IMPROVED REGULARITY IN BUMPY LIPSCHITZ DOMAINS CARLOS KENIG AND CHRISTOPHE PRANGE Abstract. This paper is devoted to the proof of Lipschitz regularity, down to the microscopic scale, for solutions of an elliptic system with highly oscillating coefficients, over a highly oscillating Lipschitz boundary. The originality of this result is that it does not assume more than Lipschitz regularity on the boundary. In particular, we bypass the use of the classical regularity theory. Our Theorem, which is a significant improvement of our previous wor on Lipschitz estimates in bumpy domains, should be read as an improved regularity result for an elliptic system over a Lipschitz boundary. Our progress in this direction is made possible by an estimate for a boundary layer corrector. We believe that this estimate in the Sobolev-Kato class is of independent interest.. Introduction This paper is devoted to the proof of Lipschitz regularity, down to the microscopic scale, for wea solutions u ε = u ε (x R N of the elliptic system { A(x/ε u ε =, x Dψ ε (,, (. u ε =, x ε ψ (,, over a highly oscillating Lipschitz boundary. x d = εψ(x /ε + Dψ ε (, ε ψ (, x d = εψ(x /ε Figure. The domain D ε (, and the portion of the boundary ε ψ (, and Throughout this wor, ψ is a Lipschitz graph, D ε ψ (, := {x (, d, εψ(x /ε < x d < εψ(x /ε + } R d ε ψ (, := {x (, d, x d = εψ(x /ε} is the lower highly oscillating boundary on which homogeneous Dirichlet boundary conditions are imposed (see Figure. The University of Chicago, 5734 S. University Avenue, Chicago, IL 6637, USA. E-mail address: ce@math.uchicago.edu. Partially supported by NSF Grants DMS-96847 and DMS-6549. Université de Bordeaux, 35 cours de la Libération, 3345 Talence, and The University of Chicago, 5734 S. University Avenue, Chicago, IL 6637, USA. E-mail address: christophe.prange@math.cnrs.fr. Partially supported by NSF Grant DMS-5893.
.. Statement of our results. Our main theorem is the following. Theorem. There exists C > such that for all ψ W, (R d, for all matrix A = A(y = (A αβ ij (y Rd N, elliptic with constant λ, -periodic and Hölder continuous with exponant ν >, for all < ε < /, for all wea solutions u ε to (., for all r ε, /] εψ(x /ε+r εψ(x (. u ε dx d dx Cr d /ε+ u ε dx d dx, ( r,r d εψ(x /ε with C = C(d, N, λ, A] C,ν, ψ W,. (, d εψ(x /ε The uniform estimate of Theorem should be read as an improved regularity result. Indeed, estimate (. can be seen as a Lipschitz estimate down to the microscopic scale O(ε. For an elliptic system over a slowly oscillating boundary x d = ψ(x for ψ merely Lipschitz, it is of course not possible to get an improved regularity estimate. What we manage to prove here is that the highly oscillating Lipschitz boundary x d = εψ(x /ε being close in the limit to the flat boundary, system (. inherits some regularity properties up to the scale O(ε of the limit system when ε. Theorem represents a considerable improvement of a recent result obtained by the two authors, namely Result B and Theorem 6 in KP5]. This first wor dealt with uniform Lipschitz regularity over highly oscillating C,ν boundaries. As is classical in the proof of uniform estimates by compactness methods, we need to build (interior and boundary correctors. Our breathrough is made possible by estimating a boundary layer corrector v = v(y solution to the system { A(y v =, yd > ψ(y (.3, v = v, y d = ψ(y, in the Lipschitz half-space y d > ψ(y with non localized Dirichlet boundary data v, without resorting to regularity theory. Theorem. Assume ψ W, (R d and v H / (Rd i.e. sup v H / (ξ+(, d <. ξ Z d Then, there exists a unique wea solution v of (.3 such that (.4 sup v dy d dy C v <, ξ Z d ξ+(, d ψ(y H / with C = C(d, N, λ, A] C,ν, ψ W,. This estimate enables to bypass the classical regularity theory for elliptic systems over C,ν boundaries, which is heavily relied on in the paper KP5]. Originality of our results. Let us describe two main aspects of our wor: ( the lac of regularity of the boundary, which is merely Lipschitz, ( the lac of structure of the oscillations of the bumpy boundary. The originality of Theorem lies in the fact that no smoothness of the boundary, which is just assumed to be Lipschitz, is needed for it to hold. Previous results in this direction, in particular our previous wor KP5], always relied on some smoothness of the boundary, typically ψ C,ν with ν >, or ψ Cω with ω a modulus of continuity satisfying a Dini type condition, i.e. ω(t/tdt <. The difficulty of dealing with Lipschitz boundaries lies in the fact that zooming in close to the boundary does not yield any improvement of flatness. Therefore, classical regularity theory (for instance Schauder theory is not effective. The theory for boundary value problems in Lipschitz domains, not based on regularity, is very dissimilar from the theory in C,ν domains. Wor on boundary value
problems in Lipschitz domains, in particular the development of potential theory, has started in the late 7 s and the 8 s with seminal wors by Dahlberg Dah77, Dah79], Dahlberg and Kenig DK87] and Jerison and Kenig JK8]. Recent progress toward uniform estimates for systems with oscillating coefficients has been achieved by Kenig and Shen KS], and Shen She5a]. Notice in particular that estimate (. of Theorem implies εψ(x /ε+ε εψ(x u ε dx d dx /ε+ Cε u ε dx d dx, ( /,/ d εψ(x /ε (, d εψ(x /ε with a constant C uniform in ε. This is the so-called Rellich estimate (over a bumpy Lipschitz boundary, which is the eystone of the potential theory in Lipschitz domains. Pioneering wor on uniform estimates in homogenization has been achieved by Avellaneda and Lin in the late 8 s AL87a, AL87b, AL89a, AL89b, AL9]. The regularity theory for operators with highly oscillating coefficients has recently attracted a lot of attention, and important contributions have been made to relax the structure assumptions on the oscillations AS4a, AS4b, GNO4]. Our wor is in a different vein. It is focused on the boundary behavior of solutions. Of course, one can flatten the boundary, and put the oscillations of the boundary into the coefficients. Here, nothing is prescribed on the boundary (except that it is Lipschitz and bounded. Notably, we do not prescribe any structure assumption on the oscillations of the boundary: ψ is neither periodic, nor quasiperiodic, nor stationary ergodic. It oscillates in a completely unprescribed way. This is the main difference with the recent developments on interior estimates in homogenization, which always assume some structure on the oscillations of the coefficients. To conclude, it is remarable that the existence of an appropriate