Improved Regularity in Bumpy Lipschitz Domains
IIMPROVED REGULARITY IN BUMPY LIPSCHITZ DOMAINS
CARLOS KENIG ∗ AND CHRISTOPHE PRANGE † Abstract.
This paper is devoted to the proof of Lipschitz regularity, down to themicroscopic 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 doesnot assume more than Lipschitz regularity on the boundary. Our Theorem, which is asignificant improvement of our previous work on Lipschitz estimates in bumpy domains,should be read as an improved regularity result for an elliptic system over a Lipschitzboundary. Our progress in this direction is made possible by an estimate for a boundarylayer corrector. We believe that this estimate in the Sobolev-Kato class is of independentinterest. Introduction
This paper is devoted to the proof of Lipschitz regularity, down to the microscopic scale,for weak solutions u ε = u ε ( x ) ∈ R N of the elliptic system(1.1) (cid:26) −∇ · A ( x/ε ) ∇ u ε = 0 , x ∈ D εψ (0 , ,u ε = 0 , x ∈ ∆ εψ (0 , , over a highly oscillating Lipschitz boundary. Throughout this work, ψ is a Lipschitz graph, D εψ (0 ,
1) := { x (cid:48) ∈ ( − , d − , εψ ( x (cid:48) /ε ) < x d < εψ ( x (cid:48) /ε ) + 1 } ⊂ R d and ∆ εψ (0 ,
1) := { x (cid:48) ∈ ( − , d − , x d = εψ ( x (cid:48) /ε ) } is the lower highly oscillating boundary on which homogeneous Dirichlet boundary condi-tions are imposed. Our main theorem is the following. Theorem 1.
There exists
C > such that for all ψ ∈ W , ∞ ( R d − ) , for all matrix A = A ( y ) = ( A αβij ( y )) ∈ R d × N , elliptic with constant λ , -periodic and Hölder continuous withexponant ν > , for all < ε < / , for all weak solution u ε to (1.1) , for all r ∈ [ ε, / (1.2) ˆ ( − r,r ) d − ˆ εψ ( x (cid:48) /ε )+ rεψ ( x (cid:48) /ε ) |∇ u ε | ≤ Cr d ˆ ( − , d − ˆ εψ ( x (cid:48) /ε )+1 εψ ( x (cid:48) /ε ) |∇ u ε | , with C = C ( d, N, λ, [ A ] C ,ν , (cid:107) ψ (cid:107) W , ∞ ) . The uniform estimate of Theorem 1 should be read as an improved regularity result.Indeed, estimate (1.2) can be seen as a Lipschitz estimate down to the microscopic scale O ( ε ) . Its originality lies in the fact that no smoothness of the boundary, which is justassumed to be Lipschitz, is needed for it to hold. Previous results in this direction alwaysrelied on some smoothness of the boundary, typically ψ ∈ C ,ν with ν > , or ψ ∈ C ω with ω a modulus of continuity satisfying a Dini type condition, i.e. ´ ω ( t ) /tdt < ∞ .Pioneering work on uniform estimates in homogenization has been achieved by Avel-laneda and Lin in the late 80’s [AL87a, AL87b, AL89a, AL89b, AL91]. The regularity ∗ The University of Chicago,
S. University Avenue, Chicago, IL , USA.
E-mail address: [email protected] . † The University of Chicago,
S. University Avenue, Chicago, IL , USA.
E-mail address: [email protected] . a r X i v : . [ m a t h . A P ] A p r heory for operators with highly oscillating coefficients has recently attracted a lot of at-tention, and important contributions have been made to relax the structure assumptionson the oscillations [AS14a, AS14b, GNO14]. Our work is in a different vein. It is focusedon the boundary behavior of solutions.Theorem 1 represents a considerable improvement of a recent result obtained by the twoauthors, namely Result B and Theorem 16 in [KP15]. This first work dealt with uniformLipschitz regularity over highly oscillating C ,ν boundaries. To the best of our knowledge,an improved regularity result up to the boundary such as the one of Theorem 1 is new. Ourbreakthrough is made possible by estimating a boundary layer corrector v = v ( y ) solutionto the system(1.3) (cid:26) −∇ · A ( y ) ∇ v = 0 , y d > ψ ( y (cid:48) ) ,v = v , y d = ψ ( y (cid:48) ) , in the Lipschitz half-space y d > ψ ( y (cid:48) ) with non localized Dirichlet boundary data v . Theorem 2.
Assume ψ ∈ W , ∞ ( R d − ) and v ∈ H / uloc ( R d − ) i.e. sup ξ ∈ Z d − (cid:107) v (cid:107) H / ( ξ +(0 , d − ) < ∞ . Then, there exists a unique weak solution v of (1.3) such that (1.4) sup ξ ∈ Z d − ˆ ξ +(0 , d − ˆ ∞ ψ ( y (cid:48) ) |∇ v | dy d dy (cid:48) ≤ C (cid:107) v (cid:107) H / uloc < ∞ , with C = C ( d, N, λ, [ A ] C ,ν , (cid:107) ψ (cid:107) W , ∞ ) . Overview of the paper.
In section 2 we recall several results related to Sobolev-Katospaces, homogenization and uniform Lipschitz estimates. These results are of constant usein our work. Then the paper has two main parts. The first aim is to prove Theorem 2 aboutthe well-posedness of the boundary layer system in a space of non localized energy over aLipschitz boundary. The key idea is to carry out a domain decomposition. Subsequently,there are three steps. Firstly, we prove the well-posedness of the boundary layer system overa flat boundary, namely in the domain R d + . This is done in section 3. Secondly, we defineand estimate a Dirichlet to Neumann operator over H / uloc . This key tool is introduced insection 4. Thirdly, we show that proving the well-posedness of the boundary layer systemover a Lipschitz boundary boils down to analyzing a problem in a layer { ψ ( y (cid:48) ) < 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 work, we are able to prove Theorem1 using a compactness scheme. Framework and notations.
Let λ > and < ν < be fixed in what follows. Weassume that the coefficients matrix A = A ( y ) = ( A αβij ( y )) , with ≤ α, β ≤ d and ≤ i, j ≤ N is real, that(1.5) A ∈ C ,ν ( R d ) , that A is uniformly elliptic i.e.(1.6) λ | ξ | ≤ A αβij ( y ) ξ αi ξ βj ≤ λ | ξ | , for all ξ = ( ξ αi ) ∈ R dN , y ∈ R d and periodic i.e.(1.7) A ( y + z ) = A ( y ) , for all y ∈ R d , z ∈ Z d . We say that A belongs to the class A ν if A satisfies (1.5), (1.6) and (1.7). or easy reference, we summarize here the standard notations used throughout the text.For x ∈ R d , x = ( x (cid:48) , x d ) , so that x (cid:48) ∈ R d − denotes the d − first components of the vector x . For ε > , r > , let D εψ (0 , r ) := (cid:8) ( x (cid:48) , x d ) , | x (cid:48) | < r, εψ ( x (cid:48) /ε ) < x d < εψ ( x (cid:48) /ε ) + r (cid:9) , ∆ εψ (0 , r ) := (cid:8) ( x (cid:48) , x d ) , | x (cid:48) | < r, x d = εψ ( x (cid:48) /ε ) (cid:9) ,D (0 , r ) := (cid:8) ( x (cid:48) , x d ) , | x (cid:48) | < r, < x d < r (cid:9) , ∆ (0 , r ) := (cid:8) ( x (cid:48) , , | x (cid:48) | < r (cid:9) , R d + := R d − × (0 , ∞ ) , Ω + := { ψ ( y (cid:48) ) < y d } , Ω (cid:91) := { ψ ( y (cid:48) ) < y d < } , Σ k := ( − k, k ) d − , where | x (cid:48) | = max i =1 ,... d | x i | . We sometimes write D ψ (0 , r ) and ∆ ψ (0 , r ) in short for D ψ (0 , r ) and ∆ ψ (0 , r ) ; in that situation the boundary is not highly oscillating because ε = 1 . Let also ( u ) D εψ (0 ,r ) := − ˆ D εψ (0 ,r ) u = 1 | D εψ (0 , r ) | ˆ D εψ (0 ,r ) u. 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 functionof a set E . The notation η usually stands for a cut-off function. Ad hoc definitions aregiven when needed. Unless stated otherwise, the duality product (cid:104)· , ·(cid:105) := (cid:104)· , ·(cid:105) D (cid:48) , D alwaysdenotes the duality between D ( R d − ) = C ∞ ( R d − ) and D (cid:48) . In the sequel, C > is alwaysa constant uniform in ε which may change from line to line.2. Preliminaries
On Sobolev-Kato spaces.
For s ≥ , we define the Sobolev-Kato space H suloc ( R d − ) of functions of non localized H s energy by H suloc ( R d − ) := (cid:40) u ∈ H sloc ( R d ) , sup ξ ∈ Z d − (cid:107) u (cid:107) H s ( ξ +(0 , d − ) < ∞ (cid:41) . We will mainly work with H / uloc . The following lemma is a useful tool to compare the H / uloc norm to the H / norm of a H / ( R d − ) function. Lemma 3.
Let η ∈ C ∞ c ( R d − ) and v ∈ H / uloc ( R d − ) . Assume that Supp η ⊂ B (0 , R ) , for R > . Then, (2.1) (cid:107) ηv (cid:107) H / ≤ CR d − (cid:107) v (cid:107) H / uloc , with C = C ( d, (cid:107) η (cid:107) W , ∞ ) . For a proof, we refer to the proof of Lemma 2.26 in [DP14].2.2.
Homogenization and weak convergence.
