Generic simplicity for the solutions of a nonlinear plate equation A. L. Pereira * Instituto de Matem´ atica e Estat´ ıstica da USP R. do Mat˜ ao, 1010 - CEP 05508-900 - S˜ ao Paulo - SP BRASIL. and M. C. Pereira † Escola de Artes, Ciˆ encias e Humanidades da USP Av. Arlindo B´ etio, 1000. CEP: 03828-080 - S˜ ao Paulo - SP BRASIL Abstract In this work we show that the solutions of the Dirichlet problem for a semilinear equation with the Bilaplacian as its linear part are generically simple in the set of C 4 -regular regions. AMS subject classification: 35J40,35B30. 1 Introduction Perturbation of the boundary for boundary value problems in PDEs have been inves- tigated by several authors, from various points of view, since the pioneering works of Rayleigh ([11]) and Hadamard ([1]). In particular, generic properties for solutions of boundary value problems have been considered in [4], [5], [6], [7], [8], [10], [12] and [14]. More recently several works appeared in a related topic, generally known as ‘shape analysis’ or ‘shape optimization’, on which the main issue is to determine conditions for a region to be optimal with respect to some cost functional. Among others, we mention [13] and [15]. Many problems of this kind have also been considered by D. Henry in [2] where a kind of differential calculus with the domain as the independent variable was developed. This approach allows the utilization of standard analytic tools such as Implicit Function Theorems and the Lyapunov-Schmidt method. In his work, Henry also formulated and * Research partially supported by FAPESP-SP-Brazil, grant 2003/11021-7. E-mail address: [email protected]† Research partially supported by CNPq - Conselho Nac. Des. Cient´ ıfico e Tecnol´ ogico - Brazil. E-mail address: [email protected]1
Transcript
Generic simplicity for the solutions
of a nonlinear plate equation
A. L. Pereira ∗
Instituto de Matematica e Estatıstica da USP
R. do Matao, 1010 - CEP 05508-900 - Sao Paulo - SP BRASIL.
andM. C. Pereira †
Escola de Artes, Ciencias e Humanidades da USP
Av. Arlindo Betio, 1000. CEP: 03828-080 - Sao Paulo - SP BRASIL
Abstract
In this work we show that the solutions of the Dirichlet problem for a semilinearequation with the Bilaplacian as its linear part are generically simple in the set ofC4-regular regions.
AMS subject classification: 35J40,35B30.
1 Introduction
Perturbation of the boundary for boundary value problems in PDEs have been inves-tigated by several authors, from various points of view, since the pioneering works ofRayleigh ([11]) and Hadamard ([1]).
In particular, generic properties for solutions of boundary value problems have beenconsidered in [4], [5], [6], [7], [8], [10], [12] and [14].
More recently several works appeared in a related topic, generally known as ‘shapeanalysis’ or ‘shape optimization’, on which the main issue is to determine conditions fora region to be optimal with respect to some cost functional. Among others, we mention[13] and [15].
Many problems of this kind have also been considered by D. Henry in [2] where akind of differential calculus with the domain as the independent variable was developed.This approach allows the utilization of standard analytic tools such as Implicit FunctionTheorems and the Lyapunov-Schmidt method. In his work, Henry also formulated and
∗Research partially supported by FAPESP-SP-Brazil, grant 2003/11021-7. E-mail address:[email protected]
†Research partially supported by CNPq - Conselho Nac. Des. Cientıfico e Tecnologico - Brazil.E-mail address: [email protected]
1
proved a generalized form of the Transversality Theorem, which is the main tool we usein our arguments.
where f(x, λ, y, µ) is a C4 real function in Rn × R× Rn × R with f(x, 0, 0, 0) ≡ 0 for allx ∈ Rn.
We show that, for a residual set of regions Ω ⊂ Rn (in a suitable topology), thesolutions u of (1) are all simple, that is , the linearisation
L(u) : u → ∆2u +∂f
∂µ(·, u,∇u, ∆u)∆u
+∂f
∂y(·, u,∇u, ∆u) · ∇u +
∂f
∂λ(·, u,∇u, ∆u)u
is an isomorphism.Our results can be seen as an extension of similar results for reaction-diffusion equa-
tions obtained by Saut and Teman ([12]) and Henry ([2]).This paper is organized as follows: in section 2 we collect some results we need from
[2]. In section 3 we prove that the differential operator
L = ∆2 + a(x)∆ + b(x) · ∇+ c(x) x ∈ Rn
is, generically, an isomorphism in the set of C4-regular regions Ω ⊂ Rn. This result isused in section 4 to prove our main result, the generic simplicity of solutions of (1). Themost difficult point there is the proof that a certain (pseudo differential) operator is notof finite range. This was proved in a separate work ([9]).
2 Preliminaries
The results in this section were taken from the monograph of Henry [2], where full proofscan be found.
2.1 Differential Calculus of Boundary Perturbation
Given an open bounded, Cm region Ω0 ⊂ Rn, consider the following open subset ofCm(Ω, Rn)
Diffk(Ω) = h ∈ Ck(Ω, Rn) | h is injective and 1/|det h′(x)| is bounded in Ω
We introduce a topology in the collection of all regions h(Ω) | h ∈ Diffk(Ω) bydefining (a sub-basis of) the neighborhoods of a given Ω by
When ‖h−iΩ‖Cm(Ω,Rn) is small, h is a Cm imbedding of Ω in Rn, a Cm diffeomorphismto its image h(Ω). Micheletti shows in [4] that this topology is metrizable, and the setof regions Cm-diffeomorphic to Ω may be considered a separable metric space which wedenote by Mm(Ω), or simply Mm. We say that a function F defined in the space Mm
with values in a Banach space is Cm or analytic if h 7→ F (h(Ω)) is Cm or analytic as amap of Banach spaces (h near iΩ in Cm(Ω, Rn)). In this sense, we may express problemsof perturbation of the boundary of a boundary value problem as problems of differentialcalculus in Banach spaces. More specifically, consider a formal non-linear differentialoperator u → v
v(x) = f(x, u(x), Lu(x)
), x ∈ Rn
where
Lu(x) =(u(x),
∂u
∂x1(x), ...,
∂u
∂xn(x),
∂2u
∂x21
(x),∂2u
∂x1∂x2(x), ...