boundary layer corrector can be proved in such generality, when the existence of bounded interior correctors in homogenization relies on some structure: see for instance Koz78, She5b, AGK5] for almost periodic structures and GNO5] for random structures. Overview of the paper. In section we recall several results related to Sobolev-Kato spaces, homogenization and uniform Lipschitz estimates. These results are of constant use in our wor. Then the paper has two main parts. The first aim is to prove Theorem about the well-posedness of the boundary layer system in a space of non localized energy over a Lipschitz boundary. The ey idea is to carry out a domain decomposition. Subsequently, there are three steps. Firstly, we prove the well-posedness of the boundary layer system over a flat boundary, namely in the domain R d +. This is done in section 3. Secondly, we define and estimate a Dirichlet to Neumann operator over H /. This ey tool is introduced in section 4. Thirdly, we show that proving the well-posedness of the boundary layer system over a Lipschitz boundary boils down to analyzing a problem in a layer {ψ(y < y d < } close to the boundary. The energy estimates for this problem are carried out in section 5. Eventually in section 6, and this is the last part of this wor, we are able to prove Theorem using a compactness scheme. Framewor and notations. Let λ > and < ν < be fixed in what follows. We assume that the coefficients matrix A = A(y = (A αβ ij (y, with α, β d and i, j N is real, that (.5 A C,ν (R d, that A is uniformly elliptic i.e. (.6 λ ξ A αβ ij (yξα i ξ β j λ ξ, and periodic i.e. for all ξ = (ξ α i R dn, y R d (.7 A(y + z = A(y, for all y R d, z Z d. 3
We say that A belongs to the class A ν if A satisfies (.5, (.6 and (.7. For easy reference, we summarize here the standard notations used throughout the text. For x R d, x = (x, x d, so that x R d denotes the d first components of the vector x. For ε >, r >, let D ε ψ (, r := { (x, x d, x < r, εψ(x /ε < x d < εψ(x /ε + r }, ε ψ (, r := { (x, x d, x < r, x d = εψ(x /ε }, D (, r := { (x, x d, x < r, < x d < r }, (, r := { (x,, x < r }, R d + := R d (,, Ω + := {ψ(y < y d }, Ω := {ψ(y < y d < }, Σ := (, d, where x = max i=,... d x i. In the whole paper, we always use the max norm of R d or R d, so that all the balls are square, not round. We sometimes write D ψ (, r and ψ (, r in short for Dψ (, r and ψ (, r; in that situation the boundary is not highly oscillating because ε =. Let also (u D ε ψ (,r := u = u. Dψ ε (,r (, r Dψ ε (,r The Lebesgue measure of a set is denoted by. For a positive integer m, let also I m denote the identity matrix M m (R. The function E denotes the characteristic function of a set E. The notation η usually stands for a cut-off function. Ad hoc definitions are given when needed. Unless stated otherwise, the duality product, :=, D,D always denotes the duality between D(R d = C (Rd and D. The space of measurable functions ψ such that ψ L (R d + ψ L (R d < is denoted by W, (R d. In the sequel, C > is always a constant uniform in ε which may change from line to line. D ε ψ. Preliminaries.. On Sobolev-Kato spaces. For s, we define the Sobolev-Kato space H s (Rd of functions of non localized H s energy by { } H s (Rd := u H s loc (Rd, sup u H s (ξ+(, d < ξ Z d We will mainly wor with H /. The following lemma is a useful tool to compare the H/ norm to the H / norm of a H / (R d function. Lemma 3. Let η C c (R d and v H / (Rd. Assume that Supp η B(, R, for R >. Then, (. ηv H / CR d v / H, with C = C(d, η W,. For a proof, we refer to the proof of Lemma.6 in DP4]... Homogenization and wea convergence. We recall the standard wea convergence result in periodic homogenization for a fixed domain Ω. As usual, the constant homogenized matrix A = A αβ M N (R is given by (. A αβ := A αβ (ydy + A αγ (y yγ χ β (ydy, T d T d 4.
where the family χ = χ γ (y M N (R, y T d, solves the cell problems (.3 y A(y y χ γ = yα A αγ, y T d and χ γ (ydy =. T d Theorem 4 (wea convergence. Let Ω be a bounded Lipschitz domain in R d and let u H (Ω be a sequence of wea solutions to A (x/ε u = f (H (Ω, where ε and the matrices A = A (y L satisfy (.6 and (.7. Assume that there exist f (H (Ω and u H (Ω, such that f f strongly in (H (Ω, u u strongly in L (Ω and u u wealy in L (Ω. Also assume that the constant matrix A defined by (. with A replaced by A converges to a constant matrix A. Then and A (x/ε u A u wealy in L (Ω A u = f (H (Ω. For a proof, which relies on the classical oscillating test function argument, we refer for instance to KLS3, Lemma.]. This is an interior convergence result, since no boundary condition is prescribed on u..3. Uniform estimates in homogenization and applications. We recall here the boundary Lipschitz estimate proved by Avellaneda and Lin in AL87a]. Theorem 5 (Lipschitz estimate, AL87a, Lemma ]. For all κ >, < µ <, there exists C > such that for all ψ C,ν (R d W, (R d, for all A A ν, for all r >, for all ε >, for all f L d+κ (D ψ (, r, for all F C,µ (D ψ (, r, for all u ε L (D ψ (, r wea solutions to { A(x/ε u ε = f + F x D ψ (, r, u ε =, x ψ (, r, the following estimate holds (.4 { } u ε L (D ψ (,r/ C r u ε L (D ψ (,r + r d/(d+κ f L d+κ (D ψ (,r + r µ F C,µ (D ψ (,r. Notice that C = C(d, N, λ, κ, µ, ψ W,, ψ] C,ν, A] C,ν. As stated in our earlier wor KP5], this estimate does not cover the case of highly oscillating boundaries, since the constant in (.4 involves the C,ν semi-norm of ψ. In this wor, we rely on Theorem 5 to get large-scale pointwise estimates on the Poisson ernel P = P (y, ỹ associated to the domain R d + and to the operator A(y. Proposition 6. For all d, there exists C >, such that for all A A ν, we have: ( for all y R d +, for all ỹ R d {}, we have (.5 (.6 P (y, ỹ Cy d y ỹ d, y P (y, ỹ ( for all y, ỹ R d {}, y ỹ, (.7 y P (y, ỹ Notice that C = C(d, N, λ, A] C,ν. C y ỹ d, C y ỹ d. The proof of those estimates starting from the uniform Lipschitz estimate of Theorem 5 is standard (see for instance AL87a]. 5
