ginzburg-Landau) energy functional. It plays animportant role in many areas of sciences and engineering. Ω is a large domain, say a boxwith sides of size 2L or a ball of radius L or Rd.
Note that E(ϕ) ≥ 0 and has two absolute minimizers, −1 and +1, so that E(−1) =E(+1) = 0. These minimizers correspond to two homogeneous substance or phases, whichwe will call the −1 and +1 phase. We are interested in minimizers for which these phasesco-exist. To obtain such minimizers we to impose constraints and/or boundary conditions.For instance, if the total amount of the −1 phase is fixed and Ω = Rd, then imposing∫
Ω
(1− u(x))ddx = α, (19.2)
would guarantee that the total amount of the −1 phase is finite and determined by α.
Before addressing these questions, we mention that the Euler-Lagrange equation forcritical points of E(u) is
−∆u+ λG′(u) = µ, (19.3)
where µ 6= 0 if condition (19.2) is imposed. In the latter case µ is the correspondingLagrange multiplier and is chosen so that (19.2) holds. For µ 6= 0, this is the staticCahn-Hilliard equation and for µ = 0, the static Allen - Cahn equation (also known asthe Ginzburg-Landau equation).
Exercise 7. Find the Euler-Lagrange equation for the functional (19.1) with side condi-tion (19.2).
Now we apply the direct method of variational calculus in order to find minimizersof the Allen - Cahn energy functional (19.1), with the function G(u) described in theparagraph after (19.1).
188 Lectures on Applied PDEs
Planar interface. We want to consider the situation where the minimizers describethe planar interface. Assume this interface is the plane x1 = 0 (by a translation anda rotation, we can always reduce to this case). Then our unknown u depends on onevariable only - on x = x1 - and the problem becomes the one dimensional one and thefunctional (19.1) becomes
E(u) =
∫R
(1
2|∇u|2 + λG(u)
)dx, (19.4)
and can interpreted as the energy per unit area of the interface x1 = 0.We have four distinct boundary conditions (BC): u(x1)→ ±1 as x1 →∞ and u(x1)→
±1, as x1 → −∞. Consequently, we are led to consider E on the following four spaces:
M±,± = u ∈ H1(Ω) : one of the above BC holds.
Clearly, E attains its strict minimum on M+,+ and M−,− at u+(x1) ≡ 1 and u−(x1) ≡ −1,respectively. In the phase separation model described by the Allen - Cahn (Ginzburg–Landau) functional, these minimizers describe homogeneous phases. Next, a minimizeron M+,− is obtained from a minimizer on M−,+ by reflection u(x1) → u(−x1). Thus itsuffices to consider only minimizers on M−,+. Observe that since E has the reflectionsymmetry, we can simplify our task by looking for odd minimizers, i.e. we pass fromM−,+ to
Modd−,+ := u ∈M−,+ : u(−x1) = −u(x1).
To fix ideas, we let G(u) = 14(u2 − 1)2. Our result here is the following
Theorem 10. The Ginzburg-Landau energy functional (19.4), defined on Modd−,+, has a
minimizer.
Proof. We identify M−,+ with the space M−,+ = χ + H10 (R), where χ is a smooth (and
odd) function satisfying χ(x) = 1 for x ≥ δ and χ(x) = −1 for x ≤ −δ. Then, sinceH1
0 (R) is weakly closed, Modd−,+ is weakly closed in H1(R).
Now, we would like to show that E on H1g (R) is w.l.s.c. and coercive, and therefore
on Modd−,+. Since G ≥ 0, the w.l.s.c. follows from Corollary 4.
The coercivity of E(u) on M(g)α is a more subtle question. Define v := χ− u ∈ H1
0 (R).We write
G(u) =1
4(1− χ2)2 − 2(1− χ2)v
(χ− 1
2v
)+ v2
(χ− 1
2v
)2
.
Assume |v| ≤ 1 for |x| ≥ δ. Then using that χ(x) = ±1 for |x| ≥ δ, we find G(u) ≥ 14v2.
This gives∫G(u)dx ≥ 1
4
∫|x|≥δ v
2dx ≥ 14
∫R v
2dx− Cδ. Therefore
E(1− v) ≥ 1
4
∫R(|∇v|2 + v2)dx− C ≡ 1
4‖v‖2
H1 − C.
Lectures on Applied PDEs 189
This implies the coercivity of E on H1g (R).
Now, by the properties established above and the key theorem on minimization, E(u)has a minimizer u∗ on the set
Modd−,+ := u ∈Modd
−,+ | |χ− u| ≤ 1.
Is this minimizer also a minimizer on Modd−,+? Observe that
E(u) ≥ E(max(min(u, 1),−1)).
umaxminu
+1
-1
Indeed, one can see that replacing the function u by the function max(min(u, 1),−1)decreases both gradient and potential term. Hence, if un is a minimizing sequence, thenso is wn := max(min(un, 1),−1) and the minimizer, w∗, satisfies |w∗| ≤ 1. Similarly, wecan show that
E(u) ≥ E(w), w := |u|χx≥0 − |u|χx<0.
So we have w∗ ≥ 0 for x ≥ 0 and w∗ ≤ 0 for x ≤ 0. The last two properties imply|χ − w∗| ≤ 1. Hence a minimizer on Modd
−,+ belongs to and therefore Modd−,+ is also a
minimizer on Modd−,+.
The minimizers w∗(x) are called kinks and w∗(−x), anti-kinks. In what follows, wedenote them by χ(x) are and χ(−x). They describe planar interfaces. Of course, byshifting χ(x) to χ(x − h), we obtain a one–parameter family of minimizers, the kinkscentred at different points of R. The kinks are solutions the Euler - Lagrange equationfor (19.4), i.e. (19.3), with µ = 0, the static Allen - Cahn equation,
−∆χ+ λG′(χ) = 0. (19.5)
Lamellar phase. In this situation, layers of oil and water coexist in a periodic array.To get a solution to (19.3), glue together a kink at z1 and an antikink at z2. There isno exact solution of this form: the kink and the antikink interact at any distance. Theyattract each other and as a result, they move toward each other and collapse. This meansthat if ϕ is a function consisting of a kink and an antikink glued together at a distance R,then E(ϕ) is monotonically increasing as a function of R. Here, ϕ is a function consisting
190 Lectures on Applied PDEs
of a kink and an antikink glued together at a distance R. However, presumably one canconstruct a periodic solution corresponding to an array of kinks and antikinks.
C
CCCCC
CCCCCC
C
CCCCC
b
a
Idea of proving the existence of the lamellar solutions. Consider the variational prob-lem of minimizing the functional
E(u) =
∫ c
−c
(1
2|∇u|2 + λG(u)
)dx, (19.6)
under constraint that∫ c−c udx = α, i.e. on the space u ∈ H1([−c, c]) : u(±c) =
1,∫ c−c udx = α, for α ∈ (−2c, 2c). For α = 2c, the minimizer is trivial, u∗ = 1, and
for α = −2c, there is no minimizer. The parameters a and b satisfy a + b = c and theirration is determined by α. Then the minimizer u∗ is extended periodically to the entirereal line.
Spherical drops and cylinder solutions. Here Ω is either the ball, BR, of radius R,centered at the origin, or the cylinder, CR of radius R or Rd, d = 2, 3. We minimize theenergy functional E , as was described above, on the set
M (g)α := u ∈ H1
g (Ω) |∫
Ω
(1− u) = α
where g ≡ 1 on ∂B, for some α > 0.
Theorem 11. Let G(u) = 14(u2 − 1)2 and Ω be bounded. The Ginzburg-Landau energy
functional E(u) defined on M(g)α , α > 0 and g ≡ 1, has a minimizer.
The proof of this theorem is similar to the one of Theorem 10.
Exercise 8. Check that the set M(g)α is weakly closed and that the functional E(u) defined
on M(g)α is w.l.s.c.
Lectures on Applied PDEs 191
One can show that this minimizer, u∗, is smooth. What is the shape of u∗? Is itspherically symmetric? One can show that the minimizer must have exactly one zero, i.e.,it is of the second type as shown on the figure below.
C
CCCCC
C
CCCCC
or
\\
+1
-1
+1
-1 R
The parameter R - the radius of the drop - is found from the condition (19.2).One can also show that u∗ is spherically symmetric. If one does not want to work
hard, then one can look directly for spherically symmetric minimizers, i.e. minimizers ofE(u) of the form
u(x) = ψ(|x|), (19.7)
subject to the side condition (19.2). But then one would have only minimizer amongspherically symmetric functions.
19.2 Vortices in Superfluids
Macroscopically, equilibrium states of superfluids and Bose-Einstein condensates are de-scribed by a function ψ : Ω → C, Ω ⊂ Rd called the order parameter, which satisfies thenonlinear differential equation
−∆ψ + (|ψ|2 − 1)ψ = 0 in Ω (19.8)
called the Gross-Pitaevskii or Ginzburg-Landau equation, with the boundary condition
|ψ| → 1 as |x| → ∂Ω. (19.9)
In 1958, V.L Ginzburg and L. Pitaevskii conjectured that for d = 2 and Ω = R2 (thecommon ‘cylindrical’ geomemtry) this equation has solutions of the form
ψn(x) = fn(r)einθ (19.10)
where (r, θ) are polar coordinates of x ∈ R2 and n is an integer. It was conjecturedthat these solutions describe vortices observed in superfluids. (In 1947, L. Onsager hasconjectured that these vortices are analogous to normal fluid vortices, except for the factthat the vortices of the superfluid were quantized while normal vortices were arbitrary.)
The rigorous result about existence and uniqueness of vortices came much later andis stated in the following
192 Lectures on Applied PDEs
Theorem 12. Let d = 2 and Ω be either BR or R2. Then, for all n, there exists a solutionof form (19.10) unique up to symmetry transformation. The function fn(r) can be takento be positive and it vanishes at r = 0 as
fn(r) = arn
for some a > 0 and is monotonically increasing to 1.
1
xr
fnpsin
Portraits of vortices:
s1s1
s2 nt
We prove this theorem by reformulating the b.v problem (19.8) - (19.9) as a variationalproblem of finding a minimizer of an appropriate functional. For a bounded domain Ω,(19.8) is the Euler-Lagrange equation for the functional
E(ψ) =
∫Ω
1
2|∇ψ|2 +
1
4(|ψ|2 − 1)2,
called the Gross-Pitaevskii or Ginzburg-Landau energy functional, on the space
M = ψ ∈ H1g (Ω) |
∫Ω
(|ψ|2 − 1)2 <∞
for some function satisfying |g| = 1. For Ω = BR, we choose
g(x) = einθ
Lectures on Applied PDEs 193
Renormalized Ginzburg-Landau functional. For d = 2 and Ω unbounded, say,Ω = R2, E(ψ) = ∞ for functions ψ of interest. We explain this more carefully for themost important case of Ω = R2.
For C1 complex functions (vector-fields) ψ satisfying the boundary condition |ψ| → 1as |x| → ∞, we define the degree by
deg ψ =1
2π
∫|x|=R
d(argψ)
for R sufficiently large. Here arg ψ = 1i
ln(ψ/|ψ|).
Exercise 9. Show that if f(r)→ 1 as r →∞, then deg(f(r)einθ) = n.
It is shown in [?] that if ψ : R2 → C is a C1 vector-field such that |ψ(x)| → 1 as|x| → ∞ and degψ 6= 0, then E(ψ) = ∞. Thus the Ginzburg-Landau energy functionalE(ψ) is not defined in infinite domains in the most interesting cases (i.e., in the presenceof vortices). To go around this problem, [?] have introduced the renormalized Ginzburg-Landau energy functional defined as follows
Eren(ψ) =1
2
∫|∇ψ|2 − (degψ)2
r2χ+
1
2(|ψ|2 − 1)2
where χ is a cut-off function with the properties: χ ∈ C∞(R2) and
χ(x) =
1 for |x| ≥ 2,0 for |x| ≤ 1.
It is shown in [?] that Eren(ψ) is defined on a large class of functions which include n–vortices and their combinations and that
dEren(ψ) = −∆ψ + (|ψ|2 − 1)ψ,
i.e. Eren(ψ) produce the correct Euler-Lagrange equations (19.8). However, as should be,the functional Eren(ψ) is not bounded below for |deg ψ| ≥ 2. Indeed, consider a functionψ describing n := deg ψ ≥ 2 single vortices. Then moving these vortices apart to infinitywould decrease Eren(ψ) indefinitely. The energy functional Eren(ψ) is suitable and naturalfor the variational approach to the vortex problem in an unbounded domain.
Now, we can state the following variational problem: Minimize Eren(ψ) among func-tions ψ with a fixed degree deg ψ = n. This the variational problem with the topologicalconstraint.
For simplicity, we prove the theorem only in the case Ω = BR. A proof in the caseΩ = R2 uses the renormalized Ginzburg-Landau energy and can be found in [?]. We provea somewhat stronger statement.
194 Lectures on Applied PDEs
Proof of Theorem 12 for Ω = BR. We minimize the functional E(ψ) on the functions ofthe form ψ = f(x)einθ for some fixed n, where f(x) is a real valued function satisfyingthe boundary condition f(x) = 1 for |x| = R. In other words, we minimize the functional
En(f) := E(feinθ)
on a space of functions f(r) satisfying f(x) = 1 for |x| = R.In what follows, we abbreviate BR = B. Write out this functional explicitly:
En(f) =
∫B
(1
2|∇f |2 +
n2
2r2f 2 +
1
4(f 2 − 1)2
where we have used that∇(feinθ) = (∇f + in∇θ)einθ,
and so for f real|∇(feinθ)|2 = |∇f |2 + n2|∇θ|2f 2,
and that |∇θ| = 1/rn.
Lemma 3. For all n, en(f) has a minimizer on the set
M = f real and En(f) <∞.and all minimizers are radially symmetric.
Proof. Let ξ = 1−f and define the functional F (ξ) = en(1−ξ). Explicitly this functionalis given as
F (ξ) =
∫B
1
2|∇ξ|2 +G(ξ)
where
G(ξ) =n2
r2(1− ξ)2 + ξ2(1− 1
2ξ)2.
Observe that ξ → 0 as |x| → R. We consider F (ξ) on the set
M = ξ ∈ H10 (B) | ξ real, G(ξ) <∞.
Since G(ξ) ≥ 0, we have that F (ξ) is w.l.s.c. on H10 (B). The functional F (ξ) is also
coercive by the difficult part of Theorem ?? One can avoid using this difficult part (whichfollows if B = R2) proceeding as follows: we have
G(ξ) ≥ 1
4ξ2 if ξ ≤ 1
and therefore F (ξ) is coercive under the additional condition that ξ ≤ 1. Now we canshow that we can make this condition automatically satisfied for minimizing sequences.Indeed, let ξn be a minimizing sequence for the functional F (ξ). Define a new sequenceξ′n(x) := min(ξn(x), 1). Clearly,
|∇ξn| ≥ |∇ξ′n| and G(ξn) ≥ G(ξ′n)
Lectures on Applied PDEs 195
xnp
1
xn
Exercise 10. Check these statements.
Hence F (ξn) ≥ F (ξ′n) so that ξ′n is also a minimizing sequence. Of course, ξ′n ≤ 1so F (ξ) is coercive on this sequence and the proof of the Key Theorem can be modifiedso as to show that the functional F (ξ) has a minimizer on the set M .
Let now f0 be a minimizer of the functional En(f). We show that f0 must be radiallysymmetric. Introduce the function
u = (f 20 )
12 ,
where g(r) :=∫ 2π
0g(r, θ) dθ
2π. Then f 2
0 = u2 and
f 40 ≥ u4 and |∇rf0|2 ≥ |∇ru|2. (19.11)
Indeed, the first of these inequalities follows from the Cauchy-Schwartz inequality and thesecond is obtained as follows: for r such that u(r) 6= 0,∣∣∣∣∣∇r
(∫ 2π
0
f 20 (r, θ)
dθ
2π
)1/2∣∣∣∣∣2
=
∣∣∣∫ 2π
0f0∇rf0
dθ2π
∣∣∣2∫ 2π
0f 2
0dθ2π
≤∫ 2π
0f 2
0dθ2π
∫ 2π
0|∇rf0|2 dθ2π∫ 2π
0f 2
0dθ2π
by the Cauchy-Schwartz inequality again. Inequality (19.11) implies that∫|∇f0|2 d2x ≥
∫ ∞0
|∇ru|2 rdr
with the equality taking place only if f0 is independent of θ. Hence
En(f0) ≥ En(u)
and the equality holds only if f0 is radially symmetric. Since f0 is a minimizer, we canconclude that it is radially symmetric. We omit the proof of monotonicity of f0 and referthe reader to [?] for this proof.
Since f0 is a minimizer of En(f) and is radially symmetric, it satisfies the Euler-Lagrange equation
−∆rf +n2
r2f + (f 2 − 1)f = 0 in Ω,
196 Lectures on Applied PDEs
where ∆r is radial Laplacian in R2 : ∆r = 1r∂r(r
∂∂r
). Since f0 is radially symmetric, wehave that ∇f0 · ∇θ = 0. This together with the relations ∆θ = 0 and |∇θ| = 1/r implies
∆(f0einθ) = (∆rf0 −
n2
r2f0)einθ.
Therefore, the function ψn(x) = f0(r)einθ satisfies Ginzburg-Landau equations (19.8) and(19.9).
19.3 The Ginzburg-Landau Equations
(See also Section 14, we repeat some definition from that section.) The Ginzburg-Landau theory gives a macroscopic description of superconducting materials and servesas an integral and paradigmatic part of particle physics. It is formulated in terms of apair (Ψ, A) : Rd → C× Rd, d = 1, 2, 3, satisfying the system of nonlinear PDE called theGinzburg-Landau equations:
−∆AΨ = κ2(1− |Ψ|2)Ψcurl2A = Im(Ψ∇AΨ)
(19.12)
where ∇A = ∇ − iA, and ∆A = ∇2A, the covariant derivative and covariant Laplacian,
respectively, and κ > 0 is a parameter, coupling constant. For d = 2, curlA := ∂1A2−∂2A1
is a scalar, and for scalar B(x) ∈ R, curlB = (∂2B,−∂1B) is a vector.
Superconductivity. The complex-valued function Ψ(x) is called an order parameter, |Ψ(x)|2gives the local density of (Cooper pairs of) superconducting electrons, and the vector fieldA(x)is the magnetic potential, so that B(x) := curlA(x) is the magnetic field. The param-eter κ > 0 is called the Ginzburg-Landau parameter, it depends on the material propertiesof the superconductor. The vector quantity J(x) := Im(Ψ∇AΨ) is the superconductingcurrent. (See eg. [33, 32]).
Particle physics. In the Abelian-Higgs model, ψ and A are the Higgs and U(1) gauge(electro-magnetic) fields, respectively. Geometrically, one can think of A as a connectionon the principal U(1)-bundle Rd × U(1), d = 2, 3.
Cylindrical geometry. In the commonly considered idealized situation of a superconductoroccupying all space and homogeneous in one direction, one is led to a problem on R2 andso may consider Ψ : R2 → C and A : R2 → R2. This is the case we deal with in thiscontribution.
Symmetries of the equations The Ginzburg-Landau equations (19.12) admit severalsymmetries, that is, transformations which map solutions to solutions.
Gauge symmetry: for any sufficiently regular function γ : R2 → R,
T gaugeγ : (Ψ(x), A(x)) 7→ (eiγ(x)Ψ(x), A(x) +∇γ(x)); (19.13)
T rotρ : (Ψ(x), A(x)) 7→ (Ψ(ρ−1x), ρ−1A((ρ−1)Tx)), (19.15)
One of the analytically interesting aspects of the Ginzburg-Landau theory is the factthat, because of the gauge transformations, the symmetry group is infinite-dimensional.
Abrikosov lattices In 1957, A. Abrikosov discovered a class of solutions, (Ψ, A), to(14.1), presently known as Abrikosov lattice vortex states (or just Abrikosov lattices),whose physical characteristics, density of Cooper pairs, |Ψ|2, the magnetic field, curlA,and the supercurrent, JS = Im(Ψ∇AΨ), are double-periodic w.r. to a lattice L. By alattice we understand here the Bravais lattice, i.e.
L = mν1 + nν2 : m,n ∈ Z,
where (ν1, ν2) is a basis of L.Denote by Ω a fundamental cell of the lattice L, say xν1 + yν2 : x, y ∈ [0, 1), where
(ν1, ν2) is a basis of L. For Abrikosov states, for (Ψ, A), the magnetic flux,∫
ΩcurlA,
through a lattice cell, Ω, is quantized,
Lemma 19.1.1
2π
∫Ω
curlA = deg Ψ = n, (19.16)
for some integer n.
Proof. The periodicity of ns = |Ψ|2 and J = Im(Ψ∇AΨ) imply that ∇ϕ − A, whereΨ = |Ψ|eiϕ, is periodic, provided Ψ 6= 0 on ∂Ω. This, together with Stokes’s theorem,∫
ΩcurlA =
∮∂ΩA =
∮∂Ω∇ϕ and the single-valuedness of Ψ, imply that
∫Ω
curlA = 2πnfor some integer n.
Using the reflection symmetry of the problem, one can easily check that we can alwaysassume n ≥ 0.
Abrikosov lattices as gauge-equivariant states. We say a state (Ψ, A) is gauge- equivariant (with respect to a lattice L, or L−equivariant) iff there exists (possiblymultivalued) function ηs : R2 → R, s ∈ L, such that
T transs (Ψ, A) = T gauge
ηs (Ψ, A). (19.17)
A key point in proving both theorems is to realize that a state (Ψ, A) is an Abrikosovlattice if and only if (Ψ, A) is gauge - equivariant.
198 Lectures on Applied PDEs
Lemma 19.2. A state (Ψ, A) is an Abrikosov lattice if and only if (Ψ, A) is gauge-equivariant.
Proof. If state (Ψ, A) satisfies (19.17), then all associated physical quantities are L−periodic,i.e. (Ψ, A) is an Abrikosov lattice. In the opposite direction, if (Ψ, A) is an Abrikosovlattice, then curlA(x) is periodic w.r.to L, and therefore A(x + s) = A(x) + ∇ηs(x),for some functions ηs(x). Next, we write Ψ(x) = |Ψ(x)|eiφ(x). Since |Ψ(x)| and J(x) =|Ψ(x)|2(∇φ(x)−A(x)) are periodic w.r.to L, we have that so is ∇φ(x)−A(x), which, to-gether with the relation A(x+s) = A(x)+∇ηs(x), gives that∇φ(x+s) = ∇φ(x)+∇ηs(x),which implies that φ(x+ s) = φ(x) + ηs(x) + cs, for some constants cs.
Since T transs is a commutative group, we see that the family of functions gs has the
important cocycle property
ηs+t(x)− ηs(x+ t)− ηt(x) ∈ 2πZ. (19.18)
This can be seen by evaluating the effect of translation by s+ t in two different ways. Wecall gs(x) the gauge exponent.
Ginzburg-Landau energy. The Ginzburg-Landau equations (19.12) are the Euler-Lagrange equations for critical points of the Ginzburg-Landau energy functional (writtenhere for a domain Q ∈ R2)
EQ(Ψ, A) :=
∫Q
|∇AΨ|2 + (curlA)2 +
κ2
2(|Ψ|2 − 1)2
. (19.19)
Superconductivity: In the case of superconductors, the functional E(ψ,A) gives the dif-ference in (Helmholtz) free energy (per unit length in the third direction) between thesuperconducting and normal states, near the transition temperature.
Particle physics: In the particle physics case, the functional E(Ψ, A) gives the energy ofa static configuration in the U(1) Yang-Mills-Higgs classical gauge theory.
Existence of Abrikosov lattices by variational techniques
Theorem 19.3. For any lattice L, there exists a smooth Abrikosov lattice solution u∗ =(Ψ∗, A∗) (i.e. satisfying (19.17)) for this L.
Proof. We want to solve the Ginzburg-Landau equations (19.12) for functions satisfyingthe condition (19.17). Notice that these equations are the Euler-Lagrange equations forthe energy functional EQ(Ψ, A), see (19.19), on the space H1(Ω), satisfying (19.17).
We say two states u′ and u are gauge-equivalent, iff there is differentiable real functionχ s.t. u′ = T gauge
χ u. We begin with the following statement whose proof can be found in[34]
Lectures on Applied PDEs 199
Lemma 19.4. Any L-equivariant state, (Ψ′, A′), is gauge-equivalent to a L-equivariantstate, (Ψ, A), satisfying divA = 0.
Now, we identify the quotient R2/L with a fundamental cell, Ω, of the lattice L andintroduce the space
H1equiv(Ω) := u ∈ H1
loc(R2) : u satisfies (19.17) and divA = 0,
equipped with the norm ‖u‖H1equiv(Ω) := ‖u‖H1(Ω). The space H1
equiv(Ω) is weakly closed in
H1(Ω). Indeed, if un → u∗ weakly in H1(Ω), then
un → u∗ pointwise on Ω. (19.20)
Using that R2 = ∪s∈L(Ω + s), we define u∗ on R2 by
u∗(x+ s) := es(x)u∗(x), ∀x ∈ Ω, s ∈ L, (19.21)
where es(x) := T gaugegs . By (19.17), es(x) obeys et(x+ s)es(x) = es+t(x). Since
which can be rewritten as u∗(y + t) = et(y)u∗(y), ∀y ∈ Ω + s, s, t ∈ L, we see that u∗satisfies (??). The equations (19.20) and (19.21) imply un(x) → u∗(x) pointwise on R2.Moreover, since un → u∗ weakly in H1(Ω) and divAn = 0, we have that divA∗ = 0. Thusu∗ ∈ H1
equiv(Ω).Now, we analyze EΩ(Ψ, A). Since divA = 0, (|Ψ|2−1)2 = |Ψ|4−2|Ψ|2 + 1 ≥ 1
2|Ψ|4−1
and Ω has a finite area, |Ω| <∞, the energy EΩ(Ψ, A) is obviously coercive,
EΩ(Ψ, A) ≥∫
Ω
|∇AΨ|2 + (curlA)2 +
κ2
4|Ψ|4
− κ2
2|Ω|
≥ c‖(Ψ, A)‖2H1(Ω) − C|Ω|, (19.22)
for come constants c, C > 0. Finally, since G = κ2
2(|Ψ|2−1)2 ≥ 0, the w.l.s.c. follows from
Corollary 4. Hence EΩ(Ψ, A) has a minimizer on the space H1equiv(Ω) and this minimizer
satisfies the Ginzburg-Landau equations (19.12).
200 Lectures on Applied PDEs
20 The Keller-Segel Equations of Chemotaxis
Chemotaxis is the directed movement of organisms in response to the concentration gra-dient of an external chemical signal and is common in biology. The chemical signals cancome from external sources or they can be secreted by the organisms themselves. Thelatter situation leads to aggregation of organisms and to the formation of patterns.
Chemotaxis is believed to underly many social activities of micro-organisms, e.g. socialmotility, fruiting body development, quorum sensing and biofilm formation. A classicalexample is the dynamics and the aggregation of Escherichia coli colonies under starvationconditions [?]. Another example is the Dictyostelium amoeba , where single cell bacteri-vores, when challenged by adverse conditions, form multicellular structures of ∼ 105 cells[?, ?]. Also, endothelial cells of humans react to vascular endothelial growth factor toform blood vessels through aggregation [?].
Consider organisms moving and interacting in a domain Ω ⊆ Rd, d = 1, 2 or 3.Assuming that the organism population is large and the individuals are small relative tothe domain Ω, Keller and Segel derived a system of reaction-diffusion equations governingthe organism density ρ : Ω × R+ → R+ and chemical concentration c : Ω × R+ → R+.The equations are of the form
∂tρ = Dρ∆ρ−∇ (f(ρ)∇c)∂tc = Dc∆c+ αρ− βc. (20.1)
Here Dρ, Dc, α, β are positive functions of x and t, ρ and c, and f(ρ) is a positivefunction modelling chemotaxis (positive chemotaxis). Assuming a closed system, one isled to impose no flux boundary conditions on ρ and c:
∂νρ = 0 and ∂νc = 0 on ∂Ω, (20.2)
where ∂νg is the normal derivative of g.The equations (20.1) have the family of homogeneous static solutions, (ρ∗, c∗) : αρ∗ −
βc∗ = 0. A simple calculation shows that these solutions are linearly unstable: thelinearized operator has two spectral bands, one positive and the other negative (they areseparated by the gap). Due to the positive chemotaxis in the system, one expects thatthe system evolves to a non-uniform state describing organism aggregation (the organismssecrete the chemical and move towards areas of higher chemical concentration). One refersto this process as (chemotactic) collapse.
In this case, the density may become infinite and form a Dirac delta singularity. Moreprecisely, we say that a solution ρ(x, t) undergoes collapse at a point x0 ∈ Rd in finitetime T <∞ if it exists for 0 ≤ t < T and
limt↑T
∫|x−x0|≥ε
ρ(x, t) dx = 0, ∀ε > 0.
Lectures on Applied PDEs 201
Since the total mass is conserved, this implies that limt↑T ρ(x0, t) =∞.The common approximations made in the literature for system (20.1) is based on the
fact that, in practically all situation, the coefficients in (20.1) are constant and satisfy
ε :=Dρ
Dc
1, α :=α
Dc
= O (1) and β :=β
Dc
1. (20.3)
The first of these conditions says that the chemical diffuses much faster than the organismsdo. As a result of this relation, one drops the ∂tc term in (20.1) (after rescaling timet → t/Dρ, this term becomes ε∂tc). Furthermore, one takes f(ρ) to be a linear functionf(ρ) = Kρ and neglects the term βc in (20.1) compared with αρ, as one expects that itwould not effect the blow-up process where ρ 1 (it is also small due to the last relationin (20.3)). These approximations, after rescaling, lead to the system
∂ρ
∂t= ∆ρ−∇ · (ρ∇c) ,
0 = ∆c+ ρ,(20.4)
with ρ and c satisfying the no-flux Neumann boundary conditions. Equation (20.4) is thesimplest model of positive chemotaxis considered in the literature. This is the equationstudied in these lectures. We note that Eq. (20.4) in three dimensions also appear inthe context of stellar collapse (see [?, ?, ?, ?]); similar equations—the Smoluchowski ornonlinear Fokker-Planck equations—model non-Newtonian complex fluids (see [?, ?, ?, ?].
20.1 Properties Keller-Segel equations
Equations (20.4) have the following properties
• (20.4) preserves positivity: if the initial condition ρ0(x) is positive, then so is ρ(x, t)(by the maximum principle).