), x ∈ Rn
More precisely, suppose Lu(·) has values in Rp and f(x, λ) is defined for (x, λ) insome open set O ⊂ Rn × Rp. For subsets Ω ⊂ Rn define FΩ by
FΩ(u)(x) = f(x, Lu(x)), x ∈ Ω (1)
for sufficiently smooth functions u in Ω such that (x, Lu(x)) ∈ O for any x ∈ Ω.Let h : Ω −→ Rn be Cm imbedding. We define the composition map (or pull-back)
h∗ of h byh∗u(x) = (u h)(x) = u(h(x)), x ∈ Ω
where u is a function defined in h(Ω). Then h∗ is an isomorphism from Cm(h(Ω)) toCm(Ω) with inverse h∗−1 = (h−1)∗. The same is true in other function spaces.
The differential operator
Fh(Ω) : DFh(Ω) ⊂ Cm(h(Ω)) −→ C0(h(Ω))
given by (1) is called the Eulerian form of the formal operator v 7→ f(·, Lv(·)), whereas
h∗Fh(Ω)h∗−1 : h∗DFh(Ω) ⊂ C
m(Ω) −→ C0(Ω)
is called the Lagrangian form of the same operator.The Eulerian form is often simpler for computations, while the Lagrangian form is
usually more convenient to prove theorems, since it acts in spaces of functions that donot depend on h, facilitating the use of standard tools such as the Implicit Function orthe Transversality theorem. However, a new variable, h is introduced. We then need tostudy the differentiability properties of the map
(u, h) 7→ h∗Fh(Ω)h∗−1u (2)
This has been done in [2] where it is shown that, if (y, λ) 7→ f(y, λ) is Ck or analyticthen so is the map above, considered as a map from Diffm(Ω)×Cm(Ω) to C0(Ω) (otherfunction spaces can be used instead of Cm). To compute the derivative we then need
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only compute the Gateaux derivative that is, the t-derivative along a smooth curvet 7→ (h(t, .), u(t, .)) ∈ Diffm(Ω)×Cm(Ω). For this purpose, it is convenient to introducethe differential operator
Dt =∂
∂t− U(x, t)
∂
∂x, U(x, t) = (
∂h
∂x)−1 ∂h
∂twhich is is called the anti-convective derivative. The results below (theorems 2.3, 2.6)are the main tools we use to compute derivatives.
Theorem 1 Suppose f(t, y, λ) is C1 in an open set in R × Rn × Rp, L is a constant-coefficient differential operator of order ≤ m with Lv(y) ∈ Rp (where defined). For opensets Q ⊂ Rn and Cm functions v on Q, let FQ(t)v be the function
y −→ f(t, y, Lv(y)), y ∈ Q.
where defined.Suppose t −→ h(t, ·) is a curve of imbeddings of an open set Ω ⊂ Rn, Ω(t) = h(t,Ω)
and for |j| ≤ m, |k| ≤ m + 1 (t, x) −→ ∂t∂jxh(t, x), ∂k
xh(t, x), ∂kxu(t, x) are continuous
on R×Ω near t = 0, and h(t, ·)∗−1u(t, ·) is in the domain of FΩ(t). Then, at points of Ω
Dt(h∗FΩ(t)(t)h∗−1)(u) = (h∗FΩ(t)(t)h∗
−1)(u) + (h∗F ′Ω(t)(t)h
∗−1)(u) ·Dtu
where Dt is the anti-convective derivative defined above,
FQ(t)v(y) =∂f
∂t(t, y, Lv(y))
andF ′
Q(t)v · w(y) =∂f
∂λ(t, y, Lv(y)) · Lw(y), y ∈ Q
is the linearisation of v −→ FQ(t)v.
EXAMPLE. Let f(x, λ, y, µ) be a smooth function in Rn × R× Rn × R and considerthe nonlinear differential operator
FΩ(v)(x) = ∆2v(x) + f(x, v(x),∇v(x),∆v(x))
which does not depend explicitly on t. Suppose also that h(t, x) = x + tV (x) + o(t) ina neighborhood of t = 0 and x ∈ Ω.Then, since ∂
∂t
(FΩ(u)
)= 0 and F ′
Ω(u) · w = L(u)w,we have, by Theorem 1∂
∂t
(h∗Fh(Ω)h
∗−1(u))
= Dt
(h∗Fh(Ω)h
∗−1(u))∣∣∣
t=0− h−1
x ht∇(h∗Fh(Ω)h
∗−1(u))∣∣∣
t=0
= h∗F ′h(Ω)h
∗−1(u) ·Dt(u)∣∣∣t=0
− h−1x ht∇
(h∗Fh(Ω)h
∗−1(u))∣∣∣
t=0
= L(u)(∂u
∂t− V · ∇u
)−V · ∇
(∆2u + f(x, u(x),∇u(x),∆u(x))
)where
L(u) = ∆2 +∂f
∂µ(·, u,∇u, ∆u)∆ +
∂f
∂y(·, u,∇u, ∆u) · ∇+
∂f
∂λ(·, u,∇u, ∆u).
4
2.2 Change of origin
We can always transfer the ‘origin’ or reference region from any Ω ⊂ Rn, to anotherdiffeomorphic region. Indeed, if H : Ω → Ω is a diffeomorphism we define, for anyimbedding h : Ω → Rn, another imbedding h = h H−1 : Ω → Rn.If x = H(x), u = H−1 ∗ u NΩ(x) = NH(Ω)(H(x)) = Ht
xNΩ(x)
‖HtxNΩ(x)‖ then h(Ω) = h(Ω),
h∗Fh(Ω)h∗−1u(x) = h∗Fh(Ω)(h
∗)−1u(x),
h∗Bh(Ω)h∗−1u(x) = h∗Bh(Ω)(h
∗)−1u(x),
using the normal
Nh(Ω)(h(x)) =(h−1)t
xNΩ(x)‖(h−1)t
xNΩ(x)‖
=(h−1)t
xNΩ(x)‖(h−1)t
xNΩ(x)‖= Nh(Ω)(h(x)).
This ‘change of origin’ will be frequently used in the sequel, as it allow us to computederivatives with respect to h at h = iΩ, where the formulas are simpler.
2.3 The Transversality Theorem
A basic tool for our results will be the Transversality Theorem in the form below, dueto D. Henry [2]. We first recall some definitions.