We recall the standard weak conver-gence result in periodic homogenization for a fixed domain Ω . As usual, the constanthomogenized matrix A = A αβ ∈ M N ( R ) is given by(2.2) A αβ := ˆ T d A αβ ( y ) dy + ˆ T d A αγ ( y ) ∂ y γ χ β ( y ) dy, where the family χ = χ γ ( y ) ∈ M N ( R ) , y ∈ T d , solves the cell problems(2.3) − ∇ y · A ( y ) ∇ y χ γ = ∂ y α A αγ , y ∈ T d and ˆ T d χ γ ( y ) dy = 0 . heorem 4 (weak convergence) . Let Ω be a bounded Lipschitz domain in R d and let u k ∈ H (Ω) be a sequence of weak solutions to −∇ · A k ( x/ε k ) ∇ u k = f k ∈ ( H (Ω)) (cid:48) , where ε k → and the matrices A k = A k ( y ) ∈ L ∞ satisfy (1.6) and (1.7) . Assume thatthere exist f ∈ ( H (Ω)) (cid:48) and u k ∈ W , (Ω) , such that f k −→ f strongly in ( H (Ω)) (cid:48) , u k → u strongly in L (Ω) and ∇ u k (cid:42) ∇ u weakly in L (Ω) . Also assume that theconstant matrix A k defined by (2.2) with A replaced by A k converges to a constant matrix A . Then A k ( x/ε k ) ∇ u k (cid:42) A ∇ u weakly in L (Ω) and ∇ · A ∇ u = f ∈ ( H (Ω)) (cid:48) . For a proof, which relies on the classical oscillating test function argument, we refer forinstance to [KLS13, Lemma 2.1]. This is an interior convergence result, since no boundarycondition is prescribed on u k .2.3. Uniform estimates in homogenization and applications.
We recall here theboundary Lipschitz estimate proved by Avellaneda and Lin in [AL87a].
Theorem 5 (Lipschitz estimate, [AL87a, Lemma 20]) . For all κ > , < µ < , thereexists 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 (0 , r )) , for all F ∈ C ,µ ( D ψ (0 , r )) , for all u ε ∈ L ∞ ( D ψ (0 , r )) weak solution to (cid:26) −∇ · A ( x/ε ) ∇ u ε = f + ∇ · F x ∈ D ψ (0 , r ) ,u ε = 0 , x ∈ ∆ ψ (0 , r ) , the following estimate holds (2.4) (cid:107)∇ u ε (cid:107) L ∞ ( D ψ (0 ,r/ ≤ C (cid:110) r − (cid:107) u ε (cid:107) L ∞ ( D ψ (0 ,r )) + r − d/ ( d + κ ) (cid:107) f (cid:107) L d + κ ( D ψ (0 ,r )) + r µ (cid:107) F (cid:107) C ,µ ( D ψ (0 ,r )) (cid:111) . Notice that C = C ( d, N, λ, κ, µ, (cid:107) ψ (cid:107) W , ∞ , [ ∇ ψ ] C ,ν , [ A ] C ,ν ) . As stated in our earlier work [KP15], this estimate does not cover the case of highlyoscillating boundaries, since the constant in (2.4) involves the C ,ν semi-norm of ∇ ψ .In this work, we rely on Theorem 5 to get large-scale pointwise estimates on the Poissonkernel P = P ( y, ˜ y ) 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:(1) for all y ∈ R d + , for all ˜ y ∈ R d − × { } , we have | P ( y, ˜ y ) | ≤ Cy d | y − ˜ y | d , (2.5) |∇ y P ( y, ˜ y ) | ≤ C | y − ˜ y | d , (2.6) (2) for all y, ˜ y ∈ R d − × { } , y (cid:54) = ˜ y , (2.7) |∇ y P ( y, ˜ y ) | ≤ C | y − ˜ y | d . Notice that C = C ( d, N, λ, [ A ] C ,ν ) . The proof of those estimates starting from the uniform Lipschitz estimate of Theorem5 is standard (see for instance [AL87a]). . Boundary layer corrector in a flat half-space
This section is devoted to the well-posedness of the boundary layer problem(3.1) (cid:26) −∇ · A ( y ) ∇ v = 0 , y d > ,v = v ∈ H / uloc ( R d − ) , y d = 0 , in the flat half-space R d + . Theorem 7.
Assume v ∈ H / uloc ( R d − ) . Then, there exists a unique weak solution v of (5.1) such that (3.2) sup ξ ∈ Z d − ˆ ξ +(0 , d − ˆ ∞ |∇ v | dy d dy (cid:48) ≤ C (cid:107) v (cid:107) H / uloc < ∞ , with C = C ( d, N, λ, [ A ] C ,ν ) . The proof is in three steps: (i) we define a function v and prove it is a weak solution to(3.1), (ii) we prove that the solution we have defined satisfies the estimate (3.2), (iii) weprove uniqueness of solutions verifying (3.2).3.1. Existence of a weak solution.
Let η ∈ C ∞ c ( R ) a cut-off function such that(3.3) η ≡ on ( − , , ≤ η ≤ , (cid:107) η (cid:48) (cid:107) L ∞ ≤ . Let y ∗ ∈ R d + be fixed. Notice that η ( |·− y (cid:48)∗ | ) ∈ C ∞ c ( R d − ) , η ( |·− y (cid:48)∗ | ) ≡ on B ( y (cid:48)∗ , , ≤ η ( |·− y (cid:48)∗ | ) ≤ and (cid:107)∇ ( η ( |·− y (cid:48)∗ | )) (cid:107) L ∞ ≤ . We define(3.4) v ( y ∗ ) := v (cid:93) ( y ∗ ) + v (cid:91) ( y ∗ ) , where for y ∈ R d + , v (cid:93) ( y ) := ˆ R d − ×{ } P ( y, ˜ y )(1 − η ( | ˜ y (cid:48) − y (cid:48)∗ | )) v (˜ y (cid:48) ) d ˜ y, and v (cid:91) = v (cid:91) ( y ) ∈ H ( R d + ) is the unique weak solution to (cid:26) −∇ · A ( y ) ∇ v (cid:91) = 0 , y d > ,v (cid:91) = η ( | y (cid:48) − y (cid:48)∗ | ) v ( y (cid:48) ) ∈ H / ( R d − ) , y d = 0 , satisfying(3.5) ˆ R d + |∇ v (cid:91) | dy (cid:48) dy d ≤ C (cid:107) ηv (cid:107) H / , with C = C ( d, N, λ ) . First of all, one has to prove that the definition of v does not dependon the choice of the cut-off η . Let η , η ∈ C ∞ c ( R ) be two cut-off functions satisfying (3.3).We denote by v ( y ∗ ) and v ( y ∗ ) the associated vectors defined by v ( y ∗ ) := ˆ R d − ×{ } P ( y ∗ , ˜ y )(1 − η ( | ˜ y (cid:48) − y (cid:48)∗ | ) v (˜ y (cid:48) ) d ˜ y + v (cid:91) ( y ∗ ) ,v ( y ∗ ) := ˆ R d − ×{ } P ( y ∗ , ˜ y )(1 − η ( | ˜ y (cid:48) − y (cid:48)∗ | )) v (˜ y (cid:48) ) d ˜ y + v (cid:91) ( y ∗ ) . Substracting, we get(3.6) v ( y ∗ ) − v ( y ∗ ) = ˆ R d − ×{ } P ( y ∗ , ˜ y )( η ( | ˜ y (cid:48) − y (cid:48)∗ | ) − η ( | ˜ y (cid:48) − y (cid:48)∗ | )) v (˜ y (cid:48) ) d ˜ y + v (cid:91) ( y ∗ ) − v (cid:91) ( y ∗ ) . ow since y (cid:55)−→ ˆ R d − ×{ } P ( y, ˜ y )( η ( | ˜ y (cid:48) − y (cid:48)∗ | ) − η ( | ˜ y (cid:48) − y (cid:48)∗ | )) v (˜ y (cid:48) ) d ˜ y is the unique solution to (cid:26) −∇ · A ( y ) ∇ v (cid:91) = 0 , y d > ,v (cid:91) = ( η ( | y (cid:48) − y (cid:48)∗ | ) − η ( | y (cid:48) − y (cid:48)∗ | )) v ( y (cid:48) ) ∈ H / ( R d − ) , y d = 0 , the difference in (3.6) has to be zero, which proves that our definition of v is independentof the choice of η .It remains to prove that v = v ( y ) defined by (3.4) is actually a weak solution to (3.1).Let ϕ (cid:5) = ϕ (cid:5) ( y (cid:48) ) ∈ C ∞ c ( R d − ) and ϕ d = ϕ d ( y d ) ∈ C ∞ c ((0 , ∞ )) . We choose η ∈ C ∞ c ( R ) satisfying (3.3) and such that η ( | · | ) ≡ on Supp ϕ (cid:5) + B (0 , . We aim at proving ˆ R d + v ( y ) ( −∇ · A ∗ ( y ) ∇ ( ϕ (cid:5) ϕ d )) dy = 0 . This relation is clear for v (cid:91) . For v (cid:93) , by Fubini and then integration by parts ˆ R d + v (cid:93) ( y ) ( −∇ · A ∗ ( y ) ∇ ( ϕ (cid:5) ϕ d )) dy = ˆ Supp ϕ (cid:5) × Supp ϕ d ˆ R d − ×{ } P ( y, ˜ y )(1 − η (˜ y )) v (˜ y (cid:48) ) ( −∇ · A ∗ ( y ) ∇ ϕ (cid:5) ϕ d ) d ˜ ydy = ˆ R d − ×{ } ˆ Supp ϕ (cid:5) × Supp ϕ d P ( y, ˜ y ) ( −∇ · A ∗ ( y ) ∇ ( ϕ (cid:5) ϕ d )) dy (1 − η (˜ y )) v (˜ y (cid:48) ) d ˜ y = ˆ R d − ×{ } (cid:104)−∇ · A ( y ) ∇ P ( y, ˜ y ) , ϕ (cid:5) ϕ d (cid:105) (1 − η (˜ y )) v (˜ y (cid:48) ) d ˜ y = 0 . Gradient estimate.