• (20.4) is scaling invariant, that is, if a pair ρ(x, t) and c(x, t) is a solution to (20.4),then for any λ > 0 so is the pair
1
λ2ρ
(1
λx,
1
λ2t
)and c
(1
λx,
1
λ2t
). (20.5)
• With the no flux boundary conditions, the total number of organisms in Ω is con-served: ∫
Ω
ρ(x, t) dx =
∫Ω
ρ(x, 0) dx.
• (20.4) is a gradient system, ∂tρ = −gradF(ρ), with the energy
F(ρ) =
∫R2
[− 1
2ρ(−∆)−1ρ+ ρ ln ρ
]dx. (20.6)
and the metric 〈v, w〉J := 〈v, J−1w〉L2 , where J := −∇ · ρ∇ > 0. In particular,
202 Lectures on Applied PDEs
• the functional F(ρ) decreases under the evolution and
• its critical points under the constraint that∫ρ = M = const are static solutions of
(20.4).
(The first term of F(ρ) can be thought of as the internal energy of the system andthe remaining terms are the entropy.)
• For d = 2, the system (20.4) has the radially symmetric static solution, R(r), wherer = |x|, with the total mass, M =
∫R(|x|)d2x = 8π, it is given explicitly by
R(r) :=8
(1 + r2)2. (20.7)
To prove that ∂tρ = −gradF(ρ), we compute the formal Gateaux derivative dF(ρ)φ =∫(−∆−1 ρ + ln ρ)φ, and therefore the gradient in the metric 〈v, w〉J := 〈v, J−1w〉L2 is
which is the negative of the r.h.s. of the first equation in (20.4) with c = −∆−1ρ. Hencethe equation (20.4) can be written as ∂tρ = −gradF(ρ).
Under this scaling, the total mass changes as∫1
λ2ρ
(1
λx, t
)= λ(d−2)
∫ρ (x, t) .
Thus one does not expect collapse for d = 1 and that collapse is possible for d ≥ 2 withcritical collapse for d = 2 and supercritical collapse for d > 2. (Equation (20.4) in d = 2is said to be L1−critical, etc.)
By the scaling invariance, for d = 2, the system (20.4) has the one-parameter familyof radially symmetric static solutions, Rλ(r) = 1
λ2R( 1
λr).
Existence vs blowup dichotomy. The dimension of the interest for us is the criticaldimension d = 2. Recall that for d = 2, system (20.4) has a radially symmetric staticsolution (20.7). We note that
∫R2 Rdx = 8π. This mass turns out to be the threshold
separating a regular behavior and a breakdown of the solution. It is shown that for theinitial condition ρ0 ≥ 0,
• (Blanchet, Dolbeault, Perthame) If the initial total mass satisfies M :=∫R2 ρ0 dx ≤
8π, then the solution to (20.4) exists globally and, for M :=∫R2 ρ0 dx < 8π, con-
verges to 0, as t→∞;
• (Biler) If the initial total mass satisfies M :=∫R2 ρ0 dx > 8π, then the solution to
(20.4) blows up in finite time.
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We sketch some key ideas in proving the first statement. To prove the global existence,we have to control some appropriate positive quantity, like a Sobolev norm. In the presentcase, this is the entropy,
∫ρ ln ρ.
If the blowup takes place along the family, Rλ(r) = λ2R(λr), of static solutions withλ → ∞, then the entropy would blow up,
∫ρλ ln ρλ =
∫ρ ln ρ + 2 lnλ
∫ρ → ∞ as
λ→∞, for any ρ with finite entropy and the total mass.Hence, if the entropy stays bounded during the evolution, this would indicate that ρ
does not blow up by the compression as in ρλ, λ → ∞. To check this, we start withcomputing the change in the entropy
∂t
∫ρ ln ρ = −4
∫|∇√ρ|2︸ ︷︷ ︸
entropy dissipation
+
∫ρ2︸︷︷︸
entropy production
. (20.8)
Depending on whether the entropy dissipation or the entropy production wins we expecteither dissipation of the solution or the collapse (blowup). The Nirenberg - Gagliardoinequality,
‖f‖24 ≤ cgn‖∇f‖2‖f‖2
shows that the dissipation wins if Mc2gn ≤ 4.
To sharpen this result one uses that the free energy decreases together with the loga-rithmic Hardy-Littlewood-Sobolev inequality,∫
f ln f ≥ 1
(M/8π)
1
2
∫f(−∆)−1f − C(M),
where M :=∫f (the dimension n = 2) and C(M) := M(1 + log π − logM), which gives
the following lower bound on the free energy,
F(ρ) ≥ (1
(M/8π)− 1)
1
2
∫ρ(−∆)−1ρ− C(M).
Combining this inequality together with the fact that the free energy is decreases, F(ρ) ≤F(ρ0), one finds the bound on the entropy
(1−M/8π)
∫ρ ln ρ ≤ F(ρ0)− 1
4πC(M),
which is used to prove the global existence for M ≤ 8π.To obtain the control of ρ one uses, instead of F(ρ) ≤ F(ρ0), its quantatative version
(the free energy production or generalized Fisher information)
∂tF(ρ) = −∫ρ|∇ ln ρ−∇c|2 (20.9)
(this can be thought of as an entropy monotonicity formula).
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Remark (Hardy-Littlewood-Sobolev inequality.). The logarithmic Hardy-Littlewood-Sobolevinequality follows from the generalized Hardy-Littlewood-Sobolev inequality,∣∣∣∣∫
The equation (20.11) defines the weak Lp space, Lpw := h is measurable and ‖h‖w,p <∞. Indeed, we have ‖h‖w,p ≤ ‖h‖p, so that Lp ⊂ Lpw. This follows from the followingexpression
Virial relation. To see how the critical mass M∗ = 8π enters here, we consider thesecond moment of mass
W :=
∫R2
|x|2ρ(x, t) dx.
We have the following virial relation
∂tW = 4M(1− 1
8πM). (20.14)
If M > 8π, then the right hand side is constant and negative, and hence, W becomesnegative in finite time. When this happens we have a contradiction since W is by definitionalways positive (recall that if ρ0 ≥ 0 then ρ(t) ≥ 0 by the maximum principle). Thus, ifM > 8π, then the solution ρ exists only for a finite time (t < t∗, where t∗ is the point oftime when W vanishes).
This result tells us nothing about how the solution break down in finite time. Thelatter process is investigated in the subsequent sections.
20.2 Rescaling
Recall that (20.4) has the manifold of static solutions Mstat := λ−2R(|x + h|/λ) | λ >0, h ∈ R2. Assuming this manifold is stable, one can slide along it either in the directionλ→∞ (dissipation) or in the direction λ→ 0 (collapse).
Lectures on Applied PDEs 205
Our goal is to understand which scenario takes place. The first key step is to passto the reference frame collapsing with the solution, by introducing the adaptive (blowup)variables,
u(y, τ) = λ2ρ(x, t), c(x, t) = v(y, τ), where y = λ−1x and τ =
∫ t
0
λ−2(s) ds,
(20.15)where λ : [0, T ) → [0,∞), T > 0, is a positive differentiable function (compression ordilatation parameter). The advantage of passing to blowup variables is
(a) if the solution ρ blows up at a finite time T , we expect λ to adjust in such a waythat λ(t)→ 0, while u stays bounded, as t ↑ T , and similarly for the dispersion;
(b) in the case of the blowup in a finite time T , we expect that τ →∞, as t ↑ T , andtherefore the blowup time, T , gets eliminated from consideration (it is mapped to ∞).
Writing (20.4) in blowup variables, we find the equation for the rescaled mass function
∂τu = ∆u−∇ · (u∇v)− a∇ · (yu), (20.16)
where a := −λλ and, recall −∆v = u. Now, the blowup problem for (20.4) is mappedinto the problem of asymptotic dynamics of solitons for the equation (20.16). One canforget about λ and consider (20.16) as an equation for u and a and then find λ, givenλ(0) = λ0, according to the formula
λ2(t) = λ20 − 2
∫ t
0
a(s) ds. (20.17)
The equation (20.16) retains all but the symmetry properties of (20.4): it is (a) posi-tivity preserving (in fact, improving), (b) mass conserving (
∫u = ρ = const) and (c) is a
gradient system, ∂tu = −gradFa(u), with the energy
Fa(u) =
∫R2
[− 1
2u(−∆)−1u+ u lnu− a
2|y|2u
]dx. (20.18)
and the metric 〈v, w〉J := 〈v, J−1w〉L2 , where J := −∇ · u∇ > 0.
20.3 M > 8π
We are interested in behaviour of solutions for initial condition with M > 8π. We beginwith considering the linearized stability of the static solution R(|x|) of (20.4). ∗ ∗ ∗ Notethat the linearized analysis does not differentiate between M > 8π and M < 8π andtherefore, between the blowup and stability. It would have to be supplemented by addi-tional information. (Extending the virial relation (??) to the equation (20.16) points outat a possible connection between M and a. Namely, let W resc :=
∫R2 |y|2u(y, τ) dy. Then
proceeding as in the derivation of (??)nd using∫R2 |y|2∇ · (yu) dy = −2
∫R2 |y|2u(y, τ) dy,
we find ∂tWresc = 4M(1− 1
8πM) + 2aW resc.)
206 Lectures on Applied PDEs
Linearizing the r.h.s., F (u) := ∆u − ∇ · (u∇v), of (20.4) on R(|x|) and denotingL := −dF (R), we find
Lξ = −∆ξ −∇ ·(ξ∇∆−1R−R∇∆−1ξ
). (20.19)
The operator L is the hessian of F(u) at R defined on the space of the linearized constraint∫ξ = 0 and therefore it is self-adjoint in the inner product 〈v, w〉J := 〈v, J−1w〉L2 ,
where, recall, J := −∇ · u∇ > 0. By a standard technique (see e.g. [12]), one hasσess(L) = [0,∞).
Similarly, the rescaled equation (20.16), linearized around the rescaled stationary so-lution R(|y|), leads to the operator La := −dFa(R), where Fa(u) := ∆u − ∇ · (u∇v) −a∇ · (yu). Explicitly,
Laξ = Lξ + a∇ · (yξ), (20.20)
The operator La is the hessian of Fa(u) at R defined on the space of the linearizedconstraint
∫ξ = 0 and therefore it is self-adjoint in the inner product 〈v, w〉J :=
〈v, J−1w〉L2 . Since the operator −∆ − a∇ · y on the space L2(R2, e−a|y|2/2dy) is unitary
equivalent to −∆ + 14a2|y|2 − a on the space L2(R2, dy), its spectrum is purely discrete
with gaps of the order O(a).∗ ∗ ∗ The information about the total mass M (of the initial condition), i.e. whether
M < 8π or M > 8π, will enter through the choice of the modalities of the behaviour ofa: for M < 8π, we expect a < 0 and converges to a nonzero limit and for M > 8π, weexpect a > 0 and, for M close to 8π, a→ 0, as τ →∞.
For a < 0 (more precisely, a = 1), the operator La was studied in detail in a numberof papers, see e.g. [?, ?].
We address the case a > 0 and very small. In the case, we can think of La and a smallperturbation of L. So, first, we examine the operator L.
Since R(|x|) breaks the translational and scaling symmetries of (20.4), the operator Lhas the the translational and scaling zero modes
∂jR =32xj
(1 + |x|2)3and ∇ · (xR) =
34
(1 + |x|2)2− 32
(1 + |x|2)3.
Since L commutes with rotations, it can be decomposed into spherical harmonics (theFourier series in the polar angular variable θ) as
L = ⊕m≥0Lm,
where the operator Lm acts on the space L2([0,∞), rdr) of radial functions and can bewritten out explicitly. Then ϕ0 := ∇ · (xR) becomes the zero eigenfunction of L0 and
∂jR give the zero eigenfunction, ϕ1 := 32|x|(1+|x|2)3
, of L1. Though ϕ0 and ϕ1 are positive,since the operators Lm are non-local, we cannot use the Perron - Frobenius argument, toconclude that 0 is the lowest and simple eigenvalue of L0 and L1.
Lectures on Applied PDEs 207
Returning to the operator La, we see that it has 3 zero or almost zero eigenvaluesoriginating from the triple degenerate eigenvalue 0 of L. Note that, since L∂jR = 0, wehave
〈∂jR,La∂jR〉 = a〈∂jR,∇ · (y∂jR)〉 = a‖∂jR‖2.
Since La commutes with rotations, it can be decomposed into spherical harmonics (theFourier series in the polar angular variable θ) as
La = ⊕m≥0Lam,
where the operator Lam acts on the space L2([0,∞), rdr) of radial functions and is relatedto the operator Lm above as Lam = Lm + a(r∂r + 2).
For a > 0 and sufficiently small, the radially symmetric part, La0, was investigated in[?], where it was shown that it has the following spectrum in the interval . a:
• one negative eigenvalue −2a + aln 1a
+ O(a ln−2 1
a
)(corresponding to the scaling
mode—for a fixed parabolic scaling it is connected to possible variation of the blowuptime)
• one near zero eigenvalue (due to the shape instability),
• the third eigenvalue, 2a+ 2aln 1a
+ O(a ln−2 1
a
), is positive, but vanishing as a→ 0.
The paper [?] also isolated the correct perturbation (adiabatic) parameter— 1ln 1a
. Finally,
we record the more precise information about the eigenvalues obtained in [?]:
λn =
µ+a
ln 1a
+K+γ+ O
(a ln−3 1
a
)n = 0
2na+ 2aln 1a
+K+γ−Hn−1−µ+a2an
+ O(a ln−3 1
a
)n ≥ 1,
(20.21)
where K := ln 2−1−2γ (here γ = −Ψ(1) = 0.577216 . . . is the Euler-Mascheroni constant)and Hn :=
∑nk=1 1/k.
We proceed with the radially symmetric case and make comments about the generalcase later.
Modification of the leading term. Non-positive EVs La ⇒Mstat := 1λ2R(x/λ) | λ >
0 is unstable and we have to construct a one-parameter deformation of it. For technicalreasons it is convenient to use a two-parameter family, Rbc(|y|)
Rbc(|y|) :=8b
(c+ |y|2)2, (20.22)
with b > 1 and both parameters b and c are close to 1, with an extra relation between theparameters a, b and c. The family Rbc(|y|) gives a two-parameter family of approximate
208 Lectures on Applied PDEs
solutions to (20.16) (see (20.7)) and forms the deformation (or almost center-unstable)manifold M := Rbc(|y|/λ) | λ > 0, p.
Introduce the deformation (or almost center-unstable) manifold
It a key point that the tangent vectors ∂bRb(τ)c(τ) and ∂cRb(τ)c(τ), spanning the tangentspace toMstat deform at Rbc are the approximate eigenfunctions of the operator La0 corre-sponding to the negative and almost zero eigenvalues displayed above.
This manifold absorbs all unstable/neutral degrees of freedom. The previous resultgives the linear stability of Mstat deform.
Splitting the solution. We expect that the solution to the rescaled KS approachesthis manifold as τ →∞. Hence we decompose the solution u(y, τ) to the rescaled KS asthe leading term, Rb(τ)c(τ)(y), and the fluctuation, φ(y, τ),
u(y, τ) = Rb(τ)c(τ)(y)︸ ︷︷ ︸leading term, finite dim
+ φ(y, τ)︸ ︷︷ ︸fluctuation, infinite dim
, (20.24)
and require that the fluctuation φ(y, τ) is orthogonal to the tangent space of Mstat deform
at Rb(τ)c(τ)(y),〈∂b,cRb(τ)c(τ)(·), φ(·, τ)〉 = 0.
The leading term, Rb(τ)c(τ)(y), and the fluctuation, φ(y, τ), evolve on a different spatialscales, as Rbc can rewritten as Rbc(y) = R b
c4,1( y√
c).
Collapse dynamics. In parametrizing solutions as above, we split the dynamics of(20.4) into a finite-dimensional part describing motion over the manifold, M, and aninfinite-dimensional fluctuation (the error between the solution and the manifold approx-imation) which is supposed to stay small.
Substituting the splitting u = Rbc+φ into the rescaled KS and projecting the resultingequation ontoMstat deform and the orthogonal complement (to the tangent space) (this isthe Lyapunov-Schmidt decomposition we used extensively in the previous lectures), wearrive at the equations for parameters b and c cτ = 2a− 4(b−1)
ln( 1a
)+R(φ, a, b, c),
bτa
= −2(b−1)
ln( 1a
)+R(φ, a, b, c),
(20.25)
and for the fluctuation φ.Using the equation for the fluctuation φ and differential inequalities for Lyapunov func-
tionals, we estimate the remainders in (20.25) in the linear approximation as |R(φ, a, b, c)| .
Lectures on Applied PDEs 209
aln2( 1
a)
1ln( 1
a)[(b− 1)‖φ‖L2 + ‖(1 + |y|2)−1φ‖2
L2 ]. As a result, these equations give the differ-
ential equation for a
aτ = − 2a2
ln( 1a), (20.26)
with an error term O( a2
ln2( 1a
)), given an a priori estimate of φ.
We solve the above equation in the leading order. First, we have 1a
(ln 1
a+O(1)
)= 2τ
which results in ln 1a(τ)
= ln 2τ− ln ln 2τ+ ln ln 2τln 2τ
+O(
1ln 2τ
). Now, recalling that λ(t)λ(t) =
a(τ) and using that λ(t)−1λ(t) = λ(τ(t))−1∂τλ(τ(t)), we obtain, after some derivations,the law
λ(t) = (T − t)− 12 e|
12
ln(T−t)|12 (c1 + o(1)). (20.27)
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗
20.4 Discussion of static solutions
The static solutions of (20.4) are critical points of the energy functional F , given in (20.6),under the constraint that
∫ρ = M = const.. Thus, they satisfy E ′(ρ) = C, where C is a
constant. Explicitly E ′(ρ) = C reads
log(ρ) +1
∆ρ = C ⇔ ∆ log(ρ) + ρ = 0 ⇔ ∆u+ eu = 0, (20.28)
where u = log(ρ). Solutions to (20.28) can be written in the form of ’Gibbs states’ρ = M ec∫
ec, with the concentration c considered as a negative potential (remember that
∆c = −ρ). In two dimensions, this equation has the solution for M = 8π, given by (20.7).This solution is a minimizer of E under the constraint that
∫ρ = 8π.
The equations (20.4) have the family of homogeneous static solutions, (ρ∗, c∗) : ρ∗ =0, c∗ =constant. These solutions have no organisms present (zero total mass) and constantconcentration of the chemoattractant. A simple calculation shows that the linearizedoperator has the spectrum filling in the positive semi-axis. This indicates that thesestatic solutions might be stable. However, any initial condition ρ0 ≥ 0, ρ0 is not identical0 will have positive mass and therefore, since the total mass is conserved, could convergeto (ρ∗, c∗) only in a weak sense with the loss of the mass.∗ ∗ ∗ ∗ ∗ ∗ ∗∗Consider the RKS with the degradation term for the chemical allowing for the ho-
mogeneous static solutions; investigate the stability of the latter and connection to theTuring instability.
210 Lectures on Applied PDEs
20.5 Appendix: Gradient formulation
The Keller-Segel models (20.1) and (20.4) are gradient systems. We begin by formulatinga normalized version of (20.1),
∂tρ = ∆ρ−∇ · (f(ρ)∇c)ε∂tc = ∆c+ ρ− γc, (20.29)
as a gradient system. This system is obtained from (20.1) by setting unimportant con-stants to 1.
Define the energy (or Lyapunov) functional
Ef (ρ, c) :=
∫Ω
1
2|∇c|2 − ρc+
γ
2c2 +G(ρ) dx, (20.30)
where G(ρ) :=∫ ρg(s) ds and g(ρ) :=
∫ ρ 1f(s)
ds. The L2-gradient of Ef (ρ, c) is
gradL2 Ef (ρ, c) =
(−c+ g(ρ)−∆c− ρ+ γc
),
and hence, if we define U = (ρ, c), then (20.29) can be written in the form ∂tU = IE ′f (U),where
I =
(∇ · f(ρ)∇ 0
0 −1ε
).
The operator I is non-positive and may be degenerate, however, assuming it is invertible,the operator I defines the metric 〈v, w〉I := −〈v, I−1w〉L2⊕L2 . In this metric, gradF(U) =−IF ′(U) and hence
∂tU = −grad Ef (U).
This shows that (20.29) has the structure of a gradient system. A consequence of this isthat the energy decreases on solutions of the KS system. Indeed, if f > 0, then
∂tEf (ρ, c) = −∥∥∥f(ρ)
12∇ (c− g(ρ) )
∥∥∥2
L2− 1
ε‖∆c+ ρ− c‖2
L2 .
The free energy F in (20.6) is obtained from (20.30) by dropping the quadratic term12c2, replacing c with −∆−1ρ in the remaining terms and using that f(ρ) = ρ.
20.6 Appendix 2: Criterion of break-down in the dimensiond ≥ 3
For d ≥ 3, we have the following result
Theorem 20.1. Take ρ0 ≥ 0. If d ≥ 3 and∫Rd x
2ρ0dx∫Rd ρ0dx
is sufficiently small (this means that
ρ0 is concentrated at x = 0), then the solution to (20.4) blows up in finite time.
Lectures on Applied PDEs 211
Proof. We now prove the case when d ≥ 3. We proceed as above and derive an upperbound on ∂tW which is negative for certain initial conditions. To this end we derive alower bound on J . Let γ = d− 2. We estimate M2 from above using J by first rewritingM2:
M2 =
∫R2d
u(x)u(y) dx dy =
∫R2d
(u(x)u(y)
|x− y|γ)α (
u(x)u(y)|x− y| αγ1−α
)1−αdx dy.
Holder’s inequality then gives
M2 ≤(∫
R2d
u(x)u(y)
|x− y|γ dx dy
)α(∫R2d
u(x)u(y)|x− y| αγ1−α dx dy
)1−α
.
The first integral is Jα. We choose αγ1−α = 2 or α = 2
γ+2so that
M2 ≤(∫
R4
u(x)u(y)
|x− y|γ dx dy
) 2γ+2(∫
R2d
u(x)u(y)|x− y|2 dx dy) γ
γ+2
= J2
γ+2
(∫R2d
u(x)u(y)|x− y|2 dx dy) γ
γ+2
.
Expanding the square we obtain that∫R2d
u(x)u(y)|x− y|2 dx dy =
∫R2d
u(x)u(y)(x2 + y2 − 2x · y
)dx dy
=
∫R2d
u(x)u(y)(x2 + y2 + 2|x||y|
)= 2WM + 2
(∫Rd|x|u(x) dx
)2
.
By Holder’s inequality again(∫Rd|x|u 1
2u12 dx
)2
≤∫Rdx2u dx
∫Rdu dx = WM.
Combining the estimates, one obtains that
M2 ≤ J2
γ+2 (4WM)γγ+2
orJ ≥ CM
γ+42 W− γ
2 .
Substituting into (5.18) gives that
∂tW ≤ 2(dM − CM γ+42 W
−γ2 ).
212 Lectures on Applied PDEs
Thus, if initially M/W (0) 1, then ∂tW is negative and W will decrease. Since itdecreases, the time derivative ∂tW becomes more negative and hence W continues todecrease and will becomes negative in a finite time (since the time derivative will alwaysbe less that the initial value of ∂tW ). This is again a contradiction since W is by definitionalways positive.
Lectures on Applied PDEs 213
21 PDEs of quantum mechanics and statistics
21.1 Hartree, Hartree - Fock and Gross-Pitaevski equations
Even for a few particles the Schrodinger equation is prohibitively difficult to solve. Henceit is important to have approximations which work in various regimes. One such approx-imation, which has a nice unifying theme and connects to a large areas of physics andmathematics, is the one approximating solutions of n-particle Schrodinger equations byproducts of n one-particle functions (i.e. functions of 3 variables). This results in a singlenonlinear equation in 3 variables, or several coupled such equations. The trade-off here isthe number of dimensions for the nonlinearity. This method, which goes under differentnames, e.g. the mean-field or self-consistent approximation, is especially effective whenthe number of particles, n, is sufficiently large.
For simplicity we consider a system of n identical, spinless bosons. It is straightforwardto include spin. To extend our treatment to fermions requires a simple additional step(see discussion below). The Hamiltonian of the system of n identical bosons of mass m,interacting with each other, is
Hn :=n∑j=1
(− ~2
2m∆xi) +
1
2
∑i 6=j
gv(xi − xj), (21.1)
acting on the state space sn1L
2(Rd), d = 1, 2, 3. Here ∆x is the Laplacian acting onthe variable x, g > 0 is a parameter called the coupling constant, v is the interactionpotential, and s is the symmetric tensor product. As we know, the quantum evolutionis given by the Schrodinger equation
i~∂Ψ
∂t= HnΨ.
This is an equation in 3n + 1 variables, x1, ..., xn and t, and it is not a simple matter tounderstand properties of its solutions.
The Hartree equation is the Euler-Lagrange equation for stationary points of the actionfunctional
S(Ψ) :=
∫ −~
2Im〈Ψ, ∂tΨ〉 −
1
2〈Ψ, HnΨ〉
dt,
considered on the set of functions
Ψ := ⊗n1ψ|ψ ∈ H1(R3).
Here (⊗n1ψ) is the function of 3n variables defined by (⊗n1ψ)(x1, ..., xn) := ψ(x1)...ψ(xn).The Euler-Lagrange equation for S(Ψ) on the set above is the following nonlinear equation
i~∂ψ
∂t= (h+ v ∗ |ψ|2)ψ. (21.2)
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This nonlinear evolution equation is called the Hartree equation (HE).If the interparticle interaction, v, is significant only at very short distances (one says
that v is very short range, which technically can be quantified by assuming that the“particle scattering length” a is small), we can replace v(x)→ 4πaδ(x), and Equation (??)becomes
i~∂ψ
∂t= hψ + 4πa|ψ|2ψ (21.3)
(with the normalization (A.9)). This equation is called the Gross-Pitaevski equation(GPE) or nonlinear Schrodinger equation. It is a mean-field approximation to the originalquantum problem for a system of n bosons. The Gross-Pitaevski equation is widely usedin the theory of superfluidity, and in the theory of Bose-Einstein condensation.
For (spinless) fermions, we consider the action S(Ψ) on the following function space
Ψ := det[ψi(xj)]|ψi ∈ H1(R3) ∀i = 1, ..., nwhere [ψi(xj)] stands for the n × n matrix with the entries indicated. Then the Euler-Lagrange equation for S(Ψ) on the latter set gives a system of nonlinear, coupled evolutionequations
i~∂ψj∂t
= (h+ v ∗∑i
|ψi|2)ψj −∑i
(v ∗ ψiψj)ψi, (21.4)
for the unknowns ψ1, ..., ψn. This systems plays the same role for fermions as the Hartreeequation does for bosons. It is called the Hartree-Fock equations (HFE).
Reconstruction of solutions to the n-particle Schrodinger equation. How do solutions of(HE) or (GPE) relate to solutions of the original many-body Schrodinger equation? Onecan show rigorously (see a review in [?]) that the solution of the Schrodinger equation
i~∂Ψ
∂t= HnΨ, Ψ|t=0 = ⊗n1ψ0
satisfies, in some weak sense and and in the mean-field regime of n→∞ and g → 0, withng fixed
Ψ−⊗n1ψ → 0
where ψ satisfies (HE) with initial condition ψ0.
(HE), (HFE) and (GPE) are invariant under the time shifts and the gauge transfor-mations,
ψ(x)→ eiαψ(x), α ∈ R.Consequently, the energy, E(ψ), and the number of particles, N(ψ), (see below) areconserved quantities.
To fix ideas, we will hereafter discuss mainly (GPE). For (HE) and (HFE) the resultsshould be appropriately modified. For (GPE) the energy functional is
E(ψ) :=
∫R3
~2
2m|∇ψ|2 + V |ψ|2 + 2πa|ψ|4
dx.
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The number of particles for (GPE) and (HE) is given by
N(ψ) :=
∫R3
|ψ|2dx
while for (HFE), by N(ψ) :=∑
i
∫R3 |ψi|2dx. Note that the energy and number of parti-
cle conservation laws are related to the time-translational and gauge symmetries of theequations, respectively.
Moreover, (HFE) is invariant under time-independent unitary transformations of ψ1, ..., ψn.As a result, (HFE) conserves the inner products, 〈ψi, ψj〉, ∀i, j.
The last item shows that the natural object for (HFE) is the subspace spanned byψi, and the equation can be rewritten as an equation for the corresponding projectionγ :=
∑i |ψi〉〈ψi|.
The above notions of the energy and number of particles are related to correspondingnotions in the original microscopic system. Indeed, let Ψ := 1√
n⊗n1 ψ. Then
〈Ψ, HnΨ〉 = E(ψ) +O( 1
n
)where E(ψ) is the energy for (HE) and
n
∫| Ψ(x1, ..., xn) |2 dx1...dxn =
∫| ψ(x) |2 dx.
The notion of bound state can be extended to the nonlinear setting as follows. Thebound states are stationary solutions of (HE) or (GPE) of the form
ψ(x, t) = φµ(x)eiµt
where the profile φµ(x) is in H2(R3). Note that the profile φµ(x) satisfies the stationaryGross-Pitaevski equation:
hφ+ 4πa|φ|2φ = −~µφ (21.5)
(we consider here (GPE) only). Thus we can think of the parameter −µ as a nonlineareigenvalue.
A ground state is a bound state such that the profile φµ(x) minimizes the energy fora fixed number of particles:
φµ minimizes E(ψ) under N(ψ) = n
(see Chapter ?? which deals with variational, and in particular minimization problems).Thus the nonlinear eigenvalue µ arises as a Lagrange multiplier from this constrainedminimization problem. In Statistical Mechanics µ is called the chemical potential (theenergy needed to add one more particle/atom, see Section ??).
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Remark 1. 1. Mathematically, the ground state can be also defined as a stationarysolution with a positive (up to a constant phase factor) profile, ψ(x, t) = φµ(x)eiµt
with φµ(x) > 0. Let δ(µ) := ‖φµ‖2. Then we have (see [?])
δ′(µ) > 0 =⇒ φµ minimizes E(ψ) under N(ψ) = n.
2. The Lagrange multiplier theorem in Section 6.4 implies that the ground state profileφµ is a critical point of the functional
Eµ(ψ) := E(ψ) + ~µN(ψ).
In fact, φµ is a minimizer of this functional under the condition N(ψ) = n.
If φµ is the ground state of (GPE), then⊗n1φµ is close to the ground state of the n−bodyHamiltonian describing the Bose-Einstein condensate (see [?] for a review, and [?, ?] andthe Appendix below for rigorous results).