A map T ∈ L(X, Y ) where X and Y are Banach spaces is a semi-Fredholm map ifthe range of T is closed and at least one (or both, for Fredholm) of dim N (T ), codimR(T ) is finite; the index of T is then
index(T ) = ind(T ) = dimN (T )− codimR(T ).
We say that a subset F of a topological space X is rare if its closure has empty interiorand meager if it is contained in a countable union of rare subsets of X. We say that Fis residual if its complement in X is meager. We also say that X is a Baire space if anyresidual subset of X is dense.
Let f be a Ck map between Banach spaces. We say that x is a regular point of f if thederivative f ′(x) is surjective and its kernel is finite-dimensional. Otherwise, x is called acritical point of f . A point is a critical value if it is the image of some critical point of f .
Let now X be a Baire space and I = [0, 1]. For any closed or σ-closed F ⊂ X and anynonnegative integer m we say that the codimension of F is greater or equal to m (codimF≥ m) if the subset φ ∈ C(Im, X) | φ(Im) ∩ F is non-empty is meager in C(Im, X).We say codim F = k if k is the largest integer satisfying codim F ≥ m. codim F ≥ m.
Theorem 2 Suppose given positive numbers k and m; Banach manifolds X, Y, Z of classCk; an open set A ⊂ X × Y ; a Ck map f : A → Z and a point ξ ∈ Z. Assume for each(x,y) ∈ f−1(ξ) that:
5
1. ∂f∂x (x, y) : TxX −→ TξZ is semi-Fredholm with index < k.
2. (α) Df(x, y) : TxX × TyY −→ TξZ is surjective
or
(β) dimR(Df(x,y))
R( ∂f∂x (x,y))
≥ m + dim N (∂f
∂x (x, y)).
3. (x, y) 7→ y : f−1(ξ) −→ Y is σ-proper, that is f−1(ξ) is a countable union of sets Mj
such that (x, y) 7→ y : Mj −→ Y is a proper map for each j.[Given (xn, yn) ∈ Mj
such that yn converges in Y , there exists a subsequence (or subnet) with limit inMj.]
We note that 3 holds if f−1(ξ) is Lindelof [ every open cover has a countable subcover]or, more specifically, if f−1(ξ) is a separable metric space, or if X, Y are separable metricspaces.
Let Ay = x|(x, y) ∈ A and
Ycrit = y | ξ is a critical value of f(·, y) : Ay 7→ Z.
Then Ycrit is a meager set in Y and, if (x,y) 7→ y : f−1(ξ) 7→ Y is proper, Ycrit is alsoclosed. If ind ∂f
∂x ≤ −m < 0 on f−1(ξ), then (2(α)) implies (2(β)) and
Ycrit = y | ξ ∈ f(Ay, y)
has codimension ≥ m in Y.[ Note Ycrit is meager iff codim Ycrit ≥ 1 ].
3 Genericity of the isomorphism property for a classof linear differential operators
Let a : Rn → R, b : Rn → Rn and c : Rn → R be functions of class C3 and consider the(formal) differential operator
L = ∆2 + a(x)∆ + b(x) · ∇+ c(x) x ∈ Rn.
We show in this section that the operator
LΩ : H4 ∩H20 (Ω) → L2(Ω) (3)
u → Lu
is, generically, an isomorphism in the set of open, connected, bounded C3-regular regionsof Rn. More precisely, we show that the set
I = h ∈ Diff4(Ω) | the operator h∗Lh(Ω)h∗−1 from H4 ∩H2
0 (Ω) into L2(Ω)is an isomorphism (4)
6
is an open dense set in Diff4(Ω). Observe that the operator h∗Lh(Ω)h∗−1 is an isomor-
phism if, and only if the operator Lh(Ω) from H4 ∩H20 (h(Ω)) to L2(h(Ω)) is an isomor-
phism, since h∗ and h∗−1 are isomorphisms from L2(h(Ω)) to L2(Ω) and H4 ∩H20 (Ω) to
H4 ∩H20 (h(Ω)) respectively. Consider the differentiable map
K : H4 ∩H20 (Ω)×Diff4(Ω) → L2(Ω)
(u, h) → h∗Lh(Ω)h∗−1u. (5)
Proposition 3 Let a : Rn → R, b : Rn → Rn, c : Rn → R be C2 functions, Ω ⊂ Rn
an open, connected bounded C4-regular region, and h ∈ Diff4(Ω). Then zero is a regularvalue of the map (application)
Kh : H4 ∩H20 (Ω) −→ L2(Ω)
u −→ h∗Lh(Ω)h∗−1u,
if and only if h∗Lh(Ω)h∗−1 is an isomorphism.
Proof. First observe that Kh is a Fredholm operator of index 0 since Lh(Ω) is Fredholmof index 0 and h∗, h∗−1 are isomorphisms. If 0 is a regular value then the linearisationof Kh at 0, which is again Kh, must be surjective. Being of index 0 it is also injectiveand therefore an isomorphism by the Open Mapping Theorem. Reciprocally, if Kh is anisomorphism, it is surjective at any point.
From 3 and the Implicit Function Theorem it follows that I is open . We thus onlyneed to show density. For that we may work with more regular regions.
It would be very convenient for our purposes to have the following ‘unique continua-tion’ result.
If u is a solution of LΩu = 0 with ∂2u∂N2 = 0 in a open set of ∂Ω, then u ≡ 0.
Such a result is not available, to the best of our knowledge, but the following ‘genericunique continuation result’ will be sufficient for our needs. We will not prove it heresince the argument is very similar to the one of 11 below.
Lemma 4 Let Ω ⊂ Rn be an open, connected, bounded C5-regular region with n ≥ 2 andJ an open nonempty subset of ∂Ω. Consider the differentiable map
G : BM ×Diff5(Ω) → L2(Ω)×H32 (J)
defined byG(u, λ, h) =
(h∗Lh(Ω)h
∗−1u, h∗∆h∗−1u∣∣∣J
)where BM = u ∈ H4 ∩H2
0 (Ω)− 0 | ‖u‖ ≤ M. Then, the set
CJM = h ∈ Diff5(Ω) | (0, 0) ∈ G(BM , h)
is meager and closed in Diff5(Ω).
