Periodic Homogenization For Elliptic NonlocalEquations... And Beyond
(Recent Trends in Differential Equations: Analysis andDiscretisation Methods– Bielefeld)
Russell Schwab
Carnegie Mellon University
17 November, 2010
Homogenization (in my words...)
The act of replacing a complicated model (a PDE) which contains lotsof microscopic detail by a simpler model (another PDE) with much less
microscopic detail in the hope of gaining macroscale observations andunderstanding which may be intractable due to the abundance of detail
in the original model. If the original model has certain “averaging”properties and the difference between the micro and macro scales is
large, then the replacement should be possible.
The Set-UpNotation for Some Relevant Operators
• The σ/2 fractional Laplace
− (−∆)σ/2 u(x) = Cn,σ
∫Rn
(u(x + y) + u(x − y)− 2u(x)) |y |−n−σ dy
• A linear operator comparable to the fractional Laplace
L(u, x) = Cn,σ
∫Rn
(u(x + y) + u(x − y)− 2u(x)) K (x , y)dy
withλ
|y |n+σ≤ K (x , y) ≤ Λ
|y |n+σ
• A fully nonlinear operator comparable to the fractional Laplace
F (u, x) =
infα
supβ
Cn,σ
∫Rn
(u(x + y) + u(x − y)− 2u(x))Kαβ(x , y)dy.
The Set-Up
Family of Oscillatory Nonlocal Equations:
F (uε,
x
ε) = 0 in D
uε = g on Rn \ D
F (u,x
ε) =
infα
supβ
f αβ(
x
ε) + Cn,σ
∫Rn
(u(x + y) + u(x − y)− 2u(x))Kαβ(x
ε, y)dy
.(
Think of a more familiar 2nd order equation:
F (D2u,x
ε) = inf
αsupβ
f αβ(
x
ε) + aαβij (
x
ε)uxixj (x)
)
The Set-Up
Periodic Nonlocal Operator F
for all z ∈ Zn
F (u, x + z) = F (u(·+ z), x)
Our F will be periodic when f αβ and Kαβ are periodic in x .
Translation Invariant Nonlocal Operator F
F is translation invariant if for any y ∈ Rn,
F (u, x + y) = F (u(·+ y), x).
just think of
L(u, x) = Cn,σ
∫Rn
(u(x + y) + u(x − y)− 2u(x)) K (y)dy
The Set-Up
Family of Oscillatory Nonlocal Equations:
F (uε,
x
ε) = 0 in D
uε = g on Rn \ D
Translation Invariant Limit Nonlocal Equations
F (u, x) = 0 in D
u = g on Rn \ D.
GOAL
Prove there is a unique nonlocal operator F so that uε will be very closeto u as ε→ 0. (Homogenization takes place.)
Main Theorem
Theorem (Homogenization of Nonlocal Equations– To Appear SIAMJ. Math. Anal.)
If F is periodic and uniformly elliptic, plus technical assumptions, thenthere exists a translation invariant elliptic nonlocal operator F with thesame ellipticity as F , such that uε → u locally uniformly and u is theunique solution of
F (u, x) = 0 in D
u = g on Rn \ D.
The Set-Up– Assumptions on F
“Ellipticity”
λ
|y |n+σ≤ Kαβ(x , y) ≤ Λ
|y |n+σ
Scaling
Kαβ(x , λy) = λ−n−σKαβ(x , y).
Symmetry
Kαβ(x ,−y) = Kαβ(x , y)
Recent Background– Nonlocal Elliptic Equations
Existence/Uniqueness (Barles-Chasseigne-Imbert)
Given basic assumptions on Kαβ and f αβ, there exist unique solutions tothe Dirichlet Problems F (uε, x/ε) = 0, F (u, x) = 0.
Regularity (Silvestre, Caffarelli-Silvestre)
uε are Holder continuous, depending only on λ, Λ, ‖f αβ‖∞, dimension,and g . (In particular, continuous uniformly in ε.)
Nonlocal Ellipticity– Good Definition (Caffarelli-Silvestre)
If u and v are C 1,1 at a point, x , then
M−(u − v)(x) ≤ F (u, x)− F (v , x) ≤ M+(u − v)(x).