It is known that for natural classes of nonlinearities and potentials V (x) there is aground state. Three cases of special interest are
1. h := − ~22m
∆ + V (x) has a ground state, and 2m~2 n|a| 1
2. V has a minimum, 2m~2 n|a| 1, and a < 0
3. V (x)→∞ as |x| → ∞ (i.e. V (x) is confining) and a > 0.
(The first and third cases are straightforward and the second case requires some work[?, ?, ?].)
Stability. We discuss now the important issue of stability of stationary solutions undersmall perturbations. Namely, we want to know how solutions of our equation with initialconditions close to a stationary state (i.e. small perturbations of φµ(x)) behave. Arethese solutions stay close to the stationary state in question, do they converge to it, or dothey depart from it? This is obviously a central question. This issue appeared implicitlyin Section ?? (and in a stronger formulation in Chapter ??) but has not been explicitlyarticulated yet. This is because the situation in the linear case that we have dealt with sofar is rather straightforward. On the other hand, in the nonlinear case, stability questionsare subtle and difficult, and play a central role.
We say that a stationary solution, φµ(x)eiµt, is orbitally (respectively, asymptotically)stable if for all initial conditions sufficiently close to φµ(x)eiα (for some constant α ∈R), the solutions of the evolution equation under consideration stay close (respectively,converge in an appropriate norm) to a nearby stationary solution (times a phase factor),φµ′(x)ei(µ
′t+β(t)). Here µ′ is usually close to µ, and the phase β depends on time, t. Thephase factors come from the fact that our equations have gauge symmetry: if ψ(x, t) is asolution, then so is eiαψ(x, t) for any constant α ∈ R. One should modify the statement
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above if other symmetries are present. The notion of orbital stability generalizes theclassical notion of Lyapunov stability, well-known in the theory of dynamical systems, tosystems with symmetries.
For the linear Schrodinger equation, all bound states, as well as stationary states corre-sponding to embedded eigenvalues, are orbitally stable. But they are not asymptoticallystable in general. For most nonlinear evolution equations in unbounded domains, themajority of states are not even Lyapunov/orbitally stable.
For (GPE), if V →∞ as |x| → ∞ (i.e. V is confining), the ground states are orbitallystable, but not asymptotically stable. If V → 0 as |x| → ∞, the ground states can beproved to be asymptotically stable in some cases (see [?, ?, ?, ?, ?, ?] and referencestherein).
21.2 Appendix II: Heuristic derivation of the Hartree equation
We give a heuristic derivation of the mean-field approximation for this equation. Arigorous derivation is scetched in [12]. First, we observe that the potential experiencedby the i-th particle is
W (xi) := V (xi) +∑j 6=i
v(xi − xj).
Assuming v(0) is finite, it can be re-written, modulo the constant term v(0), which weneglect, as W (xi) = V (xi) + (v ∗ ρmicro)(xi). Here, recall, f ∗ g denotes the convolution ofthe functions f and g, and ρmicro stands for the (operator of) microscopic density of then particles, defined by
ρmicro(x, t) :=∑j
δ(x− xj).
Note that the average quantum-mechanical (QM) density in the state Ψ is
〈Ψ, ρmicro(x, t)Ψ〉 = ρQM(x, t)
where ρQM(x, t) := n∫| Ψ(x, x2, ..., xn) |2 dx2...dxn, the one-particle density in the
quantum state Ψ.
In the mean-field theory, we replace ρmicro(x, t) with a continuous function, ρMF (x, t),which is supposed to be close to the average quantum-mechanical density, ρQM(x, t), andwhich is to be determined later. Consequently, it is assumed that the potential experiencedby the i-th particle is
WMF (xi) := V (xi) + (v ∗ ρMF )(xi).
Thus, in this approximation, the state ψ(x, t) of the i-th particle is a solution of thefollowing one-particle Schrodinger equation i~∂ψ
∂t= (h + v ∗ ρMF )ψ where, recall, h =
− ~22m
∆x + V (x). Of course, the integral of ρmicro(x, t) is equal to the total number of
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particles,∫R3 ρ
micro(x, t)dx = n. We require that the same should be true for ρMF (x, t):∫R3 ρ
MF (x, t)dx = n. We normalize the one-particle state, ψ(x, t), in the same way∫R3
|ψ(x, t)|2dx = n. (21.6)
Consider a situation in which we expect all the particles to be in the same state ψ. Thenit is natural to take ρMF (x, t) = |ψ(x, t)|2. In this case ψ solves the Hartree equation(21.2).
21.3 Quantum statistics
We formalize the theory of quantum statistics by making the following postulates (fordetails see [12]):
• States: positive trace-class operators on a Hilbert space H (as usual, up tonormalization);
• Evolution equation : i~∂ρ∂t
= [h, ρ], where h is a self-adjoint operator on H;
• Observables : self-adjoint operators on H;
• Averages : 〈A〉ρ := Tr(Aρ).
We call the theory described above quantum statistics. Assuming H = L2(Rn), the lasttwo items lead to the following expression for the probability density for the coordinates:
• ρ(x;x) - probability density for coordinate x;
and similarly for the momenta. In particular, if ρ = Pψ, then
ρ(x;x) = |ψ(x)|2,
as should be the case according to our interpretation.Note that the state space here is not a linear space but a positive cone in a linear
space. It can be identified with the space of all positive (normalized) linear functionalsA→ ω(A) := Tr(Aρ) on the space of bounded observables. Denote the spaces of boundedobservables and of trace class operators on H as L∞(H) and L1(H), respectively. Thereis a duality between density matrices and observables
〈ρ,A〉 = Tr(Aρ)
for A ∈ L∞(H) and ρ ∈ L1(H).Quantum mechanics is a special case of this theory, and is obtained by restricting the
density operators to be rank-one orthogonal projections.Another special case of quantum statistics is probability.
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21.4 Self-consistent approximation
Consider a system of n particles in an external potential V (x), interacting via a pairpotential v(x). Assuming the particles are identical and in the same state, in the self-consistent approximation, one replace the many body potential,
∑j v(x− xj) =
∫v(x−
y)nexact(y)dy = (v∗nexact)(x), where nexact(y) :=∑
j δ(x−xj), created by n−1 particles at
x and affecting the remaining one, by the ‘mean-field’ potential∑
j
∫v(x− y)nγ(y)dy =
(v ∗ nγ)(x). This leads to the time-dependent generalized Hartree or the Hartree-vonNeumann equation,
i∂γ
∂t= −[hγ, γ] (21.7)
withhγ = h+ v ∗ nγ and nγ(x) = γ(x, x). (21.8)
Here h = −∆+V acting H = L2(Rn). We describe some basic properties of this equation.If the external potential V is zero (or independent of time), then the equation (21.7)
is invariant, under spatial (or time) translations
T transh : γ 7→ UhγU
−1h , (21.9)
for any h ∈ Rd (space translations) or h ∈ R (time translations), and rotations
T rotρ : γ 7→ UργU
−1ρ , (21.10)
for any ρ ∈ SO(d). Here U translh and U rot
ρ are the standard translation and rotationtransforms U transl
h : φ(x) 7→ φ(x+ h) and U rotρ : φ(x) 7→ φ(ρ−1x).
Since the equation (21.7) is a hamiltonian system (see an appendix to this section),these symmetries lead to the conservation laws. In particular, we have
• Time translation invariance → conservation of energy,
E(γ) := Tr((h+
1
2v ∗ nγ)γ
)(21.11)
= Tr(hγ) +1
2
∫nγv ∗ nγdx. (21.12)
(The half compensates for the differentiating the quadratic term.)
• Unitarity of the Schrodinger evolution (gauge invariance) and the and cyclicity ofthe trace → conservation of number of particles (total charge),
Tr γ = const. (21.13)
• Unitarity of the Schrodinger evolution → conservation of positivity: if γ0 ≥ 0, thenfor all times γ ≥ 0,
γ0 ≥ 0 ⇒ γ ≥ 0. (21.14)
220 Lectures on Applied PDEs
• Unitarity of the Schrodinger evolution → conservation of the eigenvalues γj’s of γ,
∂tγj = 0. (21.15)
Indeed, for (21.13), we have
−i∂t Tr γ = Tr([hg, γ]
)= 0. (21.16)
To prove (21.14), we let U(γ) be the evolution generated by the self-adjoint, time-dependent operator hγ. We can rewrite the equation (21.7) as
γ = U(γ)∗γ0U(γ). (21.17)
Then the positivity of γ0 implies the positivity of γ. ((21.17) implies also (21.13).)
Eq. (21.15) follows from the isospectral properties of (21.7): γ can be written in theform (21.17). (A different proof is given as follows. Using that γj = 〈φj, γφj〉, we find∂tγj = 〈∂tφj, γφj〉 + 〈φj, γ∂tφj〉 + 〈φj, (∂tγ)φj〉. The first two terms give γj〈∂tφj, φj〉 +γj〈φj, ∂tφj〉 = γj∂t〈φj, φj〉 = 0, while the third terms gives 〈φj, i[hγ, γ]φj〉 = 0. Thisproves (21.15).)
Proposition 21.1. γ satisfies the equation (21.7) iff the eigenfunctions φj of γ satisfythe equations
i∂tφj = hγφj, (21.18)
where hγ is given in (21.8) and can be expressed in terms of φj’s using
nγ(x) = γ(x, x) =∑j
|φj(x)|2. (21.19)
Proof. To prove (21.18), we note that, since the spectrum of γ is discrete, we can write itin terms of the projections on the eigenfunctions its φj (associated with the eigenvaluesγj):
Hence, if (21.18) are satisfied then so is (21.7). On the other hand, if (21.7) is satisfied,then multiplying (21.21) scalarly by φk we obtain
γk(i∂t − hγ)φk =∑j
γjφj〈(i∂t − hγ)φj, φk〉
=∑j
γjφj〈φj, (i∂t − hγ)φk〉 = γ(i∂t − hγ)φk.
Assuming the eigenvalues γk are non-degenerate, this implies that there are real numbersµk s.t. (i∂t − hγ)φk = µkφk and therefore (i∂t − hγ)(eiµktφk) = 0.
Remark. Solutions to the equations (21.18) are parametrized by the numbers γj’s.
21.5 Equilibrium states and entropy
Clearly, if, for some function f , the operator γ satisfies the equation γ = f(hγ), thenit also satisfies (21.7), namely it is a static solution to (21.7). However, not all suchsolutions are of physical interest. We select those which are using the entropy principle,namely, requiring that they minimize the energy for the fixed entropy (and the numberof particles).
We have already defined the energy functional E(γ) on trace class operators γ, see(21.11). Now, we take the entropy of γ to be of the form
S(γ) = Tr g(γ), (21.22)
where
g(x) := −1
2(x lnx+ (1− x) ln(1− x)). (21.23)
We are interested in minimizing the internal energy E(γ) on the set S∗ := 0 ≤ γ ≤1 ∩ the entropy, S(γ), and the number of particles, N(γ) := Tr γ, are fixed. As usual,we define the free energy on the convex set S := 0 ≤ γ ≤ 1 as
FTµ(γ) = E(γ)− TS(γ)− µN(γ), (21.24)
fixing the chemical potential, µ, rather than the number of particles, N(γ). We defineFTµ(γ) on the Sobolev space space H1,1(H) defined as follows. Let b := (c1+h)1/2, wherec > 0 is such that c1 + h ≥ 1. Define the Sobolev space Hs,1(H) as
Hs,1(H) := γ ∈ L1(H) : bs/2γbs/2 ∈ L1(H). (21.25)
Define hγµ := hγ − µ. We begin with the following lemma proven in Appendix 21.8,
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Lemma 21.2. Minimizers, γ, of the internal energy E(γ) on the convex set S, withS(γ) and N(γ) fixed, are critical points of FTµ(γ), i.e. they satisfy the Euler-Lagrangeequations
dγFTµ(γ) = 0, (21.26)
for some T and µ (the latter are determined by fixing S(γ) and Tr(γ)).
For S given in (21.22), the equation (21.26) becomes
γ = g#(1
Thγµ), (21.27)
where g#(h) := (g′)−1(h). Indeed, the Gateaux derivatives, ∂γFT (γ), is given by (seeAppendix 21.8)
∂γFTµ(γ) = hγµ − Tg′(γ), (21.28)
where hγµ = hγ − µ. This together with (21.26) gives
hγµ = Tg′(γ), (21.29)
which upon inverting the function g′ gives (21.27).Note that for g(x) := −1
2(x lnx+ (1− x) ln(1− x)), we have
g′(x) = −1
2ln
x
1− x and g#(h) = (1 + e2h)−1. (21.30)
21.6 Local and global existence
Let Y := H1,1(H), where Hs,1(H) is the Sobolev space space defined in (21.25). Recall,that we think of γ as a path γ : t ∈ I → u(t) ∈ Y in Y and we are looking for weaksolutions to (21.7), i.e. for solutions of this equation in the integral form, in the spaceC([0, T ], Y ), for some T > 0. Define the ball BR,T := γ ∈ C([0, T ], Y ) : ‖γ‖C([0,T ],Y ) ≤R. We have the following result (cf. J. M. Chadam, J. M. Chadam and R. T. Glassey, A.Bove, G. Da Prato and G. Fano ([?], [?], [?]). (needs checking and filling in details)
Theorem 13. (i) Assume v ∈ L2 ∩ L1. There are functions LR and MR, R > 0, s.t. forγ0 ∈ Y , R > 2K‖γ0‖Y and T < (KLR)−1, R/2KMR, the equation (21.7) has a uniqueweak solution γ ∈ BR,T . The solution γ depends continuously on the initial condition γ0.Furthermore, either the solution is global in time or blows up in Y in a finite time (i.e.either ‖γ(t)‖Y <∞, ∀t, or ‖γ(t)‖Y <∞ for t < t∗ and ‖γ(t)‖Y →∞ as t→ t∗ for somet∗ <∞).
(ii) If in addition v∗ is positive definite, then the solutions are global.
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Proof. We will use Theorem ??. Note that the equation (21.7) is in the form (??) withthe linear operator A acting on density operators as A(ρ) := i[h, ρ] and f(γ) := [v ∗nγ, γ].We have to check that the conditions (??) - (??) are satisfied.
The operator A is an operator of adjoint representation and it generates the oneparameter group eAt(γ) = eihtγe−iht, which, by the cyclicity of the trace, unitarity of e−iht
and the fact that b commutes with e−iht, satisfies (??), for K = 1.Next, it is straightforward to show that f(γ) satisfies (??) - (??) for some MR, LR <∞.
Then Theorem ?? implies the first statement of the theorem.To prove that the solutions are global one uses the conservation of the energy and
number of particles.
21.7 Existence of ground states and Gibbs states
(needs checking and filling in details)
Theorem 14. Assume v ∈?? and g is strictly convex and satisfies ?? and g∗ ≥ 0 (g∗ isthe Legandre transform of g). Then the equation (21.7) has equilibrium states satisfying(21.27) (see Lemma 21.2).
Proof. We can minimize the free energy functional (21.31) on the convex set S := γ ∈H1,1(H) : 0 ≤ γ ≤ 1, where Hs,1(H) is the Sobolev space space defined in (21.25),directly. However, we prefer to follow, under additional condition that
v∗ = v(−i∇) = (−∆)−1,
an elegant proof of Markowich, Rein and Wolansky ([?]) In this proof we pass from thefunctional FTµ(γ) to the dual one
ΦTµ(V ) = − infγFTµ(γ, V ), (21.31)
where
FTµ(γ, V ) := Tr(hV γ
)− T Tr g(γ)− µTr γ − 1
2
∫|∇V |2, (21.32)
with hV := h + V . Notice that F (γ, vγ) = FTµ(γ), where vγ := v ∗ nγ. Using thatdγFTµ(γ, V ) = hV−Tg′(γ)−µ and
[Tr((hV−µ)γ
)−T Tr g(γ)
]∣∣hV −µ−Tg′(γ)=0
= Tg∗(1T
(hV−µ)), we compute
ΦTµ(V ) =1
2
∫|∇V |2 − T Tr g∗
( 1
T(hV − µ)
). (21.33)
Now, one can show easily that
224 Lectures on Applied PDEs
• If V∗ is a critical point of ΦTµ(V ), then γ∗ := g′∗(
1T
(hV∗ − µ))
is a critical pointof FTµ(γ). (Indeed, since dV ΦTµ(V ) = −∆V − nγ, where γ := g′∗
(1T
(hV − µ)), a
critical point of ΦTµ(V ) satisfies V = (−∆)−1nγ =: Vγ, where γ := g′∗(
1T
(hV − µ)),
or γ := g′∗(
1T
(hV − µ))
satisfies γ := g′∗(
1T
(hVγ − µ)), which is (21.27).)
• The dual functional ΦTµ(V ) is coercive and weakly lower semi-continuous and therefore has a minimizer. Moreover, it is strictly convex and so the minimizer is unique.
The last two statements imply the desired result.
21.8 Appendix: Proof of Lemma 21.2
Proof of Lemma 21.2. To prove (21.27), we observe that if γ ∈ S is a minimizer, then forany γ′, we have E((1 − s)γ + sγ′) − E(γ) ≥ 0, provided S((1 − s)γ + sγ′) − S(γ) = 0and Tr((1− s)γ + sγ′)− Tr(γ) = 0. Dividing this by s > 0 and taking s→ 0, we obtainE ′(γ)(γ′−γ) ≥ 0 for any γ′, satisfying S ′(γ)(γ′−γ) = 0 and Tr(γ′−γ) = 0. Hence γ′ = γminimizes E ′(γ)(γ′ − γ) on the set S, provided S ′(γ)(γ′ − γ) = 0 and Tr(γ′ − γ) = 0.Since E ′(γ)(γ′ − γ) is a linear function of γ′, this is possible iff γ satisfies
E ′(γ)− TS ′(γ)− µN ′(γ) = 0, ?? (21.34)
for some T and µ. On the other hand, by the definition (21.31), we have
Another way to introduce a hamiltonian structure is as follows. For a functional A(ρ)we define the gradient operator, ∂ρA(ρ), in the trace metric by the equation Tr(∂ρA(ρ)ξ) =∂sA(ρ+ sξ)|s=0. On the space of classical field observables we define the Poisson bracketby
The Jacobi identity for (21.41) is proven in Appendix ??. We observe that
A(ρ), B(ρ)|ρ=Pψ = i
∫(∂ψ(x)A∂ψ(x)B − ∂ψ(x)A∂ψ(x)B)(ψ, ψ)dx. (21.42)
The r.h.s. is the standard Poisson bracket, A,B(ψ, ψ), for the Hartree equation (see[?]). Indeed, ∂ρA(ρ) and ∂ρB(ρ) are operators on L2(R3) (1−particle observables) andtherefore
Tr((∂ρB)(Pψ)Pψ(∂ρA)(Pψ)) = 〈(∂ρA)(Pψ)∗ψ, (∂ρB)(Pψ)ψ〉.Since, as it is easy to see, (∂ρB)(Pψ)ψ = ∂ψ(x)B(ψ, ψ) and (∂ρA)(Pψ)∗ψ = ∂ψ(x)A(ψ, ψ),
this gives Tr((∂ρB)(Pψ)Pψ(∂ρA)(Pψ)) =∫∂ψ(x)A∂ψ(x)B, which implies the desired rela-
tion.We define the Hamiltonian functional on L1(H) as
H (ρ) := Tr(hρρ) +1
2
∫nρv ∗ nρdx, (21.43)
The resulting Hamilton equation,
∂tρ = H(ρ), ρ, (21.44)
is exactly the Hartree-von Neumann equation considered above. Indeed, this fact followsfrom the equation
H(ρ), ρ = −i[hρ, ρ], (21.45)
where hρ := −∆ + V (x) + (v ∗ nρ). To show the latter equation we use the definition ofthe Poisson bracket and the relation Tr(∂ρρξ) = ξ, which follows from the definition of∂ρA(ρ) above, to obtain H, ρ = −i (∂ρH(ρ)ρ− ρ∂ρH(ρ)) . Next, computing ∂ρH(ρ), weconclude that H, ρ is equal to the r.h.s. of (21.45).
226 Lectures on Applied PDEs
Remark 21.3. Equation (21.44) should be understood in the weak sense: for all a ∈ A1,
∂t Tr(aρ) = Tr(a H(ρ), ρ). (21.46)
Using the linearity of the Poisson brackets in the second factor, we obtain
Tr(aH, ρ) = H(ρ),Tr(aρ). (21.47)
Thus equation (21.44) is equivalent to the equation
∂t Tr(aρ) = H(ρ),Tr(aρ). (21.48)
The Hamiltonian (21.43) and the Poisson brackets (21.41) generate the Hartree-vonNeumann flow ϕt on one-particle density matrices. This flow induces the flow on gener-alized classical observables:
A(ρ)→ A(ϕt(ρ)). (21.49)
21.10 Appendix: Hilbert Space Approach
Quantum statistical dynamics can be put into a Hilbert space framework as follows.Consider the space HHS of Hilbert-Schmidt operators acting on the Hilbert space H.These are the bounded operators, K, such that K∗K is trace-class (see Section ??).There is an inner-product on HHS, defined by
〈F,K〉 := Tr(F ∗K). (21.50)
Exercise 1. Show that (21.50) defines an inner-product.
This inner-product makes HHS into a Hilbert space (see [?, 26]). On the space HHS,we define an operator L via
LK =1
~[H,K],
where H is the Schrodinger operator of interest. The operator L is symmetric. Indeed,
~〈F,LK〉 = Tr(F ∗[H,K]).
Using the cyclicity of the trace, the right hand side can be written as
and so 〈F,LK〉 = 〈LF,K〉 as claimed. In fact, for self-adjoint Schrodinger operators, H,of interest, L is also self-adjoint.
Lectures on Applied PDEs 227
Now consider the Landau-von Neumann equation
i∂k
∂t= Lk (21.51)
where k = k(t) ∈ HHS. Since k(t) is a family of Hilbert-Schmidt operators, the operatorsρ(t) = k∗(t)k(t) are trace-class, positive operators. Because k(t) satisfies (21.51), theoperators ρ(t) obey the equation
i∂ρ
∂t= Lρ =
1
~[H, ρ]. (21.52)
If ρ is normalized – i.e., Tr ρ = 1 – then ρ is a density matrix satisfying the Landau-vonNeumann equation (21.52). The stationary solutions to (21.51) are just eigenvectors ofthe operator L with eigenvalue zero.
To conclude, we have shown that instead of density matrices, we can consider Hilbert-Schmidt operators, which belong to a Hilbert space, and dynamical equations which areof the same form as for density matrices. Moreover, these equations can be written inthe Schrodinger-type form (21.51), with self-adjoint operator L, sometimes called theLiouville operator.
22 Existence of bubbles and Lyapunov - Schmidt de-
composition
For a field equation describing dynamics of interfaces, by a bubble we mean a smoothspherically symmetric static solution, u(x) = v(|x|), with a single interface (or moreprecisely a boundary layer) across which the solution changes from approximately constantvalue to another.
We will consider two field equations describing interfaces, the Allen-Cahn equation,(17.24), and the Cahn-Hilliard equation, introduced below. In this case, a bubble is asmooth spherically symmetric static solution, u(x) = v(|x|), with the property that
v(r)→ +1, r → 0, v(r)→ −1, r →∞. (22.1)
In additions, we make a technical assumption that v′(r)→ −1, r →∞.First, we consider the Allen-Cahn equation, (16.3), which describes the phase separa-
tion phenomena. We consider the Allen-Cahn equation in the entire space. We reproduceit here
∂u
∂t= ε2∆u− g(u), (22.2)
where u : Rd × R+ → R and ε is a small parameter, with the initial condition u|t=0 =u0(x), x ∈ Rd. Here g : R → R is the derivative, g = G′, of a double-well potential:
228 Lectures on Applied PDEs
G(u) ≥ 0 and has two non-degenerate global minima with the minimum value 0 (seeFigure 2). Specifically, we take g(u) = u3 − u and G(u) = 1
2(u2 − 1)2.
We consider static solutions of this equation in the entire space, Rd. They satisfy thestatic Allen-Cahn equation,
− ε2∆u+ g(u) = 0. (22.3)
Theorem 15. Let dimensions d ≥ 2. The Allen-Cahn equation (22.3) has no bubblesolutions.
Proof. We use the well-known approach which goes under names of virial relation, Der-rick’s theorem or Pohozaev identity.
By rescaling, we reduce to the case of ε = 1. Let r = |x| and u = v(r) be a sphericallysymmetric, C1 stationary solution to the Allen-Cahn equation:
− ∂2rv −
d− 1
r∂rv + g(v) = 0, (22.4)
with v|r=0 = 1, and v|r=∞ = −1. We multiply (22.4) by ∂rv = x · ∇v and integrate in rto obtain
−1
2(∂rv)2|∞0 − (d− 1)
∫ ∞0
(∂v
∂r
)21
rdr +G(v)|∞0 = 0.
Since the last term is zero, this equation implies
1
2(∂rv)2|r=0 = (d− 1)
∫ ∞0
(∂v
∂r
)21
rdr.
If ∂v∂r|r=0,∞ = 0,, then
∫∞0
(∂v∂r
)2 1rdr = 0 and therefore v must be a constant; this however
contradicts the boundary conditions on v. If ∂v∂r|r=0 6= 0 (by the continuity, this condition
must hold in a neighborhood of the origin), then the above relation leads to a contradictionsince the right side is infinite and the left side is finite. This completes the proof of thefirst part of the theorem.
The reason the Allen-Cahn equation has no bubbles, or that bubble solutions collapseto a point, is that the mass is not conserved and the surface tension shrinks the interfaceto a point. In a variant of the Allen-Cahn equation with the mass conservation - theCahn-Hilliard equation - the bubbles do exist as shown below.
Now, we consider the Cahn-Hilliard equation, which play a central role in materialscience. In a sense, this is a version of the Allen-Cahn equation with mass conservation.As the Allen-Cahn equation, (22.2), it presents a key model with many generalizationsand extensions. In Rd, it is of the form
ut = −∆(ε2∆u− g(u)), (22.5)
Lectures on Applied PDEs 229
where g is the same as for the Allen-Cahn equation. In particular, we can take g(u) =u3 − u. The equation (22.5) is derived from the conservation law of mass
∂u
∂t= − div J, (22.6)
where J flux of the material and Fick’s law,
J = −D∇µ, (22.7)
connecting J to the chemical potential µ coming from thermodynamic consideration andthe expression of the latter in terms of the free energy, E ,
µ = dE(u). (22.8)
(Recall, that dE(u) is the Gateaux derivative of E(u).) If we take the standard expression
E(u) :=
∫Rd
1
2(ε2|∇u|2 +G(u)), (22.9)
with G′ = g, for the free energy E and D constant, say D = 1, then the above implies theCahn-Hilliard equation, (22.5).
It follows from Gauss’ theorem that∫Rdu dx = constant
along solutions to (22.5), in agreement with the conservation of the average mass fractionof the components of the alloy.
We consider static solutions of the Cahn-Hilliard equation in Rd. They satisfy thestatic Cahn-Hilliard equation, which, after rescaling, is given by
−∆u+ g(u) = µ, (22.10)
It has a (homogeneous) solution u ≡ u for any constant u ∈ g−1(µ). This gives a staticsolution of (22.5).
Theorem 22.1. [Alikakos and Fusco] Let dimensions d ≥ 2. The Cahn-Hilliard equation,(22.5), has a one-parameter family of static bubble solutions parametrized by their radii.
First, we present our main strategy. To solve the equation (22.10) equation for u we
(i) Construct a family, (φR(r), µR) of approximate solutions to (22.10) parametrized byR > 0 (radii of the bubbles). (Here we reparameterized our problem from µ to R.)
(ii) Solve the equation (22.10) near (φR(r), µR) by using the Lyapunov - Schmidt de-composition.
230 Lectures on Applied PDEs
Ideas and sketch of the proof of Theorem 22.1. Recall that the static equation (22.10) withµ = 0, i.e. the Allen-Cahn equation, has the kink solutions:
χx0e(x) = χ((x− x0) · e
), where x0, e ∈ Rn, (22.11)
where χ(s) → ±1 as s → ±∞ (see Section 2.1 and Figure 2). Recall that for the keyg(u) = u3 − u, the function χ is given explicitly as
χ(s) = tahn(s√2
). (22.12)
Denote F (u, µ) := −∆u+ g(u)− µ. Then the equation (22.10) can be rewritten as
F (u, µ) = 0. (22.13)
Let r = |x|. As an approximate solution, we try the shifted kink
χR(r) := χ(r −R) (22.14)
We run into two problems. First, using that χR is a spherically symmetric function andtherefore
∆χR = ∂2rχR +
d− 1
r∂rχR,
we obtain F (χR, µ) = −∂2rχR − d−1
r∂rχR + g(χR)− µ. Using that χR satisfies
− ∂2rχR + g(χR) = 0, (22.15)
we find furthermore
F (χR, µ) = −d− 1
r∂rχR − µ.
The r.h.s. is small (if µ is small and R is large) but is not L2: as r → ∞, we have∂rχR → 0 and therefore F (χR, µ)→ −µ. Hence we have to modify χR(r) at infinity.
We ignore this for now and look for a solution u(x) of (22.13) in the form u(x) =χR(r) +α(r), expecting α to be small (the perturbation theory). We rewrite the equation(22.13) as an equation for α, by expanding F (u, µ) in (22.13) as
F (χR + α, µ) = F (χR, µ) + LRα +N(α, µ), (22.16)
where LR = duF (u, µ)|u=χR and N(α, µ) is defined by this equation, explicitly, N(α) :=g(χR + α)− g(χR)− g′(χR)α, so that the equation F (χR + α, µ) = 0 becomes
LRα = −(F (χR, µ) +N(α, µ)).
We try to solve this equation for the fluctuation term α, by inverting the operator LRand reducing it to the fixed point problem α = H(α), where H(α) := −L−1
R (F (χR, µ) +N(α, µ)).
To show that H is a contraction, we need that the linearized map LR = duF (u, µ)|u=χR
is invertible and its inverse is not too large. The second problem is that, as we indicatebelow,
Lectures on Applied PDEs 231
• LR has an eigenvalue of the size O(
1R
)and therefore it is either non-invertible or
invertible with the inverse of the size O (R).
Indeed, using that the shifted kink χR satisfies the equation −∂2rχR + g(χR) = 0 and
differentiating this equation w.r.to R to obtain
(−∂2r + g′(χR))χ′R = 0, (22.17)
where χ′R ≡ ∂RχR. Thus we conclude that χ′R is the zero eigenfunction of the operatorL0 := −∂2
r + g′(χR). Using this and LR = L0 − d−1r∂r, we compute
LR(−χ′R) =d− 1
rχ′′R. (22.18)
Now we use that χ′′R is concentrated near r = R, more precisely, that
|χ(k)R (r)| ≤ Ce−2|r−R| (22.19)
to obtain the estimate
LR(−χ′R) = O
(1
R
)χ′′R. (22.20)
(It is easy to check (22.19) for χ for the specific nonlinearity given in (22.12), but it holdsfor the general nonlinearities described in (22.11), though with a different exponent.) Bythe spectral theory (see Appendix E (to be done) and [12, 16]), this implies thisimplies the statement above.
