Applications of the unfolding method to some open problems in homogenization

Transcription

1 Applications of the unfolding method to some open problems in homogenization Doina Cioranescu Laboratoire J.L.Lions International Conference on Partial Differential Equations: Theory, Control and Approximations Dedicated to the Scientific Heritage of JACQUES-LOUIS LIONS Shanghai, May 28 - June 1, 2012

2 Hora: 18:00 19:30 Introduction n de Actos de la Facultad de Ciencias Homogenization: technics allowing to predict the macroscopic (or overall) behaviour of highly heterogeneous media, knowing (and using) the microscopic structure. cado de estos conceptos en Matemáticas y su evolución a lo largo d, en matemáticas, existen técnicas que permiten predecir fuertemente heterogéneos, conociendo su estructura microscópic Diseño Gráfico: Antonio José Gonz

3 Introduction In a composite material, oscillations are due to heterogeneities. In a porous material, they come also from the geometry, i.e., from the presence of perforations. One has also complicated structures with heterogeneities and perforations simultaneously, and at different scales. From the mathematical point of view, it means for example, that one has to treat PDE s with highly oscillating coefficients (related to the heterogeneities at different scales), stated in domains perforated periodically with period ε by holes of size of the order of εholes and by small holes (petits trous 1 ) of size εr(ε) ε,... We are interested in the asymptotic behavior as ε 0 of the solutions of these equations (with the hope to be able to characterize their limit, like in the case of fixed domains, as solution of a limit homogenized system). 1 Serge Gainsbourg, Le poinçonneur de Lilas: J fais des trous, des p tits trous, encore des p tits trous...

4 Introduction The aim is to answer some open questions in homogenization by using the periodic unfolding method, namely give an elementary proof to classical periodic homogenization problems for cases with several micro-scales, J. L. Lions, D. Lukkassen, L.E. Persson and P. Wall, Reiterated homogenization of monotone operators, Chinese Annals of Mathematics, (Ser. B) 22 (1), 2001 treat PDE s with oscillating coefficients and small holes, consider PDE s in domains with non smooth holes with non homogeneous Neumann boundary condition on the boundary of holes (Neumann problem), homogenization of a problem with a Fredholm alternative. The results presented here are contained in joint papers with Alain Damlamian Patrizia Donato Georges Griso Daniel Onofrei Rachad Zaki

5 Difficulties with perforated domains Perforated domains Given Ω a fixed domain in R n, together with a reference hole S, a basis of the R n whose vectors are macroscopic periods, the perforated domain Ω ε, is obtained by removing from Ω all the ε-periodic translates of εs. S

6 Difficulties with perforated domains The classical methods in homogenization are the multiple scale method which consists in looking (formally) for solutions in the form of asymptotic expansions with respect to the parameter vε (books of Benssousan, Lions and Papanicolaou, Sanchez-Palencia, Oleinik and Shamaev), the oscillating test functions (also known as energy method) due to Tartar, a mathematical method based on the construction, for each problem to be studied, of appropriate oscillating test functions. The procedure includes a priori estimates and convergence results, the two-scale convergence method (Nguetseng and Allaire), based on a new notion of convergence that is tested on special test functions. These methods are all based on the existence of the two scales x giving the position of a point in Ω, y = x/ε describing what happens in the magnified cell Y.

7 Difficulties with perforated domains The Neumann problem Let M(α, β, O) be the set of the N N matrices A ε (x) = (a ε ij (x)) 1 i,j N a.e. on O, and satisfying λ R n, (A ε (x)λ, λ) α λ 2, A ε (x)λ β λ. Let A ε = (aij ε ) 1 i,j N M(α, β, Ω ε) and f L 2 (Ω). The Neumann problem is div (A ε u ε ) + b ε u ε = f in Ω ε, A ε u ε n ε = 0 on Ω ε \ S ε, A ε u ε n ε = g ε on S ε, where b ε is measurable, positive a.e. in Ω, essentially bounded as well as its inverse, g ε is given in L 2 ( S ε ). Variational formulation: Find u ε H 1 (Ω ε) such that A ε u ε v dx = f v dx + g ε v dσ(x), Ω ε v H 1 (Ω ε). Ω ε S ε Ω

8 Difficulties with perforated domains Suppose that one has the a priori estimate u ε H 1 0 (Ω ε ) C. The space H 1 (Ω ε) strongly depending on ε the question now is: what kind of convergence can be expected for {u ε }? Traditional approach is to use uniformly bounded extension operators P ε from H 1 (Ω ε) to H 1 (Ω). This is the case for sufficiently smooth holes not intersecting the boundary of Ω. Such extension operators are constructed on the unit cell but under restrictive regularity conditions on the hole S. Without such extension operators, one cannot speak about convergence of u ε. For homogeneous Neumann condition on the boundary of the holes, a first attempt in this direction was made in G. Allaire and F. Murat, Homogenization of the homogeneous Neumann problem with nonisolated holes, Asymptotic Analysis (1993).