3. Boundary layer corrector in a flat half-space This section is devoted to the well-posedness of the boundary layer problem { A(y v =, yd >, (3. v = v H / (Rd, y d =, in the flat half-space R d +. Proposition 7. Assume v H / (Rd. Then, there exists a unique wea solution v of (3. such that (3. sup ξ Z d with C = C(d, N, λ, A] C,ν. ξ+(, d v dy d dy C v <, H / The proof is in three steps: (i we define a function v and prove it is a wea solution to (3., (ii we prove that the solution we have defined satisfies the estimate (3., (iii we prove uniqueness of solutions verifying (3.. 3.. Existence of a wea solution. Let η C c (R a cut-off function such that (3.3 η on (,, η, η L. Let y R d + be fixed. Notice that η( y C c (R d, η( y on B(y,, η( y and (η( y L. We define (3.4 v(y := v (y + v (y, where for y R d +, v (y := R d {} P (y, ỹ( η( ỹ y v (ỹ dỹ, and v = v (y H (R d + is the unique wea solution to { A(y v =, y d >, v = η( y y v (y H / (R d, y d =, satisfying (3.5 R d + v dy dy d C ηv H /, with C = C(d, N, λ. First of all, one has to prove that the definition of v does not depend on the choice of the cut-off η. Let η, η Cc (R be two cut-off functions satisfying (3.3. We denote by v (y and v (y the associated vectors defined by v (y := P (y, ỹ( η ( ỹ y v (ỹ dỹ + v(y, v (y := Substracting, we get (3.6 v (y v (y = R d {} R d {} R d {} P (y, ỹ( η ( ỹ y v (ỹ dỹ + v (y. P (y, ỹ(η ( ỹ y η ( ỹ y v (ỹ dỹ + v (y v (y. 6
Now since y R d {} P (y, ỹ(η ( ỹ y η ( ỹ y v (ỹ dỹ is the unique solution to { A(y v =, y d >, v = (η ( y y η ( y y v (y H / (R d, y d =, the difference in (3.6 has to be zero, which proves that our definition of v is independent of the choice of η. It remains to prove that v = v(y defined by (3.4 is actually a wea solution to (3.. Let ϕ = ϕ (y Cc (R d and ϕ d = ϕ d (y d Cc ((,. We choose η Cc (R satisfying (3.3 and such that η( on Supp ϕ + B(,. We aim at proving v(y ( A (y (ϕ ϕ d dy =. R d + This relation is clear for v. For v, by Fubini and then integration by parts v (y ( A (y (ϕ ϕ d dy R d + = = = Supp ϕ Supp ϕ d R d {} R d {} R d {} P (y, ỹ( η(ỹv (ỹ ( A (y ϕ ϕ d dỹdy Supp ϕ Supp ϕ d P (y, ỹ ( A (y (ϕ ϕ d dy( η(ỹv (ỹ dỹ A(y P (y, ỹ, ϕ ϕ d ( η(ỹv (ỹ dỹ =. 3.. Gradient estimate. Let ϕ = ϕ (y Cc (R d and ϕ d = ϕ d (y d Cc ((,. We choose R > such that Supp ϕ + B(, B(, R. Our goal is to prove d v(yϕ ϕ d (ydy CR v / H ϕ L ϕ d L, R d + with C = C(d, N, λ, A] C,ν. This estimate clearly implies the bound (3.. Let η Cc such that (3.3 η( on B(, R and Supp η( B(, R. Combining (3.5 and the result of Lemma 3, we get v dy dy d CR d v, H / R d + (R with C = C(d, N, λ. It remains to estimate v (yϕ ϕ d (ydy = R d + R d v (yϕ ϕ d (ydy dy d + R d v (yϕ ϕ d (ydy dy d. To estimate these terms we rely on the the bound (.6: for all y R d +, ỹ R d {}, with C = C(d, N, λ, A] C,ν. y P (y, ỹ C y ỹ d = C (y d + y ỹ d/, 7
We begin with two useful estimates. We have on the one hand for all y R d such that y + B(, B(, R, (3.7 R d y ỹ d ( η( ỹ v (ỹ dỹ = R d ỹ d ( η( y ỹ v (y ỹ dỹ R d \B(, ξ Z d \{} and on the other hand for all (y, y d R d +, (3.8 R d (yd + y ỹ d/ ( η( ỹ v (ỹ dỹ ỹ d v (y ỹ dỹ ξ d v L C v L (y d + y ỹ d/ v (ỹ dỹ R d R d (y d + y ỹ d/ dỹ v L C y d v L. Using (3.7, we get v (yϕ ϕ d (ydy dy d = R d y P (y, ỹ( η( ỹ v (ỹ dỹϕ ϕ d (ydy dy d R d R d {} ( / ( / η( ỹ η( ỹ R d R d {} y ỹ d dỹ R d {} y ỹ d v (ỹ dỹ ϕ ϕ d (y dy dy d ( / C v L R d r ϕ ϕ d (y dy dy d C v L ϕ ϕ d (y dy dy d CR d v L ϕ R d L ϕ d L. Using (3.8, we infer v (yϕ ϕ d (ydy dy d R d ( / C R d R d {} (yd + y ỹ dỹ d/ ( / R d {} (yd + y ỹ d/ v (ỹ dỹ ϕ ϕ d (y dy dy d C v L ϕ d (y d dy d ϕ (y dy CR d v y L ϕ d R d L ϕ d L. 3.3. Uniqueness. By linearity, it is enough to prove uniqueness for v = v(y wea solution to { A(y v =, yd >, such that (3.9 sup ξ Z d ξ+(, d v =, y d =, v dy d dy C <. 8
Let be a fixed integer. A rescaled version of the Lipschitz estimate of AL87a] for the flat half-space reads for all n > v v dy d dy. D(,n D(, Since v satisfies the bound (3.9, we get v dy Cn d /n d = /n n. D(,n Therefore, v dy =. D(, We conclude that v = on D(, by using Poincaré s inequality. 4. Estimates for a Dirichlet to Neumann operator The Dirichlet to Neumann operator DN is crucial in the proof of the well-posedness of the elliptic system in the bumpy half-space (see section 5. The ey idea there is to carry out a domain decomposition. The Dirichlet to Neumann map is the tool enabling this domain decomposition. Since we are woring in spaces of infinite energy to be useful DN has to be defined on H /. Similar studies have been carried out in ABZ3] (context of water-waves, GVM] (d Stoes system, DP4] (3d Stoes-Coriolis system. We first define the Dirichlet to Neumann operator on H / (R d : DN : H / (R d D (R d, such that for any v H / (R d, for all ϕ C c (R d, DN(v, ϕ D,D := A(y v e d, ϕ D,D, where v is the unique wea solution to { A(y v =, yd >, (4. v = v H / (R d, y d =. Proposition 8. ( For all ϕ Cc (R d +, (4. DN(v, ϕ yd = D,D = A(y v e d, ϕ yd = D,D = ( For all ϕ Cc (R d, (4.3 DN(v, ϕ yd = D,D = For y, ỹ R d {}, let R d {} R d {} K(y, ỹ := A(y y P (y, ỹ e d R d + A(y v ϕdy. A(y y P (y, ỹ e d v (ỹdỹϕ(ydy. be the ernel appearing in (4.3. Estimate (.7 of Proposition 6 implies that K(y, ỹ C y ỹ d, for any y, ỹ R d {}, y ỹ with C = C(d, N, λ, A] C,ν. Both formulas in Proposition 8 follow from integration by parts. Because of (4., it is clear that for all v H / (R d, for all ϕ C c (R d, (4.4 DN(v, ϕ C v H / ϕ H /, 9
with C = C(d, N, λ, so that DN(v extends as a continuous operator on H / (R d into H / (R d. Another consequence of (4. is the following corollary. Corollary 9. For all v H / (R d, DN(v, v = where v is the unique solution to (4.. R d + A(y v vdy, Our next goal is to extend the definition of DN to v H / (Rd. We have to mae sense of the duality product DN(v, ϕ. As for the definition of the solution to the flat half-space problem (see section 3, the basic idea is to use a cut-off function η to split the definition between one part DN(ηv, ϕ where ηv H / (R d, and another part DN(( ηv, ϕ which does not see the singularity of the ernel K(y, ỹ. For R >, there exists η Cc (R such that η, η on ( R, R, Supp η ( R, R +, η L. Let v H / (Rd. Let R > and ϕ Cc (R d such that Supp ϕ+b(, B(, R. We define the action of DN(v on ϕ by (4.5 DN(v, ϕ D,D := DN(η( v, ϕ H /,H / + K(y, ỹ( η( ỹ v (ỹ ϕ(y dỹdy. R d {} R d {} The fact that this definition does not depend on the cut-off η Cc (R follows from Proposition 8. The argument is similar to the one used in section 3.. The first term in the right-hand side of (4.5 is estimated using (4.4 and the bound of Lemma 3 between the H / norm of η( v and the H / norm of v. That yields DN(η( v, ϕ C η( v H / ϕ H / CR d v / H ϕ H /, with C = C(d, N, λ. We deal with the integral part in the right hand side of (4.5 in a way similar to the proof of the estimate (3.7. Using the fact that the supports of ( η( y v (y on the one hand and ϕ on the other hand are disjoint, we have K(y, ỹ( η( ỹ v (ỹ ϕ(y dỹdy R d {} C C C R d {} R d {} ( R d {} R d {} R d {} y ỹ d ( η( ỹ v (ỹ ϕ(y dỹdy ( / R d {} y ỹ d ( η( ỹ dỹ / y ỹ d ( η( ỹ v (ỹ dỹ ϕ(y dy ( / r dr ϕ(y dy v L R d {} CR d v L ϕ L, with C = C(d, N, λ, A] C,ν. These results are put together in the following proposition.