Thus we have to devise a way to overcome these problems. To tackle the first one,we modify the shifted kink χR(r) := χ(r − R) to find a better approximate solution. Tohandle the second problem, we use the Lyapunov - Schmidt reduction and contractionmapping techniques. We ignore for the moment the first problem, treating χR as a goodapproximate solution, and concentrate on the second one.
To begin with, the Lyapunov-Schmidt reduction consists of the following steps:(a) Parameterization of solutions u(x). We parameterize our solution u(x) by a pair
(R, ξ), where R > 0 and the function α(r), so that u(x) can be represented uniquely as
u(x) = χR(r) + α(r), α ⊥ χ′R. (22.21)
The condition α⊥χ′R ≡ ∂RχR (in L2(Rd)) determines R. Indeed, applying the implicitfunction theorem to the equation f(R, v) := 〈v − χR, χ
′R〉 = 0, we obtain the unique
solution R = R(v).To clarify the geometric meaning of th decomposition (22.21), we define the kink
manifoldMkink := χR(r)| R > 0 and and observe that (22.21) is equivalent to projectingu(x) to this manifold, which gives the point χR(r) onMkink and the function α(r) whichis orthogonal to this manifold, α ⊥ TχRMkink. Since the tangent space toMkink at χR(r)
232 Lectures on Applied PDEs
is spanned by χ′R(r), this leads to (22.35). The function α(r) is called the fluctuation ofu(x) around χR(r).
(b) Decomposition of the equation. Plug the decomposition (22.21) into (22.13) andproject the resulting equation onto the tangent space TχRMkink and its orthogonal com-plement, (TχRMkink)⊥ to obtain the equations for two unknowns R and α:
PRF (χR + α, µ) = 0, (22.22)
P⊥R F (χR + α, µ) = 0, (22.23)
where PR be the orthogonal projector onto χ′R, namely, PRf := χ′R∫χ′Rf/norm, and
P⊥R = 1l− PR. These are two equations for two unknowns ξ and h.(c) Solution of the second equation (22.23) for α. We plug the expansion (22.16) into
(22.23) and use the relation α = P⊥R α and the notation L⊥R := P⊥RLRP⊥R , restricted to
Ran P⊥R , to obtain the equation
P⊥R F (χR, µ) + L⊥Rα + P⊥RN(α, µ) = 0. (22.24)
Now, we use the key fact (see Corollary 22.3, below) that the operator LR, restrictedto the orthogonal complement, Ran P⊥R = (TχRMkink)⊥, of χ′R, is invertible, with theuniformly bounded inverse. To prove this one needs a fair amount of spectral theory, see[12, 16]. (For the relevant definitions and facts, see Appendix E (under construction).)As a result, we can rewrite (22.24) as
Now, it not hard to show that it is a contraction for R sufficiently large, which impliesthat
• For R−1 and µ sufficiently small, the equation (22.23) has a unique solution forα = α(R, µ) and this solution satisfies the estimate
α(R, µ) = O(Rd−12 (R−2 + µ)). (22.26)
We prove a similar result (for the correct approximate solution) later.(d) Derivation of the reduced equation. Plug the solution α = α(R, µ) of (22.23) into
This is a scalar equation for a single unknown R (the reduced equation). Solve thereduced equation for R. Then χR+α(R, µ), where R is a solution to (22.27), is a solutionto the stationary Cahn-Hilliard.
Lectures on Applied PDEs 233
(e) Solution of the reduced equation for R. Since PRf := χ′R∫χ′Rf/norm, we can
rewrite (22.27) as 〈χ′R, F (χR +α, µ)〉 = 0. Remembering the expansion (22.16), this gives
To be specific, in what follows we take g(u) = u3 − u. Then the function χ is givenexplicitly by (??) and one can easily compute the terms 〈χ′R, F (χR, µ)〉, 〈χ′R, LRα〉 andχ′R, N(α)〉 (see Appendix 22.1, below) and use the estimate (22.26), to derive from (22.28)the following equation for R
− d− 1
2|Sd−1|R−1 + O(R−4 +R−2µ+ µ2) = 0. (22.29)
This equation has the solution of the form R = O(µ−2).(f) Summing up, we expect that if R is large enough, there exists a unique spherically
symmetric solution to the stationary Cahn-Hilliard equation of the form χR + α(R, µ),
where χR is defined in (22.14) and α(R, µ) = O(Rd−12 (R−2 + µ)).
Finally, we explain how to construct the family, φR(r), R > 0, of approximate solu-tions. We assume R is sufficiently large and µ > 0, sufficiently small. To construct thefamily, φR(r), R > 0, of approximate solutions to (22.13), we look for a spherically sym-metric solution to (22.10), satisfying (approximately) the boundary conditions v(0) = −1and v(∞) = 1.
As was discussed above, the shifted kink χR(r) := χ(r − R), where r = |x|, is not asuitable approximate solution and we have to modify it. We write the family, φR(r), R >0, of approximate solutions to (22.13) as follows
φR(r) = χR(r) + ηR(r), (22.30)
where ηR(r) := η(r −R). To be specific, we take g(u) = u3 − u. We show that ηR(r) canbe choosen so that
F (φR, µ)→ 0, as r →∞, (22.31)
F (φR, µ) is bounded, as r → 0. (22.32)
Namely, we take η(∞) = ηR(∞) = δ∞, with the number δ∞ satisfying δ3∞−3δ2
∞+2δ∞ = µ,which gives δ∞ = O(µ), and η′(−R) = η′R(0) = χ′R(0) = χ′(−R). Indeed, using that χRsatisfies the equation (2.5), we compute
F (φR, µ) = −d− 1
r∂rχR − ∂2
rηR −d− 1
r∂rηR − ηR + 3ηRχ
2R + 3η2
RχR + η3 − µ. (22.33)
Since ∂rχR → 0 and χR(r)→ −1, as r →∞, the first relation requires that
−ηR + 3ηR − 3η2R + η3
R − µ→ 0,
234 Lectures on Applied PDEs
as r → ∞. Hence, it suffices to choose η(∞) = ηR(∞) = δ∞, with the number δ∞satisfying δ3
∞ − 3δ2∞ + 2δ∞ = µ. This gives δ∞ = O(µ) and F (φR, µ)→ 0, as r →∞.
For the second relation, (22.32), we require that ∂rχR + ∂rηR = O(r), as r → 0, andtherefore choose
η′(−R) = η′R(0) = χ′R(0) = χ′(−R),
which gives (22.32). Finally, for simplicity, we take ηR(r) = 0, for R/2 ≤ r ≤ 3R/2.Using the above expression, the estimate (22.19) and a convenient choice of ηR, we
obtain the simple estimates (this requires a fair amount of work not displayed here)|F (φR, µ)| . e−δR, for r ≤ R/2 or r ≥ 3R/2, and |F (φR, µ)| . (R−1 + µ), for R/2 ≤ r ≤3R/2. Using these estimates, we find
‖F (φR, µ)‖Hr . Rd−12 (R−1 + µ). (22.34)
(The term Rd−12 comes from the volume of the shell of radius R and thickness 1.)
This completes the construction of the approximate solution φR. The Lyapunov-Schmidt reduction based on this solution follows the outline above for the decomposition(22.21), i.e. for the rough approximate solution χR and is done in Appendix to thissection.
22.1 Appendix. Details of the Lyapunov-Schmidt reduction
Our goal is to solve the equation (22.10) near φR(r), for some R depending on µ to bechosen later. We look for a solution u(x) of (22.13) in the form u(x) = φR(r) + ξ(r), andtry to solve for the fluctuation term ξ. We follow the approach sketched above for thedecomposition (22.21), i.e. for the rough approximate solution χR. We begin with thefollowing proposition
Proposition 22.2. The operator LR = −∆r + g′(χR) has the following properties:
1. The smallest point, inf σ(LR), of the spectrum of LR is a non-degenerate, isolatedeigenvalue of size O
(1R
)and with an approximate eigenfunction χ′R(r);
2. For large enough R, the gap between the smallest eigenvalue and the rest of thespectrum is O(1).
Proving this proposition, as well as the corollary below, requires a fair amount of thespectral theory, see e.g. [12, 16] and Appendix E (to be expanded), and the proof isgiven below. The property 2 shows that
Corollary 22.3. The operator LR is invertible on the orthogonal complement of the sub-space spanned by the vector χ′R(r) and its inverse on this orthogonal complement is uni-formly bounded in R.
Lectures on Applied PDEs 235
Now, we follow closely the analysis in the main text. It consists of the following steps:
(a) Parameterization of solutions u(x). We parameterize our solution u(x) by a pair(R, ξ), where R > 0 and the function ξ(r), so that u(x) can be represented uniquely as
u(x) = φR(r) + ξ(r), ξ ⊥ φ′R. (22.35)
where φ′R ≡ ∂RφR. The condition ξ⊥φ′R (in L2(Rd)) determines R. Indeed, observingthat ∂RφR ≈ −φ′R and applying the implicit function theorem to the equation f(R, v) :=〈v − φR, φ′R〉 = 0, we obtain the unique solution R = R(v).
To clarify the geometric meaning of th decomposition (22.35), we define the kinkmanifold M := φR(r)| R > 0 and and observe that (22.35) is equivalent to projectingu(x) to this manifold, which gives the point φR(r) on M and the function ξ(r) whichis orthogonal to this manifold, ξ ⊥ TφRM. Since the tangent space to M at φR(r) isspanned by φ′R(r), this leads to (22.35). The function ξ(r) is called the fluctuation of u(x)around φR(r).
(b) Decomposition of the equation. It is convenient to replace M := φR(r)| R > 0by its approximate Mkink := χR(r)| R > 0. Plug the decomposition (22.35) into(22.13) and project the resulting equation onto the tangent space TχRM and its orthogonalcomplement, (TχRM)⊥ to obtain the equations for two unknowns R and ξ:
PRF (φR + ξ, µ) = 0, (22.36)
P⊥R F (φR + ξ, µ) = 0, (22.37)
where PR be the orthogonal projector onto χ′R, namely, PRf := χ′R∫χ′Rf/norm, and
P⊥R = 1l− PR. These are two equations for two unknowns ξ and h.
(c) Solution of the second equation (22.37) for ξ. (Here the choice of the familyφR(r), R > 0, plays a crucial role.)
Lemma 22.4. For R−1 and µ sufficiently small, the equation (22.37) has a unique solu-
tion for ξ = ξ(R, µ) and this solution satisfies the estimates ξ(R, µ) = O(Rd−12 (R−2 +µ)).
We prove this lemma later. Meantime we explain the idea of the proof. ExpandF (φR + ξ, µ) in ξ to obtain
F (φR + ξ, µ) = F (φR, µ) + duF (φR, µ)︸ ︷︷ ︸LR,µ
ξ +N(ξ, µ), (22.38)
where N(ξ, µ) is defined by this equation, explicitly,
N(ξ) := g(φR + ξ)− g(φR)− g′(φR)ξ.
236 Lectures on Applied PDEs
Plug this into (22.37) and use the notation L⊥ ≡ L⊥R,µ := P⊥R duF (φR, µ)P⊥R , restrictedto Ran P⊥R , to obtain the equation
P⊥R F (φR, µ) + L⊥ξ + P⊥RN(ξ, µ) = 0. (22.39)
By Corollary 22.3, the operator L⊥R,µ is invertible, with the uniformly bounded inverse.Hence we can rewrite (22.39) as
The estimation of the map Φ(ξ, µ) is done below and shows that it is a contraction forR sufficiently large, which implies that that (22.40) has a unique solution, ξ = ξ(R, µ),
satisfying the estimate ξ(R, µ) = O(Rd−12 (R−2 + µ)).
(d) Derivation of the reduced equation. Plug the solution ξ = ξ(R, µ) of (22.37) into(22.36) to obtain the new equation for R,
This is a scalar equation for a single unknown R (the reduced equation). Solve thereduced equation for R. Then φR + ξ(R, µ), where R is a solution to (22.42), is a solutionto the stationary Cahn-Hilliard.
(e) Solution of the reduced equation for R. Since PRf := χ′R∫χ′Rf/norm, we can
rewrite (22.41) as 〈χ′R, F (φR + ξ, µ)〉 = 0. Remembering the expansion (22.38), this gives
For ξ = ξ(R, µ), the last two estimates give by Lemma 22.4, 〈χ′R, Lξ〉 = O(Rd−3(R−2 +µ))and 〈χ′R, N(ξ)〉 = O(Rd−1(R−4 + µ2)). Inserting the estimates above into (22.42) anddividing by Rd−1, gives
− d− 1
2|Sd−1|R−1 + O(R−4 +R−2µ+ µ2) = 0. (22.43)
This equation has a solution R = O(µ−2).Summing up, we have shown that if R is large enough, there exists a unique spherically
symmetric solution to the stationary Cahn-Hilliard equation of the form φR + ξ(R, µ),
where φR is defined in (22.30) and ξ(R, µ) = O(Rd−12 (R−2 +µ)). Thus, the Cahn-Hilliard
equation has bubble solutions.
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Proof of Lemma 22.4. We estimate the map Φ(ξ, µ). To keep things specific we considerthe canonical nonlinearity g(u) = u3− u and d = 3. Then N(ξ) = 3φRξ
2 + ξ3. Using thisexpression, the Sobolev embedding theorems and the simple inequality ‖ξm − ηm‖L2 ≤m(‖ξ‖m−1
provided ‖ξ‖H1 , ‖η‖H1 ≤ ρ ≤ 1. Next, by (22.34) and the boundedness of P⊥R , we have
‖P⊥R F (φR, µ)‖Hr . Rd−12 (R−2 + µ). (22.45)
Using the last two relations, the uniform boundedness of L−1 and (22.40), we obtain
‖Φ(ξ, µ)‖H1 . Rd−12 (R−2 + µ) + ρ2,
‖Φ(ξ, µ)− Φ(η, µ)‖H1 . ρ‖ξ − η‖H1
Hence for ρ < 1 s.t. Rd−12 (R−2 + µ) + ρ2 ρ, which is easy to satisfy if R
d−12 (R−2 + µ)
sufficiently small (which means that R is large for d < 5), we have that Φ is a strictcontraction on the ball in H1 of radius ρ, centered at the origin, and it has a unique fixedpoint in this ball.
(Fill in details!)
Proof of Proposition 22.2. The proof below uses the definition and properties of the es-sential spectrum (see [12, 16] and Appendix E (to be expanded)). The first statementfollows from the spectral fact that σess(L) = σ(L∞), where L∞ is the evaluation of L atinfinity: L∞ := −∆r+g′(χR(∞)) = −∆r+g′(1), and the simple computation, which usesthe definition of the spectrum (see [12, 16]) and the Fourier transform,
σ(L∞) = σ(−∆r + g′(1)) = [g′(1),∞).
To prove the second statement, we observe that the equation (22.20) implies that〈χ′R, LRχ′R〉 = O
(1R
)which shows that the operator LR has an eigenvalue of the order
O(
1R
)below its essential spectrum. Using the fact that −χ′R is a positive function by the
Perron - Frobenius theory, we conclude that, for R large enough, the smallest eigenvalueof LR is non-degenerate and of the order O
(1R
), with the approximate eigenvector χ′R.
To prove the third statement, we perform a geometric analysis of L. Let (j0, j1) be apartition of unity, normalized as j2
0 + j21 = 1, and with supp j0 ⊂ x ∈ Rd||x| ≤ R/2 and
supp j1 ⊂ x ∈ Rd||x| ≥ R/3 and |∂mji| . R−m. We use the IMS formula
L :=∑i
jiLji −∑i
|∇ji|2, (22.46)
238 Lectures on Applied PDEs
which holds for any operator L of the form L = −∆ + V (x). To prove it, we use therelations L =
∑1i=0 j
2i L = −∑1
i=0 ji[ji,∆]+∑1
i=0 jiV ji,∑1
i=0 j2i ∆ =
∑1i=0 ji∆ji+ji[ji,∆]
and ji[ji,∆] = −ji(2∇ji · ∇+ ∆ji), to obtain
L = −1∑i=0
ji∆ji + ji(2∇ji · ∇+ ∆ji) +1∑i=0
jiV ji.
Similarly, we have
L = −1∑i=0
ji∆ji + (2∇ji · ∇+ ∆ji)ji +1∑i=0
jiV ji.
Subtracting the second relation from the first and dividing the result by 2, we obtainL = −∑1
i=0 ji∆ji +∑1
i=0 jiV ji + [ji,∇ji · ∇]. Since [ji,∇ji · ∇] = −∑i |∇ji|2, this gives(22.46). Since, by the choice of ji,
∑i |∇ji|2 = O
(1R2
), (22.46) implies that
L :=1∑i=0
jiLji +O( 1
R2
). (22.47)
We estimate each term jiLji separately, using the properties of supp ji.First we observe that χR ≈ 1 and therefore L ≈ −∆ + g′(1) ≥ g′(1) > 0 on x ∈
Rd||x| ≥ R/3. Hence j1Lj1 ≥ j1g′(1)j1 = g′(1)j2
1 .To show that the operator j0Lj0 is bounded below by cj2
0 on the subspace (χ′R)⊥, forsome c > 0 of the order O(1), we notice that the operator
L = −∂2r −
d− 1
r∂r + g′(χR(r))
on L2([0,∞), rd−1dr) is unitarily equivalent to the operator
− ∂2r +
d− 1
4
1
r2+ g′(χR(r)) (22.48)
on L2([0,∞), dr). Indeed, the transformation u 7→ rd−12 u from L2(rd−1dr) to L2(r
d−12 dr)
is unitary and maps L into (22.48) as follows from the computation
(∂2r +
d− 1
r∂r)(r
− d−12 v) = r−
d−12 (∂r −
d2 − 1
4r2)v.
Next, under the unitary shift ξ(r)→ ξ(r−R), the operator (22.48) is unitarily equivalentto
L := −∂2r +
d− 1
4(r +R)2+ g′(χ(r))
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on L2([−R,∞], dr).Now, under the unitary shift ξ(r) → ξ(r − R), supp j0 is mapped into a subset of
r ≥ −2R/3 and therefore L = L∗ + O(1/R) on r ≥ −2R/3, where L∗ := −∂2r +
g′(χ(r)). The operator L∗ is independent of R, has a non-degenerate eigenvalue 0 withthe eigenfuction χ′R and the essential spectrum σess(L∗) = [g′(−1),∞). Hence L∗ ≥ con the subspace (χ′R)⊥, for some c > 0 of the order O(1), and therefore, by the unitaryequivalence, this implies that j0Lj0 ≥ c0j
20 on the subspace (χ′R)⊥.
The estimates j0Lj0 ≥ c0j20 on the subspace (χ′R)⊥ and j1Lj1 ≥ g′(1)j2
1 , together withthe relation (22.47), imply that L ≥ c = min(c0, g
′(1)) on the subspace (χ′R)⊥, for largeenough R. This gives the assertion four.
∗ ∗ ∗ ∗ ∗
A Spaces and Operators: Review
In this section, we introduce the simplest and most commonly used spaces, Banach andHilbert spaces, and describe their most important examples.
A.1 Vector Spaces
We begin with the key definitions. A vector space, X, is a collection of elements, denoted,u, v, ..., for which the operations of addition, (u, v)→ u+ v and multiplication by a (realor complex) number, (α, u)→ αu, are defined in such a way that
(α + β)u = αu+ βu
andα(u+ v) = αu+ αv.
Recall that the operations of addition and multiplication have the following properties
u+ v = v + u (commutativity)
andα(βu) = (αβ)u (associativity).
Elements of a vector space are called vectors. As will be clear from the context most ofthe vector spaces we consider in these lectures are defined for multiplication by complexnumber (they are said to be vector spaces over complex numbers).
Let Ω ⊂ Rn be an open set of Rn. Main examples of vector spaces are:
(a) Rn = x = (x1, ..., xn)| −∞ < xj <∞ ∀ j– the Euclidean space of dimension n;
(b) Ck(Ω) – the space of k times continuously differentiable complex functions on Ω;
240 Lectures on Applied PDEs
(c) Lp(Ω) – the space of (measurable) complex functions, f on Ω s.t. |f |p is integrable,i.e.
∫Ω|f |p <∞;
(e) S(Rn) – space of C∞ functions vanishing at ∞ together with all their derivativesfaster than |x|−n for all n.
The spaces Lp(Ω), 1 ≤ p ≤ ∞, are introduced and studied in Section A.3.The sets above are vector spaces if we define addition and multiplication by real/complex
numbers in the point-wise way. Namely, for Rn we define
(x+ y)j = xj + yj and (αx)j = αxj ∀ jand, for the remaining spaces in (b) – (e), we define
Exercise A.1. Show that (a)-(e) are vector spaces. (Hint: For (c), use the inequality|f + g|p ≤ 2p (|f |p + |g|p) (see (A.3) below).)
A.2 Banach Spaces
To measure the size of vectors, one uses the notion of norm. A norm is defined to be amap
X 3 u→ ‖u‖ ∈ [0,∞)
which has the following properties:
(a) ‖u‖ = 0 ⇐⇒ u = 0;
(b) ‖αu‖ = |α|‖u‖;
(c) ‖u+ v‖ ≤ ‖u‖+ ‖v‖.
The last inequality is called the triangle inequality. (Observe an unusual notation for thenorm ‖ · ‖, not f(·) or n(·) as one would denote other maps). A vector space equippedwith a norm is called a normed vector space.
Having defined a norm, we can define the notion of (norm–) convergence as follows.Let fn ⊂ X be a sequence. We say that fn converges to f (∈ X), if and only if‖fn − f‖ → 0. We write fn → f .
A sequence fn ⊂ X is called a Cauchy sequence iff ‖fn − fm‖ → 0, as m,n→∞.A normed vector space X is called complete if and only if every Cauchy sequence
converges, i.e. if fn ⊂ X is a Cauchy sequence then fn converges (i.e. there is af ∈ X such that ||fn−f || → 0, as n→∞). Remark that the converse is always true: any
Lectures on Applied PDEs 241
convergent sequence is necessarily a Cauchy sequence. A complete normed vector spaceis called a Banach space.
Completeness is a very important property of a normed vector space, e.g. often onesolves an equation by successive approximations, and one wants to know that such ap-proximate solutions converge to an actual solution.
Examples: The vector spaces Rn and Lp(Ω) defined above are Banach spaces underthe norms
|x| := (n∑i=1
x2i )
1/2, ‖f‖p :=
(∫Ω
|f |p)1/p
if 1 ≤ p <∞.
Denote by Ckb (Ω) the subspace of Ck(Ω) consisting of all functions in Ck(Ω) which are
bounded together with all their derivatives up to the order k. We equip the space Ckb (Ω)
with the norm
‖f‖Ck = supx∈Ω
∑|α|=k
|∂αf(x)|,
where α = (α1, ..., αn) with αj non-negative integers, ∂α :=∏n
j=1 ∂αjxj and |α| = ∑n
i=1 αj.
Exercise A.2. Show that
(a) Ckb (Ω) is a vector space;
(b) ‖f‖Ck is a norm on Ckb (Ω).
It is shown in [8], Proposition 4.13 and Exercise 5.7 that the spaces Ckb (Ω) are complete,
i.e., that they are Banach spaces.
A.3 Lp–spaces
Consider an open subset Ω of the Euclidean space Rn. In particular Ω can be a boundedsubset or the entire Rn. Let dx denote the Lebesgue measure on Rn. We define theLp–space for 1 ≤ p <∞:
Lp(Ω) := f : Ω→ C | f is measurable and
∫|f |pdx <∞. (A.1)
In other words, f ∈ Lp(Ω)⇔ |f |p ∈ L1(Ω). We define the L∞-space:
L∞(Ω) := f : Ω→ C | f is measurable, and ess sup |f | <∞, (A.2)
where, recall that ess sup |f | := infsup |g| : g = f a.e.. We often use the abbreviationLp for Lp(Ω).
242 Lectures on Applied PDEs
Strictly speaking, elements of Lp(Ω) are equivalence classes of measurable functions:two functions define the same elements of Lp(Ω) if they differ only on a set of measure 0.
Lp, 1 ≤ p ≤ ∞ is a vector space. For p = ∞, this is obvious, and for 1 ≤ p < ∞, iteasily follows from the inequality
|f + g|p ≤ 2p (|f |p + |g|p) . (A.3)
The latter inequality is obtained as follows: |f + g|p ≤ (2 max(|f |, |g|))p ≤ 2p(|f |p + |g|p).We define for every f ∈ Lp:
‖f‖p :=
(∫|f |p)1/p
if 1 ≤ p <∞,ess sup|f | if p =∞.
(A.4)
Clearly, ‖f‖p = 0⇔ f = 0 a.e., and ‖cf‖p = |c| ‖f‖p, ∀c ∈ C. We have also the triangleinequality ‖f + g‖p ≤ ‖f‖p + ‖g‖p, which we prove later.
From these properties, it follows that the map f 7→ ‖f‖p is a norm. Hence Lp is anormed vector space for every 1 ≤ p ≤ ∞. In fact, it is a Banach space. The proof isgiven in [8], Theorem 6.6, Proposition 6.7 and Theorem 6.8.
There are three basic inequalities for Lp spaces:
1. Holder’s inequality: Let 1 ≤ p, q, r ≤ ∞, p−1 + q−1 = r−1, and f ∈ Lp, g ∈ Lq thenfg ∈ Lr and ‖fg‖r ≤ ‖f‖p‖g‖q.
2. Minkowski’s inequality: Let 1 ≤ p <∞ and f, g ∈ Lp, then ‖f + g‖p ≤ ‖f‖p + ‖g‖p.We prove these inequalities in Appendix A.4. Here we prove Holder’s inequality in easy,but instructive, special cases, (p−1, q−1) = (0, 1) and (p−1, q−1) = (1/2, 1/2).
For (p−1, q−1) = (0, 1), we have∫|fg| ≤ sup |f |
∫|g| = ‖f‖∞‖g‖1.
In the case (p−1, q−1) = (1/2, 1/2), we define first the unit vectors f = f/‖f‖2 andg = g/‖g‖2 (here we can assume f 6= 0 and g 6= 0 otherwise the result is trivial), so that‖f‖2 = 1 and ‖g‖2 = 1 (check this!). Integrating the inequality
|f |2 + |g|2 − 2|f g| ≥ 0,
and using that∫|f |2 = 1 =
∫|g|2, we obtain
1 ≥∫|f g|. (A.5)
Since the integral on the right hand side is∫|fg|/‖f‖2‖g‖2,
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we find after multiplying (A.5) by ‖f‖2‖g‖2,
‖f‖2‖g‖2 ≥∫|fg|
as claimed. Note that this inequality, which is a special case of the Holder inequality, hasits own name - the Schwarz inequality (with p = q = 2 and r = 1). One can prove thegeneral Holder inequality by interpolating between the (0, 1) and (1/2, 1/2) cases. But itis more elementary to prove it directly as we do in the next paragraph.
For more inequalities, see Appendix A.4.We can further generalize the C and Lp spaces to spaces of continuous and Lp integrable
functions from an open set Ω ⊂ Rn to a Banach space X. We denote such spaces asC(Ω, X) and L(Ω, X), respectively.
A.4 Supplement: Proofs of Holder’s and Minkowski’s inequali-ties
First, we prove Holder’s inequality. Observe first that it suffices to prove the Holderinequality for the case r = 1 (this follows easily from ‖fg‖r = (
∫|fg|r)1/r = (
∫|f |r|g|r)1/r,
and e.g. calling f r = f1 and gr = g1).We notice that the result is trivial if ||f ||p = 0 or ||g||p = 0 (for then f = 0 a.e. or
g = 0 a.e.), or if ||f ||p = ∞ or ||g||p = ∞. If neither of these cases hold, then we candefine
a =
∣∣∣∣f(x)
‖f‖p
∣∣∣∣p , b =
∣∣∣∣g(x)
‖g‖q
∣∣∣∣q and λ =1
p.
Below, we will show that for any a, b > 0, and 0 < λ < 1:
aλb1−λ ≤ λa+ (1− λ)b. (A.6)
Applying this to our case, we get
|f(x)g(x)|‖f‖p‖g‖q
≤ |f(x)|pp∫|f |p +
|g(x)|qq∫|g|q .
Integrating this inequality, and using p−1 + q−1 = 1, we arrive at Holder’s inequality.We finish the proof by showing (A.6). We can assume b 6= 0, so we can divide (A.6)
by b on both sides. (A.6) is then equivalent to the inequality (where t = a/b)
tλ − λt− 1 + λ ≤ 0.
Since 0 < λ < 1, the maximum of the function on the l.h.s. is reached at t = 1, and isequal to 0.
244 Lectures on Applied PDEs
Now we prove Minkowski’s inequality. If either p = ∞ or p = 1, then the result isobvious. Now assume p > 1 and f + g 6= 0. Then we estimate
|f + g|p ≤ (|f |+ |g|)|f + g|p−1
Integrating this inequality and applying Holder inequality with exponents p and p′ (1/p+1/p′ = 1) we obtain
∫|f + g|p ≤
∫|f ||f + g|p−1 +
∫|g||f + g|p−1
≤ ‖f‖p‖ |f + g|p−1‖p′ + ‖g‖p‖ |f + g|p−1‖p′
Taking into account that (p− 1)p′ = p, we find furthermore that∫|f + g|p ≤ (‖f‖p + ‖g‖p)‖f + g‖
pp′p
Now remember that∫|f + g|p = ‖f + g‖pp and observe that p/p′ = p− 1. Hence dividing
both sides of the latter inequality by ‖f + g‖p−1p gives Minkowski’s inequality.
Remark 2. Inequality (A.6) is a special case of the very useful Jensen’s inequality : letϕ be a convex function on [a, b] (i.e., ϕ satisfies ϕ(tx + (1 − t)y) ≤ tϕ(x) + (1 − t)ϕ(y)for all t ∈ [0, 1] and x, y ∈ [a, b]), and pk positive numbers satisfying
∑n1 pk = 1. We can
think about pk as probabilities. Then
ϕ(n∑1
pktk) ≤n∑1
pkϕ(tk), (A.7)
for all tk ∈ [a, b]. This inequality indeed implies (A.6) for ϕ(t) = et, n = 2, p1 = λ, t1 =ln a and t2 = ln b.
Exercise A.3. Prove Jensen’s inequality (A.7).