9 Open question: small holes in the general case A strange term coming from nowhere (terme étrange venu d ailleurs) Consider the problem { u ε = f in Ω ε, u ε = 0 on Ω ε. Obvious that ũ ε 0 weakly in H 1 0 (Ω). Suppose that the holes (still ε periodic) are of size εr(ε) ε (petits trous). In the case where the holes are balls: V. A. Marcenko and E. Ja. Hrouslov, Boundary value problems in domains with fine-grained boundaries (in Russian), For a precise size of the balls, ũ ε u weakly in H0 1 (Ω), u satisfying u + Cu = f, where C is a constant depending on the capacity of the reference hole. The proof uses essentially explicit estimates (starting with the Green function).

10 Open question: small holes in the general case D. C and F. Murat, Un terme étrange venu d ailleurs, in College de France Seminar, Pitman, Boston (1982). A strange term coming from nowhere, in Topics in the Mathematical Modelling of Composite Materials, Birkäuser (1997). Theorem Suppose there is a sequence of fonctions {w ε } and a distribution µ W 1, (Ω) such that w ε H 1 (Ω) and w ε = 0 on S ε, w ε 1 weakly in H 1 (Ω), For each sequence {v ε } such that v ε = 0 on S ε and v ε v weakly in H 1 0 (Ω), w ε, φv ε H 1 (Ω),H0 1(Ω) µ, φv H 1 (Ω),H0 1 (Ω), φ D(Ω).

11 Open question: small holes in the general case Then, for each n there is a critical size r n such that ũ ε u weakly in H0 1 (Ω) with if r(ε) r n, then u = 0, if r(ε) r n, u satisfies u = f in Ω, if r(ε) = r n, there exists a µ W 1, (Ω) such that { u + µu = f in Ω, u = 0 on Ω. For n 3, r n ε n n 2, in particular r3 = 3. The computation is based on the construction for balls, of explicit test functions (radial) w ε satisfying the hypotheses of the theorem.

12 Open question: small holes in the general case What happens in a general situation? Aim: to study the asymptotic behaviour of the solution of u εδ H0 1 (Ω ε δ ), A ε u εδ φ dx = f φ dx, f L 2 (Ω), Ω εδ Ω εδ φ H0 1 (Ω εδ ).

13 Multiscale problems Z δ Ω ε Y1 Y Now instead of the operator Aε one has the following one: x x A1 Y1 for ε ε εδ x A (x) = x A2 for Y2. εδ ε

14 Multiscale problems Ω ε,δ1 δ 2 Other type of multiscale domains δ 1 B.. B δ 2 T T Yδ 1 δ 2

15 Open question: a problem with the Fredholm alternative About the Fredholm alternative Consider a pure Neumann variational problem (i.e., without zero order term) in a domain O, Find u H 1 (O) A u v dx = O O fv dx + gv dσ, O v H 1 (O, where f is in L 2 (O), g in L 2 ( O) and A(x) = (a ij (x)) 1 i,j n is an elliptic matrix field. It is well-known that this problem has a unique solution (up to a constant), provided O is bounded, connected with Lipschitz boundary and if and only if f (x) dx + g dσ = 0 (Fredholm alternative). O O

16 Open question: a problem with the Fredholm alternative One way to obtain existence and uniqueness of a solution is to apply Lax-Milgram theorem to a space V (O) which is Hilbert for. the norm w V (O) = w L 2 (O). One candidate is V (O) =. {w H 1 (O), M O (w) = 0}, where M O (w) denotes the average 1 w dx (which makes O O sense only if O is of finite measure). V (O) is a Hilbert space the Poincaré-Wirtinger inequality holds for H 1 (O), v H 1 (O) v M O v dx L 2 (O) C v L 2 (O), which requires that O be connected. It is always satisfied for O bounded with Lipschitz boundary, but holds in much more general cases, for example for John domains (many of them do not have Lipschitz boundaries).