7
Theorem 5 Let a : Rn → R, b : Rn → Rn and c : Rn → R be functions of class C3.Then the operator LΩ defined in (3) is generically an isomorphism in the set of open,connected C4-regular regions Ω ⊂ Rn, n ≥ 2. More precisely, if Ω ⊂ Rn (n ≥ 2) is anopen, connected C4-regular region, than the set I defined in (4) is an open dense set inDiff4(Ω).
Proof. By proposition 3, all we need to show is that 0 is a regular value of Kh ina residual subset of Diff4(Ω). Since our spaces are separable and Kh is Fredholm ofindex 0, this would follow from the Transversality Theorem if we could prove that 0 isa regular value of K. Let us suppose that this is not true, that is, there exists a criticalpoint (u, h) ∈ K−1(0). As explained in (2.2 ), we may suppose that h = iΩ. Since weonly need to prove density, we may also suppose that Ω is C5-regular. Then, there existsv ∈ L2(Ω) such that∫
Ω
vDK(u, iΩ)(u, h) = 0 for all (u, h) ∈ H4 ∩H20 (Ω)× C5(Ω, Rn) (6)
where DK(u, iΩ) from H4 ∩H20 (Ω)× C5(Ω, Rn) to L2(Ω) is given by
DK(u, iΩ)(u, h) = LΩ(u− h · ∇u).
Choosing h = 0 and varying u in H4 ∩H20 (Ω), we obtain∫
Ω
vLΩu = 0 for all u ∈ H4 ∩H20 (Ω)
and v is therefore a weak, hence strong, solution of
L∗Ωv = 0 in Ω (7)
where L∗Ω from H4 ∩H20 (Ω) to L2(Ω) is given by
L∗Ωv = ∆2v + a∆v + (2∇a− b) · ∇v + (c + ∆a− div b)v.
By regularity of solutions of strongly elliptic equations, v is also a strong solution, thatis, v ∈ H4 ∩ H2
0 (Ω) ∩ C4,α(Ω) for some α > 0 and satisfies (7) [Note that u ∈ H5(Ω),since Ω is C5-regular.]
Choosing u = 0 and varying h in C5(Ω, Rn), we obtain
0 = −∫
Ω
vLΩ(h · ∇u) =∫
Ω
(h · ∇u)L∗Ωv − vLΩ(h · ∇u) =∫
∂Ω
h ·N∆v∆u,
for all h ∈ C5(Ω, Rn) since ∆u|∂Ω = ∂2u∂N2
∣∣∣∂Ω
. Thus ∆v∆u ≡ 0 in ∂Ω. This is not a
contradiction (or at least it is not clear that it is). We show now, however, that it is acontradiction generically; it can only happen in an ‘exceptional’ set of Diff4(Ω) . Theresult then follows by reapplying the argument above outside this exceptional set. To bemore precise, consider the map
0 (Ω)2 | ‖u‖, ‖v‖ ≤ M. We show, using the TransversalityTheorem that the set
HM = h ∈ Diff5(Ω) | (0, 0, 0) ∈ H(B2M , h)
is meager and closed in Diff5(Ω) for all M ∈ N. We may, by ‘changing the origin’ ifnecessary assume that the ‘generic uniqueness property’ stated in Lemma 4 holds in ΩWe then apply the Transversality Theorem again for the map K, with h restricted to thecomplement of HM , obtaining another subset HM of Diff5(Ω). such that 0 is a regularvalue of Kh for any h ∈ HM Taking intersection for M ∈ N, the desired result follows.
To show thatHM is a meager closed set we apply Henry’s version of the TransversalityTheorem for the map H. Since our spaces are separable and ∂K
∂u (u, i∂Ω) is Fredholm itremains only to prove that the map (u, v, h) 7→ h : H−1(0, 0, 0) → Diff5(Ω) is proper andthe hypothesis (2β). We first show that (u, v, h) → h : H−1(0, 0, 0) → Diff5(Ω) is proper.Let (un, vn, hn)n∈N ⊂ H−1(0, 0, 0) be a sequence with hn → iΩ in Diff5(Ω) (the generalcase is similar). Since (un, vn)n∈N ⊂ B2
M , we may assume, taking a subsequence thatthere exists (u, v) ∈ H2
0 (Ω)2 such that un → u and vn → v in H20 (Ω). We have, for all
n ∈ N
h∗n
(∆2 + a∆ + b · ∇+ c
)h∗n
−1un = 0 ⇐⇒ h∗n∆2h∗n−1un = −h∗n
(a∆ + b · ∇+ c
)h∗n
−1un.
Since ∆2 is an isomorphism, it follows that
un = −h∗n(∆2)−1(a∆ + b · ∇+ c
)h∗n
−1un (8)
By results on section 2.1, the right-hand side of (8) is analytic as an application fromH2
0 (Ω) × Diff5(Ω) to H4 ∩ H20 (Ω). Taking the limit as n → +∞, we obtain that u ∈
H4 ∩H20 (Ω) and satisfies ∆2u + a∆u + b · ∇u + cu = 0. By Lemma 10 of [10] we have,
for h ∈ Diff5(Ω) and v ∈ H4 ∩H20 (Ω)
h∗∆2h∗−1(v) = ∆2(v) + Lh(v) with ‖Lhu‖L2(Ω) ≤ ε(h)‖u‖H4(Ω)
and ε(h) → 0 as h → iΩ in C4(Ω, Rn).Since un → u in H2
0 (Ω) and hn → iΩ in Diff5(Ω) as n → +∞, we obtain
It follows from (9) and (10) that ‖∆2(un − u)‖L2(Ω) → 0 as n → +∞. Since ∆2 is anisomorphism from H4∩H2
0 (Ω) to L2(Ω), we obtain un → u in H4∩H20 (Ω) and, therefore
‖u‖H4∩H20 (Ω) ≤ M , for all n ∈ N. Similarly, we prove that vn → v in H4 ∩H2
0 (Ω) and‖v‖H4∩H2
0 (Ω) ≤ M from which we conclude that the map (u, v, h) → h : H−1(0, 0, 0) →Diff5(Ω) is proper.
It remains only to prove (2β), which we do by showing that
dimR(DH(u, v, h))
R(
∂H∂u (u, v, h)
)= ∞ for alll (u, v, h) ∈ H−1(0, 0, 0).