Theorem 16 (Hausdorff-Young inequality). Let 1 ≤ p ≤ 2 and p−1 + q−1 = 1. Then‖f‖q ≤ ‖f‖p.
We sketch the proof of the Hausdorff-Young inequality. Clearly, ‖f‖∞ ≤ ‖f‖1. More-over, we have shown that ‖f‖2 = ‖f‖2. For 1 < p < 2, the result follows from aninterpolation theorem (see e.g. [8]).
Note that the Hausdorff-Young inequality implies that F extends to a bounded oper-ator from Lp to Lq. We omit the proof of this statement.
Theorem 17 (Riemann-Lebesgue Lemma). Suppose that g is s.t. g ∈ L1. Then
i) g is bounded and continuous,
Lectures on Applied PDEs 245
ii) g decays at infinity: lim|x|→∞ g(x) = 0.
Proof. i) The boundedness is easily seen: ∀x,
|g(x)| =∣∣∣∣∫ eikxg(k)
∣∣∣∣ ≤ ∫ |g(k)| = ‖g‖L1 .
Next, we show continuity. Since g ∈ L1, then
limh→0
(g(x+ h)− g(x)) = limh→0
∫eik·x
(eik·h − 1
)g(k)dk = 0,
by the dominated convergence theorem (|eik·h− 1||g| ≤ 2|g|). This shows that g is contin-uous. Next, let us show ii). Since the Schwartz space S is dense in L1, there is a sequenceϕj ∈ S such that ||ϕj − g||1 → 0 as j → 0. Thus
||ϕj − g||∞ ≤∫|ϕj(k)− g(k)| = ||ϕj − g||1 → 0,
which shows that ϕj → g uniformly on Rn. But ϕj ∈ S, so ϕj → 0 as |x| → ∞, andtherefore g → 0 as |x| → ∞.
A.5 Hilbert Spaces
To measure angles between vectors one introduces the inner product (sometimes calledthe scalar product ).
The map (f, g)→ 〈f, g〉 ∈ C is called an inner product if and only if
• 〈f, g〉 is linear in the second argument, i.e. 〈f, αg + βh〉 = α 〈f, g〉 + β 〈f, h〉, forany α, β ∈ C,
• 〈f, g〉 = 〈g, f〉 (this together with the above implies that 〈f, g〉 is anti-linear in thefirst argument: 〈αf + βh, g〉 = α 〈f, g〉+ β 〈h, g〉),• 〈f, f〉 ≥ 0 with equality if and only if f = 0.
We remark that sometimes, the scalar product is taken to be linear in the first argumentand anti-linear in the second one.
The space L2(Ω) is a Hilbert space if we define the inner product as
〈f, g〉 :=
∫Ω
fg. (A.8)
Exercise A.4. Show that (A.8) is an inner product.
applied to u = ±f/‖f‖ and v = g/‖g‖ and to u = ±if/‖f‖ and v = g/‖g‖.
Exercise A.5. Check that (A.9) defines a norm. (Hint: to prove the triangle inequality,use the Schwarz inequality).
Thus a space with an inner product, or an inner product space, is also a normed space.A complete inner product space is called a Hilbert space. Clearly a Hilbert space is also aBanach space. An inner product, in addition to measuring sizes and distances, measuresrelative directions, e.g.,
f ⊥ g ⇔ 〈f, g〉 = 0.
Given a norm ‖f‖, how can we tell whether this norm comes from an inner product?The answer to this question is that ‖f‖ comes from an inner product if and only if itsatisfies the parallelogram law
‖f + g‖2 + ‖f − g‖ = 2‖f‖2 + 2‖g‖2.
In this case the inner product is given by (in the real case)
〈f, g〉 =1
2
(‖f + g‖2 − ‖f‖2 − ‖g‖2
)(A.11)
(see (A.10)).
Exercise A.6. Check that (A.11) defines a (real) inner product.
The space Rn has an inner product defined by x · y =∑n
i=1 xiyi.
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A.6 Sobolev spaces
Now we want to introduce an additional structure on Lp–spaces which measures smooth-ness, similar to the smoothness properties of functions in Ck. We do so only for p = 2,i.e. for the space L2(Rn). This is the most useful space among the Lp–spaces as it has aninner product:
〈f, g〉 :=
∫fg
and therefore it is a Hilbert space (i.e., recall, an inner product space which is complete withrespect to the norm ‖f‖ :=
√〈f, f〉 induced by the inner product). Another advantage
of the L2–space is that the Fourier transform leaves it invariant (i.e. f ∈ L2 ⇒ f ∈ L2).We now define for s integer, s ≥ 0, the new spaces
Hs(Rn) = f ∈ L2(Rn) : ∂αf ∈ L2(Rn) ∀α s.t. |α| ≤ s. (A.12)
This definition is very similar to the definition of the Cs(Rn)–spaces: in fact, byreplacing L2(Rn) in (A.12) by C(Rn), one obtains the definition of Cs(Rn). But thereis one crucial difference: in the Cs(Rn)–case, the functions f are assumed to be s timescontinuously differentiable, but in the Hs–case, they are not. Namely, the derivatives ∂αfin the above definition are understood in the distributional sense:
∂αf(ϕ) := (−1)|α|f(∂αϕ),
where f(ϕ) =∫fϕ, and ϕ ∈ S(Rn). In other words ∂αf is a linear functional defined
by the r.h.s. of the above equation and the relation in (A.12) says that ∀α with |α| ≤ s,there exists a gα ∈ L2(Rn) s.t. ∂αf(ϕ) =
∫gαϕ ∀ϕ.
The space Hs(Rn) equipped with the inner product
〈f, h〉(s) =∑|α|≤s
〈∂αf, ∂αg〉 (A.13)
is a Hilbert space.There is another way of defining the spaces Hs(Rn) using the Fourier transform defined
in the next appendix:
Hs(Rn) = f ∈ L2(Rn) : 〈k〉sf(k) ∈ L2(Rn), (A.14)
where 〈k〉 = (1 + |k|2)1/2. The definition (A.14) has the advantage that it makes sense foran arbitrary s ∈ R. Besides, it does not require extra explanations. Of course we haveto show that definitions (A.12) and (A.14) are equivalent for positive integers s. One canshow easily that the definitions (A.12) and (A.14) are equivalent for positive integers s.
The following result is often used in applications
Theorem A.7 (Sobolev embedding theorem). Hk(Rn) ⊂ Cα(Rn), with α < k − n2.
The definitions and properties above could extended to any open domain in Rn.
248 Lectures on Applied PDEs
A.7 Linear operators
Linear operators or simply operators are linear maps from one vector space Y into anothervector space X. We denote linear operators usually by capital roman letters, A,B, . . .and use the notation
A : Y → X (A.15)
and Au to denote an operator A mapping Y into X and application of A to a vectoru ∈ Y , respectively. To define an operator A, we have to give a rule that prescribes toeach element of Y an element of X (the image of u). We require this rule to be linear,i.e. ∀u, v ∈ Y , and α, β ∈ C:
A(αu+ βv) = αAu+ βAv. (A.16)
To fix ideas here and in what follows, we consider vector spaces over the complex numbersC, i.e. complex vector spaces. All the material of this section, except for spectral theory,remains unchanged if we substitute R for C.
If the space Y in (A.15) is a subset of the space X then sometimes one calls Y thedomain of A (in X) and denotes it D(A) ≡ Y . In this case we say that A is defined in Xwith domain D(A)(= Y ). The range (or image) of A is defined as
Ran (A) := Au : u ∈ Y ≡ AY.
Ran (A) is a vector space (show this). We may assume that D(A) is dense in X, i.e. forany u ∈ X, there is a sequence un ⊂ D(A) s.t. un → u as n → ∞. Indeed, if D(A) isnot dense to begin with, we consider instead of the space X simply the space Y := D(A),the closure of D(A), which is obtained by adding to Y limits of all possible sequencesun convergent in X.
Examples.1) The identity operator 1l : Lp → Lp;2) The multiplication operator Mf : L→Lp, u 7→ fu for a fixed f ∈ L∞;3) The differentiation operator ∂
∂xjin L2(Rn) with the domain D( ∂
∂xj) = H1(Rn);
4) The Laplacian ∆ :=∑n
1∂2
∂x2jin L2(Ω) with the domain D(∆) = H2(Ω);
5) Integral operators, i.e., operators of the form
(Ku)(x) =
∫K(x, y)u(y)dy,
for some function K(x, y) (called the kernel or integral kernel). The domain and range ofthe integral operator K depend on the properties of the kernel K(x, y).
6) The Fourier transform F : L1(Rn)→ L∞(Rn),
F : u(x)→ (2π)−n/2∫
e−ik·xu(x)dx;
Lectures on Applied PDEs 249
7) Convolution operator Cf : Lp → Lp, Cf : u→ f ∗ u for fixed f ∈ L1, where
(f ∗ u)(x) =
∫f(x− y)u(y)dy;
8) The wavelet transform
Wψ : f →∫ψabfdx,
where ψab = 1√|a|ψ(x−ba
)for a fixed function ψ satisfying
∫ |ψ(k)|2k
dk <∞.
Note that the Fourier transform and convolution are integral operators with the inte-gral kernels (2π)−n/2eik·x and f(x− y), respectively.
In fact also in examples 1)-4), the operators can be represented as integral operators,but with distributional kernels, e.g. K(x, y) = f(x)δ(x − y) for Mf , and K(x, y) =−δ′(xj − yj)
∏i 6=j δ(xi − yi) for ∂
∂xj.
For an operator A : Y → X we define the norm
‖A‖ ≡ ‖A‖Y→X = sup‖u‖Y =1
‖Au‖X . (A.17)
If Y = X and ‖A‖ <∞, then A is said to be bounded (in X). Observe that our definitionimplies that
‖Au‖ ≤ ‖A‖‖u‖ (A.18)
for all u ∈ Y . Now, if there is a constant C (independent of u) such that
‖Au‖ ≤ C‖u‖, (A.19)
for all u ∈ D(A), and if the space X is complete (i.e. a Banach space), then the operator Acan be extended by continuity to the whole space X as a bounded operator. The smallestconstant C satisfying (A.19) is the norm ‖A‖ of A, and so ‖Au‖ ≤ ‖A‖ ‖u‖ (so boundedoperators form a Banach algebra).
Examples of bounded operators are the identity operator, 1, of example 1) above (infact, ‖1‖ = 1), the multiplication operator, Mf , of example 2), the integral operator, K,of example 6) with a kernel K(·, ·) ∈ L2(Rn×Rn), as an operator from L2(Rn) to L2(Rn).
Exercise A.8. Show that
(1) ‖AB‖ ≤ ‖A‖‖B‖;
(2) ‖w‖ = sup‖v‖=1 | 〈w, v〉 |, and therefore
‖A‖ = sup‖u‖,‖v‖=1
| 〈Au, v〉 |;
250 Lectures on Applied PDEs
(3) ‖Mf‖ = ‖f‖∞ (Example 2 above);
(4) for integral operators (Example 5 above)
‖K‖L2→L2 ≤(∫|K(x, y)|2dxdy
)1/2
provided the r.h.s. is finite.
Exercise A.9. Show that the differentiation operator in example 3) is not bounded inL2(Rn) by finding a sequence fn of functions from D( ∂
∂xj) such that ‖fn‖ ≤ 1, ∀n, and
‖ ∂∂xjfn‖2 →∞, as n→∞.
It is easy to see that F : L1(Rn)→ L∞(Rn),∣∣∣∣(2π)−n/2∫eix·kf(x)dx
∣∣∣∣ ≤ (2π)−n/2∫|f(x)|dx.
It is considerably more difficult to show that F extends from L1(Rn)∩L2(Rn) to a boundedoperator on L2(Rn). In fact F is an isometry in the sense that
‖Ff‖L2 = ‖f‖L2
(the Plancherel theorem, see Appendix C).Moreover, by the Hausdorff-Young inequality (see Theorem 16 of Appendix A.4), F
extends to a bounded operator from Lp to Lq, for 1 ≤ p ≤ 2 and p−1 + q−1 = 1. Moreprecisely,
‖Ff‖q ≤ ‖f‖p.We say an operator A : Y → X is invertible if and only if there is an operator
A−1 : X → Y such that A−1A = 1lY and AA−1 = 1lX , where 1lX and 1lY are the identityoperators in X and Y respectively.
Exercise A.10. Show that:1) A is invertible if and only if for every f ∈ X, the equation Au = f has a uniquesolution u(= A−1f) ∈ Y , i.e. if and only if A is one–to–one (Au = 0⇒ u = 0) and onto(Ran A = X).2) if A is just one-to-one (i.e. not necessarily onto), then A is invertible as an operatorfrom Y to Ran A ⊂ X, i.e. the equation Au = f has a unique solution u = A−1f for anyf ∈ Ran A.3) if A and B are invertible then so is AB and (AB)−1 = B−1A−1 (generalize to anarbitrary number of factors).
Example A.1. We show invertibility of several important operators we encounteredabove.
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1. 1 is clearly invertible.
2. Mf is invertible for essinf |f | > 0 and M−1f = M1/f .
3. ddx
(see discussion below).
4. F is invertible with F−1 given in (C.3) (see Theorem C.1(f)).
5. −∆+1 is invertible with (−∆+1)−1f = G∗f where G := F−1( 1|k|2+1
) and ∗ denotesconvolution.
6. curl is invertible with curl−1w =∫ (x−y)2
|x−y|2 w(y)dy in 2d (Biot-Savart formula).
Consider the operator A = ddx
: H1(R) → L2(R). Then A is one-to-one: Au = 0 andu ∈ H1(R) imply u ≡ 0. Thus A has the right inverse B: f(x) →
∫ xx0f(y)dy for some
fixed x0 ∈ R: AB = 1l. However, B does not map L2(R) into H1(R). In fact it is notdefined on the entire L2(R) but only on L2(R)∩L1(R) and maps this space into L∞. Putdifferently, the operator A is not onto: Ran A = f ∈ L2(R)|
∫∞−∞ fdx = 0 6= L2(R).
Hence A in not invertible.The above illustrates the importance of finding inverses of operators: existence of an
inverse for A : Y → X is equivalent to the equation Au = f having a unique solution forevery f ∈ X. The following simple statement gives a powerful criterion for existence ofinverses.
Theorem A.11. Assume an operator A : X → X is invertible and an operator B : X →X is bounded with the norm satisfying the inequality
‖B‖ < ‖A−1‖−1
Then the operator A+B (defined on the domain of A) is invertible. Moreover, its inverseis given by the absolutely convergent series
(A+B)−1 =∞∑n=0
A−1(−BA−1)n, (A.20)
called the Neumann series (for A+B).
Exercise A.12. Prove this theorem. Hint: Show that the series (A.20) is absolutelyconvergent and gives the inverse to A + B and use that if Tn is a Cauchy sequence ofoperators from X to X, then there is an operator T : X → X s.t. Tn → T (one has to usethe fact that the space of bounded operators equipped with the operator norm is complete,i.e. is a Banach space).
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A.8 Special classes of operators
The adjoint. Consider an operator A on a Hilbert space X. With it we associate itsadjoint A∗ defined by the relation 〈A∗u, v〉 = 〈u,Av〉, for all v ∈ D(A), and for all u’ssuch that supv∈D(A),‖v‖=1 |〈u,Av〉| < ∞ (those u’s form the domain of the operator A∗,D(A∗)).
If A is a bounded operator then sup‖v‖=1 |〈u,Av〉| ≤ ||u||||A|| < ∞ and it suffices tocheck only the relation 〈A∗u, v〉 = 〈u,Av〉.
Exercise A.13. Show that if operators A and B are bounded, then(a) A∗ is a bounded operator and ‖A‖ = ‖A∗‖ (Hint: use that ‖A‖ = sup||u||,||v||=1 |〈u,Av〉|,
An important class of operators on a Hilbert space is the class of self–adjoint operators.By definition, an operator A is called self–adjoint if and only if A∗ = A. By definition,every self–adjoint operator is symmetric, i.e. 〈Au, v〉 = 〈u,Av〉, for all u, v ∈ D(A). Noticethat the converse is not true. If an operator A is symmetric then all we know is thatD(A) ⊂ D(A∗) (show this!). However, a symmetric operator A obeying D(A) = D(A∗) isalso self-adjoint. Thus every symmetric bounded operator is self–adjoint.
Consider the examples of the operators 1)–5) above. We have the following: Mf issymmetric if and only if f is a real function; ∂
∂xjis anti–symmetric, so the differentiation
operator is not symmetric, but −i ∂∂xj
is symmetric; the identity operator is obviously
symmetric; ∆ is symmetric and so is −∆ + V (x) for V (x) real; the integral operatoris symmetric if K(x, y) = K(y, x) (cf with matrices!). In addition, if we know thatK(x, y) ∈ L2(Rn × Rn), then the operator K is self-adjoint. A point is that while thesymmetry property is easy to verify, the self-adjointness property is hard. See [?] or [16]for a proof of self-adjointness of −i ∂
∂xj, ∆ and −∆ + V (x).
We observe that for any operator A on a Hilbert space X, we can write
X = NullA⊕ Ran A∗ (A.21)
Here, the null space is defined as NullA := u ∈ X : Au = 0.
Exercise 11. Show that for a bounded operator A, NullA is a closed set, and show (A.21).
Projections. A bounded operator P on X is called a projection operator (or simply aprojection) if and only if it satisfies
P 2 = P.
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This relation implies ‖P‖ ≤ ‖P‖2, i.e. ‖P‖ ≥ 1. We have
v ∈ Ran P ⇐⇒ Pv = v and v ∈ (Ran P )⊥ ⇐⇒ P ∗v = 0. (A.22)
Indeed, if v ∈ Ran P , then there is a u ∈ X s.t. v = Pu, so Pv = P 2u = Pu = v; thesecond statement is left as an
Exercise 12. Prove that (a) P ∗v = 0 if and only if v ⊥ Ran P , (b) Ran P is closed and(c) P ∗ is also a projection.
Examples. The following are projection operators :1) on a space of functions u(x), χx∈E : u(x) 7→ χE(x)u(x), where χE(x) = 1 if x ∈ E
and χE(x) = 0 if x /∈ E,2) on a space of functions u(x), χp∈E = F−1χx∈E F acting as u(x) 7→ (χE(k)u(k)) (x),3) on any Hilbert space, where ϕ, ψ are two fixed elements s.t. 〈ϕ, ψ〉 = 1: u 7→ 〈ϕ, u〉ψ4) as in 3), but where now ψi is an orthonormal set (i.e. 〈ψi, ψj〉 = δi,j): u 7→∑N
1 〈ψi, u〉ψi.
A projection P is called an orthogonal projection if and only if it is self-adjoint, i.e. ifand only if P = P ∗. Let P be an orthogonal projection, then (A.22) implies that
v ⊥ Ran P ⇐⇒ Pv = 0, i.e., NullP = (Ran P )⊥. (A.23)
The projections in Examples 1), 2) and 4) above are orthogonal. The projection inExample 3) is orthogonal if and only if ϕ = ψ.
Exercise 13. Let P be an orthogonal projection. Show that(a) ||P || ≤ 1, and therefore ||P || = 1 (Hint: Use (A.22)),(b) 1l − P is also an orthogonal projection and Ran (1l − P )⊥Ran P and Null 1l− P =Ran P ,(c) X = Ran P ⊕ NullP .
Remark. Orthogonal projections on X are in one-to-one correspondence with closedsubspaces of a Hilbert space X. This correspondence is obtained as follows. Let V =Ran P . Then V is a closed subspace of X. To show that V is closed, let vn ⊂ V ,and vn → v ∈ X, and show that v ∈ V . Since P is a projection, we have vn = Pvn, so||v−Pv|| = ||v− vn−P (v− vn)|| ≤ ||v− vn||+ ||P || ||v− vn|| → 0, as n→∞. Thereforev = Pv, so v ∈ V , and V is closed.
Conversely, given a closed subspace V , define a projection operator P by
Pu = v, where u = v + v⊥ ∈ V ⊕ V ⊥. (A.24)
Exercise 14. Show that P defined in (A.24) is an orthogonal projection with Ran P = V .For any given V , show that there is only one orthogonal projection (the one given in(A.24)) such that Ran P = V .
(*Galerkin approximation (A→ PAP )*)
254 Lectures on Applied PDEs
The space of bounded linear operators L(X, Y ) We assume that X and Y arenormed vector spaces over C, and consider the set of all bounded linear operators fromX into Y , i.e. each such operator is defined on the entire space X, and its range lies inY . This set of operators is denoted by L(X, Y ).
For A,B ∈ L(X, Y ), we define a new operator, called A+ B, by setting (A+ B)u :=Au + Bu, for all u ∈ X. Also, for λ ∈ C and A ∈ L(X, Y ), we define a new operatorλA as (λA)u = λAu, for all u ∈ X. If in addition to these two operations on operators,we equip the set L(X, Y ) with the norm introduced in (A.17), then L(X, Y ) is a normedvector space.
Exercise 15. Show that L(X, Y ) is a vector space.
An important question is: when is L(X, Y ) a Banach space? The answer is given inthe following theorem, which is not difficult to prove (see e.g. [8], Proposition 5.3):
Theorem 18. If Y is a Banach space, then L(X, Y ) is a Banach space.
The dual space. In the special case when Y = C, the space L(X, Y ) is called the dualspace of X (or simply the dual, or adjoint space or conjugate space of X), and it is denotedas X ′. Hence the elements of X ′ := L(X,C) are linear maps from X to C, and they arecalled linear functionals. Remark also that since C is complete, then the last theoremshows that X ′ is always a Banach space, whether X is complete or not.
The operator norm induces a norm on X ′: if l ∈ X ′, then
||l|| = sup||x||=1
|l(x)|.
If X is a space of functions, then X ′ can be identified with either a space of functions ora space of distributions or a space of measures. Here are some examples of dual spaces:
1) (Lp)′ = Lq, where 1/p+ 1/q = 1, if 1 ≤ p <∞ (space of functions),
2) (L∞)′ is a space of measures which is much larger than L1,
3) (Hs)′ = H−s (space of distributions if s > 0).
Note that (Lp)′ ⊃ Lq, for 1 ≤ p <∞ follows from the Holder inequality. In fact, givenf ∈ Lq, define lf (u) :=
∫fu. Since |lf (u)| ≤ ||f ||q||u||p, we see that lf is a bounded linear
functional on Lp. It can be shown that in fact any bounded linear functional on Lp canbe represented by lf for some f ∈ Lq.
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B Gateaux and Frechet derivative
Our goal is is to develop a differential calculus of maps between Banach spaces. Let Xand Y be Banach spaces and U , an open subset of X. We consider a map F : U → Y .The map F is called Gateaux differentiable at u ∈ U if and only if, for any ξ ∈ X, thelimit
dF (u)ξ :=∂
∂λ
∣∣∣∣λ=0
F (u+ λξ), (B.1)
exists and is bounded as a map from X to Y in the sense that sup‖ξ‖≤r |dF (u)ξ| . 1. Themap dF (u) is called the Gateaux derivative or sometimes, the variational derivative.
The map F is called continuously differentiable, or C1 (written F ∈ C1) in U if andonly if it is Gateaux differentiable for very u in U , and u 7→ dF (u) is a continuous mapfrom U ⊂ X to the space L(X, Y ) of bounded operators from X to Y ; i.e. if un → u inX, then dF (un)→ dF (u) in L(X, Y ).
Example B.1 (formal). 1) If F (u) = Lu, where L is a linear map, then dF (u) = L(independently of u). Indeed, dF (u)ξ = ∂
∂λL(u+λξ)|λ=0 = ∂
∂λ(Lu+λLξ)|λ=0 = Lξ.
Thus if L is bounded, then F is C1.
2) If F (u) = f u (composition map), for a fixed C1–function f : R → R, andu : Rn → R, then dF (u) is the multiplication operator by f ′(u). Indeed, dF (u)ξ =∂∂λF (u+λξ)|λ=0 = ∂
∂λf(u(x)+λξ(x))|λ=0 = f ′(u)ξ. So if f ′(u) is a bounded function,
say for some u ∈ Lp(Rn), then F : Lp(Rn)→ Lp(Rn) is differentiable at u.
Exercise B.1. Compute dF (u) for F : Rn → Rm, and for the mean curvature map
F (u) = div
(∇u√
1 + |∇u|2
).
Exercise B.2. Let Ω be a bounded domain in Rn with a smooth boundary and f ∈Ck+1(R). Show that
(a) the map F (u) = f u satisfies F : Ck(Ω) → Ck(Ω), and F is C1 with dF (u)ξ =f ′(u)ξ and
(b) the map F (u) = −∆u + f u satisfies F : Ck(Ω) → Ck−2(Ω), and F is C1 withdF (u)ξ = (−∆ + f ′(u))ξ.
As usual, the symbol o(‖ξ‖) stands for a map R : X → Y s.t.
‖R‖‖ξ‖ → 0, as ‖ξ‖ → 0.
The following properties of the Gateaux derivative play an important role in applications.
Theorem B.3. (check) Let F be C1 in U . Then
256 Lectures on Applied PDEs
(a) dF (u) is linear;
(b) (The fundamental theorem of calculus)
F (u+ ξ)− F (u) =
∫ 1
0
dsdF (u+ sξ)ξ. (B.2)
(c) (The chain rule) If F and G are Gateaux differentiable at u ∈ U and F (u) ∈ U ,respectively, then we have d(G F )(u) = dG(F (u))dF (u).
(d) Let K be a convex subset of a Banach space X (i.e. if u, v ∈ K, then su+(1−s)v ∈K, for all s ∈ [0, 1]). If F : K → K satisfies ‖dF (ψ)‖ ≤ α, ∀ψ ∈ K, then F isLipschitz:
‖F (ψ)− F (ϕ)‖ ≤ α‖ψ − ϕ‖, ∀ψ, ϕ ∈ K.
(e) If F is C1 at u ∈ X, then as ‖ξ‖ → 0
F (u+ ξ) = F (u) + dF (u)ξ + o(‖ξ‖). (B.3)
Proof. We begin with (b). For u, ξ ∈ X fixed, define the function g : [0, 1]→ Y by
g(t) = F (u+ tξ),
According to the definition of the Gateaux derivative (B.1), we have
g′(t) = dF (u+ tξ)ξ.
By Fundamental Theorem of Calculus,
g(1)− g(0) =
∫ 1
0
g′(t)dt. (B.4)
Since g′(t) = dF (u+ tξ)ξ, this gives the first statement.
Exercise B.4. Show the properties (c) and (e).
For (e), using (B.2) and writing
F (u+ ξ)− F (u)− dF (u)ξ =
∫ 1
0
ds(dF (u+ sξ)− dF (u))ξ, (B.5)
and letting ‖ξ‖ = r, we find
‖F (u+ ξ)− F (u)− dF (u)ξ‖
≤ supu,u′,ξ′:‖u′−u‖≤r,‖ξ′‖≤r
‖(dF (u′)− dF (u))ξ′‖∫ 1
0
ds1. (B.6)
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Since dF (u) is continuous in u, we have supu,u′,ξ′:‖u′−u‖≤r,‖ξ′‖≤r ‖(dF (u′)−dF (u))ξ′‖ = o(r),which gives (B.3).
Finally, for (a), it follows from the derivative definition that dF (u) is positive homo-geneous of degree 1. To show that it is additive, we set u = 0, for simplicity of notation.Then (e) ((B.3)) implies that F (ξ + η)−F (0) = dF (0)(ξ + η) + o(‖ξ + η‖). On the otherhand, F (ξ + η)− F (0) = F (ξ + η)− F (ξ) + F (ξ)− F (0) = dF (ξ)η + o(‖η‖) + dF (0)ξ +o(‖ξ‖) = dF (0)ξ + dF (0)η + o(‖ξ‖) + o(‖η‖). These two relations yield the additivity,dF (0)(ξ + η) = dF (0)ξ + dF (0)η.
We conclude this section with some useful rigorous results about Gateaux derivativesof composition operators F (u) = f u, where f is a fixed function and u belongs tothe space of differentiable functions. Such operators appear often in applications. Thestatements below are useful in this context. An important result in this direction is thefollowing.
Proposition B.5. Let Ω ⊂ Rn. Let F (u) = f u with f ∈ C2(R) and obeying theestimates
|f (k)(u)| ≤ c|u|p−k for k = 0, 1, 2, (B.7)
for some p ≥ 2. Then F : Hr(Ω)→ L2(Ω) and is C1, provided r > n/2.
Proof. Let u ∈ Hr(Ω). Then by the Sobolev embedding theorem (see Section A.6) u ∈L2(p+1)(Ω) for r > n
2− n
2p, and ‖u‖L2p . ‖u‖Hr . Hence, by (B.7), with k = 0, we have
‖F (u)‖2L2 =
∫Ω
|f(u)|2 ≤ c
∫Ω
|u|2p (B.8)
= c‖u‖2pL2p ≤ c′‖u‖2p
Hr . (B.9)
This shows that F : Hr(Ω) → L2(Ω). To show that F has a Gateaux derivative, wecompute formally dF (u)ξ = f ′(u)ξ. By (B.7), with k = 1, and the Sobelev embeddingtheorem, we have f ′(u) ∈ L∞ (indeed, ‖f ′(u)‖L∞ . ‖u‖p−1
L∞ . ‖u‖p−1Hr , provided r > n/2)
and therefore dF (u) is a bounded operator.To show that F ∈ C1, we use the mean value theorem to estimate
‖dF (u′)− dF (u)‖L2 ≤ ‖f ′(u′)− f ′(u)‖L∞ (B.10)
≤ ‖f ′′(u)‖L∞‖u′ − u‖L∞ , (B.11)
where u := θu′ + (1 − θ)u for some θ ∈ [0, 1]. Using this, (B.7), with k = 2 and theSobolev embedding theorem, we find, for r > n/2,
Corollary B.6. Assume f : C→ C satisfies estimates (B.7) with 1 ≤ p < 2(n−2r)+
, r > 0.
Define F (u) = −∆ + f(u). Then F : Hr(Ω)→ L2(Ω) and is C1.
Exercise B.7. Prove that, for r > n/2, under the conditions of Proposition B.5, F = fu(a) maps Hr(Ω) into H1(Ω) and (b) is Gateaux differentiable, as a map from Hr(Ω) intoL2(Ω).
Furthermore, we have
Theorem B.8. Let F (u) = f u, Ω ⊂ Rn be a bounded domain with smooth boundary,and let r > n/2. If f ∈ Cr+1(R), then F : Hr(Ω)→ Hr(Ω), and F is C1.
Exercise B.9. Prove this theorem for n = 1 and r = 1.
For a complete proof of Theorem B.8, see [20], page 221.
