Homogeniza*ons in Perforated Domain. Ki Ahm Lee Seoul Na*onal University

Transcription

1 Homogeniza*ons in Perforated Domain Ki Ahm Lee Seoul Na*onal University

2 Outline 1. Perforated Domain 2. Neumann Problems (joint work with Minha Yoo; interes*ng discussion with Li Ming Yeh) 3. Dirichlet Problems 3.1. Ellip*c Case (joint works with L. Caffarelli) 3.2 Parabolic Case (joint works with Sunghoon Kim) 3.3. Nonlinear Eigen Value Problem (joint works with Sunghoon Kim) 3.4 Porous Medium Equa*on (joint works with Sunghoon Kim)

3 1. Perforated Domain Ω ε Ω ε = Ω \ T aε a ε A cell

4 1.1 Linear Equations (Dirichlet Problems) Find the solution such that Δ = 0 in Ω ε = 0 in Ω ε Ω = g(x) on Ω (Neumann Problems) Find the solution such that Δ = 0 in Ω ε ν = 0 in Ω ε Ω = g(x) on Ω

5 1.2 Nonlinear Equations (Dirichlet Problems) Find the solution such that F(D 2 ) = 0 in Ω ε = 0 in Ω ε Ω = g(x) on Ω (Neumann Problems) Find the solution such that F(D 2 ) = 0 in Ω ε ν = 0 in Ω ε Ω = g(x) on Ω

6 1.3 Questions : (1). What's the uniform estimate satisfied by to extract convergent subsequence? (2). If u is a limit of, what is the equation (or Homogenized Equation, Effective Equation) satisfied by u? (Rough Idea) Homogenized Equation satisfied by u is the correctibility condition on u so that u can be corrected to so that the corrected satisfies ε - problem locally. So the condition for the existence of corrector is almost equal to the homogenized equation.

7 2. Neumann Problem 2.1 Viscosity Method F(D 2 ) = 0 in Ω ε B ν,u ε = 0 in Ω ε Ω = g(x) on Ω for an outward normal direction ν to Ω ε. Conditions : (1) F(M) is uniformly elliptic i.e. for symmetric matrices M and N λ N + Λ N F(M + N) F(M) Λ N + λ N where N = N + N and N +,N 0 (2) B(q + q,z) B(q,z) and B(q,z + z ) B(q,z) for q' 0,z' 0 (3) F( ), B(, ) are C 1.

8 Def. is a viscosity super - solution if F(D 2 ) 0 in Ω ε u B ε ν,u ε 0 in Ω ε Ω g(x) on Ω in viscosity sense. Def. is a viscosity sub - solution if F(D 2 ) 0 in Ω ε u B ε ν,u ε 0 in Ω ε Ω g(x) on Ω in viscosity sense.

9 Lemma : (Comparison Principle) If u + ε and u ε are super - and sub - solutions repectively such that u + ε u ε on Ω, then u + ε u ε in Ω. Cor. (Existence)

10 Main Theorem Let be the viscosity solution of F(D 2 ) = 0 in Ω ε ν = 0 in Ω ε Ω = g(x) on Ω. where F is homogeneous of degree one. Then (1) There is a Lipschitz function u(x) and η(ε) > 0 such that (x) u(x) < η(ε) for all x Ω ε and η(ε) 0 as ε 0. (2) There is a uniformly elliptic operator F( ) such that u(x) is a viscosity solution of F(D 2 u) = 0 in Ω u = g(x) on Ω.

11 2.2 The Concept of Convergence (1) Discrete Gradient Estimate Lemma. Set Δ e = (x +hεe) (x ) hε for h Z n. Then Δ e < C uniformly (Idea of Proof) Set Z = sup x Ω ε Δ e 2. a. Interior estimate L[Z] = a ij (x)d ij Z 0 in Ω ε for a ij = 1 F ij (sd ij u(x + hεe) + (1 s)d ij u(x), x ε ) ds. 0 Z ν = 0 on Ω ε Ω So there is no maximum in Ω ε \ Ω.

12 b. Boundary We need a super - solution such that h ε = 0 on B r h ε = 1 on B R F(D 2 h ε ) 0 in (B R \ B r ) ε h ε ν 0 in (B R \ B r ) ε (B R \ B r ) and h ε (x) < Cd(x, B r ) (Linear growth) It is not easy to construct such barrier. (Step1) Choose R' > R and r' < r. Consider the homogenization h' ε in D = (B R' \ B R ) (B R \ B r ) ε (B r \ B r' ) : a perforated in compact subset Homogenized equation will be uniformly elliptic and gives the upper bound gradient of the limit h' and then uniform upper bound of h' ε. That upper bound is also independent of r r'.