Proposition. ( For v H / (R d, for any ϕ C c (R d, we have DN(v, ϕ C v H / ϕ H /, with C = C(d, N, λ. ( For v H / (Rd, for R > and any ϕ C c we have Supp ϕ + B(, B(, R, (4.6 DN(v, ϕ CR d v / H ϕ H /, with C = C(d, N, λ, A] C,ν. (R d such that 5. Boundary layer corrector in a bumpy half-space This section is devoted to the well-posedness of the boundary layer problem { A(y v =, yd > ψ(y, (5. v = v H / (Rd, y d = ψ(y, in the bumpy half-space Ω + := {y d > ψ(y }. For technical reasons, the boundary ψ W, (R d is assumed to be negative, i.e. ψ(y < for all y R d. We prove Theorem of the introduction which asserts the existence of a unique solution v in the class sup ξ Z d ξ+(, d ψ(y v dy d dy <. Ω y d = y d = ψ(y Figure. Splitting of the half-space Ω + The idea is to split the bumpy half-space into two subdomains (see Figure : a flat half-space R d + on the one hand and a bumpy channel Ω := {ψ(y < y d < } on the other hand. Both domains are connected by a transparent boundary condition involving the Dirichlet to Neumann operator DN defined in section 4. Therefore, solving (3. is equivalent to solving (5. A(y v =, > y d > ψ(y, v = v H / (Rd, y d = ψ(y, A(y v e d = DN(v yd =, y d =. This fact is stated in the following technical lemma.
Lemma. If v is a wea solution of (5. in Ω such that (5.3 sup v dy d dy <, v yd = H/ (Rd, ξ Z d ξ+(, d ψ(y then ṽ, defined by ṽ(y := v(y for ψ(y < y d < and ṽ R d is the unique solution to (3. + with boundary condition ṽ yd = + = v y d = given by Proposition 7, is a wea solution to (5.. Moreover, the converse is also true. Namely, if v is a wea solution to (5. in Ω + such that v dy d dy <, sup ξ Z d ξ+(, d ψ(y then v {ψ(y <y d <} is a wea solution to (5.. The main advantage of the domain decomposition is to mae it possible to wor in a channel, bounded in the vertical direction, in which one can rely on Poincaré type inequalities. Therefore our method is energy based, which maes it possible to deal with rough boundaries. We now turn to the existence of a solution of (5. satisfying (5.3. We lift the boundary condition v. There exists V such that sup ξ Z d ξ+(, d ψ(y V + V dy d dy C v, H / with C = C(d, N, ψ W, and such that the trace of V is v. Thus, w := v V solves the system A(y w = F, > y d > ψ(y, (5.4 w =, y d = ψ(y, A(y w e d = DN(w yd = + f, y d =, where F := A(y V, f := DN(V yd = A(y V e d. Notice that the source terms satisfy the following estimates: (5.5 sup F dy d dy C v, ξ Z d ξ+(, d ψ(y H / with C = C(d, N, λ, ψ W, and for all ϕ Cc (R d such that B(, R Supp ϕ B(, R for some R >, (5.6 f, ϕ CR d v / H ϕ H /, with C = C(d, N, λ, A] C,ν, ψ W,. There are three steps in the proof of the well-posedness of (5.. Firstly, for n N we build approximate solutions w n = w n (y solving (5.7 A(y w n = F, > y d > ψ(y, w n =, {y d = ψ(y } { y = n}, A(y w n e d = DN(w n yd = + f, y d =, on Ω,n := {y ( n, n d, > y d > ψ(y } and extend w n by on Ω \ Ω,n. We have that w n H (Ω. For n fixed, this construction is classical. Indeed, using the positivity of the Dirichlet to Neumann operator (see (4. an easy energy estimate yields the bound Ω,n w n dx n d, so that a Galerin scheme maes it possible to conclude that w n exists.
Ω T = ξ + ( m, m d ξ y d = Ω T y d = ψ(y y y + ( r, r d Ω,y,r y d = y d = ψ(y Ω, Σ := (, d y d = y d = ψ(y Ω y d = y d = ψ(y Figure 3. The channel Ω, the subdomains Ω, and Ω,y,r, and the midsize box Ω T Secondly, we aim at getting estimates uniform in n on w n in the norm (5.8 sup w n dy d dy. ξ Z d ξ+(, d ψ(y This is done by carrying out so-called Saint-Venant estimates in the bounded channel. We close this step by using a hole-filling argument. The method has been pioneered by Ladyžensaja and Solonniov LS8] for the Navier-Stoes system in a bounded channel. Here the situation is more involved because of the non local operator DN on the upper boundary. The situation here is closer to GVM, DGV] (d Stoes system and DP4] (3d Stoes-Coriolis system. Finally, one has to chec that wea limits of w n are indeed solutions of (5.4. This step is straightforward because of the linearity of the equations. Uniqueness follows from the Saint-Venant estimate of the second step, with zero source terms. We focus on the second step, which is by far the most intricate one. We first introduce some notations for subdomains of Ω, which are displayed on Figure 3. Let r >, y Rd and Ω,y,r := { y y < r, > y d > ψ(y }. Let w r H (Ω be a wea solution to A(y w r = F r, > y d > ψ(y, (5.9 w r =, y d = ψ(y, A(y w r e d = DN(w r yd = + f r, y d =, 3
R d ξ T = ξ + ( m, m d ξ T = ξ + ( m, m d Σ = (, d Σ +m = ( m +, + m d Figure 4. Two midsize cubes T and T of volume m d belonging to C,m such that w r = on Ω \ Ω,y,r, and where F r := F Ω,y,r, f r := f B(y,r. Notice that w n defined above (see (5.7 is equal to w r solution of (5.9 for r := n and y =. The reason w r is introduced is that in section 5. we will need to translate the origin: w r will be a translate of w n. All estimates are carried out on w r solving the system (5.9 so that we have the uniformity of constants both in r and y. For N, let Ω, := {y (, d, > y d > ψ(y }. Notice that Ω, = Ω,, with the notation above for Ω,y,r. Our goal is to estimate, E := w r dy. Ω, In order to deal with the non local character of the Dirichlet to Neumann operator, we rely on a careful splitting of R d into midsize boxes of volume m d. In the following, for, m N,, m, m = m even, Σ := (, d, and the set C,m (see Figure 4 denotes the family of cubes T of volume m d contained in R d \ Σ +m with vertices in Z d, i.e. } C,m := {T = ξ + ( m, m d, ξ Z d and T R d \ Σ +m. Let also C m be the family of all the cubes of volume m d with vertices in Z d { C m := T = ξ + ( m, m d, ξ Z d }. Notice that for m, C,m C,m C m,m C m. For T C,m, (5. E T := w r dy, Ω T Ω T := {y T, > y d > ψ(y }. 4