Example B.2. For an exercise, we derive directly relation (B.3) for the map F (u) = gu.We have
f(u+ ξ)− f(u) = f ′(u)ξ +R(ξ)
where R(ξ) =∫ 1
0(f ′(u+ tξ)− f ′(u))ξ dt. Now we estimate by the mean value theorem
||R(ξ)||2 ≤∫ 1
0
‖f ′′(u+ tξ)ξ2‖2t dt
for some 0 ≤ t ≤ t. Using the estimate |f ′′(u)| ≤ c|u|p−2 and the triangle and Holderinequalities we derive furthermore
‖R(ξ)‖2 ≤ c
∫ 1
0
‖|u+ tξ|p−2ξ2‖2t dt
.(‖|u|p−2ξ2‖2 + ‖ξp‖2
)≤
(‖u‖p2p‖ξ‖2
2p + ‖ξ‖p2p).
Hence ‖R(ξ)‖2 → 0 and therefore ‖f(u + ξ) − f(u) − f ′(u)ξ‖2 → 0 as ‖ξ‖ → 0. Thisshows (B.3) and consequently F is C1.
Though the Gateaux derivative is straightforward to compute, for theoretical consid-erations, one needs often a stronger notion of derivative: the Frechet derivative. Beforewe define the Frechet derivative, let us remark that equation (B.1) is equivalent to
F (u+ λξ)− F (u) = λdF (u)ξ + o(λ), (B.13)
were o(λ) is a vector in Y satisfying limλ→0 ‖o(λ)‖/λ = 0. Notice that in general, o(λ)depends on ξ.
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The map F is called Frechet differentiable at u ∈ X if and only if there exists abounded linear map dF (u) ∈ L(X, Y ) s.t. (B.3) holds as ‖ξ‖ → 0. The operator dF (u)satisfying (B.3) is called the Frechet derivative of F at the point u.
From this definition and equation (B.13), it is clear that if F is Frechet differentiable atu, with Frechet derivative dF (u), then F is Gateaux differentiable at u with the Gateauxderivative given by the same operator dF (u). In the opposite direction, if F ∈ C1, thenthe preceding theorem implies
Theorem B.10. If F is continuously Gateaux differentiable at u ∈ X, with Gateauxderivative dF (u), then F is Frechet differentiable at u, and the Frechet derivative is givenby the same operator dF (u).
For a detailed discussion of Frechet and Gateaux derivatives, we refer to [1, ?].In everything that follows, by the derivative dF (u) we understand the Gateaux deriva-
tive. We point out that in most of our applications, we deal with C1 maps, in which casethe Frechet and Gateaux derivatives coincide, according to the last theorem.
C Fourier transform
In this section, we describe one of the most powerful tools in analysis – the Fouriertransform. This transform allows us to analyze a fine structure of functions and to solvedifferential equations. The Fourier transform takes functions of time to functions offrequencies, functions of coordinates to functions of momenta, and vice versa.
Initially, we define the Fourier transform on the Schwartz space S(Rn) = S:
S = f ∈ C∞(Rn) : 〈x〉N |∂αf(x)| is bounded ∀N and ∀α, (C.1)
where 〈x〉 = (1 + |x|2)1/2 and α = (α1, ..., αn), with αj non-negative integers, ∂α :=∏nj=1 ∂
αjxj and |α| = ∑n
i=1 αj. On S, we define the Fourier transform F : f 7→ f by
f(k) := (2π)−n/2∫f(x)e−ik·xdx. (C.2)
Define also the inverse Fourier transform of f(k) as
f(x) := (2π)−n/2∫f(k)eix·kdk. (C.3)
Some key properties of the Fourier transform are collected in the following
Theorem C.1. Assume f, g ∈ S(Rn). Then we have:
(a) (−i∂)αf 7→ kαf , and xαf 7→ (−i∂)αf ,
260 Lectures on Applied PDEs
(b) fg 7→ (2π)−n/2f ∗ g, and f ∗ g 7→ (2π)n/2f g,
(c) (f ) = f = (f ) ,
(d) f = f ,
(e)∫f g =
∫fg,
(f)∫f g =
∫fg.
Properties (a) - (f) hold (possibly, with signs changed in (a)) also whenˆ is replaced by .
We give a formal proof. Integrating by parts, we compute
−i(∂xjf ) (k) = (2π)−n/2∫
(−i)∂xjf(x)e−ik·xdx
= (2π)−n/2∫f(x)i∂xje
−ik·xdx
= kj f(k).
Exercise 16. Prove the remaining relations in (a)
Now we prove the second relation in (b). Using e−ik·x = e−ik·(x−y)e−ik·y and changingthe variable of integration as x′ = x− y, we obtain
f ∗ g (k) := (2π)−n/2∫e−ik·x(
∫f(x− y)g(y)dy)dx
= (2π)−n/2∫
(
∫e−ik·(x−y)f(x− y)dx) e−ik·yg(y)dy
= (2π)n/2f(k′)f(k)g(k).
Exercise 17. Prove the first relation in (b) from the second one and (c).
The proof of (c) is more subtle. We use an approximation of unity ϕt(x) = t−nϕ(x/t)and compute ϕt ∗ (f ) . Let us define ϕx(y) := ϕ(x− y). Using property (b), we find
ϕt ∗ (f ) =
∫ϕxt · (f ) dy =
∫(ϕxt ) fdy =
∫((ϕxt ) ) fdy.
Exercise 18. Show (formally, without justification of the interchange of the order ofintegration etc.) that
((ϕxt ) ) = ((ϕt) )x = t−n(ϕ)
(x− yt
).
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Thus we haveϕt ∗ (f ) = ((ϕ) )t ∗ f (C.4)
We can choose ϕ such that (ϕ) ∈ L1, and∫
(ϕ) (x)dx = 1. Indeed, take e.g. ϕ(x) =
(4π)−n/2e−|x|2
and use the fact that ((e−|x|2) ) = e−|x|
2. With this in mind, we take the
limit t→ 0 in (C.4) and use the properties of the approximation of identity to get
ϕt ∗ (f ) → (f ) and ((ϕ) )t ∗ f → f as t→ 0
to obtain (f ) = f . Similarly one shows that (f ) = f .
Exercise 19. Prove the relations in (d) – (f).
By definition of the Dirac δ–function, we obtain
F : δ(x− x0)→ (2π)−n/2e−ik·x0 .
Hence the property (c) implies that F−1 : (2π)−n/2e−ik·x0 → δ(x−x0), and, by taking thecomplex conjugate (remember that F(f) = F∗(f) = F−1(f)), we arrive at
F : (2π)−n/2e−ik0·x 7→ δ(k − k0). (C.5)
Exercise 20. Using (C.5), prove formally that (f ) = f = (f ) , and that (fg) =(2π)−n/2f ∗ g.
Statement (f) is called the Plancherel Theorem. The adjoint F∗ of the Fourier trans-form is defined by 〈F∗u, v〉 = 〈u,Fv〉 for all u, v ∈ S(Rn), where 〈·, ·〉 is the standardinner product in L2(Rn). Then (d) and (e) show that F∗ = F−1. This together with (e)implies that FF∗ = id = F∗F on S, which is a restatement of the Plancherel theorem.
Corollary C.2. F extends to a unitary operator on L2, i.e. to a bounded operatorsatisfying F∗ = F−1.
The next theorem gives the important example of the Fourier transform - the Fouriertransform of a Gaussian :
Theorem C.3. Let A be a n× n matrix s.t. ReA := (A+A∗)/2 is positive definite (i.e.x · ReAx > 0 if x 6= 0). Then we have
F : e−x·Ax/2 7→ (detA)−1/2e−k·A−1k/2 (C.6)
Proof. We prove the theorem only for positive definite matrices. If A is positive definite(i.e. if x · Ax > 0 for x 6= 0), then there is an orthogonal matrix U (i.e. U is real andUUT = UTU = id) s.t. A := UTAU is diagonal, sayA = diag(λ1, . . . , λn). Letting x = Uyand noticing that x · Ax = y · UTAUy, and that detU = 1, we get∫
e−x·Ax/2e−ik·xdx =
∫e−y·Ay/2eik
′·ydy =n∏1
∫e−λjy
2j /2eik
′jyjdyj,
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where k′ = UTk, and we have used k ·Uy = UTk · x. It is left as an exercise to show thatfor n = 1,
F : e−λx2/2 7→ λ−1/2e−k
2/2λ. (C.7)
The last two relations imply the desired statement.
Exercise 21. Show (C.7).
The function e−x·Ax is called a Gaussian. It is one of the most common functionsin applications. There is another important function whose Fourier transform can beexplicitly computed:
F : |x|−α 7→Cn,α|k|−n+α if α 6= n,Cn,n ln |k| if α = n.
(C.8)
The coefficients are given for α = 2 by
Cn,2 =
((2− n)σn−1)−1 for n 6= 2,−(σn−1)−1 = −(2π)−1 for n = 2,
(C.9)
where σn is the volume of the n–dimensional unit sphere Sn = x ∈ Rn+1 : |x| = 1.One can easily deduce formula (C.8) modulo the constants (C.9). Indeed, since |x|−α isrotationally invariant, then so is its Fourier transform (see Exercise C.5 below). Also,since |x|−α is homogeneous of degree −α, then its Fourier transform is homogeneous ofdegree −n+α (see Exercise C.5 below). Hence (C.8) follows. Though it is easy to computethe Fourier transform of |x|−α, it is not easy to justify it. Indeed, the function |x|−α israther singular and definitely does not belong to S(Rn).
Exercise C.4. For n = 1, compute the Fourier transform of the characteristic functionχ(−a,a)(x), using definition (C.2).
As an example we show the following relation
((|k|2 + µ2)−1) =e−µ|x|
4π|x| , for n = 3, (C.10)
which appears often in applications. Indeed, let f(k) = (|k|2 +µ2)−1 and n = 3. We have
f(x) = (2π)−3/2
∫R3
eik·x
|k|2 + µ2dk
= limR→∞
(2π)−1/2
∫ R
0
∫ 1
−1
eir|x|vr2
r2 + µ2drdv
= limR→∞
(2π)−1/2
i|x|
∫ R
−R
eir|x|r
r2 + µ2dr.
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In the second equality, we change to spherical coordinates v = cosφ = k·x|k||x| with r = |k|.
Now, the last integral can be computed by changing to a contour integral over a rectanglein the upper half complex plane with top vertices at −R+ i
√R and R+ i
√R and taking
the limit as R→∞. By the Residue theory we have (C.10) (show this).
Exercise C.5. Let fh(x) := f(x − h) for h ∈ Rn, f (λ)(x) := λn/2f(λx) for λ ∈ R+ andf (g)(x) := f(gx) for ga ∈ SO(n) be the translation, dilation and rotation of f(x). Showthat
F : fh(x) 7→ e−ik·hf(k), (C.11)
F : f (λ)(x) 7→ f (1/λ)(k). (C.12)
F : f (g)(x) 7→ f (g−1)(k). (C.13)
Define the unitary operators Th : f(x) 7→ f(x− h) (the operator of translation by h)and Sλ : f(x) 7→ λn/2f(λx) (the operator of dilation by λ). Then (C.11) and (C.12) imply
F Th = Me−ik·h F ,where we recall that Me−ik·h is the operator of multiplication by e−ik·h, and
F Sλ = S1/λ F .Note the following important property of the Fourier transform, called the uncertaintyprinciple: let f ∈ S(Rn) be a Schwartz function on Rn, s.t. ‖f‖L2(Rn) = 1. Then(∫
|x|2|f(x)|2dnx)1/2(∫
|k|2|f(k)|2dnk)1/2
≥ 1/2. (C.14)
Exercise C.6. Prove inequality (C.14) (Hint: in the case n = 1, notice that
〈f, (−i∂x)x− x(−i∂x)f〉 = −i‖f‖2,
where the scalar product and the norm are in L2(R). On the other hand,
〈f, (−i∂x)x− x(−i∂x)f〉 = 2iIm 〈(−i∂x)f, xf〉 .Use these two observations to show that ‖f‖2 ≤ 2‖(−i∂x)f‖ ‖xf‖, and finish the proof byinvoking Plancherel’s theorem).
Let w ∈ L2(Rn) be a given function. Then the integral
(2π)−n/2∫w(x− y)f(x)e−ik·xdnx (C.15)
is called the windowed Fourier transform of f (with the window function w). In signalanalysis, i.e. for n = 1, one often uses the Gaussian (2π)−1/2e−x
2/(2α) for w. In this case,the transform (C.15) is called the Gabor transform.
For more details see [18], Chapter 5.
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D Linear evolution and semigroups
Let X be a Banach space and A be a linear (closed) operator on X with dense domainY = D(A). Our goal is to solve the initial value problem
∂u
∂t= Au, u|t=0 = u0 ∈ Y
for u ∈ C(R, Y ) ∩ C1(R, X). We use the following terminology.
• The family, U(t), t ≥ 0, of operators on X is called a (strongly continuous) semigroupif and only if
(a) U(t) are bounded ∀t ≥ 0.
(b) U(0) = 1 and U(t+ s) = U(t)U(s).
(c) t→ U(t)ϕ is continuous ∀ϕ ∈ X.
• The family U(t) is called a contraction semigroup if and only if U(t) is a semigroupand ||U(t)|| ≤ 1.
• The (closed) operator
Au := lims→0
1
s(U(s)− 1)u (D.1)
on X with the domain
D(A) := u ∈ X | the limit on the r.h.s of (F.14) exists is called the generator of U(t). If A is the generator of the semigroup U(t), then wewrite
U(t) = eAt.
Theorem D.1. If A is the generator of U(t), then U(t)D(A) ⊂ D(A) and ∀u0 ∈ D(A),u := U(t)u0 solves the equation
∂u
∂t= Au (D.2)
with the initial condition u|t=0 = u0.
Proof. If u ∈ D(A), then
AU(t)u = lims→0
1
s(U(s)− 1)U(t)u = U(t) lim
s→0
1
s(U(s)− 1)u exists.
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Hence U(t)u ∈ D(A). Furthermore,
AU(t)u = lims→0
1
s(U(t+ s)u− U(t)u) ≡ ∂
∂tU(t)u.
Corollary D.2. If an operator A is the generator of a semigroup U(t), then the initialvalue problem
∂u
∂t= Au, u|t=0 = u0
has a solution for any u0 ∈ D(A) and this solution is given by the formula u = U(t)u0.
Thus the main question here is: when does an operator A generate a semigroup? Firstof all bounded operators generate semigroups. Indeed, for a bounded operator B we define
U(t) ≡ etB :=∞∑n=0
(tB)n
n!. (D.3)
The series on the r.h.s. converges absolutely since
∞∑n=0
∥∥∥∥(tB)n
n!
∥∥∥∥ ≤ ∞∑n=0
tn
n!||B||n = et||B|| <∞.
Exercise D.3. Show that equation (D.3) defines a semigroup and that this semigroup isgenerated by B.
For unbounded operators the answer to the question above is given by the following.
Theorem D.4 (Hille-Yosida). Let A be a closed operator such that (a) (0,∞) ⊂ ρ(A)and (b) ||(A − λ)−1|| ≤ 1/λ for any λ > 0. Then A generates a unique semigroup andthis semigroup is contractive.
Proof. The idea is very simple: we approximate the operator A by bounded operators Aλso that Aλu→ Au ∀u ∈ D(A) as λ→∞; construct the semigroup, Uλ(t), for Aλ by theformula
Uλ(t) =∞∑n=0
1
n!(tAλ)
n (D.4)
(see equation (D.3)); define the semigroup, U(t), for A as the limit
U(t)u = limλ→∞
Uλ(t)u (D.5)
for any u ∈ D(A) and then, by continuity, extend U(t) to the entire space X.
266 Lectures on Applied PDEs
We define Aλ as
Aλ := Aλ(λ− A)−1.
(Note A−λ for λ > 0 is invertible by the condition that (0,∞) ⊂ ρ(A).) Then ∀u ∈ D(A),by (b)
λ(λ− A)−1u− u = (λ− A)−1Au→ 0,
as λ→∞, and
‖λ(λ− A)−1‖ ≤ 1.
Hence, by an ε/3 argument,
λ(λ− A)−1u→ u ∀u ∈ X.
This implies ∀u ∈ D(A),
Aλu = λ(λ− A)−1Au→ Au.
Now we consider the semigroup Uλ(t) defined in (D.4). Due to (b) and the relationAλ = λ2(λ− A)−1 − λ, we have
‖eAλt‖ ≤ e−λt∞∑n=0
1
n!‖(tλ2(λ− A)−1)n‖
≤ e−λt∞∑n=0
tn
n!λ2n‖(λ− A)−1‖n
≤ e−λt∞∑n=0
tn
n!λn. (D.6)
Hence
||eAλt|| ≤ 1, (D.7)
i.e., eAλt is the contractive semigroup.Now we show that eAλt, λ > 0 is a Cauchy family in the sense that
||(eAλ′ t − eAλt)u|| → 0 (D.8)
as λ, λ′ → ∞ for any u ∈ X. To prove this we represent the operator acting on u insidethe norm as an integral of a derivative
Lectures on Applied PDEs 267
eAλ′ t − eAλt =
∫ t
0
∂
∂seAλ′seAλ(t−s)ds.
Using the equation
∂
∂seAλs = Aλe
Aλs = eAλsAλ
gives
eAλ′ t − eAλt =
∫ t
0
eAλ′seAλ(t−s)(Aλ′ − Aλ)ds.
The last equation together with (D.7) yields
||(eAλ′ t − eAλt)u|| ≤∫ t
0
||eAλ′seAλ(t−s)(Aλ′ − Aλ)u||ds
≤∫ t
0
||(Aλ′ − Aλ)u||ds = t||(Aλ′ − Aλ)u||
and therefore (D.8) follows for any t ≥ 0 first ∀u ∈ D(A) and then by continuity ∀u ∈ X.Now equation (D.8) implies that the limit on the r.h.s. of (D.5) exists ∀t ∈ [0,∞)∀u ∈
X and satisfies
||U(t)|| ≤ 1.
Equations (D.5) and (D.7) imply also that U(t) is the semigroup: U(t + s) = U(t)U(s)and U(0) = 1. It remains to prove that U(t) is strongly continuous and is generated bythe operator A. Using the relation Uλ(s)− 1 =
∫ s0dtUλ(t)Aλ, we find for u ∈ D(A)
lims→0||(U(s)− 1)u|| = lim
s→0limλ→∞||(Uλ(s)− 1)u||
= lims→0
limλ→∞||∫ s
0
dtUλ(t)Aλu||
≤ lims→0
limλ→∞
∫ s
0
dt||Aλu||
= lims→0
s||Au|| = 0.
Hence U(s)u → u as s → 0 ∀u ∈ D(A). Since U(t) is bounded uniformly in t weconclude that it is strongly continuous. Similarly, using the relation Uλ(s) − 1 − Aλ =∫ s
0dt(Uλ(t)−1)Aλ and the strong continuity of Uλ(t) shown above, we conclude that U(t)
is generated by A.
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How do we check the conditions of the Hille-Yosida theorem? Usually this is a hardbusiness. However, there are several cases where this can be easily done. Below, X is aHilbert Space.A) A = −A∗ (A is anti-self-adjoint). Then σ(A) ⊂ iR and ‖(A− λ)−1‖ ≤ λ−1 for λ > 0.B) A = A∗ ≤ 0 (A is non-positive). Then σ(A) ⊂ (−∞, 0] and ‖(A − λ)−1‖ ≤ λ−1 forλ > 0.C) Perturbations of generators. For example, A = A0+B where A0 generates a semigroupand ‖B(A0 − λ)−1‖ ≤ β with β < 1 ∀λ ≥ λ0, for some λ0 > 0. Indeed, in this case wehave for λ > λ0
A− λ = [1 + Tλ](A0 − λ) (D.9)
where Tλ = B(A0−λ)−1. By the condition above ||Tλ|| ≤ β ∀λ ≥ λ0. Since β < 1, 1 +Tλis invertible and therefore so is the r.h.s. of (D.9) ∀λ ≥ λ0. Therefore (λ0,∞) ⊂ ρ(A).Moreover, (D.9) implies that
(A− λ)−1 = (A0 − λ)−1(1 + Tλ)−1
and therefore
||(A− λ)−1|| ≤ ||(A0 − λ)−1|| ||(1 + Tλ)−1||.
Since A0 generates a semigroup, we have that ||(A0 − λ)−1|| ≤ 1/λ. Since ||Tλ|| ≤ β < 1we have that ||(1 + Tλ)
−1|| ≤ (1 − β)−1. Collecting the last three estimates we concludethat ∀λ ≥ λ0
||(A− λ)−1|| ≤ (1− β)−1λ−1.
This is not quite what we need (remember (b)). However, for the operator Aµ = A − µwith µ = min(λ0, (1− β)−1), we obtain
Thus the operator Aµ generates a contraction semigroup eAµt, ||eAµt|| ≤ 1. Now, eAt :=eAµteµt gives a semigroup for the operator A and this semigroup satisfies the estimate
||eAt|| ≤ eµt.
Exercise D.5. Check the last statement.
Examples.
1) The Schodinger equation. In this case A = −iH where H is a self-adjoint operator(e.g., a Schrodinger operator H := −∆ + V (x) on L2(Rd)).
Lectures on Applied PDEs 269
2) The heat equation. In this case A = A∗ ≤ 0 or more generally A = −A0 − A1 withA0 = A∗0 > 0 and
‖A−1/20 A1A
−1/20 ‖ < 1.
For example A = −∑ ∂xiaij(x)∂xj +∑bi(x)∂xi + c(x) with the matrix (aij(x)) ≥ δ1,
(∑ |bi(x)|2)1/2 ≤ γ and c(x) ≥ γ2/δ.
3) The wave equation. In this case
A =
(0 1−H 0
)with H = H∗ ≥ 0. (D.10)
Indeed, if v satisfies the wave equation
∂2v
∂t2= −Hv with H ≥ 0,
then the element u = (v, ∂v/∂t) satisfies the equation
∂u
∂t= Au
with the operator A given in (D.10). On the Hilbert space X = D(H)⊕L2 with the innerproduct
〈u,w〉X = 〈u1, Hw1〉+ 〈u2, w2〉where u = (u1, u2) and w = (w1, w2), the operator (D.10) is anti-self-adjoint, A = −A∗,and therefore it generates a unique contraction semigroup.
Example. The acoustical wave equation.
∂2v
∂t2= c2ρ∇ · 1
ρ∇v.
The operator H := −c2ρ∇ · 1ρ∇ is self-adjoint and, in fact, non-negative on the space
L2(R3, (c2ρ)−1dx).
Exercise D.6. Show that H is symmetric, i.e.,
〈Hu, v〉 = 〈u,Hv〉 ∀u, v ∈ D(H).
4) Maxwell equations. In a vacuum, Maxwell’s equations for the electric and magneticfields, E(x, t) and H(x, t), read
270 Lectures on Applied PDEs
curlE = −µ∂H∂t
, curlH = ε∂E∂t
div(εE) = 0, div(µH) = 0
where ε and µ are dielectric constant and magnetic permeability, respectively. Theseequations can be written as
∂tu = JAu,
where u = (E,H), J =
(0 1−1 0
)and
A =
(µ−1curl 0
0 ε−1curl
)on the Hilbert space (we use that ε and µ are independent of x)
H1 = (E,H) |E,H ∈ H1(R3,R3), divE = 0, divH = 0.One can show that the operator A with the domain H1 ⊂ L2(R3,R3) ⊕ L2(R3,R3) isself-adjoint (show that it is symmetric) and σess(H) = [0,∞). Hence the operator JAgenerates a contraction semigroup according to criterion D) above.
E Elements of spectral theory
E.1 Definitions
Consider an operator A acting on a Banach space X with a domain D(A) (i.e., A :D(A)→ X). The spectrum, σ(A), of an operator A is the set in C defined by
σ(A) := z ∈ C : A− z1l is not invertible . (E.1)
For notational convenience, the operator “multiplication by z ∈ C” will be simply writtenas z instead of z1l. Clearly, eigenvalues of A belong to σ(A) (in fact, if λ is an eigenvalue,then Auλ = λuλ for some nonzero uλ ∈ X (uλ is called an eigenvector), so (A−λ)uλ = 0,and A− λ is not invertible). In general, the spectrum can also contain continuous piecesand it can take very peculiar forms.
Exercise E.1. Referring to the examples in Appendix A.7, show that(a) the spectrum of the multiplication operator introduced in example 1) is σ(Mf ) =
Ran f ;(b) the differentiation operator ∂
∂xjhas spectrum σ( ∂
∂xj) = iR;
(c) the identity operator (see 3)) has the spectrum consisting of one point σ(1l) = 1;
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(d) the Laplacian ∆ :=∑n
1∂2
∂x2jon L2(Ω) with the domain D(∆) = H2(Ω) has the
spectrum [0,∞);(e) the Laplacian ∆ on [−a, a]n with (i) Dirichlet boundary conditions (i.e. u = 0
on the boundary) or (ii) the periodic boundary conditions has only eigenvalues of finitemultiplicities; find these eigenvalues.
The study of the spectra of operators is called spectral analysis.Classification of the spectrum. Inverting A on (NullA)⊥, perturbation theory (the
continuous dependence of eigenvalues and their multiplicities of the perturbation param-eter).
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗The complement of the spectrum is called the resolvent set ρ(A):
ρ(A) := C\σ(A).
One can show (see [12]) that the set σ(A) is closed (and, consequently, the set ρ(A) isopen).
We have
Theorem E.2. The set σ(A) is closed (and, consequently, the set ρ(A) is open).
Proof. We prove the equivalent statement that the set ρ(A) is open. Let z0 ∈ ρ(A). ThenA− z0 is invertible. We write
A− z = A− z0 + z0 − z = (A− z0)[1l + (z0 − z)(A− z0)−1].
If |z0 − z| < ‖(A − z0)−1‖−1, then the operator in the square brackets on the right isinvertible by the Neumann theorem (see Theorem A.11). Therefore the operator A− z isinvertible for |z − z0| < ‖(A− z0)−1‖−1 as a product of two invertible operators.
For z ∈ ρ(A) the operator A− z has a bounded in X inverse. Denote this inverse as
RA(z) := (A− z)−1.
It is called the resolvent of A at z ∈ ρ(A). It plays an important role in analysis ofoperators. The proof of the theorem above shows that the resolvent is an analytic operatorvalued function in z ∈ ρ(A) in the sense that for any z0 ∈ ρ(A) and for any z such that|z − z0| < ‖RA(z0)‖−1, the resolvent RA(z) can be expanded in the series
RA(z) = RA(z0)∞∑n=0
((z − z0)RA(z0))n (E.2)
which is absolutely convergent is the sense that
272 Lectures on Applied PDEs
∞∑n=0
‖(z − z0)RA(z0)‖n =∞∑n=0
|z − z0|n‖RA(z0)‖n <∞.
Indeed the series above is just the Neumann series for the inverse of the operator A− z =(A− z0)[1l + (z0 − z)(A− z0)−1] (see Theorem A.11).
The resolvent satisfies two equations:
RA(z)−RA(w) = (z − w)RA(z)RA(w) (E.3)
and
RA(z)−RB(z) = RA(z)(B − A)RB(z), (E.4)
called the first and second resolvent equations. The first equation follows from the secondone with B = (z − w)1l and the second equation is equivalent to the identity 1
A− 1
B=
1A
(B − A) 1B
which can be easily verified.
Proposition E.3. If A is a bounded operator then σ(A) ⊂ z ∈ C | |z| ≤ ‖A‖.
Proof. Expand formally by the Neumann series
1
A− z =−1
z
1
1− A/z =−1
z
∞∑n=0
(A
z
)n.
This shows that z ∈ ρ(A) (i.e., (A− z)−1 is bounded) if and only if ‖A/z‖ < 1. Equiva-lently, z ∈ ρ(A) if and only if |z| > ‖A‖. Therefore, z ∈ σ(A) if and only if |z| ≤ ‖A‖.
E.2 Location of the essential spectrum
(under construction)
E.3 Perron-Frobenius Theory
Consider a bounded operator T on the Hilbert space X = L2(Ω).
Definition E.4. An operator T is called positivity preserving/improving if and only ifu ≥ 0, u 6= 0 =⇒ Tu ≥ 0/Tu > 0.
Note if T is positivity preserving, then T maps real functions into real functions.
Theorem E.5. Let T be a bounded positive and positivity improving operator and let λbe an eigenvalue of T with an eigenvector ϕ. Thena) λ = ‖T‖ ⇒ λ is simple and ϕ > 0 (modulo a constant factor).b) ϕ > 0 and ‖T‖ is an eigenvalue of T ⇒ λ is simple and λ = ||T ||.
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Proof. a) Let λ = ||T ||, Tψ = λψ and ψ be real. Then |ψ| ± ψ ≥ 0 and thereforeT (|ψ| ± ψ) > 0. The latter inequality implies that |Tψ| ≤ T |ψ| and therefore
〈|ψ|, T |ψ|〉 ≥ 〈|ψ|, |Tψ|〉 ≥ 〈ψ, Tψ〉 = λ||ψ||2.Since λ = ||T || = sup||ψ||=1〈ψ, Tψ〉, we conclude using variational calculus (see e.g. [12]or [?]) that
T |ψ| = λ|ψ| (E.5)
i.e., |ψ| is an eigenfunction of T with the eigenvalue λ. Indeed, since λ = ‖T‖ =sup‖ψ‖=1〈ψ, Tψ〉, |ψ| is the maximizer for this problem. Hence |ψ| satisfies the Euler-Lagrange equation T |ψ| = µ|ψ| for some µ. This implies that µ||ψ||2 = 〈|ψ|, T |ψ|〉 =λ||ψ||2 and hence µ = λ. Equation (E.5) and the positivity improving property of Timply that |ψ| > 0.
Now either ψ = ±|ψ| or |ψ| + ψ and |ψ| − ψ are nonzero. In the latter case they areeigenfunctions of T corresponding to the eigenvalue λ : T (|ψ| ± ψ) = λ(|ψ| ± ψ). By thepositivity improving property of T this implies that |ψ|±ψ > 0 which is impossible. Thusψ = ±|ψ|.
If ψ1 and ψ2 are two real eigenfunctions of T with the eigenvalue λ then so is aψ1 +bψ2
for any a, b ∈ R. By the above, either aψ1 + bψ2 > 0 or aψ1 + bψ2 < 0 ∀a, b ∈ R \ 0,which is impossible. Thus T has a single real eigenfunction associated with λ.
Let now ψ be a complex eigenfunction of T with the eigenvalue λ and let ψ = ψ1 +iψ1
where ψ1 and ψ2 are real. Then the equation Tψ = λψ becomes
Tψ1 + iTψ2 = λψ1 + iλψ2.