Suppose this is not true for some (u, v, h) ∈ H−1(0, 0, 0). Assuming, as we may, that h =iΩ it follows that there exist θ1, ..., θm ∈ L2(Ω)2×L1(∂Ω) such that, for all h ∈ C5(Ω, Rn)there exist u, v ∈ H4 ∩H2
DH1(u, v, iΩ)(u, v, h) = LΩ(u− h · ∇u)DH2(u, v, iΩ)(u, v, h) = L∗Ω(v − h · ∇v)
DH3(u, v, iΩ)(u, v, h) =
∆v∆(u− h · ∇u) + ∆u∆(v − h · ∇v)
+h ·N ∂
∂N(∆u∆v)
∣∣∣∂Ω
It follows from (11) that
LΩ(u− h · ∇u) =m∑
i=1
ciθ1i (12)
L∗Ω(v − h · ∇v) =m∑
i=1
ciθ2i (13)
m∑i=1
ciθ3i =
∆v∆(u− h · ∇u) + ∆u∆(v − h · ∇v)
+h ·N ∂
∂N(∆u∆v)
∣∣∣∂Ω
. (14)
10
Let u1, ..., ul be a basis for the kernel of LΩ and consider the operators
AL : L2(Ω) → H4 ∩H10 (Ω)
CL : H52 (∂Ω) → H4 ∩H1
0 (Ω)
defined byw = AL(z) + CL(g)
where Lw − z belongs to a (fixed) complement of R(LΩ) in L2(Ω), ∂w∂N = g on ∂Ω and∫
Ωwφ = 0 for all φ ∈ N (L∗Ω). Let also v1, ..., vl be a basis for the kernel of L∗Ω and
consider the operatorsAL∗ : L2(Ω) → H4 ∩H1
0 (Ω) and
CL∗ : H52 (∂Ω) → H4 ∩H1
0 (Ω)
similarly defined. We have shown in [9] that these operators are well defined.From equations (12) and (13), we obtain
u− h · ∇u =l∑
i=1
ξiui +m∑
i=1
ciALθ1i − CL(h ·N∆u) (15)
since ∂∂N (u− h · ∇u)
∣∣∣∂Ω
= −h ·N ∂2u∂N2
∣∣∣∂Ω
= −h ·N∆u|∂Ω and
v − h · ∇v =l∑
i=1
ηivi +m∑
i=1
ciAL∗θ1i − CL∗(h ·N∆v) (16)
since ∂∂N (v − h · ∇v)
∣∣∣∂Ω
= −h ·N ∂2v∂N2
∣∣∣∂Ω
= −h ·N∆v|∂Ω.
Substituting (15) and (16) in (14), we obtain thath ·N ∂
∂N(∆u∆v)−
[∆u∆CL∗(h ·N∆v) + ∆v∆CL(h ·N∆u)
]∣∣∣∂Ω
(17)
remains in a finite dimensional space when h varies in C5(Ω, Rn).The set U = x ∈ ∂Ω | ∆u(x) 6= 0 is nonempty since we have assumed that ‘generic
unique continuation’ holds in Ω. Therefore, we must have ∆v|U ≡ 0. If h ≡ 0 in ∂Ω−U ,then h ·N∆v ≡ 0 in ∂Ω and, therefore
∆u∆CL∗(h ·N∆v) = ∆u∆CL∗(0)
belongs to the finite dimensional space [∆u∆v1, ...,∆u∆vl] where v1, ..., vl is a basisfor the kernel of N (L∗Ω). It follows that
h ·N ∂
∂N(∆u∆v)−∆v∆CL(h ·N∆u)
∣∣∣U
= h ·N ∂
∂N(∆u∆v)
∣∣∣U
= h ·N∆u∂
∂N(∆v)
∣∣∣U
(18)
11
remains in a finite dimensional space, when h varies in C5(Ω, Rn) with h ≡ 0 in ∂Ω− U.
Since ∆u(x) 6= 0 for any x ∈ U , this is only possible ( dim Ω ≥ 2) if ∂∆v∂N
∣∣∣U≡ 0. But
then v ≡ 0 in Ω by Theorem 6 below, and we reach a contradiction proving the result.
During the proof of theorem 5 we have used the following uniqueness theorem, whichis a direct consequence of Theorem 8.9.1 of [3].
Theorem 6 Suppose Ω ⊂ Rn is an open connected, bounded, C4-regular domain and Bis an open ball in Rn such that B ∩ ∂Ω is a (nontrivial) C4 hypersurface. Suppose alsothat u ∈ H4(Ω) satisfies
|∆2u| ≤ C(|∆u|+ |∇u|+ |u|
)a.e. in Ω
for some positive constant C and u =∂u
∂N= ∆u =
∂∆u
∂N= 0 in B ∩ ∂Ω. Then u is iden-
tically null.
4 Generic simplicity of solutions
Let f(x, λ, y, µ) be a real function of class C4 defined in Rn × R × Rn × R satisfyingf(x, 0, 0, 0) ≡ 0 for all x ∈ Rn. We prove in this section our main result: generically inthe set of connected, bounded C4-regular regions Ω ⊂ Rn, n ≥ 2, the solutions u of
∆2u + f(·, u,∇u, ∆u) = 0 in Ωu = ∂u
∂N = 0 on ∂Ω(19)
are all simple. We choose p > n2 , so that the continuous imbedding p, W 4,p∩W 2,p
0 (Ω) →C2,α(Ω) holds for some α > 0.
It follows then, from the Implicit Function Theorem, that the set of solutions isdiscrete in W 4,p ∩W 2,p
0 (Ω) and, in particular finite if f is bounded.
Remark 7 Since we have assumed f(x, 0, 0, 0) ≡ 0 in Rn, the null function u ≡ 0 is asolution of (19) for any Ω ⊂ Rn. It follows from theorem 5 that u ≡ 0 is simple for Ω inan open dense set of Diff4(Ω). We therefore concentrate in the proof of generic simplicityfor the nontrivial solutions.
Proposition 8 Let Ω ⊂ Rn be an open, connected bounded C4-regular region and f(x, λ, y, µ)a C2 real function defined in Rn ×R×Rn ×R. Then, zero is a regular value of the map
Fh : W 2,p ∩W 1,p0 (Ω) −→ Lp(Ω)
u −→ h∗∆2h∗−1u + h∗f(·, h∗−1u,∇h∗−1u, ∆h∗−1u),
if and only if all solutions of (19) in h(Ω) are simple.