Let ϕ (cid:5) = ϕ (cid:5) ( y (cid:48) ) ∈ C ∞ c ( R d − ) and ϕ d = ϕ d ( y d ) ∈ C ∞ c ((0 , ∞ )) .We choose R > such that Supp ϕ (cid:5) + B (0 , ⊂ B (0 , R ) . Our goal is to prove (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ R d + ∇ v ( y ) ϕ (cid:5) ϕ d ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CR d − (cid:107) v (cid:107) H / uloc (cid:107) ϕ (cid:5) (cid:107) L (cid:107) ϕ d (cid:107) L , with C = C ( d, N, λ, [ A ] C ,ν ) . This estimate clearly implies the bound (3.2). Let η ∈ C ∞ c ( R ) such that (3.3) η ( | · | ) ≡ on B (0 , R ) and Supp η ( | · | ) ⊂ B (0 , R ) . Combining (3.5) and the result of Lemma 3, we get ˆ R d + |∇ v (cid:91) | dy (cid:48) dy d ≤ CR d − (cid:107) v (cid:107) H / uloc , with C = C ( d, N, λ ) .It remains to estimate ˆ R d + ∇ v (cid:93) ( y ) ϕ (cid:5) ϕ d ( y ) dy = ˆ ˆ R d − ∇ v (cid:93) ( y ) ϕ (cid:5) ϕ d ( y ) dy (cid:48) dy d + ˆ ∞ ˆ R d − ∇ v (cid:93) ( y ) ϕ (cid:5) ϕ d ( y ) dy (cid:48) dy d . To estimate these terms we rely on the the bound (2.6): for all y ∈ R d + , ˜ y ∈ R d − × { } , |∇ y P ( y, ˜ y ) | ≤ C | y − ˜ y | d = C ( y d + | y (cid:48) − ˜ y (cid:48) | ) d/ , with C = C ( d, N, λ, [ A ] C ,ν ) . e begin with two useful estimates. For y ∈ B (0 , R ) , we have on the one hand(3.7) ˆ R d − | y (cid:48) − ˜ y (cid:48) | d (1 − η ( | ˜ y (cid:48) | )) | v (˜ y (cid:48) ) | d ˜ y ≤ ˆ R d − \ B (0 , | y (cid:48) − ˜ y (cid:48) | d | v (˜ y (cid:48) ) | d ˜ y ≤ (cid:88) ξ ∈ Z d − \{ } | ξ | d (cid:107) v (cid:107) L uloc ≤ C (cid:107) v (cid:107) L uloc and on the other hand(3.8) ˆ R d − y d + | y (cid:48) − ˜ y (cid:48) | ) d/ (1 − η ( | ˜ y (cid:48) | )) | v (˜ y (cid:48) ) | d ˜ y ≤ ˆ R d − y d + | y (cid:48) − ˜ y (cid:48) | ) d/ | v (˜ y (cid:48) ) | d ˜ y ≤ ˆ R d − y d + | y (cid:48) − ˜ y (cid:48) | ) d/ d ˜ y (cid:107) v (cid:107) L uloc ≤ Cy d (cid:107) v (cid:107) L uloc . Using (3.7), we get (cid:12)(cid:12)(cid:12)(cid:12) ˆ ˆ R d − ∇ v (cid:93) ( y ) ϕ (cid:5) ϕ d ( y ) dy (cid:48) dy d (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ ˆ R d − ˆ R d − ×{ } ∇ y P ( y, ˜ y )(1 − η ( | ˜ y (cid:48) | )) v (˜ y (cid:48) ) d ˜ yϕ (cid:5) ϕ d ( y ) dy (cid:48) dy d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ ˆ R d − (cid:32) ˆ R d − ×{ } − η ( | ˜ y (cid:48) | ) | y (cid:48) − ˜ y (cid:48) | d d ˜ y (cid:48) (cid:33) / (cid:32) ˆ R d − ×{ } − η ( | ˜ y (cid:48) | ) | y (cid:48) − ˜ y (cid:48) | d | v (˜ y (cid:48) ) | d ˜ y (cid:33) / | ϕ (cid:5) ϕ d ( y ) | dy (cid:48) dy d ≤ C (cid:107) v (cid:107) L uloc ˆ ˆ R d − (cid:18) ˆ ∞ r (cid:19) / | ϕ (cid:5) ϕ d ( y ) | dy (cid:48) dy d ≤ C (cid:107) v (cid:107) L uloc ˆ ˆ R d − | ϕ (cid:5) ϕ d ( y ) | dy (cid:48) dy d ≤ CR d − (cid:107) v (cid:107) L uloc (cid:107) ϕ (cid:5) (cid:107) L (cid:107) ϕ d (cid:107) L . Using (3.8), we infer (cid:12)(cid:12)(cid:12)(cid:12) ˆ ∞ ˆ R d − ∇ v (cid:93) ( y ) ϕ (cid:5) ϕ d ( y ) dy (cid:48) dy d (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ ∞ ˆ R d − (cid:32) ˆ R d − ×{ } y d + | y (cid:48) − ˜ y (cid:48) | ) d/ d ˜ y (cid:48) (cid:33) / (cid:32) ˆ R d − ×{ } y d + | y (cid:48) − ˜ y (cid:48) | ) d/ | v (˜ y (cid:48) ) | d ˜ y (cid:33) / | ϕ (cid:5) ϕ d ( y ) | dy (cid:48) dy d ≤ C (cid:107) v (cid:107) L uloc ˆ ∞ y d | ϕ d ( y d ) | dy d ˆ R d − | ϕ (cid:5) ( y (cid:48) ) | dy (cid:48) ≤ CR d − (cid:107) v (cid:107) L uloc (cid:107) ϕ (cid:5) (cid:107) L (cid:107) ϕ d (cid:107) L . Uniqueness.
By linearity, it is enough to prove uniqueness for v = v ( y ) weak solutionto (cid:26) −∇ · A ( y ) ∇ v = 0 , y d > ,v = 0 , y d = 0 , such that(3.9) sup ξ ∈ Z d − ˆ ξ +(0 , d − ˆ ∞ |∇ v | ≤ C < ∞ . Clearly, by Poincaré’s inequality, for all a > , sup ξ ∈ Z d − ˆ ξ +(0 , d − ˆ a | v | ≤ Ca . or k ∈ N , we will take as a test function η k v ∈ H ( R d + ) for an ad hoc cut-off η k such that η k ≡ on ( − k, k ) d − × (0 , k ) and Supp η k ⊂ ( − k − , k + 1) d − × ( − , k + 1) .We want to construct η k such that (cid:107)∇ η k (cid:107) L ∞ is bounded uniformly in k . Let η ∈ C ∞ c ( B (0 , / such that ´ R d η = 1 . We define η k as folows η k ( y ) := ˆ R d ( − k − / ,k +1 / d − × ( − / ,k +1 / ( y − ˜ y ) η (˜ y ) d ˜ y = ˆ ( − k − / ,k +1 / d − × ( − / ,k +1 / η ( y − ˜ y ) d ˜ y. For y ∈ ( − k, k ) d − × (0 , k ) , Supp( η ( y − · )) ⊂ ( − k − / , k + 1 / d − × ( − / , k + 1 / ,so η k ( y ) = 1 . Moreover, Supp η k ⊂ ( − k − / , k + 1 / d − × ( − / , k + 1 /
2) + Supp η ⊂ ( − k − , k +1) d − × ( − , k +1) . Finally, convolution inequalities imply (cid:107)∇ η k (cid:107) L ∞ ≤ (cid:107)∇ η (cid:107) L .Now, testing against η k v , we get ˆ R d + A ( y ) ∇ v · ∇ ( η k v ) = ˆ R d + A ( y ) η k ∇ v · ∇ v + 2 ˆ R d + A ( y ) η k ∇ v · ( ∇ η k ) v. Therefore, letting E k := ˆ ( − k,k ) d − × (0 ,k ) |∇ v | , we have(3.10) E k ≤ C ∗ ( E k +1 − E k ) , where C ∗ = C ∗ ( d, N, λ, (cid:107)∇ η (cid:107) L ) . Using the hole-filling trick, we get for fixed k and for all n ≥ k , E k ≤ (cid:18) C ∗ C ∗ + 1 (cid:19) n − k E n . Estimate (3.9) implies E n ≤ Cn d − , so that E k ≤ C (cid:18) C ∗ C ∗ + 1 (cid:19) n − k n d − n →∞ −→ , and E k = 0 . This concludes the uniqueness proof.4. Estimates for a Dirichlet to Neumann operator
The Dirichlet to Neumann operator DN is crucial in the proof of the well-posedness ofthe elliptic system in the bumpy half-space (see section 5). The key idea there is to carryout a domain decomposition. The Dirichlet to Neumann map is the tool enabling thisdomain decomposition. Since we are working in spaces of infinite energy to be useful DN has to be defined on H / uloc . Similar studies have been carried out in [ABZ13] (context ofwater-waves), [GVM10] ( d Stokes system), [DP14] ( d Stokes-Coriolis system).We first define the Dirichlet to Neumann operator on H / ( R d − ) : DN : H / ( R d − ) −→ D (cid:48) , such that for any v ∈ H / ( R d − ) , for all ϕ ∈ C ∞ c ( R d − ) , (cid:104) DN( v ) , ϕ (cid:105) D (cid:48) , D := (cid:104) A ( y ) ∇ v · e d , ϕ (cid:105) D (cid:48) , D , where v is the unique weak solution to(4.1) (cid:26) −∇ · A ( y ) ∇ v = 0 , y d > ,v = v ∈ H / ( R d − ) , y d = 0 . Proposition 8.
1) For all ϕ ∈ C ∞ c ( R d + ) , (4.2) (cid:104) DN( v ) , ϕ | y d =0 (cid:105) D (cid:48) , D = (cid:104) A ( y ) ∇ v · e d , ϕ | y d =0 (cid:105) D (cid:48) , D = − ˆ R d + A ( y ) ∇ v · ∇ ϕ. (2) For all ϕ ∈ C ∞ c ( R d − ) , (4.3) (cid:104) DN( v ) , ϕ | y d =0 (cid:105) D (cid:48) , D = ˆ R d − ×{ } ˆ R d − ×{ } A ( y ) ∇ y P ( y, ˜ y ) · e d v (˜ y ) d ˜ yϕ ( y ) dy. For y, ˜ y ∈ R d − × { } , let K ( y, ˜ y ) := A ( y ) ∇ y P ( y, ˜ y ) · e d be the kernel appearing in (4.3). Estimate (2.7) of Proposition 6 implies that | K ( y, ˜ y ) | ≤ C | y − ˜ y | d , for any y, ˜ y ∈ R d − × { } , y (cid:54) = ˜ y with C = C ( d, N, λ, [ A ] C ,ν ) .Both formulas in Proposition 8 follow from integration by parts. Because of (4.2), it isclear that for all v ∈ H / ( R d − ) , for all ϕ ∈ C ∞ c ( R d − ) ,(4.4) |(cid:104) DN( v ) , ϕ (cid:105)| ≤ C (cid:107) v (cid:107) H / (cid:107) ϕ (cid:107) H / , with C = C ( d, N, λ ) , so that DN( v ) extends as a continuous operator on H / ( R d − ) .Another consequence of (4.2) is the following corollary. Corollary 9.