Since Tψ2 and Tψ2 and λ are real (see above) we conclude that Tψi = λψ2, i = 1, 2, andtherefore by the above ψ2 = cψ1 for some constant c. Hence ψ = (1 + ic)ψ1 is positiveand unique modulo a constant complex factor.
b) By a) and eigenfunction, ψ, corresponding to ν := ||T || can be chosen to be positive,ψ > 0. But then
*Question: Can the condition that ||T || is an eigenvalue of T (see b) beremoved?*
Now we consider the Schrodinger operator H = −∆ + V (x) with a real, boundedpotential V (x). The above result allows us to obtain the following important
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Theorem E.6. Let H = −∆ + V (x) have an eigenvalue E0 with an eigenfunction ϕ0(x)and let inf σ(H) be an eigenvalue. Then
ϕ0 > 0 ⇒ E0 = infλ|λ ∈ σ(H) and E0 is non-degenerate
and, conversely,
E0 = infλ|λ ∈ σ(H) ⇒ E0 is non-degenerate and ϕ0 > 0
(modulo multiplication by a constant factor).
Proof. To simplify the exposition we assume V (x) ≤ 0 and let W (x) = −V (x) ≥ 0. Forµ > supW we have
(−∆−W + µ)−1 = (−∆ + µ)−1
∞∑n=0
[W (−∆ + µ)−1]n (E.6)
where the series converges in norm as
‖W (−∆ + µ)−1‖ ≤ ‖W‖‖(−∆ + µ)−1‖ ≤ ‖W‖L∞µ−1 < 1
by our assumption that µ > supW = ‖W‖L∞ . To be explicit we assume that d = 3.Then the operator (−∆ + µ)−1 has the integral kernel
e−√µ|x−y|
4π|x− y| > 0
while the operator W (−∆ + µ)−1 has the integral kernel
W (x)e−√µ|x−y|
4π|x− y| ≥ 0.
Consequently, the operator
(−∆ + µ)−1f(x) =1
4π
∫e−√µ|x−y|
|x− y| f(y) dy
is positivity improving (f ≥ 0, f 6= 0⇒ (−∆ + µ)−1f > 0) while the operator
W (−∆ + µ)−1f(x) =1
4π
∫W (x)
e−√µ|x−y|
|x− y| f(y) dy
is positivity preserving (f ≥ 0 ⇒ W (−∆ + µ)−1f ≥ 0). The latter fact implies that theoperators [W (−∆+µ)−1]n, n ≥ 1, are positivity preserving (prove this!) and consequentlythe operator
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(−∆ + µ)−1 +∞∑n=1
[W (−∆ + µ)−1]n
is positivity improving (prove this!).Thus we have shown that the operator (H +µ)−1 is positivity improving. Series (E.6)
shows also that (H + µ)−1 is bounded. Since
〈u, (H + µ)u〉 ≥ (− supW + µ)‖u‖2 > 0,
we conclude that the operator (H +µ)−1 is positive (as an inverse of a positive operator).Finally, ‖(H + µ)−1‖ = supσ((H + µ)−1) = (inf σ(H) + µ)−1 is an eigenvalue by thecondition of the theorem. Hence the previous theorem applies to it. Since Hϕ0 = E0ϕ0 ⇔(H + µ)−1ϕ0 = (E0 + µ)−1ϕ0, the theorem under verification follows. *This paragraphneeds details!*.
F Hamiltonian systems
F.1 Complex Hamiltonian Systems
Now, we describe hamiltonian equations on complex Banach spaces. Let Z be a complexBanach space. As above we can identify it with the real space by writing it as Z = V +iV ,for some real vector space, and associating with it a real space Z := V ⊕ V . After thatwe can transfer to it the hamiltonian theory we developed for real spaces. Our goal is tofind a natural complex expression for this theory.
Recall that above (see Subsection 6) we defined the complex Gateaux derivatives,dψH(ψ) and dψH(ψ), and the complex gradients, ∂ψH(ψ) and ∂ψH(ψ). Now, we assumethat Z has an inner product (complex). Then we can define the symplectic form on Z as
ω(u, v) := −Im〈u, v〉 = Re〈u, iv〉. (F.1)
Now, if we define the real inner product on Z by
Re〈u, v〉 = 〈~u,~v〉, (F.2)
thenω(u, v) = 〈~u, J−1~v〉, (F.3)
where J is the standard symplectic operator (7.15), J =
(0 1−1 0
). Indeed, under the
map vect, i goes into J−1: vect(iφ) = J−1~φ. This shows that, indeed, (F.1) gives thesymplectic form on Z. We identify it as a symplectic form on the complex space Z.
Now, we consider two specific complex hamiltonian theories: the complex Klein-Gordon and Schrodinger field theories.
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Complex Klein-Gordon equation (Classical field theory). Here the phase spaceis Z = H1(Rd,C)×H1(Rd,C), identified with the real space
Z = H1(Rd,Rm)×H1(Rd,Rm), (F.4)
with m = 2, and the Hamiltonian is the same as in (7.19) and the symplectic form isgiven now by (7.17), or (7.12), with J given by (7.15).
Schrodinger Hamiltonian system. Here the phase space is the complex Hilbertspace, H1(Rd,C), the symplectic form ω(ξ, η) := Im
∫ξη and the Hamiltonian is given by
H(ψ, ψ) :=
∫Rd
|∇xψ|2 + V |ψ|2 + f(ψ)
dx.
The the symplectic form is given now by (F.3), i.e. by −Im〈u, v〉 = Re〈u, iv〉 = 〈~u, J−1~v〉,where 〈u, v〉 is the complex L2− inner product. The Hamilton equation for this system is∂tψ = i−1∂H, or in detail,
i∂tψ = (−∆x + V )ψ + ∂ψf(ψ). (F.5)
Poisson brackets. With the symplectic form on Z, Ω(v, w), we associate the the bi-linear map (f, g)→ f, g, from pairs of differentiable functions on Z on to a function onZ, called the Poisson brackets, by
f, g := ω(Xf , Xg). (F.6)
The map (F.6) has the following properties: for any functions f , g, and h from Z to R,
1. f, g = −g, f (skew-symmetry);
2. f, gh = f, gh+ gf, h (Leibniz rule);
3. f, g, h+ g, h, f+ h, f, g = 0 (the Jacobi identity).
Bilinear maps, (f, g) → f, g, having these properties, are called the Poisson brackets.The map (F.6) also obeys f, g = 0 ∀ g =⇒ f = 0. Poisson brackets with thelatter property are said to be non-degenerate. Note that a space of smooth functions (orfunctionals), together with a Poisson bracket, has the structure of a Lie Algebra.
If the symplectic form is defined by (7.12), then the Poisson bracket is given by
f, g := 〈∂f, J∂g〉. (F.7)
Indeed, by the definition, we have that f, g := ω(Xf , Xg) = 〈Xf , J−1Xg〉. Now, using
here that Xf (u) = −J∂f(u), etc., we see that f, g = 〈J∂f, J−1J∂g〉 = 〈J∂f, ∂g〉.
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Specifically, we have the following Poisson brackets:1) One-particle Classical Mechanics with the phase space Z = R3 ×R3, with the Poissonbracket
f, g = ∇xf · ∇kg −∇kf · ∇xg. (F.8)
(The symplectic operator is (7.15).)2) For the phase space Z = H1(Rd,Rm)×H1(Rd,Rm), with L2−inner product, we definethe Poisson brackets as
f, g =
∫∂πf · ∂φG− ∂φF · ∂πgdx. (F.9)
(The symplectic operator is (7.15).)3) Let V be a real, inner-product vector space, called the space of velocities, X is an opensubset of V , called the position, or configuration, space. We assume that V is reflexive,i.e. V ′′ = V . We define a Poisson brackets on Z = X × V ′ by
f, g = 〈∂πf, ∂φg〉 − 〈∂φf, ∂πg〉. (F.10)
(Remember ∂φf ∈ V ′ and ∂πf ∈ V ′′.) (The symplectic operator is (7.15).)The space Z together with a Poisson bracket on C∞(Z,R) is called a Poisson space.For a complex symplectic space, as above, with the symplectic form (F.1), the Poisson
bracket is given by f, g := Re〈∂f, (−i)∂g〉, or explicitly
f, g =
∫∂ψf∂ψg − ∂ψf∂ψgdx. (F.11)
Functions on a Poisson space, (Z, ·, ·), are called classical observables. The classicalevolution of observables is given by f(u, t) = f(ut), where ut is the solution of (7.11) withthe initial condition u. Note that f(u, t) solves the equation
d
dtf(u, t) = f,H(u, t). (F.12)
with the initial condition f(u, 0) = f(u). In the opposite direction, a solution of thisequation with an initial condition f(u) is given by f(u, t) = f(ut). Recall that Φt(u) := utis the the flow generated by (7.11) (the hamiltonian flow).
Exercise 2. Prove this.
Suppose that Z is a space of functions u(x) on Rd with a Poisson bracket f, g.Consider the evaluation functional Z 3 u 7→ u(x), which we denote (with some abuse ofnotation) as u(x), and similarly for V . Then we can take f(u) = u(x) in (F.12), whichleads to the equations
∂tut(x) = u(x), H(ut). (F.13)
If the Poisson bracket f, g corresponds to the symplectic form ω, then this equation isequivalent to the Hamilton equations (7.11).
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F.2 Conservation laws and symmetries
We say that an observable f(u) is conserved or is constant of motion, iff f(ut) is indepen-dent of t. The equation (F.12) implies
Proposition 5. f(u) is conserved, if and only if its Poisson bracket with the Hamiltonian,H, vanishes: f,H = 0.
Symmetries. Conservation laws often arise from symmetries, that is, transformationswhich map solutions to solutions. Assume that a symmetry is realized by a one-parametergroup τs, s ∈ R, of bounded, operators on Z: if u is a solution then so is τsu for every s.
We list some examples of symmetries for the Schrodinger hamiltonian system with thephase space Z = H1(Rd,C):
• Time translation invariance: (τsψ)(x, t) := ψ(x, t+ s), s ∈ R.
• Space translation invariance: (τsψ)(x, t) := ψ(x+ s, t), s ∈ Rd.
• Space rotation (and reflection) invariance: (τRψ)(x, t) := ψ(R−1x, t), R ∈ SO(Rd).
Note that in the second and third cases the families above are multi-parameter groups,but they can be reduced to one-parameter ones.
We now assume that Z is a Hilbert space and that the symplectic form is givenby (7.12) (i.e. ωJ(v, w) = 〈v, J−1w〉, with a symplectic operator, J : Z∗ → Z, i.e. alinear, invertible, bounded, anti-dual (J∗ = −J) operator). Moreover, we assume thatthe operators τs, s ∈ R, are unitary, i.e. τ ∗s = τ−1
s . Recall that for a unitary one-parametergroup τs, s ∈ R, on Z, we can define the generator as the (closed) operator A, defined by
Au := lims→0
1
s(U(s)− 1)u (F.14)
on Z with the domain D(A) := u ∈ X | the limit on the r.h.s of (F.14) exists . More-over, differentiating the relation τ ∗s = τ−1
s w.r.to s at s = 0, we see that A is anti-unitary,A∗ = −A.
Theorem 19. Assume that the symplectic form is given by (7.12). Let τs, s ∈ R, be aone-parameter group of bounded, unitary operators on Z, commuting with J . Then
• τs, s ∈ R, is a symmetry of the Hamiltonian system, (Z, ωJ , H), iff Hτs = H, ∀s ∈R (mod a constant).
• τs, s ∈ R, is a symmetry =⇒ the ’charge’ Q(u) := 12〈u, J−1Au〉 is conserved.
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Proof. By the definition, τs, s ∈ R, is a symmetry of the Hamiltonian system, (Z, ωJ , H),iff ∂tu = XH(u) implies ∂tτsu = VH(τsu), for every s ∈ R. This can be rewritten asτ−sXH(τsu) = XH(u). Assuming that the equation ∂tu = XH(u) has a unique solutionfor every initial condition u0 ∈ Z, this relation and (7.13) give τ−sJdH(τsu) = JdH(u),or τ−sdH(τsu) = dH(u), which in turn is equivalent to dH(τsu) = dH(u), which againis equivalent to H τs = H, ∀s ∈ R. Indeed, by the definition of the gradient and theunitarity of τs, we have dH(τsu)ξ = (dH)(τsu)τsξ = dH(u) = 〈(dH)(τsu), τsξ〉 = dH(u) =〈τ−s(dH)(τsu), ξ〉 = 〈dH(u), ξ〉 = dH(u)ξ. Hence dH(τsu) = dH(u).
To prove the second statement, we differentiate the relation H(τsu) = H(u) at s = 0and use the definition of the Gateaux derivative to obtain dH(u)Au = 0. Now, relatingthis expression to the gradient of H gives 〈dH(u), Au〉 = 0. We write 〈dH(u), Au〉 =〈dH(u), JJ−1Au〉 and observe that J−1Au is the gradient of Q(u) := 1
2〈u, J−1Au〉 to find
that 〈dH(u), JdQ(u)〉 = 0. Finally, using the formula (F.7), f, g := 〈df, Jdg〉, relatingthe Poisson bracket to the inner product, 〈dH(u), JdQ(u)〉 = H,Q, we conclude thatH,Q = 0, and therefore Q(u) is conserved.
We consider the Schrodinger hamiltonian system, with the phase space Z = H1(Rd,C)and symplectic form ω(u, v) := −Im〈u, v〉 = Re〈u, iv〉, so that J = −i. We list conserva-tion laws for it, implied by the symmetries considered above:
• Time translation invariance ((τsψ)(x, t) := ψ(x, t + s), s ∈ R) → conservation ofenergy, H(ψ, ψ);
• Space translation invariance ((τsψ)(x, t) := ψ(x + s, t), s ∈ Rd) → conservation ofmomentum P (ψ, ψ) :=
∫ψ(−i∇)ψdx;
• Space rotation (and reflection) invariance ((τRψ)(x, t) := ψ(R−1x, t), R ∈ SO(Rd))→ conservation of angular momentum L(ψ, ψ) :=
For the hamiltonian system of classical field theory with the phase space and symplecticform given by (7.17) (Z = H1(Rd,Rm) × H1(Rd,Rm) and ω(v, v′) :=
∫Rd(ξη
′ − ξ′ · η) =〈v, ·J−1v′〉 , where v := (ξ, η) and the symplectic operator J is given by (7.15)), we havesimilar conservation laws, e.g. for the translation invariance we have conserved the fieldmomentum (for m = 1)
P (φ, π) :=
∫π∇φdx. (F.15)
Now, we consider an abstract real dynamical system with the phase space Z = Ω×V ′,where V is a reflexive (i.e. V ′′ = V ) Banach space and Ω ⊂ V with the symplectic
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form ω(v, v′) := η′(ξ) − η(ξ′), where v := (ξ, η), given by (7.12) with J given by (7.15).Symmetries τs, s ∈ R, now act as
τ−s : (φ, π)→ (τsφ, τ′−sπ),
with τ ′s being the dual group action of τs on V ∗, defined by 〈τ ′sπ, φ〉 = 〈π, τsφ〉 (recall that〈·, ·〉 is the coupling between X and X∗). The conserved classical observable correspondingto τs, is given by Q(φ, π) := 1
2〈u, J−1Au〉 = 〈π,Aφ〉, where u = (φ, π) and A is the
generator of the group τs, ∂sτs = Aτs, and A := diag (A,−A′). It has vanishing Poissonbracket (”commutes”) with the Hamiltonian,
H,Q = ∂sH(τsφ, τ′−sπ)|s=0 = 0,
and consequently is conserved under evolution: Q(φt, πt) is a constant in t where Φt =(φt.πt) is a solution to (7.18). We formulate this as
Us is a symmetry of (7.18) → 〈π,Aφ〉 is conserved
For the Schrodinger hamiltonian system, the one-parameter group τs, s ∈ R, can bechosen to be unitary and the dual group is given by τ ′s = Cτ−sC, where C is the complexconjugation, so that, since π = ψ, we have τ ′−sπ = τsψ. Hence the operator Ts acts on H
as TsH(ψ, ψ) := H(τsψ, τsψ). The conserved classical observable is defined as Q(ψ, ψ) :=〈ψ, iAψ〉 where A is the (anti-self-adjoint) generator of the group τs, ∂sτs = Aτs, and 〈ψ, φ〉is the standard scalar product on L2(Rd,C). As above, it has vanishing Poisson bracket(”commutes”) with the Hamiltonian, H,Q = ∂sH(τsφ, τsψ)|s=0 = 0, and consequentlyis conserved under evolution: Q(ψt, ψt) is a constant in t where ψt is a solution to (F.5).
G Some facts from geometry of surfaces
G.1 Area of a hypersurface
In this appendix we derive a standard formula for the area of a surface in Euclideanspaces. Let S be a smooth n–dimensional surface in Rn+1. Such a surface is called a(smooth) hypersurface. Hypersurfaces appear as graphs or level sets of images of somefunctions.
Let U ⊂ Rn and f : U → R, be smooth. Then graph f := (u, f(u)) |u ∈ U is ahypersurface in Rn+1.
Let ϕ : Rn+1 → R be a smooth function and let 0 ∈ Ran ϕ. Then the zero level set ofϕ,
ϕ−1(0) := x ∈ Rn+1 |ϕ(x) = 0is a smooth hypersurface.
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Finally a map ψ : U ⊂ Rn → S is called a local parametrization of S, provided (fillin).
Lemma G.1. Let U ⊂ Rn be an open set and f : U → R. Then the area of the surfaceS :=graphf is given by the formula, A(S) = A(f), where
A(f) :=
∫U
√1 + |∇f |2dnu. (G.1)
Proof. Let x = (u, xn+1), where u = (x1, . . . , xn). The picture below shows the followingformula for the area element of S: ∆A = dnu
cosα, where α is the angle between the xn+1-axis
(the unit vector en+1) and the normal ν = ν(x) to S at a given point x ∈ S.
##bb
bbJJJJ
α
S∆A
ν
en+1
dnx′ = dx1 . . . dxn
Let ϕ(x) = xn+1 − f(u) so that S = x ∈ U × R | ϕ(x) = 0. Then on S:
ν(x) =∇ϕ(x)
|∇ϕ(x)| ,
and therefore for x ∈ S
cosα =∇ϕ(x) · en+1
|∇ϕ(x)| =1√
1 + |∇f |2.
Since area(S) =∫SdA, the result follows.
G.2 Mean curvature of a hypersurface
In this appendix we define the notion of the mean curvature of a hypersurface in Euclideanspaces and derive convenient expressions for it. Let S be a smooth n–dimensional surfacein Rn+1. Such a surface is called a (smooth) hypersurface. Hypersurfaces appear as graphsor level sets of images of some functions.
Let U ⊂ Rn and f : U → R, be smooth. Then graph f := (u, f(u)) |u ∈ U is ahypersurface in Rn+1.
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Let ϕ : Rn+1 → R be a smooth function and let 0 ∈ Ran ϕ. Then the zero level set ofϕ,
ϕ−1(0) := x ∈ Rn+1 |ϕ(x) = 0is a smooth hypersurface.
Finally a map ψ : U ⊂ Rn → S is called a local parametrization of S, provided (fillin).
We give now the definition of different notions of curvatures at a point x0 ∈ S. Webegin with a simple but coordinate dependant definition. Pick a coordinate system s.t.∇f(x′0) = 0, where x0 = (x′0, x
n+10 ). Then we define
• the principal curvatures at x0 as the eigenvalues of Hessf(x′0),
• the Gauss curvature at x0 as det Hessf(x′0),
• the mean curvature at x0 as h(x0) = Tr Hessf(x′0) = ∆f(x′0).
Lemma 4. Let S = graphf for some f s.t. f(x′) 6= 0. Then the mean curvature at x isgiven by (10.4).
Proof. Consider first an arbitrary coordinate system and a function f : Ω → R, s.t.S = graphf . We denote as before x = (x′, xn+1) ∈ Rn+1, x′ = (x1, . . . , xn) ∈ Ω ⊂ Rn. Aswe have shown, the unit normal vector to S at x = (x′, f(x′)), ν(x), can be expressed as
ν(x) =(−∇f(x′), 1)√1 + |∇f(x′)|2
. (G.2)
Now for a given point x0 ∈ S, let x = (x′, xn+1) ∈ Rn+1 be a special coordinate system s.t.there is a domain U ⊂ Rn and a function f : U → R s.t. S = graphf and ∇f(x′0) = 0.Then we can express the normal vector ν(x) in terms of this function as
ν(x) =(−∇f(x′), 1)√1 + |∇f(x′)|2
.
Now compute
div ν(x) = − ∆f(x′)
(1 + |∇f(x′)|2)1/2+
|∇f(x′)|2(1 + |∇f(x′)|2)3/2
,
and therefore we get div ν(x0) = −∆f(x′0). By the definition of the mean curvature atthe point x0, ∆f(x′0) = −h(x0), and therefore div ν(x0) = −h(x0), which together with(G.2) implies the lemma.
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Now we define the curvature in a coordinate-independent way. This way reveals thegeometrical meaning of this notion and on the way deals with some important conceptsin the theory of surfaces.
We define the tangent space, TxS, to S at x as
TxS := ξ ∈ Rn+1 | ∃ C1 path, γs, in S s.t γs|s=0 = x, and ∂sγs|s=0 = ξ(i.e., TxS is the space of initial velocities of curves on S starting at x). Let ν(x) bean ’outward’ unit normal to the surface S at a point x ∈ S. Denote by Sn the unitn–dimensional sphere. The map ν : S → Sn, given by
ν : x→ ν(x),
is called the Gauss map. The negative of its derivative
Ax := −∂ν(x)
at x is called the Weingarten map. The map Ax measures the rate of change in thedirection of ν(x) as it moves along S. By definition,
Remark. Let ψ : U → S be a parametrization of S at x. Take ξ = ei and η = ej in (G.3)where ei is an orthonormal basis in Rn ⊃ U . Then
(Ax)ij = 〈∂2ψ(u)
∂ui∂uj, ν(x)〉
where u = ψ−1(x). The matrix Ax is called the 2nd fundamental form. (The 1st funda-mental form of S is the metric on S induced by the Euclidean metric on Rn+1.)
Below, all differential operations, e.g. ∇,∆, are defined in the corresponding Euclidianspace (either Rn+1 or Rn).
Proposition G.3. In the level set representation, S = ϕ−1(0), we have
H(x′) ≡ H(ϕ)(x) = div
( ∇ϕ|∇ϕ|
)(x), (G.4)
where x = (x1, . . . , xn) and x′ = (x, xn+1).
Proof. Let γs be a path on S starting at x with an initial velocity ξ: γs|s=0 = x and∂sγs|s=0 = ξ. Differentiating ϕ(γs) = 0 we find ∇ϕ · ξ = 0. Hence ∇ϕ ⊥ TxS andtherefore ν(x) is equal to ∇ϕ
|∇ϕ| up to a sign,
ν(x) =∇ϕ|∇ϕ|(x). (G.5)
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To prove (G.4) we have by the definition of Ax
Axξ = ∂s|s=0∇ϕ(γs)
|∇ϕ(γs)|=∑i
∂xi
( ∇ϕ(x)
|∇ϕ(x)|
)ξi.
Place the coordinate system at the point x with en+1 = ν(x). Then (Ax)ij = ∂xi
(∂xjϕ(x)
|∇ϕ(x)|
)and therefore H(x) = div ν(x) = div
(∇ϕ(x)|∇ϕ(x)|
)as claimed.
We also give expressions for the mean curvatures for hypersurface given as graphs overother surfaces.
1) Graph representation over a plane: S = graph of f : U → R, where U is an open setin Rn ⊂ Rn+1. In this case S is the zero level set of the function ϕ(x′) = xn+1−f(x).Using (G.4) with this function, we obtain
H(x) = div
(∇f√|∇f |2 + 1
). (G.6)
Denote by Hess f the standard euclidean hessian, Hess f :=(
∂2f∂ui∂uj
). Then we can
rewrite (G.7) as
H(x) =1√
|∇f |2 + 1[∆f − ∇f Hess f∇f
|∇f |2 + 1]. (G.7)
2) Graphs over sphere: S = graph of ρ : Sn → R+, where Sn is the standard n−unitsphere in Rn+1 centred at the origin. In this case S is the zero level set of thefunction ϕ(x′) = |x′| − ρ(x′), where x = x/|x|. Using (G.4) with this function, weobtain
2) Graphs over a cylinder: S = graph of ρ : Cn → R+, where Cn is the standard n−unitcylinder in Rn+1 with the axis along xn+1−axis. In this case S is the zero level setof the function ϕ(x′) = |x| − ρ(xn+1, x), where x = x/|x|. Using (G.4) with thisfunction, we obtain
Of a special interest are hypersurfaces, S, in Rn+1 (i.e. surfaces of dimension n), whosemean curvature, H, is constant, i.e. satisfies the equation
H(x) = h. (G.8)
Such surfaces are called the constant mean curvature surfaces if h 6= 0 and the minimalsurfaces if h = 0.
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Spherically and axi - symmetric (equivariant) solutions:
a) Sphere. The n−dimensional sphere of the radius R centred at the origin can begiven either as the level set ϕ(x) = 0, where ϕ(x) := |x|2 − R2, or as graph f ,where f(u) =
√R2 − |u|2. (SR = RSn, where Sn is the unit n-sphere.) Then we
have
H(x) = div
( ∇ϕ|∇ϕ|
)= div(x) =
n
R.
b) Cylinder. In the implicit function representation the cyliner is given by ϕ = 0,where ϕ := r −R, with r =
(∑n−1i=1 x
2i
) 12 .
Properties of (G.8):
• (G.8) is invariant under rigid motions of the surface, i.e. ψ 7→ Rψ + a, whereR ∈ O(n+ 1), a ∈ Rn+1 and ψ = ψ(u) is a parametrization of St, is a symmetry of(G.8).
• (G.8), with h = 0, is invariant under the scaling, for any λ > 0,
H(x) 7→ λH(λx). (G.9)
The first two properties come from the fact that the mean curvature is invariant undertranslations, rotations and scaling. The invariance under rigid motions is obvious.
H Linear stability of self-similar surfaces
H.1 Normal hessians
Recall that the hessian of a functional E(ϕ) is defined as HessE(ϕ) := d gradVa(ϕ). Notethat unlike the Gateaux derivative, d, the gradient grad and therefore the hessian, Hess,depends on the Riemann metric on the space on which E(ϕ) is defined.
Since the tangential variations lead to reparametrization of the surface, in what followswe are dealing with normal variations, η = fν. (In physics terms, specifying normal vari-ations is called fixing the gauge.) We use the following notation for a linear operator, A,on normal vector fields on S: ANf = A(fν). For instance, HessN E(ϕ)f = HessE(ϕ)(fν)and
dNF (ϕ)f = dF (ϕ)(fν). (H.1)
We consider the hessian of the modified volume functional Va(ϕ), at a self-similar ϕ(i.e. H(ϕ) = aϕ · ν) and in the normal direction (i.e. for normal variations, η = fν) in
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the Riemann metric h(ξ, η) :=∫Sλξηρ. In what follows, we call this hessian the normal
hessian and denote it by HessN Va(ϕ).Before we proceed, we mention the following important property of Va(ϕ): the equation
H(ϕ) − aϕ · ν(ϕ) = 0 breaks the scaling and translational symmetry. Indeed, using therelations
H(λϕ) = λ−1H(ϕ), ν(λϕ) = ν(ϕ), ∀λ ∈ R+, (H.2)
H(ϕ+ h) = H(ϕ), ν(ϕ+ h) = ν(ϕ), ∀h ∈ Rn+1, (H.3)
H(gϕ) = H(ϕ), ν(gϕ) = gν(ϕ), ∀g ∈ O(n+ 1), (H.4)
using that g ∈ O(n + 1) are isometries in Rn+1and using the notation Ha(ϕ) := H(ϕ) −aϕ · ν(ϕ), we obtain
Hλ−2a(λϕ) = λ−1Ha(ϕ), ∀λ ∈ R+, (H.5)
Ha(ϕ+ h) + ah · ν(ϕ) = Ha(ϕ), ∀h ∈ Rn+1, (H.6)
Ha(gϕ) = Ha(ϕ), ∀g ∈ O(n+ 1). (H.7)
We want to address the spectrum of the normal hessian, Hess⊥ Va(ϕ). First, we notethat the tangential variations lead to zero modes of the full hessian, HessVa(ϕ). Indeed,we have
Proposition H.1. The full hessian, HessVa(ϕ), of the modified volume functional Va(ϕ),has the eigenvalue 0 with the eigenfunctions which are tangential vector fields on S
Proof. We consider a family αs of diffeormorphisms of U , with α0 = 1 and ∂sϕαs|s=0 = ξ,a tangential vector field, reparametrizing the immersion ϕ, and define the family ϕ αsof variations of ϕ. Then ϕ αs satisfies again the soliton equation, Ha(ϕ αs) = 0.Differentiating the latter equation w.r.to s at s = 0 and using that ∂sϕ αs|s=0 = ξ, is atangential vector field, we obtain
dHa(ϕ)ξ = 0, (H.8)
which proves the proposition.
Theorem H.2. The hessian, HessN Va(ϕ), of the modified volume functional Va(ϕ), at aself-similar ϕ (i.e. H(ϕ) = aϕ ·ν) and in the normal direction (i.e. for normal variations,η = fν), has
• the eigenvalue −2a with the eigenfunction ϕ · ν(ϕ),
• the eigenvalue −a with the eigenfunctions νj(ϕ), j = 1, . . . , n+ 1, and,
• the eigenvalue 0 with the eigenfunctions σjϕ · ν(ϕ), j = 1, . . . , 12n(n− 1), where σj
are generators of the Lie algebra of SO(n+ 1), unless ϕ is a sphere.
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Proof. If an immersion ϕ satisfies the soliton equation H(ϕ) = aϕ · ν(ϕ), then by (H.5),we have Hλ−2a(λϕ) = 0 for any λ > 0. Differentiating this equation w.r.to λ at λ = 1, weobtain dHa(ϕ)ϕ = −2aϕ · ν(ϕ).
Now, choosing ξ to be equal to the tangential projection, ϕT , of ϕ, and subtractingthe equation (H.8) from the last equation, we find dHa(ϕ)(ϕ · ν(ϕ))ν(ϕ) = −aϕ · ν(ϕ).Since by the definition (H.1), dHa(ϕ)fν(ϕ) = dNHa(ϕ)f , this proves the first statement.