Proof. The proof is very similar to the one of proposition 3 and will be left to thereader.
12
Proposition 9 A function u ∈ W 4,p ∩W 2,p0 (Ω) is a solution (resp. a simple solution)
of h∗∆2h∗−1u + h∗f(·, h∗−1u,∇h∗−1u, ∆h∗−1u) = 0 in Ω,u = ∂u
∂N = 0 on ∂Ω(20)
if and only if v = h∗−1u is a solution (resp. simple solution) of (19) in h(Ω).
Proof. Let u ∈ W 4,p ∩W 2,p0 (Ω). Since h∗ and h∗−1 are isomorphisms, we have
It is clear that u = 0 in ∂Ω if and only if v = 0 in ∂h(Ω). Writing y = h(x), we obtain
∂v
∂Nh(Ω)(y) = Nh(Ω)(y) · ∇y(u h−1)(y)
= Nh(Ω)(y) · (h−1)ty(y)∇xu(x)
= Nh(Ω)(y) · (h−1x )t(x)∇xu(x)
= Nh(Ω)(y) · ((hx)−1)t(x)∂u
∂N(x) NΩ(x)
=∂u
∂N(x)
1||(h−1
x )tNΩ(x)||((h−1
x )t(x)NΩ(x) · (h−1x )t(x)NΩ(x)
)where we have used that u = 0 in ∂Ω. Since h−1
x (x) is non-singular it follows that∂v
∂Nh(Ω)(y) = 0 if and only if ∂u
∂N (x) = 0. Thus u is a solution of (20) if and only if
h∗−1u is a solution of (19) in h(Ω). Finally, since h∗L(u)h∗−1 is an isomorphism inW 4,p ∩W 2,p
0 (Ω) if and only if L(v) is an isomorphism in W 4,p ∩W 2,p0 (h(Ω)) so the result
follows.
It follows from (9) and (8) that, in order to show generic simplicity of the solutionsof (19) is enough to show that 0 is a regular value of Fh, generically in h ∈ Diff4(Ω). Weshow, using the Transversality Theorem, that 0 is a regular value of
FM : BM × VM → Lp(Ω)(u, h) → h∗∆2h∗−1u + h∗f(·, h∗−1u,∇h∗−1u, ∆h∗−1u) (21)
where BM = u ∈ W 4,p ∩W 2,p0 (Ω) − 0 | ‖u‖ ≤ M and VM is an open dense set in
Diff4(Ω), for all M ∈ N. Taking the intersection of VM for M ∈ N we obtain the desiredresidual set.
Remark 10 Applying the Implicit Function Theorem to the map FM defined in (21) weobtain that the set
FM = h ∈ Diff4(Ω) | all solutions u of (20) with ‖u‖W 4,p∩W 2,p0 (Ω) < M are simple
13
is open in Diff4(Ω) for all M ∈ N. To prove density, we may work with more regular(for example C∞) regions.
If we try to apply the Transversality Theorem directly to the function F defined in‖u‖W 4,p∩W 2,p
0 (Ω)×Diff4(Ω) by (21) we do not obtain a contradiction. What we do obtainis that the possible critical points must satisfy very special properties. The idea is thento show that these properties can only occur in a small (meager and closed) set andthen restrict the problem to its complement. In our case the ‘exceptional situation’ ischaracterized by the existence of a solution u of (19) and a solution v of the problem
L∗(u)v = 0 in Ωv = ∂v
∂N = 0 on ∂Ω
satisfying the additional property ∆u∆v ≡ 0 on ∂Ω. We show in Lemma 12 that thissituation is really ‘exceptional’, that is, it can only happen if h is outside an open densesubset of Diff4(Ω) (for u and v restricted to a bounded set).
We will need the following ‘generic unique continuation result’.
Lemma 11 Let Ω ⊂ Rn n ≥ 2 be an open, connected, bounded, C5-regular domain, J anonempty open subset of Ω and f(x, λ, y, µ) a C2 real function defined in Rn×R×Rn×Rwith f(·, 0, 0, 0) ≡ 0. Consider the map
M = h ∈ Diff5(Ω) | (0, 0) ∈ G(AM , h)is a closed meager subset of Diff5(Ω).
Proof. We apply the Transversality Theorem. Observe that G is differentiable. In factit is analytic in h as observed in section 2.1, and the differentiability in u follows fromthe smoothness of f and the continuous immersion W 4,p ∩W 2,p
0 (Ω) ⊂ C2,α(Ω) for someα > 0. We compute its differential using Theorem 1 (see example 2.1 of section 2.1).
DG(u, iΩ)(u, h) =(L(u)(u− h · ∇u),∆(u− h · ∇u) + h ·N ∂∆u
∂N
∣∣∣J
).
To verify (1) and (2), we proceed as in the proof of theorem 5. We prove that (2β)holds showing that
dimR(DG(u, h))
R(
∂G∂u (u, h)
)= ∞ for all (u, h) ∈ G−1(0, 0).
14
Suppose, by contradiction this is not true for some (u, h) ∈ G−1(0, 0). By ‘changing theorigin’ we may suppose that h = iΩ. Then, there exist θ1, ..., θm ∈ Lp(Ω) ×W 2− 1
p ,p(J)for all h ∈ C5(Ω, Rn) there exists u ∈ W 4,p∩W 2,p
0 (Ω) and scalars c1, ..., cm ∈ R such that
DG(u, iΩ)(u, h) =m∑
i=1
ciθi,
that is,
L(u)(u− h · ∇u) =m∑
i=1
ciθ1i (22)
∆(u− h · ∇u) + h ·N ∂∆u
∂N
∣∣∣J
=m∑
i=1
ciθ2i (23)
where
L(u) = ∆2 +∂f
∂µ(·, u,∇u, ∆u)∆ +
∂f
∂y(·, u,∇u, ∆u) · ∇+
∂f
∂λ(·, u,∇u, ∆u). (24)
Let u1, ..., ul be a basis for the kernel of L0(u) = L(u)∣∣∣W 4,p∩W 2,p
0 (Ω)and consider
the operatorsAL(u) : Lp(Ω) → W 4,p ∩W 1,p
0 (Ω)
CL(u) : W 3− 1p ,p(∂Ω) → W 4,p ∩W 1,p
0 (Ω)
defined byw = AL(u)(z) + CL(u)(g)
if L(u)w − z belongs to a fixed complement of R(L0(u)) in Lp(Ω), ∂w∂N = g on ∂Ω and∫
Ωwφ = 0 for all φ ∈ N (L∗0(u)). (We proved these operators are well defined in [9]).Choosing h ∈ C5(Ω, Rn) such that h ≡ 0 on ∂Ω− J , we obtain from (22), that
u− h · ∇u =l∑
i=1
ξiui +m∑
i=1
ciAL(u)(θ1i ) (25)
since u− h · ∇u ∈ W 4,p ∩W 2,p0 (Ω).