17 Open question: a problem with the Fredholm alternative Aim: homogenization of this problem when O is the perforated domain Ω ε. There are several reasons why this fails, in particular it is not true in general that Ω ε is connected, the boundary of Ω ε can be so irregular that v in L 2 (Ω ε) does not imply v H 1 (Ω ε) but only v H 1 loc (Ω ε). Consequently, the problem is not well-posed for general f and g satisfying the Fredholm condition.

18 Unfolding in perforated domains Difficulties with perforated domains: a priori estimates, functions defined on oscillating domains, on oscillating boundaries, connectednes, a.s.o. In order to get rid of extension operators, the tool we use is the periodic unfolding method in perforated domains. D. C., A. Damlamian and G. Griso, The periodic unfolding method in homogenization, Note C.R.A.S. Paris, (2002)& SIAM J. of Math. Anal. (2008) D.C., A. Damlamian, G. Griso and D. Onofrei, The periodic unfolding method for perforated domains and Neumann sieve models, J. Math. Pures Appl., 89 (2008) D. C., P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Portugaliae Mathematica, (2009) D. C., A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. of Math. Anal. (2012)

19 Geometry Perforated domains Let b = (b 1,..., b n ) be a basis in R n. Denote by Y the reference cell, which in the simplest case is the open parallelotop generated by the basis b, { y R n y = n y i b i, (y 1,..., y n ) (0, 1) n}. i=1 Let S be a closed strict subset of Y and denote by Y the part occupie d by the material i.e., Y = Y \ S.

20 Geometry One can consider situations where no choice of the basis of periods gives a parallelotop Y such that Y = Y \ S is connected (condition necessary for the validity of the Poincaré-Wirtinger inequality). The method applies if there exists a reference cell Y having the paving property with respect to the period basis and such that Y is connected. There are many possible Y s that give a connected Y.

21 Geometry The perforated domain Ω ε is obtained by removing from Ω the set of holes S ε, Ω ε = (R n ) ε Ω, where (R n ) ε is R n ε-periodically perforated by εs. The following notations will be used: Ω ε = Ω ε Ω ε, Λ ε = Λ ε Ω ε, Sε = Ω ε S ε.

22 Geometry Furthermore, and even if Ω is Lipschitz, it is not necessarily so of Ω usps). For a connected domain Ω, the perforated set Ω is not necessarily b is connected. even when Ω is suchconnected, an example ε ε Moreover, even if Ω is Lipschitz, it is not necessarily so of Ω ε, there can be cusps.

23 Geometry This is not a fluke, it can happen even with a very smooth boundary! or a straight boundary: should not think this is a fluke. It can happen even with an analytic boundary aight boundary: However, we can see that the core of Ω ε is conne cell Y is itself connected and is connected with its nei below).

24 The unfolding operator in perforated domains The unfolding operator and its properties In the definition of the unfolding method, the decomposition below plays an essential role. For z R n, denote by [z] Y the unique (up to a set of measure zero) integer combination n j=1 k jb j of the periods such that z [z] Y belongs to Y. Set now {z} Y = z [z] Y Y a.e. for z R n. [ x ] { x } Consequently x = ε ε + ε Y ε, for x Y Rn.

25 The unfolding operator in perforated domains The unfolding operator Tε domain Ω ε as follows: for functions defined on the perforated For any function φ Lebesgue-measurable on Ω ε, the unfolding operator Tε is defined by ( [ x ) Tε φ ε (φ)(x, y) = ε ]Y + εy a.e. for (x, y) Ω ε Y, 0 a.e. for (x, y) Λ ε Y. This operator maps functions defined on the oscillating domain Ω ε, to functions defined on the fixed domain Ω Y. For every φ in L 1 (Ω ε) one has the integration formula 1 Tε (φ)(x, y) dx dy = φ(x) dx. Y Ω Y Ω ε

26 Convergence result in perforated domains Definition. A bounded open set O satisfies the Poincaré-Wirtinger inequality for the exponent p [1, + ] if there exists a constant C p such that u W 1,p (O), u M O (u) L p (O) C p u L p (O). (Obviously, for O to satisfy the above condition, it has to be connected.) Hypothesis (H p ) The open set Y satisfies the Poincaré-Wirtinger inequality for the exponent p (p [1, + ]), and for every vector b i, i {1,..., n} of the basis b, the interior of Y (b i + Y )) is connected.