M−u(x) = infαβ
Lαβu(x)
and M+u(x) = sup
αβ
Lαβu(x)
.
Historical Homogenization Background– Related To ThisWork (!!!!)
(The true bibliography is long! My profuse apologies to anyone left offthe list. Here is just for MAJOR references for this work...)
• Bensoussan-Lions-Papanicolaou 1978– Asymptotic analysis forperiodic structures
• Lions-Papanicolaou-Varadhan (unpublished)– PeriodicHomogenization of Hamilton-Jacobi Equations
• Evans 1992– Periodic Homogenization of Fully Nonlinear PDEs
• Arisawa-Lions 1998– On Ergodic Stochastic Control
• Caffarelli-Souganidis-Wang 2005– Stochastic Homogenization OfFully Nonlinear Elliptic Equations
• (related results) Arisawa 2009– Homogenization of a Class ofIntegro-Differential Equations with Levy Operators
Motivating The “Corrector” Equation
• Expansion (formally!!!): uε(x) = u(x) + εσv(·/ε) + o(εσ)
• Scaling: L(εσv(·/ε), x) = L(v , x/ε)
• Separation of Scales: as ε→ 0, x can be considered fixed w.r.t. x/εand x/ε = y becomes a global variable
• formal generalization of Fredholm Alternative
Motivating The “Corrector” EquationJust for the fractional Laplace sufficient for the idea...
−(−∆)σ/2uε = f (x
ε)
• Plug Expansion into equation plus scaling:
−(−∆)σ/2u(x) +−(−∆)σ/2v(x
ε) = f (
x
ε)
• Global variables:
−(−∆)σ/2u(x) +−(−∆)σ/2v(y) = f (y) + F
• Need to find such a v : Fredholm says only if
F +
∫Q1
fdy − (−(−∆)σ/2u(x)) = 0
Motivating The “Corrector” EquationJust for the fractional Laplace sufficient for the idea...
−(−∆)σ/2uε = f (x
ε)
• Plug Expansion into equation plus scaling:
−(−∆)σ/2u(x) +−(−∆)σ/2v(x
ε) = f (
x
ε)
• Global variables:
−(−∆)σ/2u(x) +−(−∆)σ/2v(y) = f (y) + F
• Need to find such a v : Fredholm says only if
F +
∫Q1
fdy − (−(−∆)σ/2u(x)) = 0
Motivating The “Corrector” EquationRestate what we just did...
True Corrector... Special Case
There exists a unique constant, F , such that there is a bounded periodicperiodic solution of
−(−∆)σ/2v(y) = f (y) + F (u, x)
Formal Solution To Homogenization
−(−∆)σ/2u(x) +−(−∆)σ/2v(x
ε) = f (
x
ε)
which reads, after ε→ 0,F (u, x) = 0
Generalizing Away From Fractional Laplace
Let’s remove some unnecessary restrictions... It was all formal anyway!
• Expansion? Why one v?? Let’s try a whole sequence of v ε souε(x) = u(x) + v ε(x) + o(εσ)
• Decay! We will need a compatibility condition on the v ε... requiresupx |v ε(x)| → 0
• Fractional Laplacian? Not Special! Only used scaling, really...upgrade to F (u, x/ε)
• Why global equation and periodic v?? How about just solve theequation in B1?
“Corrector” Equation
equation for φ + v ε
F (φ+ v ε,x
ε) =
infα
supβ
f αβ(
x
ε) +
∫Rn
(φ(x + y) + φ(x − y)− 2φ(x))Kαβ(x
ε, y)dy
+
∫Rn
(v ε(x + y) + v ε(x − y)− 2v ε(x))Kαβ(x
ε, y)dy
“frozen” operator on φ at x0
[Lαβφ(x0)](x) =
∫Rn
(φ(x0 + z) + φ(x0 − z)− 2φ(x0))Kαβ(x , z)dz
“Corrector” Equation
Analogy to 2nd order equation
aij(x
ε)(φ+ v)xixj (x) = aij(
x
ε)φxixj (x) + aij(
x
ε)vxixj (x)
and aij(xε )φxixj (x) is uniformly continuous in x .