For all v ∈ H / ( R d − ) , (cid:104) DN( v ) , v (cid:105) = − ˆ R d + A ( y ) ∇ v · ∇ v ≤ , where v is the unique solution to (4.1) . Our next goal is to extend the definition of DN to v ∈ H / uloc ( R d − ) . We have to makesense of the duality product (cid:104) DN( v ) , ϕ (cid:105) . As for the definition of the solution to the flathalf-space problem (see section 3), the basic idea is to use a cut-off function η to splitthe definition between one part (cid:104) DN( ηv ) , ϕ (cid:105) where ηv ∈ H / ( R d − ) , and another part (cid:104) DN((1 − η ) v ) , ϕ (cid:105) which does not see the singularity of the kernel K ( y, ˜ y ) .For R > , there exists η ∈ C ∞ c ( R ) such that(4.5) ≤ η ≤ , η ≡ on ( − R, R ) , Supp η ⊂ ( − R − , R + 1) , (cid:107) η (cid:48) (cid:107) L ∞ ≤ . Let v ∈ H / uloc ( R d − ) . Let R > and ϕ ∈ C ∞ c ( R d − ) such that Supp ϕ + B (0 , ⊂ B (0 , R ) .There exists η ∈ C ∞ c ( R ) satisfying the conditions (4.5). We define the action of DN( v ) on ϕ by(4.6) (cid:104) DN( v ) , ϕ (cid:105) D (cid:48) , D := (cid:104) DN( η ( | · | ) v ) , ϕ (cid:105) H − / ,H / + ˆ R d − ×{ } ˆ R d − ×{ } K ( y, ˜ y )(1 − η ( | ˜ y (cid:48) | )) v (˜ y (cid:48) ) ϕ ( y (cid:48) ) d ˜ ydy. The fact that this definition does not depend on the cut-off η ∈ C ∞ c ( R ) follows fromProposition 8.The first term in the right-hand side of (4.6) is estimated using (4.4) and the bound ofLemma 3 between the H / norm of η ( | · | ) v and the H / uloc norm of v . That yields |(cid:104) DN( η ( | · | ) v ) , ϕ (cid:105)| ≤ C (cid:107) η ( | · | ) v (cid:107) H / (cid:107) ϕ (cid:107) H / ≤ CR d − (cid:107) v (cid:107) H / uloc (cid:107) ϕ (cid:107) H / , with C = C ( d, N, λ ) . e deal with the integral part in the right hand side of (4.6) in a way similar to theproof of estimates (3.7) and (3.8). Using the fact that the supports of (1 − η ( | y (cid:48) | )) v ( y (cid:48) ) on the one hand and ϕ on the other hand are disjoint, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ R d − ×{ } ˆ R d − ×{ } K ( y, ˜ y )(1 − η ( | ˜ y (cid:48) | )) v (˜ y (cid:48) ) ϕ ( y (cid:48) ) d ˜ ydy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ R d − ×{ } ˆ R d − ×{ } | y − ˜ y | d (1 − η ( | ˜ y (cid:48) | )) | v (˜ y (cid:48) ) || ϕ ( y (cid:48) ) | d ˜ ydy ≤ C ˆ R d − ×{ } (cid:32) ˆ R d − ×{ } | y − ˜ y | d (1 − η ( | ˜ y (cid:48) | )) d ˜ y (cid:33) / (cid:32) ˆ R d − ×{ } | y − ˜ y | d (1 − η ( | ˜ y (cid:48) | )) | v (˜ y (cid:48) ) | d ˜ y (cid:33) / | ϕ ( y (cid:48) ) | dy ≤ C ˆ R d − ×{ } (cid:18) ˆ ∞ r dr (cid:19) / | ϕ ( y (cid:48) ) | dy (cid:107) v (cid:107) L uloc ≤ CR d − (cid:107) v (cid:107) L uloc (cid:107) ϕ (cid:107) L , with C = C ( d, N, λ, [ A ] C ,ν ) .These results are put in a nutshell in the following proposition. Proposition 10. (1) For v ∈ H / ( R d − ) , for any ϕ ∈ C ∞ c ( R d − ) , we have |(cid:104) DN( v ) , ϕ (cid:105)| ≤ C (cid:107) v (cid:107) H / (cid:107) ϕ (cid:107) H / , with C = C ( d, N, λ ) .(2) For v ∈ H / uloc ( R d − ) , for R > and any ϕ ∈ C ∞ c ( R d − ) such that Supp ϕ + B (0 , ⊂ B (0 , R ) , we have (4.7) |(cid:104) DN( v ) , ϕ (cid:105)| ≤ CR d − (cid:107) v (cid:107) H / uloc (cid:107) ϕ (cid:107) H / , with C = C ( d, N, λ, [ A ] C ,ν ) . Boundary layer corrector in a bumpy half-space
This section is devoted to the well-posedness of the boundary layer problem(5.1) (cid:26) −∇ · A ( y ) ∇ v = 0 , y d > ψ ( y (cid:48) ) ,v = v ∈ H / uloc ( R d − ) , y d = ψ ( y (cid:48) ) , in the bumpy half-space Ω + := { y d > ψ ( y (cid:48) ) } . For technical reasons, the boundary ψ ∈ W , ∞ ( R d − ) is assumed to be negative, i.e. ψ ( y (cid:48) ) < for all y (cid:48) ∈ R d − . We prove Theorem2 of the introduction which asserts the existence of a unique solution v in the class sup ξ ∈ Z d − ˆ ξ +(0 , d − ˆ ∞ ψ ( y (cid:48) ) |∇ v | dy d dy (cid:48) < ∞ . The idea is to split the bumpy half-space into two subdomains: a flat half-space R d + on the one hand and a bumpy channel Ω (cid:91) := { ψ ( y (cid:48) ) < y d < } on the other hand.Both domains are connected by a transparent boundary condition involving the Dirichlet o Neumann operator DN defined in section 4. Therefore, solving (3.1) is equivalent tosolving(5.2) −∇ · A ( y ) ∇ v = 0 , > y d > ψ ( y (cid:48) ) ,v = v ∈ H / uloc ( R d − ) , y d = ψ ( y (cid:48) ) ,A ( y ) ∇ v · e d = DN( v | y d =0 ) , y d = 0 . This fact is stated in the following technical lemma.
Lemma 11. If v is a weak solution of (5.2) in Ω (cid:91) such that sup ξ ∈ Z d − ˆ ξ +(0 , d − ˆ ψ ( y (cid:48) ) |∇ v | dy d dy (cid:48) < ∞ , then ˜ v , defined by ˜ v ( y ) := v ( y ) for ψ ( y (cid:48) ) < y d < and ˜ v | R d + is the unique solution to (3.1) with boundary condition ˜ v | y d =0 + = v | y d =0 − given by Theorem 7, is a weak solution to (5.1) .Moreover, the reverse is also true. Namely, if v is a weak solution to (5.1) in Ω + such that sup ξ ∈ Z d − ˆ ξ +(0 , d − ˆ ∞ ψ ( y (cid:48) ) |∇ v | dy d dy (cid:48) < ∞ , then v | { ψ ( y (cid:48) )
Let C ∗ be given by Proposition 12, and let(5.11) A := ∞ (cid:88) k =1 (cid:18) C ∗ C ∗ + 1 (cid:19) k (2 k − d − < ∞ , B := ∞ (cid:88) k =1 (cid:18) C ∗ C ∗ + 1 (cid:19) k (2 k − d − < ∞ . We now choose an integer m so that(5.12) m ≥ , m is even and − − d Bm > . Notice that m = m ( d, N, λ, [ A ] C ,ν , (cid:107) ψ (cid:107) W , ∞ , (cid:107) v (cid:107) H / uloc ) , but is independent of r and y (cid:48) .The reason for taking m even is technical; it is only used in the translation argument below.Take n = lm = 2 lm (cid:48) , with l ∈ N , l ≥ , and take w n to be the solution of (5.6).There exists T ∗ ∈ C m such that T ∗ ⊂ Σ n and E T ∗ = sup T ∈C m E T . By definition, thereis ξ ∗ ∈ Z d − for which T ∗ = ξ ∗ + ( − m (cid:48) , m (cid:48) ) d − . We want to center T ∗ at zero by simplytranslating the origin. Doing so, w ∗ n ( y ) := w n ( y (cid:48) + ξ ∗ , y d ) is a solution of (5.8) with y (cid:48) := − ξ ∗ , r = n and A ∗ ( y ) := A ( y (cid:48) + ξ ∗ , y d ) , ψ ∗ ( y (cid:48) ) := ψ ( y (cid:48) + ξ ∗ ) , v ∗ ( y (cid:48) ) := v ( y (cid:48) + ξ ∗ ) ,F ∗ ( y ) := F ( y (cid:48) + y ∗ , y d ) and f ∗ ( y (cid:48) ) := f ( y (cid:48) + ξ ∗ ) . Notice that [ A ∗ ] C ,µ = [ A ] C ,ν , (cid:107) ψ ∗ (cid:107) W , ∞ = (cid:107) ψ (cid:107) W , ∞ and (cid:107) v ∗ (cid:107) H / uloc = (cid:107) v (cid:107) H / uloc , so that w ∗ n satisfies the Saint-Venant estimate (5.10) with the same constant C ∗ . Further-more, E m (cid:48) = E T ∗ . Lemma 13.
We have the following a priori bound E m (cid:48) ≤ − d Am d − , where A is defined by (5.11) . The Lemma is obtained by downward induction, using a hole-filling type argument.Since w ∗ n is supported in Ω (cid:91), n , we start from k sufficiently large in (5.10). For k =2 n + m (cid:48) = (4 l + 1) m (cid:48) , estimate (5.10) implies E (4 l +1) m (cid:48) ≤ C ∗ C ∗ + 1 ((4 l + 1) m (cid:48) ) d − , because E T = 0 for any T ∈ C (4 l +1) m (cid:48) ,m . Then, E (2(2 l − m (cid:48) = E (4 l +1) m (cid:48) − m ≤ C ∗ C ∗ + 1 (2(2 l − d − ( m (cid:48) ) d − + (cid:18) C ∗ C ∗ + 1 (cid:19) (4 l +1) d − ( m (cid:48) ) d − . et p ∈ { , . . . l − } . We then have E (2 p +1) m (cid:48) ≤ C ∗ C ∗ + 1 (2 p + 1) d − ( m (cid:48) ) d − + . . . (cid:18) C ∗ C ∗ + 1 (cid:19) l − p (4 l + 1) d − ( m (cid:48) ) d − + 2 − d m (cid:34) C ∗ C ∗ + 1 (2 p + 1) d − + . . . (cid:18) C ∗ C ∗ + 1 (cid:19) l − p (4 l + 1) d − (cid:35) E m (cid:48) . Eventually, for p = 0 E m (cid:48) ≤ C ∗ C ∗ + 1 ( m (cid:48) ) d − + (cid:18) C ∗ C ∗ + 1 (cid:19) (3 m (cid:48) ) d − + . . . (cid:18) C ∗ C ∗ + 1 (cid:19) l − p (4 l + 1) d − ( m (cid:48) ) d − + 2 − d m (cid:34) C ∗ C ∗ + 1 + . . . (cid:18) C ∗ C ∗ + 1 (cid:19) l − p (4 l + 1) d − (cid:35) E m (cid:48) ≤ − d Am d − + 2 − d m BE m (cid:48) . Therefore, E m (cid:48) < (cid:18) − Bm (cid:19) E m (cid:48) ≤ − d Am d − , which proves Lemma 13.Finally, sup ξ ∈ Z d − ˆ ξ +(0 , d − ˆ ψ ( y (cid:48) ) |∇ w n | ≤ E m ≤ − d Am d − , which proves the a priori bound in the norm (5.7) uniformly in n .5.2. Proof of Proposition 12.