27 Convergence result in perforated domains The following compactness result holds: Theorem Under hypothesis (H p ), suppose that w ε in W 1,p (Ω ε) satisfies w ε W 1,p (Ω ε ) C. Then, there exist w in W 1,p (Ω) and ŵ in L p (Ω; Wper 1,p (Y )) with M Y (ŵ) 0, such that, up to a subsequence, { T ε (w ε ) w strongly in L p loc (i) (Ω; W 1,p (Y )), Tε (w ε ) w weakly in L p (Ω; W 1,p (Y )), (ii) Tε ( w ε ) w + y ŵ weakly in L p (Ω Y ). This result is essential for homogenization problems. Since the unfolding operator Tε transforms functions defined on the oscillating domain Ω ε into functions defined on the fixed domain Ω Y, there is no need of any extension operator to the whole of Ω. Therefore, regularity hypotheses on the boundary S insuring the existence of such extension operators, are not required (contrary to the classical methods).

28 Convergence result in perforated domains Macro-micro decomposition For the proof, a separation of scales is carried out by using a macro-micro decomposition of φ in W 1,p (Ω ε), p in [1, + ], φ = Q ε(φ) + R ε(φ), a.e. in Ω ε. The macro approximation Q ε is defined by an average at the points of Ξ ε and extended to the set Ω ε by Q 1 -interpolation (as customary in the finite element method). By construction, Q ε(φ) W 1,p (Ω) is of the same order as φ W 1,p (Ω while the norm ε) of the remainder R ε(φ) of order ε. As a consequence, the function ŵ is defined by the convergence 1 ε T ε ( R ε (w ε ) ) ŵ weakly in L p (Ω; W 1,p (Y )).

29 Neumann problem This solves one of the difficulties proper to perforated domains, passing to the limit in integral, A ε u ε v dx. Elementary in vue of main convergence theorem!! Ω ε If A ε = A(x/ε) then Tε (A ε )(x, y) = A(y)!! So A ε u ε v dx 1 Y Ω ε Ω Y A(y)T ε ( u ε ) T ε ( v) dx dy 1 A(x, y) [ u 0 (x) + y û 0 (x, y) ] v dx dy. Y Ω Y What about the other terms in the variational formulation of the Neumann problem A ε u ε v dx + b ε u ε dx = f v dx + g ε v dσ(x)? S ε Ω ε Ω ε Ω ε

30 Neumann problem We should point out that as far as we know, there is no homogenization result for general g ε. This is mainly due to the lack of uniform bounds for the solutions. What is usually done, is to suppress the holes in a fixed boundary layer in which case the unfolding approach works easily (since all unfolded functions always vanish in this layer by construction). Another approach is to assume that g ε is identically zero outside of S ε and solve, for all v H 1 (Ω ε), A ε u ε v dx + b ε u ε vdx = f v dx + g ε v dσ(x). S ε Ω ε Ω ε Ω ε

31 Boundary unfolding operators The boundary unfolding operator T b ε Assume there is a well-defined trace operator from W 1,p (Y ) to W 1 1/p,p ( S) (e.g if all components of S have a Lipschitz boundary). Definition For any function ϕ Lebesgue-measurable on Ω ε S ε, the boundary unfolding operator Tε b is defined by ( [ x ) Tε b φ ε (φ)(x, y) = ε ]Y + εy a.e. for (x, y) Ω ε S, 0 a.e. for (x, y) Λ ε S. If ϕ W 1,p (Ω ε), Tε b (ϕ) is just the trace on S of Tε (ϕ). Therefore the operator Tε b enjoys some of properties of the unfolding operator Tε.

32 Boundary unfolding operators In particular, the integration formula, which reads ϕ(x) dσ(x) = 1 Tε b (ϕ)(x, y) dx dσ(y), S ε ε Y Ω S transforms an integral on the rapidly oscillating set S ε into an integral on a fixed set Ω S. The integration formula implies T b ε (ϕ) L p (Ω S) = ε 1/p Y 1/p ϕ L p ( S ε). (Important for obtaining a priori estimates for the Neumann problem when dealing with Sε g ε v dσ(x)).