Free and frozen variables, x and x0
Uniform continuity (Caffarelli-Silvestre)
[Lαβφ(x0)](x) is uniformly continuous in x0, independent of x and αβ
“Corrector” Equation
NEW OPERATOR Fφ,x0
Fφ,x0(v ε,x
ε) = inf
αsupβ
f αβ(
x
ε) + [Lαβφ(x0)](
x
ε)
+
∫Rn
(v ε(x + y) + v ε(x − y)− 2v ε(x))Kαβ(x
ε, y)dy
New “Corrector” Equation
Fφ,x0(v ε, xε ) = F (φ, x0) in B1(x0)
v ε = 0 on Rn \ B1(x0).
“Corrector” Equation
Proposition (“Corrector” Equation)
There exists a unique choice for the value of F (φ, x0) such that thesolutions of the “corrector” equation also satisfy
limε→0
maxB1(x0)
|v ε| = 0.
(via the perturbed test function method, this proposition is equivalent tohomogenization)
Finding F ... Variational Problem
(Caffarelli-Sougandis-Wang... *In spirit)
Consider a generic choice of a Right Hand Side, l is fixed
Fφ,x0(v εl ,
xε ) = l in B1(x0)
v εl = 0 on Rn \ B1(x0).
How does the choice of l affect the decay of v εl ?
decay property
limε→0 maxB1(x0) |v ε| = 0 ⇐⇒ (v εl )∗ = (v εl )∗ = 0
Variational Problem
l very negative
p+(x) = (1− |x |2)2 · 1B1 is a subsolution of equation=⇒ (v εl )∗ > 0 and we missed the goal.
(Fφ,x0(v εl ,
x
ε) = l
)
Variational Problem
(Fφ,x0(v εl ,
x
ε) = l
)l very positive
p−(x) = −(|x |2 − 1)2 · 1B1 is a supersolution of equation=⇒ (v εl )∗ < 0 and we missed the goal, but in the other direction.
Can we choose an l in the middle that is “JUST RIGHT”?
Variational Problem
(Fφ,x0(v εl ,
x
ε) = l
)l very positive
p−(x) = −(|x |2 − 1)2 · 1B1 is a supersolution of equation=⇒ (v εl )∗ < 0 and we missed the goal, but in the other direction.
Can we choose an l in the middle that is “JUST RIGHT”?
Variational Problem
(Fφ,x0(v εl ,
x
ε) = l
)l very positive
p−(x) = −(|x |2 − 1)2 · 1B1 is a supersolution of equation=⇒ (v εl )∗ < 0 and we missed the goal, but in the other direction.
Can we choose an l in the middle that is “JUST RIGHT”?
Variational Problem
(l << 0
)l = ?????
(l >> 0
)
Can we choose an l in the middle that is “JUST RIGHT”?
Obstacle Problem
(Caffarelli-Sougandis-Wang) The answer is YES.
Information From Obstacle Problem
The obstacle problem gives relationship between the choice of l and thedecay of v εl .
Obstacle Problem
The Solution of The Obstacle Problem In a Set A
U lA = inf
u : Fφ,x0(u, y) ≤ l in A and u ≥ 0 in Rn
equation: U l
A is the least supersolution of Fφ,x0 = l in A
obstacle: U lA must be above the obstacle which is 0 in all of Rn
Lemma (Holder Continuity)
U lA is γ-Holder Continuous depending only on λ, Λ, ‖f αβ‖∞, φ,
dimension, and A.
Monotonicity and Periodicity of Obstacle Problem
If A ⊂ B, then U lA ≤ U l
B . For z ∈ Zn, U lA+z(x) = U l
A(x − z)
Obstacle Problem
The Solution of The Obstacle Problem In a Set A
U lA = inf
u : Fφ,x0(u, y) ≤ l in A and u ≥ 0 in Rn
equation: U l
A is the least supersolution of Fφ,x0 = l in A
obstacle: U lA must be above the obstacle which is 0 in all of Rn
Lemma (Holder Continuity)
U lA is γ-Holder Continuous depending only on λ, Λ, ‖f αβ‖∞, φ,
dimension, and A.
Monotonicity and Periodicity of Obstacle Problem
If A ⊂ B, then U lA ≤ U l
B . For z ∈ Zn, U lA+z(x) = U l
A(x − z)
Obstacle Problem
NOTATION
Rescaled Solution
uε,l = inf
u : Fφ,x0(u,y
ε) ≤ l in Q1 and u ≥ 0 in Rn
.