Proposition. There exists a constant C = C (d, N, λ, A] C,ν, ψ W,, v / H such that for all r >, y Rd, for all, m N, m 3 and m/ = m, for any wea solutions w r H (Ω of (5.9, the following bound holds (5. E C ( Notice that C is independent of r and y. d + E +m E + 3d 5 m 3d 3 sup T C,m E T The crucial point for the control of the large-scale energies in (5. is the fact that the power 3d 5 of is strictly smaller that the power 3d 3 of m. Before tacling the proof of Proposition, let us explain how to infer from (5. an a priori bound uniform in n on w n solution of (5.7. 5.. Proof of the a priori bound in the Sobolev-Kato space: downward induction. Assume that the Saint-Venant estimate of Proposition has been established. We come bac to its proof in section 5. below. Our goal is to infer from (5. an a priori bound (5. E := w n dy d Ω, uniform in n for w n solution of (5.7. Let us stress that w n is also a solution to (5.9 with r = n and y =. As explained above, this bound is all we need to get the existence of w solving (5.4. To show estimate (5., we prove (see Lemma 3 below that the energy E m contained in one elementary midsize box of volume m d is bounded uniformly in m. The number m is an auxiliary parameter chosen thans to the latitude allowed by the Saint-Venant estimate (5.. Proving an a priori bound would be much easier if the Saint-Venant estimate (5. did not involve the term 3d 5 m 3d 3 sup E T, T C,m whose reason for being there is the nonlocality of the Dirichlet to Neumann operator. For the sae of the explanation, assume temporarily ( that the Saint-Venant relation reads (5.3 E C d + E + E, instead of (5.. The auxiliary number which is in (5. to control the non local terms has no reason to appear in (5.3. For n, E + = E, therefore Now, using the hole filling tric, we get E n E n Eventually, C C + (5.4 E E n C n d. { (n d + C n d }, C C + (n d +... { ( C C + ( C C + n C n d + (n d + ( C n ( C = C +. C + d } C n d. which is bounded uniformly in n. Let us go bac to our actual estimate (5.. The general idea of the downward induction is the same as in the simple case above. As expected yet, the non local terms mae the 5
R d ξ T = ξ + ( m, m d Σ n = ( n, n d Figure 5. The channel Ω seen from above: midsize cube T of volume m d such that Ω T concentrates most energy among the midsize cubes T C m induction tricier. Before going into the details, we fix some notation, and define the auxiliary integer m once for all. Let C be given by Proposition, and let (5.5 ( C A := (C ( C + C ( d <, B := + C ( 3d 5 <. + = These two numbers are the analogues of the terms appearing in the right hand side of (5.4. We now choose an integer m so that (5.6 m 3, m is even and 5 3d B m >. Notice that m = m(d, N, λ, A] C,ν, ψ W,, v / H, but is independent of r and y. The reason for taing m even is technical; it is only used in the translation argument below. The reason for imposing the condition = 5 3d B m > is to be able to swallow the right hand side into the left hand side at the end of the iteration. We also tae n = lm = l m, with l N, l and let w n be the solution of (5.7. How do the non local terms mae the downward induction more complicated? The trouble comes from the fact that when iterating downward the boxes Ω,, on which the energy E is computed, are always centered at. Nevertheless, at the end of the iteration (down to = m the energy E m may not be comparable in any way to sup E T, T C m,m which appears on the right hand side. The way out of this deadloc is to iterate downward taing into account another center ξ defined as follows. There exists T C m such that T Σ n and E T = sup T Cm E T (see Figure 5. Of course for all m sup E T sup E T sup E T = E T. T C,m T C m,m T C m By definition, there is ξ Z d for which T = ξ + ( m, m d. 6
In order to write the iteration, it is more convenient to first center T at zero. So we now simply translate the origin. Doing so, w n(y := w n (y + ξ, y d is a solution of (5.9 with y := ξ, r = n and Notice that A (y := A(y + ξ, y d, ψ (y := ψ(y + ξ, v (y := v (y + ξ, F (y := F (y + y, y d and f (y := f(y + ξ. A ] C,µ = A] C,ν, ψ W, = ψ W, and v / H = v / H, so that w n satisfies the Saint-Venant estimate (5. with the same constant C. Here we use the ey fact that the Saint-Venant estimate (5. is uniform in y. Furthermore, E m = E T. We are now ready to state the a priori bound and to prove it by downward induction. Lemma 3. We have the following a priori bound where A is defined by (5.5. E m d Am d, The Lemma is obtained by downward induction, using a hole-filling type argument. Since w n is supported in Ω,n, we start from sufficiently large in (5.. For = n + m = (4l + m, estimate (5. implies E (4l+ m C ((4l + m d, because E T = for any T C (4l+ m,m. Then, ( E ((l + m = E (4l+ m m C C C + ((l +d m d + Let p {,... l }. We then have C C + C (4l+ d m d. E (p+ m C + (p + d m d ( C l p ( C +... C (4l d m d l p + + C C (4l + d m d + + 5 3d C ( ] C l p m C + (p + 3d 5 +... C (4l + 3d 5 E m. + Eventually, for p = ( E m C C C + md + C (3 m d + ( C l p ( C +... C (4l d m d l p + + C C (4l + d m d + + 5 3d C ( ] C l p m C + +... C (4l + 3d 5 E m d Am d + 5 3d + m Therefore, which proves Lemma 3. E m ( < 5 3d Bm E m d Am d, 7 BE m.