To prove the second statement, we observe that the soliton equation implies, by (H.6),that Ha(ϕ + sh) + ash · ν(ϕ) = 0 and any constant vector field h. Differentiating thisequation w.r.to s at s = 0, we obtain dHa(ϕ)h = −ah · ν(ϕ). Now, choosing ξ to be equalto the tangential projection, hT , of h, and subtracting the equation (H.8) from the lastequation, we find dHa(ϕ)(h · ν(ϕ))ν(ϕ) = −ah · ν(ϕ), which together with (H.1) gives thesecond statement.
Finally, to prove the third statement, we differentiate the equation Ha(g(s)ϕ) = 0,where g(s) is a one-parameter subgroup of O(n + 1), w.r.to s at s = 0, to obtaindHa(ϕ)σϕ = 0, where σ denotes the generator of g(s). Now, choosing ξ in (H.8) tobe equal to the tangential projection, (σϕ)T , of σϕ, and subtracting the equation (H.8)from the last equation, we find dHa(ϕ)(σϕ · ν(ϕ))ν(ϕ) = 0, which together with (H.1)gives the third statement.
Remark H.3. a) For a 6= 0, the soliton equation, ϕ · ν(ϕ) = a−1H(ϕ), and Proposi-tion H.7 imply that the mean curvature H is an eigenfunction of HessN Va(ϕ) with theeigenvalue −2a.
b) Strictly speaking, if the self-similar surface is not compact, then ϕ · ν(ϕ) andνj(ϕ), j = 1, . . . , n + 1, generalized eigenfunctions of HessN Va(ϕ). In the second case,the Schnol-Simon theorem (see Appendix ?? or [12]) implies that the points −2a and −abelong to the essential spectrum of HessN Va(ϕ).
c) We show below that the normal hessian, HessNsph Va(ϕ), on the sphere of the radius√an
, given in (H.9), has no other eigenvalues below 2an
, besides −2a and −a. A similarstatement, but with n replaced by n− 1, we have for the cylinder.
We call the eigenfunction ϕ · ν(ϕ), νj(ϕ) and σjϕ · ν(ϕ), j = 1, . . . , n+ 1, the scaling,translational and rotational modes. They originate from the normal projections, (λϕ)N
and (sh)N , of scaling, translation and rotation variations.
Linear stability. Given a self-similar surface ϕ, we consider for example the manifoldof surfaces obtained from ϕ by symmetry transformations,
Mϕ := λgϕ+ z : (λ, z, g) ∈ R+ × Rn+1 × SO(n+ 1).
By the spectral theorem above, it has unstable and central manifolds corresponding tothe eigenvalues −2a, −a and 0. Hence, we can expect only the dynamical stability in thetransverse direction.
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Definition H.4 (Linear stability of self-similar surfaces). We say that a self-similarsurface φ, with a > 0, is linearly stable (for the lack of a better term) iff the nor-mal hessian satisfies HessN Va(ϕ) > 0 on the subspace
(I.e. the only unstable motions allowed are scaling, translations and rotations.)In what follows a > 0. Theorem H.2 implies that if φ is not spherically symmetric,
then 0 is an eigenvalue of HessN Va(ϕ) of multiplicity at least n + 1. This gives the firststatement in the following corollary, while
Corollary H.5. If a self-similar surface with a > 0 satisfies HessN Va(ϕ) > 0 on thesubspace span ϕ · ν(ϕ), νj(ϕ), j = 1, . . . , n+ 1⊥, then it a sphere or a cylinder.
Theorem H.6. For a self-similar surface with a > 0, HessN Va(ϕ) ≥ −2a iff H(ϕ) > 0.
To prove this theorem we will use the Perron-Frobenius theory (see Appendix E.3)and its extension as given in [24]. We begin with
Definition H.7. We say that a linear operator on L2(S) has a positivity improving prop-erty iff either A, or e−A, or (A+µ)−1, for some µ ∈ R, takes non-negative functions intopositive ones.
Proposition H.8. The normal hessian, HessN Va(ϕ), has a positivity improving property.
Proof. By standard elliptic/parabolic theory, e∆ and (−∆ + µ)−1, for any µ > 0, hasstrictly positive integra kernel and therefore is positivity improving. To lift this result toHessN Va(ϕ) := −∆− |W |2 − a1 + ϕ · ∇ we proceed exactly as in [24, 29].
Since, the operator HessN Va(ϕ) is bounded from below and has a positivity improvingproperty, it satisfies the assumptions of the Perron-Frobenius theory (see Appendix E.3)and its extension in [24, 29] to the case when the positive solution in question is not aneigenfunction. The latter theory, Proposition H.7 and Remark H.3 imply the statementof Theorem H.6.
Corollary H.9. Let ϕ be a self-similar surface. We have(a) For a < 0 (ϕ is an expander), H(ϕ) changes the sign.(b) For a = 0, inf HessN Va(ϕ) < 0.(c) For a > 0, if ϕ is an entire graph over Rn and is weakly stable, then ϕ is a
hyperplane.
Indeed, (a) follows directly from Theorems H.2 and H.6, while (b) follows from thefact that for a = 0, 0 is an eigenvalue of the multiplicity n + 2 and therefore, by thePerron-Frobenius theory, it is not the lowest eigenvalue of HessN Va(ϕ). (c) is based onthe fact that entire graphs over Rn cannot have strictly postive mean curvature. (check,references)
290 Lectures on Applied PDEs
H.2 Linear stability of spheres, cylinders and planes
We now discuss the normal hessians on the n−sphere and (n, k)−cylinder.
Explicit expressions. 1) For the n−sphere SnR of radius R =√
na
in Rn+1, the normalhessian is
HessNsph Va(ϕ) = −an
∆Sn − 2a, (H.9)
on L2(Sn), where ∆Sk is the Laplace-Beltrami operator on the standard n−sphere Sk.2) For the n−cylinder CnR = Sn−kR × Rk of radius R =
√n−ka
in Rn+1, we have
HessNcyl Va(ϕ) = −∆y − ay · ∇y −a
n− k∆Sn−k − 2a, (H.10)
acting on L2(Cn).
Spectra of Lspha := HessNsph Va(ϕ) and Lcyl
a := HessNcyl Va(ϕ). We describe the spectra
of the normal hessians on the n−sphere and (n, k)−cylinder, of the radii√
an
and√
an−1
,respectively.
It is a standard fact that the operator −∆Sn is a self-adjoint operator on L2(Sn). Itsspectrum is well known (see [30]): it consists of the eigenvalues l(l+ n− 1), l = 0, 1, . . . ,
of the multiplicities m` =
(n+ ln
)−(n+ l − 2
n
). Moreover, the eigenfunctions cor-
responding to the eigenvalue l(l+n− 1) are the restrictions to the sphere Sn of harmonicpolynomials on Rn+1 of degree l and denoted by Ylm (the spherical harmonics),
In particular, the first eigenvalue 0 has the only eigenfunction 1 and the second eigenvaluen has the eigenfunctions ω1, · · · , ωn+1.
Consequently, the operator Lspha := HessNsph Va(ϕ) = − a
n(∆Sn+2n) is self-adjoint and its
spectrum consists of the eigenvalues an(l(l+n−1)−2n) = a(l−2)+ a
nl(l−1), l = 0, 1, . . . , of
the multiplicities m`. In particular, the first n+2 eigenvectors of Lspha (those with l = 0, 1)
correspond to the non-positive eigenvalues,
Lspha ω0 = −2aω0, Lsph
a ωj = −aωj, j = 1, . . . , n+ 1, (H.12)
and are due to the scaling (l = 0) and translation (l = 1) symmetries.We proceed to the cylindrical hessian Lcyl
a := HessNcyl Va(ϕ), given in (H.10). Thevariables in this operator separate and we can analyze the operators −∆y − ay · ∇ and− an−k (∆Sn−k +2(n−k)) separately. The operator − a
n−k (∆Sn−k +2n) was already analyzed
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above. The operator −∆y − ay · ∇ is the Ornstein - Uhlenbeck generator, which can beunitarily mapped by the gauge transformation
v(y, w)→ v(y, w)e−a4|y|2
into the the harmonic oscillator Hamiltonian Hharm := −∆y + 14a2|y|2 − ka. Hence the
linear operator −∆y − ay · ∇ is self-adjoint on the Hilbert space L2(R, e−a2 |y|2dy). Since,as was already mentioned, the operator − a
n−k (∆Sn−k + 2(n − k)) is self-adjoint on the
Hilbert space L2(Cn), we conclude that the linear operator Lcyla is self-adjoint on the
Hilbert space L2(Rk × Sn−k, e−a2 |y|2dydw). Moreover, the spectrum of −∆y − ay · ∇ isa∑k
1 si : si = 0, 1, 2, 3, . . ., with the normalized eigenvectors denoted by φs,a(y), s =
(s1, . . . , sk),
(−∆y − ay · ∇)φs,a = ak∑1
siφs,a, si = 0, 1, 2, 3, . . . . (H.13)
Using that we have shown that the spectrum of − an−k (∆Sn−k + 2(n− k)) is a
n−k (l(l +
n−k−1)−2(n−k)) = an−k l(l+n−k−1)−2a, l = 0, 1, . . . , and denoting r =
∑k1 si, si =
0, 1, 2, 3, . . ., we conclude the spectrum of the linear operator Lcyla , for k = 1, is
with the normalized eigenvectors given by φr,l,m,a(y, w) := φr,a(y)Ylm(w). This equationshows that the non-positive eigenvalues of the operator Lcyl
a , for k = 1, are
• the eigenvalue −2a of the multiplicity 1 with the eigenfunction φ0,0,0,a(y) = ( a2π
)14
((r, l) = (0, 0)), due to scaling of the transverse sphere;
• the eigenvalue−a of the multiplicity n with the eigenfunctions φ0,1,m,a(y) = ( a2π
)14wm,
m = 1, . . . , n ((r, l) = (1, 1)), due to transverse translations;
• the eigenvalue 0 of the multiplicity n with the eigenfunctions φ1,1,m,a(y) = ( a2π
)14√aywm,
m = 1, . . . , n ((r, l) = (0, 1)), due to rotation of the cylinder;
• the eigenvalue −a of the multiplicity 1 with the eigenfunction φ1,0,0,a(y) = ( a2π
)14√ay
((r, l) = (1, 0));
• the eigenvalue 0 of the multiplicity 1 with the eigenfunction φ2,0,0,a(y) = ( a2π
)14 (1−
ay2) ((k, l) = (2, 0)).
The last two eigenvalues are not of the broken symmetry origin and are not covered byTheorem H.2. They indicate instability of the cylindrical collapse
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By the description of the spectra of the normal hessians of Lspha := HessNsph Va(ϕ) and
Lcyla := HessNcyl Va(ϕ), we conclude that the spherical collapse is linearly stable while the
cylindrical one is not.It is shown in [29] that indeed the spherical collapse is (nonlinearly) stable while the
cylindrical one is not. We will also show that the last two eigenvalues of Lcyla in the list
above are due to translations of the point of the neckpinch on the axis of the cylinder anddue to shape instability, respectively.
H.3 F−stability of self-similar surfaces
Another notion of stability was introduced by analogy with minimal surfaces in [?]:
Definition H.10 (F−stability of self-similar surfaces, [?]). We say that a self-similarsurface φ, with a > 0, is F−stable iff the normal hessian satisfies HessN Va(ϕ) ≥ 0 on thesubspace span ϕ · ν(ϕ), νj(ϕ), j = 1, . . . , n+ 1⊥.
Remark H.11. a) The F−stability, at least in the compact case, says that the HessN Va(ϕ)has the smallest possible negative subspace, i.e ϕ has the smallest possible Morse index.
b) The reason the F−stability works in the non-compact case is that, due to the sep-aration of variables for the cylinder = (compact surface) ×Rk, the orthogonality to thenegative eigenfunctions of the compact factor removes the entire branches of the essentialspectrum. This might not work for warped cylinders.
c) Remark H.3(c) shows that the spheres and cylinders are F−stable. However, it isshown in [9, 29] that cylinders are dynamically unstable.
The following statement follows from Remark H.3(c) and the definition of the F−stability(see also the first part of Remark H.11(5)):
There are no smooth, embedded self-similar (a > 0), F−stable surfaces in Rn+1 close
to Sk × Rn−k, where Sk is the round k−sphere of radius√
ka.
A slight but a key improvement of this result is a hard theorem:
Theorem H.12 (Colding - Minicozzi). The only smooth, complete, embedded self-similar(a > 0), F−stable surfaces in Rn+1 of polynomial growth are Sk × Rn−k, where Sk is
the round k−sphere of radius√
ka.
Remark H.13. The difference between this theorem and (trivial) Corollary H.5 is thatthe latter requires a slightly stronger condition HessN Va(ϕ) > 0 on the subspace span ϕ ·ν(ϕ), νj(ϕ), j = 1, . . . , n+ 1⊥, then the F−stability.
Theorem H.6 implies that if it is F−stable, then H(ϕ) > 0.
Theorem H.14 (Colding- Minicozzi, Huisken). The only smooth, complete, embeddedself-similar surfaces in Rn+1, with a > 0, polynomial growth and H(ϕ) > 0, are Sk×Rn−k,
where Sk is the round k−sphere of radius√
ka.
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(Compare with surfaces of constant mean curvature)Theorems H.6 and H.14 imply Theorem H.12.There is a considerable literature on stable minimal surfaces. Much of it related to
the Bernstein conjecture:The only entire minimal graphs are linear functions.It was shown it is true for n ≤ 7:
(a) Bernstein for n = 2;
(b) De Georgi for n = 3;
(c) Almgren for n = 4; .
(d) Simons for n = 5, 6, 7.
(b) and (d) used in part results of Fleming on minimal cones. These results were extendedby Schoen, Simon and Yau.
Bombieri, De Georgi and Giusti constructed a contra example for n > 7.
I Remarks on Riemann surfaces and line bundles
I.1 Riemann surfaces of higher genus
Now, the question is: Can we extend our analysis to other Riemann surfaces?What is the Riemann surface? The Riemann is a 2-dimensional holomorphic manifold.
For the formal definition see [4], page 27, or [31]. I will not give a formal definition aswe will not use it, but present a few examples, so that you would recognize a Riemannsurface if it stares at your face. Here are the examples
(1) The complex plane C;(2) Any domain (a connected, open set) in C;(3) The the Riemann sphere P1 ≡ CP1 := C ∪ ∞11;(4) The torus C/L;(5) The Poincare half-plane H := z ∈ C : Imz > 0.(6) A nonsingular algebraic curve (z, w) ∈ C2 : p(z, w) = 0;(7) If X is a (compact) Riemann surface and G is a finite group of its holomorphic
automorphisms, then X/G is a Riemann surface.We have the following key theorem in the theory of Riemann surfaces.Theorem 1. Every connected Riemann surface X is homeomorphic to one of the
following
11P1 is diffeomorphic to S2 under the map (x1, x2, x3) → x1/1 − x3 + ix1/1 − x3, if x3 6= 1 and(0, 0, 1)→∞, if x3 = 1. The notation P1 ≡ CP1 refers to the fact that P1 can be viewed as the complexprojective space, P1 = (C2/0)/C∗, where C∗ := C/0 acts on C2/0 as λ(z0, z1) = (λz0, λz1).
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• The Riemann sphere P1;
• C or C/Z = C\0 or C/L
• H/Γ, for some discrete subgroup Γ ⊂ PSL(2,R) acting freely (i.e. without fixedpoints) on H.
Here PSL(2,R) is the quotient of the group SL(2,R) of the Mobius maps,
T : z → az + b
cz + d, a, b, c, d ∈ R, ad− bc = 1, (I.1)
by the subgroup ±1. A discrete subgroup Γ of PSL(2,R) is called the Fuchsian group.Idea of the proof. A general argument gives that a general Riemann surface X can be
written as X = (universal cover)/π1(X). Then one shows the universal covers of Riemannsurfaces are either P1, or C, or H, while π1(X) are id, L and Γ, respectively.
Thus going from g = 1 to g > 1 is going from Abelian π1(X), to a non-Abelian one.The surfaces listed in the theorem above are called elliptic, parabolic, hyperbolic, re-
spectively.Metric. On the basic spaces P1; C and H, we can define the metrics ds = 2|dz|/(1 +
|z|2), ds = |dz| and ds = |dz|/Imz, which have the Gaussian curvatures 1, 0 and −1,respectively.
Genus. A compact (closed) Riemann surface is homeomorphic to a sphere with ghandles. g is called the genus of the surface. For g = 0, we have a sphere, for g = 1, atorus and the surfaces described in the third item, i.e. surfaces of the form H/Γ, where Γis a Fuchsian group, acting freely on H, have g ≥ 2. Thus, we are interested in Riemannsurfaces of higher genera, g > 1.
Thus most of Riemann surfaces are of the form H/Γ, for some Fuchsian group Γ;such surfaces are called the hyperbolic Riemann surfaces. Understanding these surfacesinvolves understanding Fuchsian groups acting freely on H.
Hyperbolic plane. To begin with we mention that in the Poincare-Lobachevski modelof the hyperbolic geometry, the Poincare half-plane H := z ∈ C : Imz > 0 is equippedwith the metric
ds = |dz|/Imz.This means that for every point z ∈ H we are given the inner product (v, v′)z :=
1|Imz|2 Re(vv′) between any two tangent vectors v and v′ at this point. (Instead of Re(vv′)′
in C, we could have taken v · v in R2, where v · v′ is the usual Euclidean dot product.)The area element on H is given by
ω :=1
|y|2dxdy = − 1
2i|Imz|2dz ∧ dz
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for z = x+ iy. (Recall that dz ∧ dz = −2idx ∧ dy = −2idxdy.)The next two theorems state important properties of the action of the group SL(2,R)
on H.Theorem. SL(2,R) is a group of isometries (in fact, unitaries) of H with respect to
the metric ds = |dz|/y.Theorem. The hyperbolic area is invariant under the transformations from SL(2,R),
i.e. ω(TA) = ω(A), where ω(A) :=∫Aω.
Fuchsian groups. Recall that a Fuchsian group is a discrete subgroup of PSL(2,R).Important examples of such a group are
(a) The modular group PSL(2,Z);(b) All hyperbolic/parabolic cyclic groups, i.e. cyclic groups, generated by hyper-
bolic/parabolic elements. (An element T in (I.1) is called hyperbolic/parabolic, if itsatisfies |TrT | = |Tr g| = |a+ d| > 2/ = 2.)
Line bundle - equivariance correspondence: the general case. The construction(I.11) and Proposition I.1 can be generalized to arbitrary Riemann surfaces. Specifically,(I.11) generalizes to
L→ X, where L := H× C/Γ and X := H/Γ, (I.2)
with the base manifold X := H/Γ and the projection p : [(x,Ψ)]→ [x], where the groupΓ acts on the space H× C by
γ : (z,Ψ)→ (γz, eigγ(z)Ψ). (I.3)
Here ρ(z, γ) = eigγ(z) is an automorphy factor satisfying the co-cycle condition
gγγ′(x)− gγ(γ′x)− gγ′(x) ∈ 2πZ. (I.4)
In fact, all line bundles over a Riemann surface X can be obtained this way. Inother words, given a line bundle L over a Riemann surface X, find an automorphy factora(z, γ) = eigγ(z) s.t. L := H × C/Γ, with the action of Γ determined by a through (I.3).A difficult question here is how to determine this automorphy factor a explicitly in termsof the connection. It will be discussed in an appendix later on.
As before, we say that a function, Ψ and a one-form, A on H are equivariant iff theysatisfy the relations
where the function ηs(x) satisfies the cocycle condition
ηs+t(x)− ηs(x+ t)− ηt(x) ∈ 2π. (I.8)
As we show below, L−equivariant functions and vector fields are in one-to-one correspon-dence with sections and covariant derivatives (or connections) of the line bundle, L, overa complex torus.
We summarize this correspondence as
• L−equivariant functions on R2 ⇐⇒ sections on L
• L−equivariant vector fields (1-forms) on R2 ⇐⇒ connections on L
This allows to reformulate the Ginzburg-Landau equations as equations on a linebundle over a complex torus, which is one of the simplest Riemann surfaces. Then weextend the latter to an arbitrary Riemann surface and explore the consequences of this.
First, for a lattice L, we define the action of L on the space R2 by
s : x→ x+ s.
Next, given a function ηs(x) satisfying the cocycle condition (I.8), we define the action ofa lattice L on the space R2 × C by
s : (x,Ψ)→ (x+ s, eiηs(x)Ψ). (I.9)
Now, we can define the spaces T := R2/L and L := R2 × C/L of the equivalence classes,[x] and [(x,Ψ)], of elements x ∈ R2 and (x,Ψ) ∈ R2 × C under the action s : x → x + sand (I.9) of the group L, e.g.
One can show that(i) the spaces L := R2 × C/L and T := R2/L are manifolds (i.e. locally, the look like
R2 × C and R2, respectively);(ii) L is not of the form L = T × V , for some vector space V , i.e. it is not a trivial
line bundle.
Lectures on Applied PDEs 297
(iii) T := R2/L is a Riemann surface. It is homeomorphic to the standard torus:
[λν1 + µν2]→ (e2πiλ, e2πiµ), (I.10)
where (ν1, ν2) is a basis in L and λ, µ ∈ R. It is called the (complex, flat) torus.With the map p : L→ T, defined as p : [(x,Ψ)]→ [x], L is a line bundle over T in the
sense of the definition below.By a (complex) line bundle (over a manifold X) one means a triple (L,X, p), where L
and X are manifolds and p is a map, p : L→ X, with the property that X can be coveredby neighbourhoods U s.t. p−1(U) are homeomorphic to U × C (i.e. locally, L looks likeU × C).12
Here we mention a couple of definitions:
• A line bundle (L,X, p) is said to be trivial if it is of the form L = X × V , for somevector space V and with p : (x, v)→ x.
• Subspaces Lx := p−1(x) are called the fibers over x. (Lx are vector spaces iso-morphic to C.) L can be presented as a disjoint union of its fibers, L = ∪x∈XLx.• A line bundle (L,X, p) is said to be U(1)-, or unitary, bundle if transition functions
are required to have values in U(1).
We mention also that all operations with vector spaces pass to the line bundles: fortwo line bundles, L and L′, one can define direct sum L ⊕ L′ and the tensor productL ⊗ L′. For instance, writing L = ∪x∈XLx, etc, we have L ⊗ L′ = ∪x∈X(Lx ⊗ L′x) , withthe natural projection p : L⊗ L′ → X given by p(x, v ⊗ v′) = x.
We can think of line bundles as manifolds with the additional structure of a localsplitting intothe product of the base and a fiber.
In our case if we restrict to a fundamental cell Ω, then L ≈ Ω× C.To sum up, we constructed the line bundle
L→ T, where L := R2 × C/L and T := R2/L, (I.11)
with the base manifold T := R2/L and the projection p : [(x,Ψ)]→ [x].
Next, we need some more definitions:
12For a quick introduction to the line bundles, one can read pages 2-7 of Michael Murray, Line Bundles,http://www.maths.adelaide.edu.au/michael.murray/line bundles.pdf.
The definition above shows that a line bundle is determined by its transition functions ϕ−1V ϕU :
(U ∩ V ) × C → (U ∩ V ) × C, where ϕU : U × C → p−1(U) are homeomorphisms alluded to above.The transition functions give also a convenient description of line bundles. For instance, a line bundle(L,X, p) is said to be C∞/holomorphic, iff the transition functions C∞/holomorphic and L⊗L′ is a linebundle with the transition functions ϕ−1
V ϕU ⊗ (ϕ′V )−1 ϕ′U .Transition functions act as bundle maps, i.e. fiberwise: ϕ−1
V ϕU : (x,w)) → (x, a(x)w). We assumethat, for each x ∈ U ∩ V , the transformations a(x) : Lx → Lx are from the unitary group U(1) acting onLx.The corresponding line bundle is called the U(1) or unitary bundle.
298 Lectures on Applied PDEs
• The sections of L are maps σ : T→ L, satisfying p σ = 1, i.e. they map points ofT to points on the fibers above them. (Sections are locally usual functions, but witha global twist (in our case provided by factoring by L), they generalize the notionof a function.)
• A covariant derivative ∇ (or connection) on L is a map from sections to one-formson T with values in L, which (i) is linear and (ii) satisfies the Leibniz rule ∇(fσ) =f∇σ + df ⊗ σ. (Covariant derivatives are locally usual derivatives, they extend thenotion of a derivative to a manifold, but also the physics notion of the covariantderivative.)
• A curvature of connection a on L is the two-form Fa := da on T with values in L.
• Gauge transformations: for any g(x) ∈ C1(T, U(1)),
ψ(x)→ g(x)ψ(x), ∇ → g(x)−1∇g(x). (I.12)
∗ ∗ ∗ If sections of L are transformed as
ψ(x)→ g(x)ψ(x), (I.13)
for g(x) ∈ C1(T, U(1)), then the connection transforms as a(x) → a(x) − g−1(x)dg(x),so that, as before, ∇g∗(x)a(x) = g(x)−1∇a(x)g(x), where g∗a := a − dgg−1, while da isinvariant. ∗ ∗ ∗
In coordinates x, we can write the covariant derivative (connection) ∇A = ∇+ iA as∇A =
∑i∇idx
i, where A =∑
iAidxi is the connection one-form and ∇i := ∂i + iAi, the
components of ∇A.
Proposition I.1. Given an automorphy factor ρ and the associated line bundle L (see(I.11)), there is one-to-one correspondence between L−equivariant functions on R2 andsections of L and L−equivariant one-form on R2 and connections on L.
In particular, given an L−equivariant function, Ψ, on R2 and and an L−equivariantone-form, A, on R2, the corresponding section and connection on L are defined as
ψ([x]) = [(x,Ψ(x))], ∇aψ([x]) = [∇AΨ(x)],
where ∇A : Ψ→ dΨ− iAΨ.
Before proceeding to the proof of this statement, we give some more definitions. Thefundamental cell of L ⊂ C is a subset, Ω, of C having the properties
(i) Ω ∩ sΩ = ∀s ∈ L/0 and(ii) ∪s∈LΩ = C.
(The second property says that Ω tessellates C.)
Lectures on Applied PDEs 299
Using the restriction π∣∣Ω
of the map π : x → [x] to Ω, one can identify T with afundamental cell Ω of L:
T ≈ Ω.
By (i), π∣∣Ω
is one-to-one (if x ∈ Ω, then sx /∈ Ω). By (ii), for every [x] ∈ T, there iss = s[x] ∈ L s.t. x ∈ sΩ and therefore s−1x ∈ Ω. Since s−1x ∈ [x], this gives the inversemap T→ Ω ([x]→ s−1
[x]x).
(π : C→ Ω is a covering map of Ω and C is its universal cover.)
Proof of Proposition I.1. With an L−equivariant function Ψ on R2 we associate the sec-tion ψ of L as
ψ([x]) = [(x,Ψ(x))].
To check that this is consistent, let x′ be another representative of [x], then there is s inL s.t. x′ = x+ s. Since Ψ(x+ s) = eiηs(x)Ψ(x), the pair (x′,Ψ(x′)) belongs to [(x,Ψ(x))].
Conversely, given a section ψ in a line bundle L → T, we associate with it theL−equivariant function Ψ on R2, constructed as follows. First, using the constructionabove, we identify T with a fundamental cell Ω of L. For x ∈ Ω, we define Ψ = Ψ(x)uniquely by the relation
ψ([x]) = [(x,Ψ)]
(there is only one Ψ in [(x,Ψ)] associated with x ∈ Ω). Now, we define Ψ(x) = Ψ. Thisdefines Ψ(x) on Ω. For x ∈ Ω + s, we define
Ψ(x+ s) = eiηs(x)Ψ(x),
where ηs(x) is an automorphy exponent, constructed below. Since sΩ tessellate C, thefunction Ψ(x) is defined on C. It is straightforward to show that, due to the cocyclecondition (14.7), Ψ(x) is L−equivarinant.L−equivariant one-forms, A, are in one-to-one correspondence with covariant deriva-
tives (or connections) on L, constructed as follows. With L-equivariant one-form A, weassociate the covariant derivative on complex functions on R2, as before, by ∇A : Ψ →dΨ − iAΨ, but now dΨ − iAΨ is a one-form. Now, we define a([x]) = [A(x)] and, forψ ↔ Ψ(x), we set13
∇aψ([x]) = [∇AΨ(x)],
13Recall that the equivalence class [∇AΨ] is defined by the relation ∇AΨ ∼ ∇A′Ψ′ ⇔ ∃χ : R2 → G :(Ψ′, A′) = T gaugeχ (Ψ, A). (We will use the notation ∇A[Ψ] := [∇AΨ] = [(d− iA)Ψ].)
∇aψ([x]) = [∇AΨ(x)],
and use the equivariance of A(x) to check that the r.h.s. is independent of the representative x of [x] wehave chosen.
Note that ∇A : X → (∇A)X , where X is a vector field on the base manifold R2/L, and
(∇A)X [Ψ]([x]) := [(dX(x) − iA(X)(x))Ψ(x)].
300 Lectures on Applied PDEs
and use the equivariance of A(x) to check that the r.h.s. is independent of the represen-tative x of [x] we have chosen.
I.2 Inner product on sections
On sections, we define the inner product by 〈f, h〉 :=∫T fhdA, where dA is the area
element on the torus T. This can be extended to p−forms, p = 1, 2 (sections can beconsidered as 0−form) as
〈f, h〉 :=
∫T(f, h)zdA, (I.14)
where (f, h)z is the inner product on p−forms, p = 1, 2, at the point z. Very briefly, thelatter is defined as follows.
For a general surface X, the inner product on the tangent space TzX to X at a pointz ∈ X gives the inner product on the dual, T ∗zX, to TzX (the cotangent space to X ata point z ∈ X). This gives the inner product on one-forms. This can be further lifted tothe inner product on two-forms. See [7] for details.
In the present case, the inner product (f, h)z is independent of z. If we identify T witha fundamental cell Ω of the lattice L, then dA = dz ∧ dz on Ω and the expression (I.16)becomes
〈f, h〉 :=
∫Ω
(f, h)zdz ∧ dz. (I.15)
Very briefly, the inner product on one-forms is given in co-ordinate x as 〈α, β〉 :=∑∫Xαiβidvol, where α =
∑i αidx
i, etc, and dvol is the volume/area element on X, onfunctions on X. It can be written as
〈α, β〉 :=
∫X
α ∧ ∗βdvol. (I.16)
Here ∗ is the Hodge star operator, which maps p−forms into (d − p)−forms. (In theorthogonal co-ordinates on X ⊂ R2 and in a conformally flat metric ds2 = λ2dx1 ∧ dx2,we define ∗ by
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