33 Boundary unfolding operators Convergence for T b ε Theorem Suppose w ε is in W 1,p (Ω ε), g ε is in L p (Ŝε) and T ε (w ε ) w strongly in L p (Ω; W 1,p (Y )), T b ε (g ε ) g weakly in L p (Ω S). Then ε g ε w ε dσ(x) 1 g(x, y) w(x, y) dxdσ(y). S ε Y Ω S

34 Application to the Neumann problem Homogenization for the Neumann problem Hypotheses There is a matrix A such that Tε ( A ε ) A a.e. in Ω Y (or in measure in Ω Y ). There exist g in L 2 (Ω S) and G in L 2 (Ω) satisfying ε (g ε ) g weakly in L 2 (Ω S), 1 ε M S(Tε b (g ε )) G weakly in L 2 (Ω). T b Two standard examples of functions g ε that satisfy these hypotheses g ε (x) = ε g({x/ε} Y ) if M S (g) 0 = g = 0, G = M S (g), g ε (x) = g({x/ε} Y ) if M S (g) = 0 = G = 0.

35 Application to the Neumann problem Neumann homogenized problem div (A 0 u) + Y Y M Y (b) u = Y f div G in Ω, Y A 0 u n = G n on Ω. Remark Strange phenomenon: the non-homogeneous Neumann condition on the boundary of the holes inside Ω contributes to a non-homogeneous Neumann condition on Ω in the limit problem. G(x) =. S ( G+M S (y M g)(x) ) Y Y Y M ( Y A(x, ) y χ 0 (x, ) ) in Ω, where y M = y M Y (y), χ 0 (Y -periodic) is defined by n i,k=1 n i,k=1 ( a ik (x, y) χ 0(x, y) ) = g in Y, y i y k a ik (x, y) χ 0(x, y) y k n i = 0 on Y.

36 Small holes We want to pass to the limit in A ε u εδ φ = knowing that Ω εδ φ H0 1 (Ω εδ ), Ω εδ f φ, ũ εδ H 1 0 (Ω) C f L 2 (Ω). f L 2 (Ω)

37 Unfolding operators continuing The unfolding operator T ε,δ depending on two parameters ε and δ Definition For φ L p (Ω), p [1, [, the unfolding operator T ε,δ : L p (Ω) L p (Ω R n ) is defined by T ε (φ)(x, δz) a.e. for (x, z) T ε,δ (φ)(x, z) = Ω ε 1 δ Y, 0 otherwise.

38 Unfolding operators continuing Theorem Suppose n 3. Then and z ( Tε,δ (u) ) 2 L 2 (Ω 1 δ Y ) ε2 δ N 2 u 2 L 2 (Ω), T ε,δ ( u M ε Y (u) ) 2 L 2 (Ω;L 2 (R n )) Cε2 δ N 2 u 2 L 2 (Ω), where C denotes the Poincaré-Wirtinger constant for H 1 (Y ). The local average MY ε : Lp (Ω) is defined as MY ε (φ)(x) = T ε (φ)(x, y) dy. Hypothesis: we suppose n 3, and ε and δ = δ(ε) are such that there exists a positive constant k satisfying k = lim ε ε 0 δ n/2 1 Y, 0 k < +.

39 Unfolding operators continuing In order to state the homogenization result, we introduce the functional space K S defined as follows: K S = {Φ L 2 (R n ) ; Φ L 2 (R N ), Φ constant on S}, and the function θ, the solution of the cell problem θ L (Ω; K S ), θ(x, S) 1, t A 0 (x, z) z θ(x, z) z Ψ(z) dz = 0 a.e. for x Ω, R n \S Ψ K S with Ψ(T ) = 0.

40 Unfolding operators continuing Theorem Suppose that, as ε 0, there exists a matrix A such that T ε ( A ε ) (x, y) A(x, y) a.e. in Ω Y. Furthermore, suppose that there exists a matrix field A 0 such that as ε and δ 0, T ε,δ ( A ε ) (x, z) A 0 (x, z) a.e. in Ω (R n \ S). Then u ε,δ u 0 weakly in H 1 0 (Ω), where u is the solution of the problem { div (A hom u 0 ) + k 2 Θu 0 = f in Ω, u = 0 on Ω, where Θ(x) =. S t A 0 (x, z) z θ(x, z) n S dσ z.