Finally, sup w n dy d dy E m d Am d, ξ Z d ξ+(, d ψ(y which proves the a priori bound in the norm (5.8 uniformly in n. 5.. Proof of Proposition : the Saint-Venant estimate. We proceed in four steps: ( we construct a cut-off η with bounds uniform in to carry out the local estimates, ( we carry out energy estimates on system (5.9, and seperate the large scales (non local effects from the small scales, (3 we show a control of the non local terms, (4 we gather the estimates to get (5.. Construction of a cut-off. Let η C (B(, / such that η and R η =. For all d N, let η = η (y be defined by η (y = /,+/] d (y ỹ η(ỹ dỹ = R d η(y ỹ dỹ. /,+/] d For all N, we have the following properties: η on, ] d, Supp η, + ] d, η C c (R d and most importantly, we have the control uniformy in. η L η L Energy estimate. Testing the system (5.9 against η w r we get (5.7 η A(y w r w r dy = Ω η A(y w r η w r dy Ω By ellipticity, we have + F r, η w r + f r, η w r + DN(w r yd =, η w r. λ η w r dy Ω η A(y w r w r dy. Ω Using that η w r vanishes on the lower oscillating boundary, the fundamental theorem of calculus yields a Poincaré type inequality (5.8 η w r L (Ω C yd (η w r L (Ω, where the constant C only depends on the height of the channel Ω, and therefore not on. The following estimate (or variations of it is of constant use: by the trace theorem and the Poincaré inequality (5.8 (5.9 ( / η 4 w r(y, dy η w r H / C { η w r L (Ω + (η w } r L (Ω Σ + C { yd (η w r L (Ω + (η w r L (Ω } C (η w r L (Ω C(E + E / + C (Ω η 4 w r dy /, 8
with C = C(d, ψ W,, η L and C = C (d. We now estimate every term on the right hand side of (5.7. We have, η A(y w r η w r dy ( / ( η Ω λ w r dy η w r dy Ω Ω ( / C η w r dy (E + E /, Ω with C = C(λ, η L. We also have, F r, η w r = F, η w r = F, (η w r C d (E+ E / + C d / ( / η 4 w r dy, Ω where C = C( v / H, η L and C = C ( v / H, and by the trace theorem and Poincaré inequality f r, η w r = f, η w r C d η w r H / C d (η w r L ( / C d (E+ E / + C d η 4 w r dy, Ω with C = C(d, ψ W,, v / H, η L and C = C ( v / H. We have now to tacle the non local term involving the Dirichlet to Neumann operator. We split this term into DN(w r yd =, η w r = DN(( η +m w r yd =, η w r By Corollary 9, + DN((η +m η w r yd =, η w r + DN(η w r, η w r. DN(η w r, η w r. Relying on Proposition and on estimate (4.6, we get DN((η+m η w r yd =, η w r C d (η +m η w r yd = / H η w r H / C(E +m E / (Ω η 4 w r dy / + C(E +m E / (E + E /, with C = C(d, N, λ, A] C,ν, ψ W,, η L. Notice that the bound (4.4 for the Dirichlet to Neumann operator in H / here is actually enough, since w r is compactly supported. However, when dealing with solutions not compactly supported, as for the uniqueness proof in section 5.3, we have to use the result of Proposition. Control of the non local term. Lemma 4. For all m 3, all m = m/, we have ( (5. R d y ỹ d ( η +m w r(ỹ, dỹ dy C 3d 5 m 3d 3 sup E T, T C,m Σ + where C = C(d. 9
Let y Σ + be fixed. We have R d y ỹ d ( η +m w r(ỹ, dỹ = R d y ỹ d (η +(j+(m η +j(m w r(ỹ, dỹ = j= j= Σ +(j+(m + \Σ +j(m y ỹ d w r(ỹ, dỹ j= T C,j,m T y ỹ d w r(ỹ, dỹ, where C,j,m is a family of disjoint cubes T = ξ+( m, m d such that T Σ +(j+(m + \ Σ +j(m and T C,j,m T = Σ +(j+(m + \ Σ +j(m. For all T C,j,m, by Cauchy-Schwarz, trace theorem and Poincaré inequality T ( y ỹ d w r(ỹ, dỹ / ( / T y ỹ dỹ w d r (ỹ, dỹ T ( / ( / C T y ỹ dỹ w d r dỹ Ω T ( ( / / C y ỹ dỹ sup E d T, T C,j,m T where Ω T and E T are defined in (5.. Notice that the constant C in the last inequality only depends on d and on ψ W,. Moreover, for any T C,j,m, ( T / y ỹ dỹ d and the number of elements of C,j,m is bounded by Σ +(j+(m + \ Σ +j(m #C,j,m = Therefore, R d m d m d ( + j(m y d, ( + j(m d m d. y ỹ d ( η +m w r(ỹ, dỹ ( / d m C sup E T T C,j,m ( + j(m y d C C ( sup E T T C,j,m ( sup T C,j,m E T / j= T C,j,m j= m d 3 ( + j(m d ( + j(m y d / ( + m d m d ( + m y, d
with C = C(d. Eventually, we get for m 3 ( Σ + R d y ỹ d ( η +m w r(ỹ, dỹ dy ( ( + m d 4 C sup E T T C,j,m m d Σ + ( + m y dy d ( ( + m d 4 ( + d 3d 5 C sup E T T C,j,m m d C (m d m 3d 3 sup E T, T C,j,m with C = C(d, the last inequality being only true on condition that m/ = m. This proves Lemma 4. In particular, by the definition of DN in (4.5, by the fact that ( η +m w r(ỹ, and η w r(y, have disjoint support, by estimate (5. and by the bound (5.9 we get DN(( η+m w r yd =, η w r C R d R d y ỹ d ( η +m (ỹ w r (ỹ, η (y w r (y, dỹ dy ( / ( C η 4 w r(y, dy η +m (ỹ Σ + R d y ỹ d w r (y, dỹ dy C 3d 5 m 3d 3 C 3d 5 m 3d 3 Σ + ( η 4 w r(y, dy Σ + / ( sup E T T C,j,m ( ] ( / (E + E / + η 4 w dy Ω with C = C(d, N, λ, A] C,ν, ψ W,. sup T C,j,m E T, End of the proof of the Saint-Venant estimate. Combining all our bounds and using /, E + E E +m E, η 4 C( η L η whenever possible, we get from (5.7 the following estimate ( / λ η w r dy C η w r dy (E +m E / +C d (E+m E / +C(E +m E Ω Ω ( / ( / + C d η w r dy + C(E +m E / η w r dy Ω Ω + C 3d 5 m 3d 3 ( ] ( / (E + E / + η w dy Ω sup T C,j,m E T with C = C(d, N, λ, A] C,ν, v / H, ψ W,. Swallowing every term of the type η w dy Ω /, in the left hand side, we end up with the Saint-Venant estimate (5.. This concludes the proof of Proposition.