Construction of a cut-off.
Let η ∈ C ∞ ( B (0 , / such that η ≥ and ´ R d η = 1 . For all k ∈ N , let η k = η k ( y (cid:48) ) be defined by η k ( y (cid:48) ) = ˆ R d − [ − k − / ,k +1 / d − ( y (cid:48) − ˜ y (cid:48) ) η (˜ y (cid:48) ) d ˜ y (cid:48) = ˆ [ − k − / ,k +1 / d − η ( y (cid:48) − ˜ y (cid:48) ) d ˜ y (cid:48) . For all k ∈ N , we have the following properties: η k ≡ on [ − k, k ] d − , Supp η k ⊂ [ − k − , k + 1] d − , η k ∈ C ∞ c ( R d − ) and most importantly, we have the control (cid:107)∇ η k (cid:107) L ∞ ≤ (cid:107)∇ η (cid:107) L uniformy in k . Energy estimate.
Testing the system (5.8) against η k w r we get(5.13) ˆ Ω (cid:91) η k A ( y ) ∇ w r · ∇ w r = − ˆ Ω (cid:91) η k A ( y ) ∇ w r · ∇ η k w r + (cid:104)∇ · F, η k w r (cid:105) + (cid:104) f, η k w r (cid:105) + (cid:104) DN( w r | y d =0 ) , η k w r (cid:105) . By ellipticity, we have λ ˆ Ω (cid:91) η k |∇ w r | ≤ ˆ Ω (cid:91) η k A ( y ) ∇ w r · ∇ w r . he following estimate (or variations of it) is of constant use: by the trace theorem andPoincaré inequality(5.14) (cid:32) ˆ Σ k +1 η k | w r ( y (cid:48) , | dy (cid:48) (cid:33) / ≤ (cid:107) η k w r (cid:107) H / ≤ C (cid:107) η k w r (cid:107) H (Ω (cid:91) ) ≤ C (cid:107)∇ ( η k w r ) (cid:107) L (Ω (cid:91) ) ≤ C ( E k +1 − E k ) / + C (cid:48) (cid:18) ˆ Ω (cid:91) η k |∇ w | (cid:19) / , with C = C ( d, (cid:107) ψ (cid:107) W , ∞ , (cid:107) η (cid:107) L ) and C (cid:48) = C (cid:48) ( d ) . We now estimate every term on the righthand side of (5.13). We have, (cid:12)(cid:12)(cid:12)(cid:12) ˆ Ω (cid:91) η k A ( y ) ∇ w r · ∇ η k w r (cid:12)(cid:12)(cid:12)(cid:12) ≤ λ (cid:18) ˆ Ω (cid:91) η k |∇ w r | (cid:19) / (cid:18) ˆ Ω (cid:91) |∇ η k | | w r | (cid:19) / ≤ C (cid:18) ˆ Ω (cid:91) η k |∇ w r | (cid:19) / ( E k +1 − E k ) / , with C = C ( λ, (cid:107) η (cid:107) L ) . We also have, |(cid:104)∇ · F, η k w r (cid:105)| = |(cid:104) F, ∇ ( η k w r ) (cid:105)|≤ Ck d − ( E k +1 − E k ) / + C (cid:48) k d − (cid:18) ˆ Ω (cid:91) η k |∇ w r | (cid:19) / , where C = C ( (cid:107) v (cid:107) H / uloc , (cid:107)∇ η (cid:107) L ) and C (cid:48) = C (cid:48) ( (cid:107) v (cid:107) H / uloc ) , and by the trace theorem andPoincaré inequality |(cid:104) f, η k w r (cid:105)| ≤ Ck d − (cid:107) η k w r (cid:107) H / ≤ Ck d − (cid:107)∇ ( η k w r ) (cid:107) L ≤ Ck d − ( E k +1 − E k ) / + C (cid:48) k d − (cid:18) ˆ Ω (cid:91) η k |∇ w r | (cid:19) / , with C = C ( d, (cid:107) ψ (cid:107) W , ∞ , (cid:107) v (cid:107) H / uloc , (cid:107)∇ η (cid:107) L ) and C (cid:48) = C (cid:48) ( (cid:107) v (cid:107) H / uloc ) . We have now totackle the non local term involving the Dirichlet to Neumann operator. We split this terminto (cid:104) DN( w r | y d =0 ) , η k w r (cid:105) = (cid:104) DN((1 − η k + m − ) w r | y d =0 ) , η k w r (cid:105) + (cid:104) DN(( η k + m − − η k ) w r | y d =0 ) , η k w r (cid:105) + (cid:104) DN( η k w r ) , η k w r (cid:105) . By Corollary 9, (cid:104)
DN( η k w r ) , η k w r (cid:105) ≤ . Relying on Proposition 10 and on estimate (4.7), we get |(cid:104)
DN(( η k + m − − η k ) w r | y d =0 ) , η k w r (cid:105)| ≤ Ck d − (cid:107) ( η k + m − − η k ) w r | y d =0 (cid:107) H / uloc (cid:107) η k w r (cid:107) H / ≤ C ( E k + m − E k ) / (cid:18) ˆ Ω (cid:91) η k |∇ w r | (cid:19) / + C ( E k + m − E k ) / ( E k +1 − E k ) / , with C = C ( d, N, λ, [ A ] C ,ν , (cid:107) ψ (cid:107) W , ∞ , (cid:107) η (cid:107) L ) . Notice that the bound (4.4) for the Dirichletto 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 proofin section 5.3, we have to use the result of Proposition 10. ontrol of the non local term. Lemma 14.
For all m ≥ , all k ≥ m (cid:48) = m/ , we have (5.15) ˆ Σ k +1 (cid:18) ˆ R d − | y (cid:48) − ˜ y (cid:48) | d (1 − η k + m − ) | w r (˜ y (cid:48) , | d ˜ y (cid:48) (cid:19) dy (cid:48) ≤ C k d − m d − sup T ∈C k,m E T , where C = C ( d ) . Let y (cid:48) ∈ Σ k +1 be fixed. We have ˆ R d − | y (cid:48) − ˜ y (cid:48) | d (1 − η k + m − ) | w r (˜ y (cid:48) , | d ˜ y (cid:48) = ∞ (cid:88) j =1 ˆ R d − | y (cid:48) − ˜ y (cid:48) | d ( η k +( j +1)( m − − η k + m − ) | w r (˜ y (cid:48) , | d ˜ y (cid:48) = ∞ (cid:88) j =1 ˆ Σ k +( j +1)( m − \ Σ k + j ( m − | y (cid:48) − ˜ y (cid:48) | d | w r (˜ y (cid:48) , | d ˜ y (cid:48) = ∞ (cid:88) j =1 (cid:88) T ∈C k,j,m ˆ T | y (cid:48) − ˜ y (cid:48) | d | w r (˜ y (cid:48) , | d ˜ y (cid:48) , where C k,j,m is a family of disjoint cubes T = ξ +( − m (cid:48) , m (cid:48) ) d − such that T ⊂ Σ k +( j +1)( m − \ Σ k + j ( m − and (cid:71) T ∈C k,j,m T = Σ k +( j +1)( m − \ Σ k + j ( m − . For all T ∈ C k,j,m , by Cauchy-Schwarz, trace theorem and Poincaré inequality ˆ T | y (cid:48) − ˜ y (cid:48) | d | w r (˜ y (cid:48) , | d ˜ y (cid:48) ≤ (cid:18) ˆ T | y (cid:48) − ˜ y (cid:48) | d d ˜ y (cid:48) (cid:19) / (cid:18) ˆ T | w (˜ y (cid:48) , | d ˜ y (cid:48) (cid:19) / ≤ C (cid:18) ˆ T | y (cid:48) − ˜ y (cid:48) | d d ˜ y (cid:48) (cid:19) / (cid:18) ˆ Ω T |∇ w | d ˜ y (cid:48) (cid:19) / ≤ C (cid:18) ˆ T | y (cid:48) − ˜ y (cid:48) | d d ˜ y (cid:48) (cid:19) / (cid:32) sup T ∈C k,j,m E T (cid:33) / , where Ω T and E T are defined in (5.9). Notice that the constant C in the last inequalityonly depends on d and on (cid:107) ψ (cid:107) W , ∞ . Moreover, for any T ∈ C k,j,m , (cid:18) ˆ T | y (cid:48) − ˜ y (cid:48) | d d ˜ y (cid:48) (cid:19) / ≤ m d − ( k + j ( m − − | y (cid:48) | ) d , and the number of elements of C k,j,m is bounded by C k,j,m = (cid:12)(cid:12) Σ k +( j +1)( m − \ Σ k + j ( m − (cid:12)(cid:12) m d − (cid:46) ( k + j ( m − d − m d − . herefore, ˆ R d − | y (cid:48) − ˜ y (cid:48) | d (1 − η k + m − ) | w r (˜ y (cid:48) , | d ˜ y (cid:48) ≤ C (cid:32) sup T ∈C k,j,m E T (cid:33) / ∞ (cid:88) j =1 (cid:88) T ∈C k,j,m m d − ( k + j ( m − − | y (cid:48) | ) d ≤ C (cid:32) sup T ∈C k,j,m E T (cid:33) / ∞ (cid:88) j =1 m d − ( k + j ( m − d − ( k + j ( m − − | y (cid:48) | ) d ≤ C (cid:32) sup T ∈C k,j,m E T (cid:33) / ( k + m − d − m d − ( k + m − − | y (cid:48) | ) d − , with C = C ( d ) . Eventually, we get for m ≥ ˆ Σ k +1 (cid:18) ˆ R d − | y (cid:48) − ˜ y (cid:48) | d (1 − η k + m − ) | w r (˜ y (cid:48) , | d ˜ y (cid:48) (cid:19) dy (cid:48) ≤ C (cid:32) sup T ∈C k,j,m E T (cid:33) ( k + m − d − m d − ˆ Σ k +1 k + m − − | y (cid:48) | ) d − dy (cid:48) ≤ C (cid:32) sup T ∈C k,j,m E T (cid:33) ( k + m − d − m d − ( k + 1) d − ( m − d − ≤ C k d − m d − sup T ∈C k,j,m E T , with C = C ( d ) , the last inequality being only true on condition that k ≥ m/ m (cid:48) . Thisproves Lemma 14.In particular, by the definition of DN in (4.6), by the fact that (1 − η k + m − ) w r (˜ y (cid:48) , and η k w r ( y (cid:48) , have disjoint support, by estimate (5.15) and by the bound (5.14) we get |(cid:104) DN((1 − η k + m − ) w r | y d =0 ) , η k w r (cid:105)|≤ C (cid:12)(cid:12)(cid:12)(cid:12) ˆ R d − ˆ R d − | y (cid:48) − ˜ y (cid:48) | d (1 − η k + m − (˜ y (cid:48) )) | w r (˜ y (cid:48) , | η k ( y (cid:48) ) | w r ( y (cid:48) , | d ˜ y (cid:48) dy (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:32) ˆ Σ k +1 η k | w r ( y (cid:48) , | dy (cid:48) (cid:33) / ˆ Σ k +1 (cid:32) ˆ R d − − η k + m − (˜ y (cid:48) ) | y (cid:48) − ˜ y (cid:48) | d | w r ( y (cid:48) , | d ˜ y (cid:48) (cid:33) dy (cid:48) / , ≤ C k d − m d − (cid:32) ˆ Σ k +1 η k | w r ( y (cid:48) , | dy (cid:48) (cid:33) / (cid:32) sup T ∈C k,j,m E T (cid:33) ≤ C k d − m d − (cid:34) ( E k +1 − E k ) / + (cid:18) ˆ Ω (cid:91) η k |∇ w | (cid:19) / (cid:35) (cid:32) sup T ∈C k,j,m E T (cid:33) , with C = C ( d, N, λ, [ A ] C ,ν , (cid:107) ψ (cid:107) W , ∞ ) . End of the proof of the Saint-Venant estimate.