13 (Step2) Choose a small r r' so that h' ε is a super - solution with a small error.

14 (2) Almost Flatness Lemma. osc Qε (m)\t a ε (m) = O(ε) uniformly. (Idea of Proof) Set v ε (y) = 1 ε ( ε y) min F(D 2 v ε ) = 0 in B 4 B 4 ε \T a ε v ε ν = 0 in T a B 4 There is y 0 B 4 and v(y 0 ) = 0 0 in B 4(0). Then By the Harnack inequality and Discrete gradient estimate, supv ε Cinf v ε Cv ε (y 0 + m) Cε B B 2 2 for y 0 + m B 2

15 Lemma (Golobal ε - Lipschitz Estimate) (x) (y) < C( x y +ε) for all x, y Ω ε Lemma There is a Lipschitz function u(x) and η(ε) > 0 such that (x) u(x) < η(ε) for all x Ω ε and η(ε) 0 as ε 0.

16 2.3 Correctors Lemma (Correctors) For any given matrix P and a vector ξ, there are ε - periodic functions w ε ( ;P,ξ), a unique constant σ (P,ξ), and bounded functions σ ε (P,ξ)(x) such that F(P + D 2 w ε ) = σ ε in R n \ T aε w ε ν = ν ξ - σ ε on T aε and σ ε σ uniformly as ε 0.

17 Let us consider w ε + F(P + D 2 w ε ) = 0 in R n \ T aε w ε ν + w = ν ξ on T ε a ε For large M depending on P and ξ, h ± (x) = ±M are super - and sub - solutions. Lemma : (Existence of Corrector) There is a ε - periodic solution w ε which is uniformly bounded.

18 Set w ˆ ε (x) = w ε (x) w ε (0) and w ˆ x ε (x) = εη ε ε Then η ε satisfies η ε (0) = 0 and ε 2 η ε + F(εP + D 2 w ε ) = εw ε (0) in R n \ T a η ε ν + εη = ν ξ - w (0) on T ε ε a By Harnack type estimate and the periodicity, Lemma : η < C uniformly. ε and osc w ε < Cε Set σ ε (P,ξ) = w ε (x) σ (P,ξ) (Uniqueness comes from a simple comparison.)

19 2.4 The proof of main theorem Set F(P,ξ) = σ (P,ξ). Theorem. When a ε = a 0 ε, u is a viscosity solution of F(D 2 u,du) = 0 in Ω u = 0 on Ω. Q P Claim : u is a sub - solution. Otherewise at x 0, P(x) = 1 2 P ijx i x j + ξ i x i + c F(P,ξ) < 2δ 0 < 0. u and then F(Q,ξ) < δ 0 < 0. Let Q ε = Q + w ε.

20 F(Q ε ) = F(Q + D 2 w ε ) = σ ε < F(Q,ξ) + δ 0 /2 < - δ 0 /2 Q ε ν ν ξ ν ξ - σ > - F(Q,ξ) δ /2 > δ /2 > 0 ε 0 0 Threfore Q ε is a super - solution of ε - problem in a small neighborhood B δ of x 0 = 0. And Q ε > on B δ from the convergence lemma. So Q ε > in B δ which is a contradiction. on T a ε

21 Lemma : 1. λ N + Λ N F(M + N,ξ) F(M,ξ) Λ N + λ N 2. F(tM, tξ) = tf(m,ξ) 3. If F( ) is convex, so is F(,ξ).

22 2.5 Discussions (1). There are two key estimates: Discrete gradient estimate and Harnack - type estimate in cell problems i.e. supv ε C(inf v ε +"data") B B 2 2 (2). For a bounded function f (x), consider F(D 2 ) = f (x) in Ω ε ν = 0 in Ω Ω, ε = g(x) on Ω. ( Discrete H o lder Estimate :) (x) (y) < C( x y α +ε) for all x, y Ω ε. (Main idea : Perturbation theory of Schauder type) (i) Discrete gradient estimate for fixed constant. (ii) Perturbation lemma (iii) Improvement approximation with scaling and Perturbation lemma

23 (3). F 1 (D 2 ) = 0 in Ω ε F 2 (D 2 ) = 0 in T ε ν + 1 ν = 0 in Ω 2 ε = g(x) on Ω. ( Harnack Estimate :) Ω (Continuity of Flux) (4). (small permeability in perforated domain) F 1 (D 2 ) = 0 F 2 (D 2 ) = 0 in Ω ε in T ε ν + ε 1 ν = 0 in Ω 2 ε Ω = g(x) on Ω.