5.3. End of the proof of Theorem : uniqueness. Extracting subsequences using a classical diagonal argument and passing to the limit in the wea formulation of (5.7 relying on the continuity of the Dirichlet to Neumann map asserted in estimate (4.4 yields the existence of a wea solution w to the system (5.4. In addition, the wea solution satisfies the bound (5. sup w dy d dy d Am d <. ξ Z d ξ+(, d ψ(y Let us turn to the uniqueness of the solution to (5.4 satisfying the bound (5.. By linearity of the problem, it is enough to prove the uniqueness for zero source terms. Assume w Hloc (Ω is a wea solution to (5.4 with f = F = satisfying (5. sup w dy d dy C <. ξ Z d ξ+(, d ψ(y Repeating the estimates leading to Proposition (see section 5., we infer that for the same constant C appearing in the Saint-Venant estimate (5. and for m defined by (5.6, for N, m/ = m, (5.3 E C ( E +m E + 3d 5 m 3d 3 sup E T T C,m The fact that w, unlie w n, does not vanish outside Ω,n does not lead to any difference in the proof of this estimate. Since sup T C m E T <, for any ε, there exists T ε C m such that (5.4 sup T C m E T ε E T ε sup T C m E T. Again, Tε := ξε +( m, m d for ξε Z d, and we can translate Tε so that it is centered at the origin as has been done in section 5.. Estimate (5.3 still holds. For any n N, E n C n d where C is defined by (5.. The idea is now to carry out a downward iteration. For any n = (l + m with l N, l fixed, for p {,... l } one can show that C ( C ( ] C l p E (p+ m C + + C +... + C E n + + 5 3d C ( ] C l p m C + (p + 3d 5 +... C (l + 3d 5 + C ( + C l p+ C C + C n d + + 5 3d C ( ] C l p m C + (p + 3d 5 +... C (l + 3d 5 + Thus, C ( + C E m C C + C + C ( + C C C + C + l (l + d m d + 5 3d m. B sup T C m E T l+ (l + d m d + 5 3d B(E m + ε. m sup T C m E T sup E T. T C m
From this we infer using (5.6 that C ( + C l+ E m C C + C (l + d m d + 6 3d + m Therefore, from equation (5.4 sup E T T C m ( + 6 3d m B ε, l Bε 6 3d m which eventually leads to sup T Cm E T =, or in other words w =. Combining this existence and uniqueness result for the system (5.4 in the bumpy channel Ω with Lemma and Proposition 7 about the well-posedness in the flat half-space finishes the proof of Theorem. 6. Improved regularity over Lipschitz boundaries The goal in this section is to prove Theorem of the introduction. Let us recall the result we prove in the following proposition. Proposition 5. For all ν >, γ >, there exist C > and ε > such that for all ψ W, (R d, < ψ < and ψ L γ, for all A A ν, for all < ε < (/ε, for all wea solutions u ε to (., for all r ε/ε, /] (6. u ε C u ε, Dψ ε (,r Dψ ε (, or equivalently, with C = C(d, N, λ, ν, γ, A] C,ν. u ε Cr u ε, Dψ ε (,r Dψ ε (, We rely on a compactness argument inspired by the pioneering wor of Avellaneda and Lin AL87a, AL89b], and our recent wor KP5]. The proof is in two steps. Firstly, we carry out the compactness argument. Secondly, we iterate the estimate obtained in the first step, to get an estimate down to the microscopic scale O(ε. A ey step in the proof of boundary Lipschitz estimates is to estimate boundary layer correctors, which is done by combining the classical Lipschitz estimate with a uniform Hölder estimate, as in AL87a, Lemma 7] or KP5, Lemma ]. Here, we are able to relax the regularity assumption on ψ. This progress is enabled by our new estimate (.4 for the boundary layer corrector, which holds for Lipschitz boundaries ψ. We begin with an estimate which is of constant use in this part of our wor. Tae ψ W, (R d and A A ν. By Cacciopoli s inequality, there exists C > such that for all ε >, for all wea solutions u ε to { A(x/ε u ε =, x Dψ ε (,, (6. u ε =, x ε ψ (,, for all < θ <, ( / ( xd u ε Dψ (,θ = xd u ε xd u ε D ψ (,θ D ψ (,θ (6.3 ( / C θ d/ u ε. ( θ Notice that C in (6.3 only depends on λ. 3 D ψ (, Bε.
Proposition 5 is a consequence of the two following lemmas. The first one contains the compactness argument. The second one is the iteration lemma. In order to alleviate the statement of the following lemma, the definition of the boundary layer v is given straight after the lemma. Lemma 6. For all ν >, γ >, there exists θ >, < µ <, ε >, such that for all ψ W, (R d, < ψ < and ψ L γ, for all A A ν, for all < ε < ε, for all wea solutions u ε to (6. we have u ε Dψ ε (, implies D ε ψ (,θ u ε (x ( xd u ε D ε ψ (,θ x d + εχ d (x/ε + εv(x/ε] θ +µ. The boundary layer v = v(y is the unique solution given by Theorem to the system { A(y v =, yd > ψ(y (6.4, v = y d χ d (y, y d = ψ(y. The estimate of Theorem implies v dy d dy C sup ξ Z d ξ+(, d ψ(y { } ψ / H (Rd + χ(, ψ( H /, (Rd with C = C(d, N, λ, A] C,ν, ψ W,. Now, by Sobolev injection W, (R d H / (R d ψ H / (Rd C ψ W, (R d, with C = C(d and by classical interior Lipschitz regularity χ(, ψ( / H C χ(, ψ( W, (R d C χ W, (R d C, with in the last inequality C = C(d, N, λ, A] C,ν. Eventually, (6.5 sup v dy d dy C, ξ Z d ξ+(, d ψ(y with C = C(d, N, λ, A] C,ν, ψ W, uniform in ε. The interior corrector χ d as well as the boundary layer corrector v are crucial in the iteration procedure, which is the second step of the method. Lemma 7. Let θ, ε and γ be given as in Lemma 6. For all ψ W, (R d, < ψ < and ψ L γ, for all A A ν, for all N, >, for all < ε < θ ε, for all wea solutions u ε to (6. there exists a ε RN satifying such that implies (6.6 D ε ψ (,θ u ε (x a ε a ε C + θ µ +... θ µ( θ d/, ( θ u ε Dψ ε (, x d + εχ d (x/ε + εv(x/ε] θ (+µ, where v = v(y is the solution, given by Theorem, to the boundary layer system (6.4. 4
The condition ε < θ ε can be seen as giving a lower bound on the scales θ for which one can prove the regularity estimate: θ > ε/ε. In that perspective, estimate (6.6 is an improved C,µ estimate down to the microscale ε/ε. For fixed < ε/ε < / and r ε/ε, /], there exists N such that θ + < r θ. We aim at estimating u ε (x Dψ ε (,r using the bound (6.6. We have (6.7 ( / ( / u ε (x u ε (x Dψ ε (,r Dψ ε (,θ ( ] / u ε (x a ε x d + εχ d (x/ε + εv(x/ε D ε ψ (,θ ( + a ε D ε ψ (,θ / ( x d + D ε ψ (,θ / ( εχ d (x/ε + D ε ψ (,θ / εv(x/ε. Let us focus on the term involving the boundary layer. Let η = η(y d Cc (R be a cut-off such that η on (, and Supp η (,. The triangle inequality yields ( D ε ψ (,θ / ( εv(x/ε D ε ψ (,θ εv(x/ε (x d + εχ d (x/εη(x d /ε / ( + D ε ψ (,θ Poincaré s inequality implies ( / εv(x/ε (x d + εχ d (x/εη(x d /ε D ε ψ (,θ θ ( D ε ψ (,θ θ ( Dψ ε (,θ + θ ε ( / εv(x/ε (x d + εχ d (x/εη(x d /ε (x d + εχ d (x/εη(x d /ε /. / / v(x/ε + ( + χ L ( θ η(x d /ε Dψ ε (,θ ( D ε ψ (,θ (x d + εχ d (x/εη (x d /ε /. Estimate (6.5 now yields v(x/ε Cεθ, Dψ ε (,θ so that eventually using ε/ε r θ, ( / εv(x/ε (x d + εχ d (x/εη(x d /ε C D ε ψ (,θ 5 (ε / θ / + θ + θ (ε + ε Cθ ε