Solution in Q1 and Solution in Q1/ε
uε,l(x) = εσU lQ1/ε
(x
ε)
Obstacle Problem
Dichotomy
(i) For all ε > 0, U lQ1/ε
= 0 for at least one point in every complete cell
of Zn contained in Q1/ε.
(ii) There exists some ε0 and some cell, C0, of Zn such thatU lQ1/ε0
(y) > 0 for all y ∈ C0.
Lemma (Part (i) of The Dichotomy)
If (i) occurs, then (v εl )∗ ≤ 0.
Lemma (Part (ii) of The Dichotomy)
If (ii) occurs, then (v εl )∗ ≥ 0
Obstacle Problem
Dichotomy
(i) For all ε > 0, U lQ1/ε
= 0 for at least one point in every complete cell
of Zn contained in Q1/ε.
(ii) There exists some ε0 and some cell, C0, of Zn such thatU lQ1/ε0
(y) > 0 for all y ∈ C0.
Lemma (Part (i) of The Dichotomy)
If (i) occurs, then (v εl )∗ ≤ 0.
Lemma (Part (ii) of The Dichotomy)
If (ii) occurs, then (v εl )∗ ≥ 0
Obstacle Problem
Proof of First Lemma (If (i) occurs, then (v εl )∗ ≤ 0)...
• Rescale back to Q1.Definition of uε,l =⇒ v εl ≤ uε,l
• (i) =⇒ uε,l = 0 at least once in EVERY cell of εZn. HolderContinuity =⇒ uε,l ≤ Cεγ .
Obstacle Problem
Proof of Second Lemma (If (ii) occurs, then (v εl )∗ ≥ 0)...
• Given any δ > 0, Periodicity, Monotonicity, and (ii) allowconstruction of a connected cube Cε ⊂ Q1 such that uε,l > 0 in Cεand |Cε| / |Q1| ≥ 1− δ.
Obstacle ProblemProof of Second Lemma continued (If (ii) occurs, then (v εl )∗ ≥ 0)
• Properties of uε,l =⇒ uε,l is a solution inside Cε.
• Comparison with v εl and boundary continuity =⇒uε,l − v εl ≤ C (δ1/n)γ .
• Upper limit in ε: (−v εl )∗ ≤ 0
• Same as (v εl )∗ ≥ 0
Choice for FChoose a special l such that l is ARBITRARILY CLOSE to values thatgive (i) and values that give (ii).
The Good Choice of F
F (φ, x0) = sup
l : (ii) happens for the family (U lQ1/ε
)ε>0
Choice for F
(l << 0
)l = F
(l >> 0
)
Needed Properties for F
Still need to show
Elliptic Nonlocal Equation
• F (u, x) is well defined whenever u is bounded and “C 1,1 at thepoint, x”.
• F (u, ·) is a continuous function in an open set, Ω, wheneveru ∈ C 2(Ω).
• Ellipticity holds: If u and v are C 1,1 at a point, x , then
M−(u − v)(x) ≤ F (u, x)− F (v , x) ≤ M+(u − v)(x).
Comparison
This follows from ellipticity and translation invariance.
True Corrector Equation
Periodic Corrector
F (φ, x0) is the unique constant such that the equation,
Fφ,x0(w , y) = F (φ, x0) in Rn
admits a global periodic solution, w .
Inf-Sup Formula
Corollary: Inf-Sup formula
F (φ, x0) = infW periodic
supy∈Rn
(Fφ,x0(W , y))
Stochastic Homogenization and ABP (Work inPreparation)
From Periodic to Stationary Ergodic
Was F (u, x + z) = F (u(·+ z), x) for all z ∈ Zn
Now F (u, x + y , ω) = F (u(·+ y), x , τyω) for all y ∈ Rn, and on someprobability space with an ergodic group of transformations, τy : Ω→ Ω.
(i) now |contact| /∣∣Q1/ε
∣∣→ c > 0 (ii) now |contact| /∣∣Q1/ε
∣∣→ 0
Stochastic Case... What About (v εl )∗ and (v εl )∗?