24 (5). Randomly Perforated domain. We need to consider obstacle problem for Neumann problem to find subaddative quantity following L. Caffarelli, P. Souganidis and L. Wang. We need the stability of this quantity through discrete nondegeneracy estimate. And we need a kind of uniform estimate to extract convergent subsequence.

25 3. Dirichlet Problem (Dirichlet Problems) Find the solution such that F(D 2 ) = 0 in Ω ε = 0 in Ω ε Ω = g(x) on Ω The Dirichlet Problem can be considered as an highly oscillating obstacle problem

26 Highly Oscilla*ng Obstacle Problems Oscilla*ng Obstacles ϕ ε = ϕ χ Ta ε ϕ

27 Possible Cases

28 3.1 Main Theorem For Laplace Equa*ons Main Theorem For Laplace Equa*ons (1). (The Concept of Convergence) There is u and p > 0 such that u in L p. And for any δ > 0, there is D δ Ω such that u uniformly in D δ and Ω \ D δ < δ (2) Let a * ε = ε α * for α * = n n 2 for n = 3 and α * = e 1 ε 2 for n = 2. (a) For c 0 α * ε α ε C 0 α * ε, u is a viscosity solution of Δu +κ Bro (ϕ u) + where κ Br o = 0 in Ω u = 0 on Ω α is the capacity of B ro if r o = lim ε * exists. ε 0 α ε

29 (b) If α ε = o(α * ε ), then u is a viscosity solution of Δu = 0 in Ω u = 0 on Ω. (c) If α * ε = o(α ε ), then u is a viscosity solution of Δu 0 in Ω u ϕ in Ω u = 0 on Ω.

30 3.1.2 Correctors Δ w ε = k in R n w ε (x) = 1 in T aε Lemma. Let a ε = c 0 ε α. There is a unique number α * = liminf w ε = for any k > 0 if α > α * liminf w ε (x) = 1 for α = α * and k = k ε s.t. k ε cap(b 1 ) liminf w ε = 0 for any k > 0 if α < α * n n 2 s.t.

31 (c) a ε = o(a ε * ) (b) a ε * = o(a ε ) (c) a ε * = ε α *

32 3.1.3 The Concept of Convergence (1) Discrete Gradient Estimate Lemma. Set Δ e = (x +hεe) (x ) hε. Then Δ e < C uniformly

33 (2) Almost Flatness Lemma. Set a ε = (a ε * ) 1/ 2. Then osc Bε (m)\ B a ε (m ) = o(ε γ ) uniformly.

34 Nonlinear Equa*ons F(D 2 u,x) = 0 is uniformly elliptic if 1 N F(M + N,x) F(M,x) Λ N Λ for n n symmetric metrices M, N s.t. N 0. (NL ε ) F(D 2 ) 0 in Ω (x) = 0 on Ω (x) ϕ ε (x) in Ω Ex) F(D 2 u) = k 1 λ + (D 2 u) + k 2 λ (D 2 u) = 0 Let u = r α. Then F(D 2 r α ) = α (k r α +2 1 (α +1) k 2 ) = 0.

35 Homogeneous Solu*ons

36 Theorem II (1). (The Concept of Convergence) There is u and p > 0 such that u in L p. And for any δ > 0, there is D δ Ω such that u uniformly in D δ and Ω \ D δ < δ (2) In (Type III), for any a ε, the limit u is a least viscosity super - solution of F(D 2 u) 0 in Ω u ϕ in Ω u = 0 on Ω.

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38 3.2. Parabolic Problems Find the smallest super - solution such that Δ t 0 in Ω (0,T] ϕ ε (x) in Ω (0,T] = 0 on Ω (0,T] = g on Ω {0}

39 Theorem. Let a * ε = ε α * for α * = n n 2 for n = 3 and α * = e ε 2 for n = 2. For c 0 α * ε α ε C 0 α * ε, u is a viscosity solution of Δu +κ Bro (ϕ u) + - u t = 0 in Q T 1 where κ Bro u = 0 on l Q T u = g(x) in Ω {t = 0} α is the capacity of B ro if r o = lim ε * exists. ε 0 α ε Idea : Q ε (x,t) = Q(x,t) + (ϕ(x 0 ) Q(x 0,t 0 ))w ε (x)

40 3.3. Nonlinear Eigen Value Problems Let us consider nonlinear eigen value problems: p Δϕ ε = λϕ ε in Ω ε ϕ ε > 0 in Ω ε ϕ ε = 0 on T aε ϕ ε = 0 on Ω for 0 < p < 1. Let λ =1.