Substituting (25) in (23), we obtain that h ·N ∂∆u
∂N
∣∣∣J
remains in a finite dimen-
sional subspace when h varies in C5(Ω, Rn). Since dim Ω ≥ 2 this is possible only if∂∆u
∂N≡ 0 on J so u satisfies
∆2u + f(·, u,∇u, ∆u) = 0 in Ωu = ∂u
∂N = 0 on ∂Ω∆u = ∂∆u
∂N = 0 on J.(26)
15
We claim that u satisfies the hypotheses of Cauchy’s Uniqueness Theorem 6. Indeed,since u ∈ W 4,p(Ω) ∩ C2,α(Ω) for some α > 0 (p > n
2 ) and is a solution of the uniformlyelliptic equation ∆2u+f(·, u,∇u, ∆u) = 0 in Ω then u ∈ W 4,p(Ω)∩C4,α(Ω). Furthermore,u = ∂u
We conclude that u ≡ 0, which gives the searched for contradiction.
Lemma 12 Let Ω ⊂ Rn n ≥ 2 be an open, connected, bounded, C5-regular domain andf(x, λ, y, µ) a C3 real function defined in Rn×R×Rn×R with f(·, 0, 0, 0) ≡ 0. Considerthe map
0 (Ω)− 0 | ‖u‖ ≤ M and p−1 + q−1 = 1 with p > n2 ,
AM,q = u ∈ W 4,q ∩W 2,q0 (Ω) − 0 | ‖u‖ ≤ M, DM = Diff5(Ω) − C∂Ω
M , C∂ΩM given by
Lemma 11 and
L∗(w) = ∆2 +∂f
∂µ(·, w,∇w,∆w)∆
+[2∇
(∂f
∂µ(·, w,∇w,∆w)
)− ∂f
∂y(·, w,∇w,∆w)
]· ∇
+∆[∂f
∂µ(·, w,∇w,∆w)
]− div
(∂f
∂y(·, w,∇w,∆w)
)+
∂f
∂λ(·, w,∇w,∆w).
ThenEM = h ∈ DM | (0, 0, 0) ∈ Q(AM )×AM,q, h)
is a meager closed subset of Diff5(Ω).
(Observe that L∗(w) is the formal adjoint of L(w) defined by (24).)
Proof. We again apply the Transversality Theorem. The differentiability of Q iseasy to establish, and its derivative can be computed using theorem 1 (see example 2.1)
16
DQ(u, v, iΩ)(u, v, h) =(L(u)(u− h · ∇u),
L∗(u)(v − h · ∇v) +(∂L∗
∂w(u) · v
)(u− h · ∇u),
∆(u− h · ∇u)∆v + ∆u∆(v − h · ∇v) + h ·N ∂
∂N(∆u∆v)
∣∣∣∂Ω
)where ∂L∗
∂w (u) · v is the second order differential operator given by(∂L∗
∂w(u) · v
)z =
( ∂2f
∂λ∂µv +
∂2f
∂y∂µ· ∇v +
∂2f
∂µ2 ∆v)∆z
+[2∇
( ∂2f
∂λ∂µv +
∂2f
∂y∂µ· ∇v +
∂2f
∂µ2∆v
)−
( ∂2f
∂λ∂yv +
∂2f
∂y2∇v +
∂2f
∂µ∂y∆v
)]· ∇z[(∂2f
∂λ2v +
∂2f
∂λ∂y· ∇v +
∂2f
∂λ∂µ∆v
)+∆
( ∂2f
∂λ∂µv +
∂2f
∂y∂µ· ∇v +
∂2f
∂µ2∆v
)− div
( ∂2f
∂λ∂yv +
∂2f
∂y2∇v +
∂2f
∂y∂µ∆v
)]z.