Combining all our bounds and using E k +1 − E k ≤ E k + m − E k , η k ≤ C ( (cid:107) η (cid:107) L ∞ ) η k henever possible, we get from (5.13) the following estimate λ ˆ Ω (cid:91) η k |∇ w r | ≤ C (cid:18) ˆ Ω (cid:91) η k |∇ w r | (cid:19) / ( E k + m − E k ) / + Ck d − ( E k + m − E k ) / + C ( E k + m − E k )+ Ck d − (cid:18) ˆ Ω (cid:91) η k |∇ w r | (cid:19) / + C ( E k + m − E k ) / (cid:18) ˆ Ω (cid:91) η k |∇ w r | (cid:19) / + C k d − m d − (cid:34) ( E k +1 − E k ) / + (cid:18) ˆ Ω (cid:91) η k |∇ w | (cid:19) / (cid:35) (cid:32) sup T ∈C k,j,m E T (cid:33) / , with C = C ( d, N, λ, [ A ] C ,ν , (cid:107) v (cid:107) H / uloc , (cid:107) ψ (cid:107) W , ∞ ) . Swallowing every term of the type ˆ Ω (cid:91) η k |∇ w | in the left hand side, we end up with the Saint-Venant estimate (5.10). This concludes theproof of Proposition 12.5.3. End of the proof of Theorem 2.
Extracting subsequences using a classical diagonalargument and passing to the limit in the weak formulation of (5.6) relying on the continuityof the Dirichlet to Neumann map asserted in estimate (4.4) yields the existence of a weaksolution w to the system (5.3). In addition, the weak solution satisfies the bound(5.16) sup ξ ∈ Z d − ˆ ξ +(0 , d − ˆ ∞ ψ ( y (cid:48) ) |∇ w | dy d dy (cid:48) ≤ − d Am d − < ∞ . Let us turn to the uniqueness of the solution to (5.3) satisfying the bound (5.16). Bylinearity of the problem, it is enough to prove the uniqueness for zero source terms. Assume w ∈ H loc (Ω (cid:91) ) is a weak solution to (5.3) with f = F = 0 satisfying(5.17) sup ξ ∈ Z d − ˆ ξ +(0 , d − ˆ ψ ( y (cid:48) ) |∇ w | ≤ C < ∞ . Repeating the estimates leading to Proposition 12 (see section 5.2), we infer that for thesame constant C ∗ appearing in the Saint-Venant estimate (5.10) and for m defined by(5.12), for k ∈ N , k ≥ m/ m (cid:48) ,(5.18) E k ≤ C ∗ (cid:32) E k + m − E k + k d − m d − sup T ∈C k,m E T (cid:33) . The fact that w , unlike w n , does not vanish outside Ω (cid:91),n does not lead to any difference inthe proof of this estimate.Since sup T ∈C m E T < ∞ , for any ε , there exists T ∗ ε ∈ C m such that(5.19) sup T ∈C m E T − ε ≤ E T ∗ ε ≤ sup T ∈C m E T . Again, T ∗ ε := ξ ∗ ε +( − m (cid:48) , m (cid:48) ) d − for ξ ∗ ε ∈ Z d − , and we can translate T ∗ ε so that it is centeredat the origin as has been done in section 5.1. Estimate (5.18) still holds. For any n ∈ N , E n ≤ C n d − where C is defined by (5.17). The idea is now to carry out a downward teration. For any n = (2 l + 1) m (cid:48) with l ∈ N , l ≥ fixed, for p ∈ { , . . . l − } one canshow that E (2 p +1) m (cid:48) ≤ (cid:34) C ∗ C ∗ + 1 + (cid:18) C ∗ C ∗ + 1 (cid:19) + . . . (cid:18) C ∗ C ∗ + 1 (cid:19) l − p (cid:35) E n + 2 − d m (cid:34) C ∗ C ∗ + 1 (2 p + 1) d − + . . . (cid:18) C ∗ C ∗ + 1 (cid:19) l − p (2 l + 1) d − (cid:35) sup T ∈C m E T ≤ C C ∗ + 12 C ∗ + 1 (cid:18) C ∗ C ∗ + 1 (cid:19) l − p +1 n d − + 2 − d m (cid:34) C ∗ C ∗ + 1 (2 p + 1) d − + . . . (cid:18) C ∗ C ∗ + 1 (cid:19) l − p (2 l + 1) d − (cid:35) sup T ∈C m E T . Thus, E m (cid:48) ≤ C C ∗ + 12 C ∗ + 1 (cid:18) C ∗ C ∗ + 1 (cid:19) l (2 l + 1) d − ( m (cid:48) ) d − + 2 − d m B sup T ∈C m E T ≤ C C ∗ + 12 C ∗ + 1 (cid:18) C ∗ C ∗ + 1 (cid:19) l +1 (2 l + 1) d − ( m (cid:48) ) d − + 2 − d m B ( E m (cid:48) + ε ) . From this we infer using (5.12) that E m (cid:48) ≤ C C ∗ + 12 C ∗ + 1 (cid:18) C ∗ C ∗ + 1 (cid:19) l +1 (2 l + 1) d − ( m (cid:48) ) d − + 2 − d m Bε l →∞ −→ − d m Bε.
Therefore, from equation (5.19) sup T ∈C m E T ≤ (cid:18) − d m B (cid:19) ε, which eventually leads to sup T ∈C m E T = 0 , or in other words w = 0 .Combining this existence and uniqueness result for the system (5.3) in the bumpy chan-nel Ω (cid:91) with Lemma 11 and Theorem 7 about the well-posedness in the flat half-spacefinishes the proof of Theorem 2.6. Improved regularity over Lipschitz boundaries
The goal in this section is to prove Theorem 1 of the introduction. Let us recall theresult we prove in the following proposition.
Proposition 15.
For all ν > , γ > , there exists C > and ε > such that for all ψ ∈ W , ∞ ( R d − ) , − < ψ < and (cid:107)∇ ψ (cid:107) L ∞ ≤ γ , for all A ∈ A ν , for all < ε < (1 / ε ,for all weak solution u ε to (1.1) , for all r ∈ [ ε/ε , / (6.1) − ˆ D εψ (0 ,r ) |∇ u ε | ≤ C − ˆ D εψ (0 , |∇ u ε | , or equivalently, − ˆ D εψ (0 ,r ) | u ε | ≤ Cr − ˆ D εψ (0 , | u ε | , with C = C ( d, N, λ, ν, γ, [ A ] C ,ν ) . We rely on a compactness argument inspired by the pioneering work of Avellaneda andLin [AL87a, AL89b], and our recent work [KP15]. The proof is in two steps. Firstly, wecarry out the compactness argument. Secondly, we iterate the estimate obtained in thefirst step, to get an estimate down to the microscopic scale O ( ε ) . key step in the proof of boundary Lipschitz estimates is to estimate boundary layercorrectors, which is done by combining the classical Lipschitz estimate with a uniformHölder estimate, as in [AL87a, Lemma 17] or [KP15, Lemma 10]. We are able to relax theregularity assumption on ψ . This progress is enabled by our new estimate (1.4) for theboundary layer corrector, which holds for Lipschitz boundaries ψ .We begin with an estimate which is of constant use in this part of our work. Take ψ ∈ W , ∞ ( R d − ) and A ∈ A ν . By Cacciopoli’s inequality, there exists C > such that forall ε > , for all weak solution u ε to(6.2) (cid:26) −∇ · A ( x/ε ) ∇ u ε = 0 , x ∈ D εψ (0 , ,u ε = 0 , x ∈ ∆ εψ (0 , , for all < θ < ,(6.3) (cid:12)(cid:12)(cid:12) ( ∂ x d u ε ) D ψ (0 ,θ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ˆ D ψ (0 ,θ ) ∂ x d u ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:32) − ˆ D ψ (0 ,θ ) | ∂ x d u ε | (cid:33) / ≤ C θ d/ (1 − θ ) (cid:32) − ˆ D ψ (0 , | u ε | (cid:33) / . Notice that C in (6.3) only depends on λ .Proposition 15 is a consequence of the two following lemmas. The first one contains thecompactness argument. The second one is the iteration lemma. In order to alleviate thestatement of the following lemma, the definition of the boundary layer v is given straightafter the lemma. Lemma 16.