with C = C(d, N, λ, A] C,ν, ψ W,. It follows from (6.7 and (6.6 that ( / u ε (x θ (+µ + Cθ Cθ Cr, Dψ ε (,r which is the estimate of Proposition 5. 6.. Proof of Lemma 6. Let < θ < /8 and u H (D (, /4 be a wea solution of { A (6.8 u =, x D (, /4, u =, x (, /4, such that u 4 d. D (,/4 The classical regularity theory yields u C (D (, /8. Using that for all x D (, θ u (x ( xd u x d = u (x u (x, ( xd u x d,θ,θ ( = D xd u (x, tx d xd u (y x d dxdt. (, θ we get (6.9 D (,θ D (,θ u (x ( xd u D (,θx d Ĉθ 4, where Ĉ = Ĉ(d, N, λ. Fix < µ <. Choose < θ < /8 sufficiently small such that (6. θ +µ > Ĉθ4. The rest of the proof is by contradiction. Fix γ >. Assume that for all N, there exists ψ W, (R d, (6. < ψ < and ψ L γ, there exists A A ν, there exists < ε < /, there exists u ε solving { A (x/ε u ε =, x D ε ψ (,, u ε =, x ε ψ (,, such that (6. and (6.3 uε D ε ψ (,θ (x ( xd u ε D ε ψ (, ε D ψ (,θ u ε ] x d + ε χ d (x/ε + ε v (x/ε > θ +µ. Notice that χ d is the cell corrector associated to the operator A (y and v is the boundary layer corrector associated to A (y and to the domain y d > ψ (y. First of all, for technical reasons, let us extend u ε by zero below the boundary, on {x (, d, x d ε ψ (x /ε }. The extended functions are still denoted the same, and u ε is a wea solution of A (x/ε u ε = on {x (, d, x d ε ψ (x /ε + }. 6
For sufficiently large, by Cacciopoli s inequality, u ε dx C u ε dx C, ( /4,/4 d ( /,/ d where C = C(d, N, λ. Therefore, up to a subsequence, which we denote again by u ε, we have (6.4 u ε u ε u, strongly in L (( /4, /4 d (, /4, u, wealy in L (( /4, /4 d (, /4. Moreover, ε ψ ( /ε converges to because ψ is bounded uniformly in (see (6.. Let ϕ Cc (D (, /4. Theorem 4 implies that A (x/ε u ε ϕdx A u ϕdx, D ε ψ (,/4 so that u is a wea solution to D (,/4 A u = in D (, /4. Furthermore, for all ϕ Cc (( /4, /4 d (,, = u ε ϕdx {x ( /4,/4 d, x d ε ψ (x /ε } ( /4,/4 d (, u ϕdx, so that u (x = for all x ( /4, /4 d (,. In particular, u = in H / ( (, /4. Thus, u is a solution to (6.8 and satisfies the estimate (6.9. It remains to pass to the limit in (6.3 to reach a contradiction. Since D ε ψ (, θ = D (, θ, we have (6.5 ( x d u ε D ε ψ (,θ ( x d u D (,θ D (, θ D ε ψ (,θ D (,θ + ( ( D ε ψ (,θ\d (,θ D (,θ\d ε ψ (,θ ( xd u ε xd u ε xd u dx ] xd u dx The first term in the right hand side of (6.5 tends to thans to the wea convergence of u ε in (6.4. The second term in the right hand side of (6.5 goes to when because of the L bound on the gradient, and the fact that ( ( D ε ψ (, θ \ D (, θ D (, θ \ D ε ψ (, θ. Therefore, D ε ψ (,θ D (,θ ( x d u ε ε D ψ (,θ Moreover, the strong L convergence in (6.4 implies u ε D ε ψ (,θ D (,θ The last thing we have to chec is the convergence D ε ψ (,θ ] x d + ε χ d (x/ε ( xd u D (,θx d. u. ε v (x/ε. 7.
Let η = η(y d Cc (R such that η on (, and Supp η (,. We have (6.6 ε v (x/ε D ε ψ (,θ D ε ψ (,θ ε v (x/ε (x d + ε χ d (x/ε η(x d /ε + (x d + ε χ d D ε (x/ε η(x d /ε. ψ (,θ The last term in the right hand side of (6.6 goes to when. Now by Poincaré s inequality, ε v (x/ε (x d + ε χ d (x/ε η(x d /ε D ε ψ (,θ Cθ D ε ψ (,θ D ε ψ (,θ v (x/ε + On the one hand by estimate (6.5 v (x/ε Cεd θ d Cε D ψ (,θ D ψ (,θ/ε sup ξ Z d ] ( (x d + ε χ d (x/ε η(x d /ε. ξ+(, d v (y dx ψ (y v dx Cε with in the last inequality C = C(d, N, λ, A] C,ν uniform in ε, and on the other hand ( (x d + ε χ d (x/ε η(x d /ε ( + χ L η(x d /ε D ε ψ (,θ D ε ψ (,θ + ε (x d + ε χ d D ε (x/ε η (x d /ε Cε, ψ (,θ with in the last inequality C = C(d, N, λ, A] C,ν. These convergence results imply that passing to the limit in (6.3 we get θ +µ ] (x ( xd u ε x d + ε χ d (x/ε + ε v (x/ε dy uε D ε ψ (,θ which contradicts (6.. ε D ψ (,θ D (,θ u (x ( xd u D (,θx d Ĉθ 4, 6.. Proof of Lemma 7. The proof is by induction on. The result for = is true because of Lemma 6. Let N,. Assume that for all ψ W, (R d such that < ψ < and ψ L γ, for all A A ν, for all N, >, for all < ε < θ ε, for all wea solutions u ε to (6. there exists a ε RN satifying such that implies (6.7 D ε ψ (,θ u ε (x a ε a ε C + θ µ +... θ µ( θ d/, ( θ u ε Dψ ε (, x d + εχ d (x/ε + εv(x/ε] dy θ (+µ. 8
This is our induction hypothesis. Given ψ W, (R d, < ψ < and ψ L γ and A A ν, < ε < θ ε and a solution u ε to (6. such that u ε Dψ ε (, we define U ε (x := { ]} θ (+µ u ε (θ x a ε θ x d + εχ d (θ x/ε + εv(θ x/ε for all x D ε/θ ψ (,. The goal is to apply the estimate of Lemma 7 to U ε. By the induction estimate (6.7, we have D ε/θ ψ (, U ε. Moreover, U ε solves the system { A(θ x/ε U ε =, x D ε/θ ψ (,, (6.8 U ε =, x ε/θ ψ (,. The boundary layer v solving (6.4 has been designed for U ε to solve (6.8. It follows that U ε satisfies the assumptions of Lemma 6. Therefore, for all ε/θ < ε, we have U ε (x ( xd U ε ε/θ D x d + ε ψ (,θ θ χd (θ x/ε + ε ] θ v(θ x/ε θ +µ. D ε/θ ψ (,θ Eventually, with D ε ψ (,θ+ u ε (x a ε + x d + εχ d (x/ε + εv(x/ε] θ (+µ(+, satisfying the estimate a ε + := aε + θµ ( xd U ε ε/θ D ψ (,θ a ε + C + θ µ +... θ µ( θ µ θ d/ + C ( θ θ d/ ( θ C + θ µ +... θ µ θ d/. ( θ This concludes the iteration step and proves Lemma 7. As a concluding remar, let us notice that Theorem the improved Lipschitz estimate of Theorem can be extended to the system (6.9 { A(x/ε u ε = f + F, x Dψ ε (,, u ε =, x ε ψ (,. The proof goes through exactly as in the paper KP5]. Theorem 8. Let < µ < and κ >. There exists C >, such that for all ψ W, (R d, for all matrix A = A(y = (A αβ ij (y Rd N, elliptic with constant λ, -periodic and Hölder continuous with exponant ν >, for all < ε < /, for all f L d+κ (Dψ ε (,, for all F C,µ (Dψ ε (,, for all uε wea solution to (6.9, for all 9
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