41 Homogenized Equa*on Theorem : (1). (The Concept of Convergence) There is u and q > 0 such that u in L q. And for any δ > 0, there is D δ Ω such that u uniformly in D δ and Ω \ D δ < δ (2) Let a * ε = ε α * for α = n * n 2 for n = 3 and α = * e ε 2 for n = 2. (a) For c 0 α * ε α ε C 0 α * ε, u is a viscosity solution of Δu -κ Br o u + = λ u p in Ω where κ Br o u > 0 in Ω u = 0 on Ω α is the capacity of B ro if r o = lim ε * exists. ε 0 α ε 1

42 Main Steps a. Discrete Gradient Estimate for the Green Function for Laplace operator. idea) G Ωε (x, y) = G Ω (x, y) + h ε (x;y) Δ h ε (x;y) = 0 in Ω ε where h ε (x;y) = G Ω (x, y) on Ω ε h ε (x;y) has discrete gradient estimate with the order of G Ω. Discrete Gradient estimate of from Green's representation. b. Almost Flatness v ε (x) = ( a ε * x) Δv ε = a ε * v ε p

43 c. Correctibility Condition (or Homogenizd Equation) - Δ w ε + 1 b (b bw ε ) p = k ε Set v ε (x) = w ε (a ε * x). -Δv ε + (a * ε ) 2 b (b bv ε ) p = k ε (a * ε ) 2 in Q ε aε * v ε =1 on D v ε = v ε = 0 on Q ε aε * \ D - Δv ε dx = (a * ε ) 2 k ε b p 1 (1 v ε ) p dx Q ε aε * \ D Q ε aε * \ D When ε 0, -κ(d) = k b p 1 k = -κ(d) + b p 1

44 P(x) is correctible P ε (x) = P(x) P(x 0 )w ε is a solution of ε - problem 0 = Δ P ε + (P ε (x)) p ΔP P(x 0 )Δw ε + P(x 0 ) p (1 w ε ) p ΔP + P(x 0 )k ΔP κ(d)p(x 0 ) + P(x 0 ) p d. Discrete Hopf Principle idea) Creat a barrier from the oscillating obstacle problem Discrete nondegeneracy of

45 3.4. Porous Medium Equa*ons in a fixed Perforated domain Δu m ε t = 0 in Ω aε (0,T] = 0 on Ω aε (0,T] = g ε on Ω aε {0} Set v ε = u m ε which is a flux. v ε 0 on a fixed boundary Ω. For v > δ > 0, we can use the results on Laplacian. 1 v 1 m Δv v = 0 in Ω ε ε t ε a ε (0,T] v ε = 0 on Ω aε (0,T] v ε = g m ε on Ω aε {0}

46 Theorem. Let a * ε = ε α * for α * = n n 2 for n = 3 and α * = e ε 2 for n = 2. For c 0 α * ε α ε C 0 α * ε, u is a viscosity solution of v 1 1 m ( Δv -κ B r o v + ) - v t = 0 in Q T 1 where κ Br o v = 0 on l Q T v = g(x) m in Ω {t = 0} is the capacity of B ro if r o = lim ε 0 α ε α ε * exists.

47 Main Steps a. Discrete Nondegeneracy ϕ idea) U ε (x,t) = ε (x) 1/(m(m 1)) (1 + t) c 1 U ε (x,t) c 2 U ε (x,t) for some 0 < c 1 < c 2 < Discrete nondegeneracy of u > 0 in Ω b.discrete Gradient Estimate for,t (x,0) 0 by applying maximum principle on Z = D e ε h u 2

48 c. Almost Flatness ϕ idea) U ε (x,t) = ε (x) 1/(m(m 1)) (1 + t) c 1 U ε (x,t) c 2 U ε (x,t) for some 0 < c 1 < c 2 < Set T a ε = { (x,t) < δ 0 }. On Ω a ε = Ω \ T a ε, the equation is uniformly parabolic. Similar method can be applicable.

49 Thank you!