For all ν > , γ > , there exists θ > , < µ < , ε > , such that for all ψ ∈ W , ∞ ( R d − ) , − < ψ < and (cid:107)∇ ψ (cid:107) L ∞ ≤ γ , for all A ∈ A ν , for all < ε < ε , forall weak solution u ε to (6.2) we have − ˆ D εψ (0 , | u ε | ≤ implies − ˆ D εψ (0 ,θ ) (cid:12)(cid:12)(cid:12) u ε ( x ) − ( ∂ x d u ε ) D εψ (0 ,θ ) (cid:104) x d + εχ d ( x/ε ) + εv ( x/ε ) (cid:105)(cid:12)(cid:12)(cid:12) dy ≤ θ µ . The boundary layer v = v ( y ) is the unique solution given by Theorem 2 to the system(6.4) (cid:26) −∇ · A ( y ) ∇ v = 0 , y d > ψ ( y (cid:48) ) ,v = y d + χ d ( y ) , y d = ψ ( y (cid:48) ) . The estimate of Theorem 2 implies sup ξ ∈ Z d − ˆ ξ +(0 , d − ˆ ∞ ψ ( y (cid:48) ) |∇ v | ≤ C (cid:110) (cid:107) ψ (cid:107) H / uloc ( R d − ) + (cid:107) χ ( · , ψ ( · )) (cid:107) H / uloc ( R d − ) (cid:111) , with C = C ( d, N, λ, [ A ] C ,ν , (cid:107) ψ (cid:107) W , ∞ ) . Now, by Sobolev injection W , ∞ ( R d − ) (cid:44) → H / ( R d − ) (cid:107) ψ (cid:107) H / uloc ( R d − ) ≤ C (cid:107) ψ (cid:107) W , ∞ ( R d − ) , with C = C ( d ) and by classical interior Lipschitz regularity (cid:107) χ ( · , ψ ( · )) (cid:107) H / uloc ≤ C (cid:107) χ ( · , ψ ( · )) (cid:107) W , ∞ ( R d − ) ≤ C (cid:107) χ (cid:107) W , ∞ ( R d ) ≤ C, with in the last inequality C = C ( d, N, λ, [ A ] C ,ν ) . Eventually,(6.5) sup ξ ∈ Z d − ˆ ξ +(0 , d − ˆ ∞ ψ ( y (cid:48) ) |∇ v | ≤ C, with C = C ( d, N, λ, [ A ] C ,ν , (cid:107) ψ (cid:107) W , ∞ ) uniform in ε . emma 17. Let θ , ε and γ be given as in Lemma 16. For all ψ ∈ W , ∞ ( R d − ) , − <ψ < and (cid:107)∇ ψ (cid:107) L ∞ ≤ γ , for all A ∈ A ν , for all k ∈ N , k > , for all < ε < θ k − ε , forall weak solution u ε to (6.2) there exists a εk ∈ R N satifying | a εk | ≤ C θ µ + . . . θ µ ( k − θ d/ (1 − θ ) , such that − ˆ D εψ (0 , | u ε | ≤ implies (6.6) − ˆ D εψ (0 ,θ k ) (cid:12)(cid:12)(cid:12) u ε ( x ) − a εk (cid:104) x d + εχ d ( x/ε ) + εv ( x/ε ) (cid:105)(cid:12)(cid:12)(cid:12) dy ≤ θ (2+2 µ ) k , where v = v ( y ) is the solution, given by Theorem 2, to the boundary layer system (6.4) . The condition ε < θ k − ε can be seen as giving a lower bound on the scales θ k for whichone can prove the regularity estimate: θ k − > ε/ε . In that perspective, estimate (6.6) isan improved C ,µ estimate down to the microscale ε/ε .For fixed < ε/ε < / and r ∈ [ ε/ε , / , there exists k ∈ N such that θ k +1 < r ≤ θ k .We aim at estimating − ˆ D εψ (0 ,r ) | u ε ( x ) | using the bound (6.6). We have(6.7) (cid:32) − ˆ D εψ (0 ,r ) | u ε ( x ) | (cid:33) / ≤ (cid:32) − ˆ D εψ (0 ,θ k ) | u ε ( x ) | (cid:33) / ≤ (cid:32) − ˆ D εψ (0 ,θ k ) (cid:12)(cid:12)(cid:12) u ε ( x ) − a εk (cid:104) x d − ψ ( x (cid:48) ) + εχ d ( x/ε ) + εv ( x/ε ) (cid:105)(cid:12)(cid:12)(cid:12) dy (cid:33) / + | a εk | (cid:32) − ˆ D εψ (0 ,θ k ) | x d | (cid:33) / + (cid:32) − ˆ D εψ (0 ,θ k ) | εχ d ( x/ε ) | (cid:33) / + (cid:32) − ˆ D εψ (0 ,θ k ) | εv ( x/ε ) | (cid:33) / . Let us focus on the term involving the boundary layer. Let η = η ( y d ) ∈ C ∞ c ( R ) be a cut-offsuch that η ≡ on ( − , and Supp η ⊂ ( − , . The triangle inequality yields (cid:32) − ˆ D εψ (0 ,θ k ) | εv ( x/ε ) | (cid:33) / ≤ (cid:32) − ˆ D εψ (0 ,θ k ) | εv ( x/ε ) − ( x d + εχ d ( x/ε )) η ( x d /ε ) | (cid:33) / + (cid:32) − ˆ D εψ (0 ,θ k ) | ( x d + εχ d ( x/ε )) η ( x d /ε ) | (cid:33) / . oincaré’s inequality implies (cid:32) − ˆ D εψ (0 ,θ k ) | εv ( x/ε ) − ( x d + εχ d ( x/ε )) η ( x d /ε ) | (cid:33) / ≤ θ k (cid:32) − ˆ D εψ (0 ,θ k ) (cid:12)(cid:12)(cid:12) ∇ (cid:16) εv ( x/ε ) − ( x d + εχ d ( x/ε )) η ( x d /ε ) (cid:17)(cid:12)(cid:12)(cid:12) (cid:33) / ≤ θ k (cid:32) − ˆ D εψ (0 ,θ k ) |∇ v ( x/ε ) | (cid:33) / + (1 + (cid:107)∇ χ (cid:107) L ∞ ) θ k (cid:32) − ˆ D εψ (0 ,θ k ) | η ( x d /ε ) | (cid:33) / + θ k ε (cid:32) − ˆ D εψ (0 ,θ k ) | ( x d + εχ d ( x/ε )) η (cid:48) ( x d /ε ) | (cid:33) / . Estimate (6.5) now yields − ˆ D εψ (0 ,θ k ) |∇ v ( x/ε ) | ≤ Cεθ − k , so that eventually using ε/ε ≤ r ≤ θ k , (cid:32) − ˆ D εψ (0 ,θ k ) | εv ( x/ε ) − ( x d + εχ d ( x/ε )) η ( x d /ε ) | (cid:33) / ≤ C (cid:18) ε / θ k/ + θ k + θ k ε ( ε + ε ) (cid:19) ≤ Cθ k with C = C ( d, N, λ, [ A ] C ,ν , (cid:107) ψ (cid:107) W , ∞ ) . It follows from (6.7) and (6.6) that (cid:32) − ˆ D εψ (0 ,r ) | u ε ( x ) | (cid:33) / ≤ θ (1+ µ ) k + Cθ k ≤ Cθ k ≤ Cr, which is the estimate of Proposition 15.6.1.
Proof of Lemma 16.