(We have written f instead of f(·, u,∇u, ∆u) to simplify the notation).The hypotheses (1) and (3) of the Transversality Theorem can be verified as in the
proof of Theorem 5. We prove (2β) by showing that
dim R(DQ(u, v, h))
R(
∂Q∂(u,v) (u, v, h)
)= ∞
for all (u, v, h) ∈ Q−1(0, 0, 0). Suppose this is not true for (u, v, h) ∈ Q−1(0, 0, 0).‘Changing the origin’, we may assume that h = iΩ. Then, there exist θ1, ..., θm ∈Lp(Ω)×Lq(Ω)×L1(∂Ω) such that for all h ∈ C5(Ω, Rn) there exists u ∈ W 4,p∩W 2,p
0 (Ω),v ∈ W 4,q ∩W 2,q
0 (Ω) and scalars c1, ..., cm ∈ R such that
DQ(u, v, iΩ)(u, v, h) =m∑
i=1
ciθi,
that is,
L(u)(u− h · ∇u) =m∑
i=1
ciθ1i (27)
L∗(u)(v − h · ∇v) +(∂L∗
∂w(u) · v
)(u− h · ∇u) =
m∑i=1
ciθ2i (28)
17
andm∑
i=1
ciθ3i =
∆(u− h · ∇u)∆v + ∆u∆(v − h · ∇v)
+h ·N ∂
∂N(∆u∆v)
∣∣∣∂Ω
. (29)
Let u1, ..., ul be a basis for the kernel of L0(u) = L(u)∣∣∣W 4,p∩W 2,p
0 (Ω), v1, ..., vl a
basis for the kernel of L∗0(u) and consider the operators
AL(u) : Lp(Ω) → W 4,p ∩W 1,p0 (Ω)
CL(u) : W 3− 1p ,p(∂Ω) → W 4,p ∩W 1,p
0 (Ω)
defined byw = AL(u)(z) + CL(u)(g)
where L(u)w − z belongs to a fixed complement of R(L0(u)) in Lp(Ω), ∂w∂N = g on ∂Ω,∫
Ωwφ = 0 for all φ ∈ N (L∗0(u)) and
AL∗(u) : Lq(Ω) → W 4,q ∩W 1,q0 (Ω)
CL∗(u) : W 3− 1q ,q(∂Ω) → W 4,q ∩W 1,q
0 (Ω)
defined byt = AL∗(u)(z) + CL∗(u)(g)
where L∗(u)t − z belongs to a fixed complement of R(L∗0(u)) in Lq(Ω), ∂t∂N = g on ∂Ω
and∫Ω
tϕ = 0 for all ϕ ∈ N (L0(u)). (We proved these operators are well defined in [9]).From (27) and (28) it follows that
u− h · ∇u =l∑
i=1
ξiui +m∑
i=1
ciAL(u)(θ1i )− CL(u)(h ·N∆u) (30)
v − h · ∇v =s∑
i=1
ηivi +m∑
i=1
ciAL∗(u)(θ2i )− CL∗(u)(h ·N∆v)
−AL∗(u)
((∂L∗
∂w(u) · v
)(u− h · ∇u)
). (31)
Substituting in (29), we obtain thath ·N ∂
∂N(∆u∆v)−∆v∆
(CL(u)(h ·N∆u)
)+∆u∆
[AL∗(u)
((∂L∗
∂w(u) · v
)CL(u)(h ·N∆u)
)− CL∗(u)(h ·N∆v)
]∣∣∣∂Ω
18
remains in a finite dimensional space when h varies in C5(Ω, Rn), that is, the operator
Υ(h) =
h ·N ∂
∂N(∆u∆v)−∆v∆
(CL(u)(h ·N∆u)
)(32)
+∆u∆[AL∗(u)
((∂L∗
∂w(u) · v
)CL(u)(h ·N∆u)
))− CL∗(u)(h ·N∆v)
]∣∣∣∂Ω
defined in C5(Ω, Rn) is of finite rangeWe proved in [9] that, if dim Ω ≥ 2, a necessary condition for Υ to be of finite range
is∂
∂N(∆u∆v) ≡ 0 on ∂Ω. (33)
Thus the functions u, v must satisfy∆2u− f(·, u,∇u, ∆u) = 0 in Ωu = ∂u
∂N = 0 on ∂Ω(34)
L∗(u)v = 0 in Ωv = ∂v
∂N = 0 on ∂Ω(35)
and also∆u∆v|∂Ω =
∂
∂N(∆u∆v)
∣∣∣∂Ω
= 0. (36)
Let U = x ∈ ∂Ω | ∆u(x) 6= 0. Observe that U is a nonempty, since iΩ ∈ D∂ΩM
(given by Lemma 11).
By equation (36), we have ∆v|U =∂∆v
∂N
∣∣∣U≡ 0. Therefore v ∈ W 4,q∩W 2,q
0 (Ω) satisfiesthe hypotheses of theorem 6. Thus v ≡ 0 in Ω and we reach a contradiction, proving theresult.
Theorem 13 Generically in the set of open, connected, bounded C4-regular regions ofRn n ≥ 2 the solutions of (19) are all simple.
Proof.Consider the differentiable map
F : BM × UM → Lp(Ω)
defined byF (u, h) = h∗∆2h∗−1u + h∗f(·, h∗−1u,∇h∗−1u, ∆h∗−1u)
where BM = u ∈ W 4,p ∩W 2,p0 (Ω) − 0 | ‖u‖ ≤ M, p > n
2 , UM = DM − EM , DM
is the complement of the meager closed set given by Lemma 11 and EM is the meagerclosed set given by Lemma 12. Observe that UM is an open dense subset of Diff4(Ω).We show, using the Transversality Theorem, that the set
h ∈ UM | u → F (u, h) has 0 as a regular value
19
is open and dense in UM . Our result then follows by taking intersection with M varyingin N.
As observed in Remark 10 we may suppose, that our regions are C5-regular. Also byRemark 7, we only need to consider the nontrivial solutions.
As in the previous results, the verification of hypotheses (1) and (3) of the Transver-sality Theorem is simple, so we just show that (2α) holds.
Suppose, by contradiction, that there exists a critical point (u, h) ∈ F−1(0) and,h = iΩ. Then, there exists v ∈ Lq(Ω) such that∫
Ω
v DF (u, iΩ)(u, h) = 0 (37)
for all (u, h) ∈ W 4,p ∩ W 2,p0 (Ω) × C5(Ω, Rn) where DF (u, iΩ) : W 4,p ∩ W 2,p
0 (Ω) ×C5(Ω, Rn) → Lp(Ω) is given by
DF (u, iΩ)(u, h) = L(u)(u− h · ∇u)
with L(u) = ∆2 + ∂f∂µ (·, u,∇u, ∆u)∆ + ∂f
∂y (·, u,∇u, ∆u) · ∇+ ∂f∂λ (·, u,∇u, ∆u).
Taking h = 0 in (37), we have∫Ω
v L(u) u = 0 ∀u ∈ W 4,p ∩W 2,p0 (Ω),
that is, v ∈ N (L∗(u)). Since ∂Ω is C5-regular and f is of class C4, it follows by regularityresults for uniformly elliptic equations that v ∈ W 5,q(Ω) ∩ C4,α(Ω) for α > 0 and satisfy
L∗(u) v = 0 in Ωv = ∂v
∂N = 0 on ∂Ω.(38)
If u = 0 and h varies in C5(Ω, Rn), we obtain
0 = −∫
Ω
v L(u)(h · ∇u)
=∫
Ω
(h · ∇u)L∗(u)v − vL(u)(h · ∇u)
= −
∫∂Ω
h ·N∆v∆u, ∀h ∈ C5(Ω, Rn).
Therefore, we have∫
∂Ω
h ·N∆v∆u = 0 ∀h ∈ C5(Ω, Rn) from which ∆v∆u ≡ 0 on ∂Ω.
Since iΩ ∈ UM , we reach a contradiction, proving the theorem.
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20
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