• (i) corresponds to positive contact limit ratio... (v εl )∗ ≤ 0 in thiscase corresponds almost exactly as in periodic. Only observation isthat for positive contact limit, ergodicity of τ implies limit must alsobe positive in any subdomain of a uniform partition of Q1/ε.
• (ii) corresponds to zero contact limit... (v εl )∗ is much harder!Requires more basic machinery... Need to say
(uεQ1− v εl )→ 0
where (uεQ1− v εl ) solves a maximal equation
M+(uε − v ε) ≥ −C (F )1contact(ε)(x),
think of
Lu(x) = Cn
∫Rn
(u(x + y) + u(x − y)− 2u(x)) K (x , y)dy
Choice for F
The Good Choice of F
F (φ, x0) = sup
l : lim
ε→0
|contact set(ε)|∣∣Q1/ε
∣∣ = 0
(l << 0
)l = F
(l >> 0
)
Stochastic Homogenization and ABP (Work InPreparation)
• Sufficient condition to conclude homogenization is that sup |hε| → 0when |Kε| → 0 and
M+(hε) ≥ −C (F )1Kε(x),
• very little is known for such an equation, even for linear equations!(This is taken care of by the Aleksandrof-Bakelman-Pucci estimatein the 2nd order setting)
Stochastic Homogenization and ABP (Work InPreparation)
Results for a restricted class of kernels
Theorem (ABP-type estimate (with N. Guillen– in prep.))
If Kαβ in M+ are of the form
λ |y |−n−σ ≤ yTAαβ(x)y |y |−n−σ−2 ≤ Λ |y |n−σ ,
then for f ≥ 0, u ≤ 0 in Rn \ B1 and a subsolution in B1 of
M+(u, x) ≥ −f (x) (think linear bad coefficients L(u, x) ≥ −f (x))
the estimate holds:
supB1
(u) ≤ C (Λ, σ, n)(‖f ‖∞)(2−σ)/2(‖f ‖Ln)σ/2.
Stochastic Homogenization and ABP (Work InPreparation)
As a consequence, the method outlined above holds, so...
Theorem (Stochastic Homogenization for Restricted Class ofKernels)
If Kαβ in M+ are of the form
λ |y |−n−σ ≤ yTAαβ(x , ω)y |y |−n−σ−2 ≤ Λ |y |n−σ ,
then stochastic homogenization holds for operators of the form
F (u, x) =
infα
supβ
∫Rn
(u(x + y) + u(x − y)− 2u(x))(yTAαβ(x , ω)y
|y |n+σ+2)dy.
Thank You!
M. Arisawa and P.-L. Lions. On ergodic stochastic control. Comm.Partial Differential Equations, 23(11-12):2187–2217, 1998.
Mariko Arisawa. Homogenization of a class of integro-differentialequations with Levy operators. Comm. Partial Differential Equations,34(7-9):617–624, 2009.
G. Barles, E. Chasseigne, and C. Imbert. On the dirichlet problem forsecond-order elliptic integro-differential equations. Indiana Univ.Math. J., 57(1):213–246, 2008.
Guy Barles and Cyril Imbert. Second-order elliptic integro-differentialequations: viscosity solutions’ theory revisited. Ann. Inst. H.Poincare Anal. Non Lineaire, 25(3):567–585, 2008.
Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou.Asymptotic analysis for periodic structures, volume 5 of Studies inMathematics and its Applications. North-Holland Publishing Co.,Amsterdam, 1978.
L. A. Caffarelli and L. Silvestre. Regularity results for nonlocalequations by approximation. Arch. Ration. Mech. Anal., To Appear.
L. A. Caffarelli and L. Silvestre. Regularity theory for fully nonlinearintegro-differential equations. Comm. Pure Appl. Math., To Appear.
Luis A. Caffarelli, Panagiotis E. Souganidis, and L. Wang.Homogenization of fully nonlinear, uniformly elliptic and parabolicpartial differential equations in stationary ergodic media. Comm.Pure Appl. Math., 58(3):319–361, 2005.
Lawrence C. Evans. Periodic homogenisation of certain fullynonlinear partial differential equations. Proc. Roy. Soc. EdinburghSect. A, 120(3-4):245–265, 1992.
Russell W. Schwab. Periodic homogenization for nonlinearintegro-differential equations. SIAM J. Math. Analysis, to appear.