Let < θ < / and u ∈ H ( D (0 , / be a weak solutionof(6.8) (cid:26) −∇ · A ∇ u = 0 , x ∈ D (0 , / ,u = 0 , x ∈ ∆ (0 , / , such that − ˆ D (0 , / | u | ≤ d . The classical regularity theory yields u ∈ C ( D (0 , / . Using that for all x ∈ D (0 , θ ) u ( x ) − (cid:16) ∂ x d u (cid:17) ,θ x d = u ( x ) − u ( x (cid:48) , − (cid:16) ∂ x d u (cid:17) ,θ x d = 1 | D (0 , θ ) | ˆ ˆ D (0 ,θ ) (cid:0) ∂ x d u ( x (cid:48) , tx d ) − ∂ x d u ( y ) (cid:1) x d dydt. we get(6.9) − ˆ D (0 ,θ ) (cid:12)(cid:12)(cid:12) u ( x ) − ( ∂ x d u ) D (0 ,θ ) x d (cid:12)(cid:12)(cid:12) dy ≤ (cid:98) Cθ , where (cid:98) C = (cid:98) C ( d, N, λ ) . Fix < µ < . Choose < θ < / sufficiently small such that(6.10) θ µ > (cid:98) Cθ . The rest of the proof is by contradiction. Fix γ > . Assume that for all k ∈ N , thereexists ψ k ∈ W , ∞ ( R d − ) ,(6.11) − < ψ < and (cid:107) ψ k (cid:107) L ∞ ≤ γ, here exists A k ∈ A ν , there exists < ε k < /k , there exists u ε k k solving (cid:26) −∇ · A k ( x/ε k ) ∇ u ε k k = 0 , x ∈ D ε k ψ k (0 , ,u ε k k = 0 , x ∈ ∆ ε k ψ k (0 , , such that(6.12) − ˆ D εkψk (0 , | u ε k k | ≤ and(6.13) − ˆ D εkψk (0 ,θ ) (cid:12)(cid:12)(cid:12)(cid:12) u ε k k ( x ) − ( ∂ x d u ε k k ) D εkψk (0 ,θ ) (cid:104) x d + ε k χ dk ( x/ε k ) + ε k v k ( x/ε k ) (cid:105)(cid:12)(cid:12)(cid:12)(cid:12) dy > θ µ . Notice that χ dk is the cell corrector associated to the operator −∇ · A k ( y ) ∇ and v k is theboundary layer corrector associated to −∇ · A k ( y ) ∇ and to the domain y d > ψ k ( y (cid:48) ) .First of all, for technical reasons, let us extend u ε k k by zero below the boundary, on { x (cid:48) ∈ ( − , d − , x d ≤ ε k ψ k ( x (cid:48) /ε k ) } . The extended functions are still denoted the same,and u ε k k is a weak solution of −∇ · A k ( x/ε k ) ∇ u ε k k = 0 on { x (cid:48) ∈ ( − , d − , x d ≤ ε k ψ k ( x (cid:48) /ε k ) + 1 } .For k sufficiently large, by Cacciopoli’s inequality, ˆ ( − / , / d |∇ u ε k k | ≤ C ˆ ( − / , / d | u ε k k | ≤ C, where C = C ( d, N, λ ) . Therefore, up to a subsequence, which we denote again by u ε k k , wehave(6.14) u ε k k k →∞ −→ u , strongly in L (( − / , / d − × ( − , / , ∇ u ε k k k →∞ (cid:42) ∇ u , weakly in L (( − / , / d − × ( − , / . Moreover, ε k ψ k ( · /ε k ) converges to because ψ k is bounded uniformly in k (see (6.11)).Let ϕ ∈ C ∞ c ( D (0 , / . Theorem 4 implies that ˆ D εkψk (0 , / A k ( x/ε k ) ∇ u ε k k · ∇ ϕ k →∞ −→ ˆ D (0 , / A ∇ u ∇ ϕ, so that u is a weak solution to −∇ · A ∇ u = 0 in D (0 , / . Furthermore, for all ϕ ∈ C ∞ c (( − / , / d − × ( − , , ˆ { x (cid:48) ∈ ( − / , / d − , − ≤ x d ≤ ε k ψ k ( x (cid:48) /ε k ) } u ε k k ϕ k →∞ −→ ˆ ( − / , / d − × ( − , u ϕ, so that u ( x ) = 0 for all x ∈ ( − / , / d − × ( − , . In particular, u = 0 in H / (∆ (0 , / .Thus, u is a solution to (6.8) and satisfies the estimate (6.9). t remains to pass to the limit in (6.13) to reach a contradiction. Since | D ε k ψ k (0 , θ ) | = | D (0 , θ ) | , we have(6.15) (cid:12)(cid:12)(cid:12)(cid:12) ( ∂ x d u ε k k ) D εkψk (0 ,θ ) − ( ∂ x d u ) D (0 ,θ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | D (0 , θ ) | (cid:34)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ D εkψk (0 ,θ ) ∩ D (0 ,θ ) (cid:0) ∂ x d u ε k k − ∂ x d u (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ˆ (cid:16) D εkψk (0 ,θ ) \ D (0 ,θ ) (cid:17) ∪ (cid:16) D (0 ,θ ) \ D εkψk (0 ,θ ) (cid:17) (cid:12)(cid:12) ∂ x d u ε k k − ∂ x d u (cid:12)(cid:12)(cid:35) . The first term in the right hand side of (6.15) tends to thanks to the weak convergenceof ∇ u ε k k in (6.14). The second term in the right hand side of (6.15) goes to when k → ∞ because of the L bound on the gradient, and the fact that (cid:12)(cid:12)(cid:12)(cid:16) D ε k ψ k (0 , θ ) \ D (0 , θ ) (cid:17) ∪ (cid:16) D (0 , θ ) \ D ε k ψ k (0 , θ ) (cid:17)(cid:12)(cid:12)(cid:12) k →∞ −→ . Therefore, − ˆ D εkψk (0 ,θ ) ∩ D (0 ,θ ) (cid:12)(cid:12)(cid:12)(cid:12) ( ∂ x d u ε k k ) D εkψk (0 ,θ ) (cid:104) x d + ε k χ dk ( x/ε k ) (cid:105) − ( ∂ x d u ) D (0 ,θ ) x d (cid:12)(cid:12)(cid:12)(cid:12) k →∞ −→ . Moreover, the strong L convergence in (6.14) implies − ˆ D εkψk (0 ,θ ) ∩ D (0 ,θ ) | u ε k k − u | k →∞ −→ . The last thing we have to check is the convergence − ˆ D εkψk (0 ,θ ) | ε k v k ( x/ε k ) | k →∞ −→ . Let η = η ( y d ) ∈ C ∞ c ( R ) such that η ≡ on ( − , and Supp η ⊂ ( − , . We have(6.16) − ˆ D εkψk (0 ,θ ) | ε k v k ( x/ε k ) | ≤ − ˆ D εkψk (0 ,θ ) | ε k v k ( x/ε k ) − ( x d + ε k χ dk ( x/ε k )) η ( x d /ε k ) | + − ˆ D εkψk (0 ,θ ) | ( x d + ε k χ dk ( x/ε k )) η ( x d /ε k ) | . The last term in the right hand side of (6.16) goes to when k → ∞ . Now by Poincaré’sinequality, − ˆ D εkψk (0 ,θ ) | ε k v k ( x/ε k ) − ( x d + ε k χ dk ( x/ε k )) η ( x d /ε k ) | ≤ Cθ (cid:34) − ˆ D εkψk (0 ,θ ) |∇ v k ( x/ε k ) | + − ˆ D ψk (0 ,θ ) (cid:12)(cid:12)(cid:12) ∇ (cid:16) ( x d + ε k χ dk ( x/ε k )) η ( x d /ε k ) (cid:17)(cid:12)(cid:12)(cid:12) (cid:35) . On the one hand by estimate (6.5) − ˆ D εkψk (0 ,θ ) |∇ v k ( x/ε k ) | ≤ Cε dk θ d ˆ D ψk (0 ,θ/ε k ) |∇ v k ( y ) | ≤ Cε k sup ξ ∈ Z d − ˆ ξ +(0 , d − ˆ ∞ ψ k ( y (cid:48) ) |∇ v k | ≤ Cε k k →∞ −→ ith in the last inequality C = C ( d, N, λ, [ A ] C ,ν ) uniform in ε , and on the other hand − ˆ D εkψk (0 ,θ ) (cid:12)(cid:12)(cid:12) ∇ (cid:16) ( x d + ε k χ dk ( x/ε k )) η ( x d /ε k ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ (1 + (cid:107)∇ χ k (cid:107) L ∞ ) − ˆ D εkψk (0 ,θ ) | η ( x d /ε k ) | + 1 ε k − ˆ D εkψk (0 ,θ ) | ( x d + ε k χ dk ( x/ε k )) η (cid:48) ( x d /ε k ) | ≤ Cε k k →∞ −→ , with in the last inequality C = C ( d, N, λ, [ A ] C ,ν ) . These convergence results imply thatpassing to the limit in (6.13) we get θ µ ≤ − ˆ D εkψk (0 ,θ ) (cid:12)(cid:12)(cid:12)(cid:12) u ε k k ( x ) − ( ∂ x d u ε k k ) D εkψk (0 ,θ ) (cid:104) x d + ε k χ dk ( x/ε k ) + ε k v k ( x/ε k ) (cid:105)(cid:12)(cid:12)(cid:12)(cid:12) dy k →∞ −→ − ˆ D (0 ,θ ) (cid:12)(cid:12)(cid:12) u ( x ) − ( ∂ x d u ) D (0 ,θ ) x d (cid:12)(cid:12)(cid:12) dy ≤ (cid:98) Cθ , which contradicts (6.10).6.2. Proof of Lemma 17.
The proof is by induction on k . The result for k = 1 is truebecause of Lemma 16. Let k ∈ N , k ≥ . Assume that for all ψ ∈ W , ∞ ( R d − ) such that − < ψ < and (cid:107)∇ ψ (cid:107) L ∞ ≤ γ , for all A ∈ A ν , for all k ∈ N , k > , for all < ε < θ k − ε ,for all weak solution u ε to (6.2) there exists a εk ∈ R N satifying | a εk | ≤ C θ µ + . . . θ µ ( k − θ d/ (1 − θ ) , such that − ˆ D εψ (0 , | u ε | ≤ implies(6.17) − ˆ D εψ (0 ,θ k ) (cid:12)(cid:12)(cid:12) u ε ( x ) − a εk (cid:104) x d + εχ d ( x/ε ) + εv ( x/ε ) (cid:105)(cid:12)(cid:12)(cid:12) dy ≤ θ (2+2 µ ) k . This is our induction hypothesis.Given ψ ∈ W , ∞ ( R d − ) , − < ψ < and (cid:107)∇ ψ (cid:107) L ∞ ≤ γ and A ∈ A ν , < ε < θ k − ε and a solution u ε to (6.2) such that − ˆ D εψ (0 , | u ε | ≤ we define U ε ( x ) := 1 θ (1+ µ ) k (cid:110) u ε ( θ k x ) − a εk (cid:104) θ k x d + εχ d ( θ k x/ε ) + εv ( θ k x/ε ) (cid:105)(cid:111) for all x ∈ D ε/θ k ψ (0 , . The goal is to apply the estimate of Lemma 17 to U ε . By theinduction estimate (6.17), we have − ˆ D ε/θkψ (0 , | U ε | ≤ . Moreover, U ε solves the system(6.18) (cid:40) −∇ · A ( θ k x/ε ) ∇ U ε = 0 , x ∈ D ε/θ k ψ (0 , ,U ε = 0 , x ∈ ∆ ε/θ k ψ (0 , . he boundary layer v solving (6.4) has been designed for U ε to solve (6.18). It followsthat U ε satisfies the assumptions of Lemma 16. Therefore, for all ε/θ k < ε , we have − ˆ D ε/θkψ (0 ,θ ) (cid:12)(cid:12)(cid:12)(cid:12) U ε ( x ) − ( ∂ x d U ε ) D ε/θkψ (0 ,θ ) (cid:104) x d + εθ k χ d ( θ k x/ε ) + εθ k v ( θ k x/ε ) (cid:105)(cid:12)(cid:12)(cid:12)(cid:12) dy ≤ θ µ . Eventually, − ˆ D εψ (0 ,θ k +1 ) (cid:12)(cid:12)(cid:12) u ε ( x ) − a εk +1 (cid:104) x d + εχ d ( x/ε ) + εv ( x/ε ) (cid:105)(cid:12)(cid:12)(cid:12) dy ≤ θ (2+2 µ )( k +1) , with a εk +1 := a εk + θ µk ( ∂ x d U ε ) D ε/θkψ (0 ,θ ) satisfying the estimate | a εk +1 | ≤ C θ µ + . . . θ µ ( k − θ d/ (1 − θ ) + C θ µk θ d/ (1 − θ ) ≤ C θ µ + . . . θ µk θ d/ (1 − θ ) . This concludes the iteration step and proves Lemma 17.
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