41 Unfolding operators continuing The function Θ equals Θ(x) = R n \S A 0 (x, z) z θ(x, z) z θ(x, z) dz, which is non-negative and can be interpreted as the local capacity of the set S. The contribution of the oscillations of the matrix A ε in the homogenized problem are reflected by the the presence of the homogenized operator A hom in the limit problem. The contribution of the perforations is the zero order strange term k 2 Θ(x) u 0. Remark 1. If k = 0, the small holes have no influence at the limit. 2. If lim δ N 2 1 =, one has u 0 = 0. ε

42 Homogenization in multiscale domains The periodic unfolding method and multiscale problems The periodic unfolding method turns out to be particularly well-adapted to multiscale problems. As an example, we treat here a problem with two different small scales. Consider two periodicity cells Y and Z, and suppose that Y is partitioned in two non-empty disjoint open subsets Y1 and Y2, i.e. such that Y1 Y2 = o/ and Y = Y 1 Y 2. Z δ Ω ε Y1 Y

43 Homogenization in multiscale domains Let A εδ be a matrix field defined by ( x ) A 1 A εδ (x) = ( ε x ) A 2 εδ in Y 1, in Y 2, where A 1 is in M(α, β, Y 1 ) and A 2 in M(α, β, Z). The two scales are ε and δ, associated respectively to the cells Y and Z. Consider the problem A εδ u εδ w dx = Ω with f in L 2 (Ω). Ω f w dx w H 1 0 (Ω),

44 Homogenization in multiscale domains The Lax-Milgram theorem immediately gives the existence and uniqueness of u εδ in H0 1 (Ω) satisfying the estimate u εδ H 1 0 (Ω) 1 α f L 2 (Ω). So, there is some u 0 such that, up to a subsequence, u εδ u 0 weakly in H 1 0 (Ω)). Using the unfolding method for the scale ε we have T ε (u εδ ) u 0 weakly in L 2 (Ω; H 1 (Y )), ( ) T ε uεδ u0 + y û in L 2 (Ω Y ). These convergences do not see the oscillations at the scale εδ.

45 Homogenization in multiscale domains In order to capture them, take the restriction to the set Ω Y 2 Obviously, v εδ (x, y). = 1 ε T ε( Rε (u εδ ) ) Y2. v εδ û Y2 weakly in L 2 (Ω; H 1 (Y 2 )). Now apply to v εδ a similar unfolding operation T y δ for the variable y, thus adding a new variable z Z, T y y ) δ (v εδ)(x, y, z) = v εδ (x, δ[ δ ]Z +δz for x Ω, y Y 2 and z Z.

46 Homogenization in multiscale domains All the estimates and weak convergence properties concerning the original unfolding T ε still hold for T y δ (x being a mere parameter). This implies that there exists some ũ L 2 (Ω Y 2, Hper(Z) 1 such that the following convergences hold: T y ( ) δ y v εδ y û Ω2 + z ũ weakly in L 2 (Ω Y 2 Z), ( ( ) ) T ε uεδ u 0 + y û + z ũ weakly in L 2 (Ω Y 2 Z). T y δ With 100 scales ε, δ 1, δ 2,..., δ 99 one would have T z 99 δ 99 (... ( T z 1 δ 1 (T ε ( uεδ ))... ) u0 + y û + z1 ũ 1... z99 ũ 99 weakly in L 2 (Ω Y 2 Z 1... Z 99 )!!

47 Homogenization in multiscale domains The limits u 0, û and ũ are characterized as follows: Theorem. The functions u 0 H 1 0 (Ω), û L 2 (Ω, H 1 per(y )/R), ũ L 2 (Ω Y 2, H 1 per(z)/r) are the unique solutions of the variational problem 1 Y Z Ω Y 2 Z + 1 Y Ω [ ][ ] A 2 (z) u 0 + y û + z ũ Ψ + y Φ + z Θ dx dy dz Y 1 [ ][ ] A 1 (y) u 0 + y û Ψ + y Φ dx dy = f Ψ dx, Ω Ψ H0 1 (Ω), Φ L 2 (Ω; Hper(Y 1 )/R), Θ L 2 (Ω Y 2, Hper(Z)/R). 1

48 Neumann problem in perforated domains with Fredholm alternative Last open problem is that related to Find u H 1 (Ω ε) A u v dx = fv dx + g v dσ, S ε Ω ε Ω ε v H 1 (Ω ε), We would like to apply to it the theorem for perforated domains. Theorem. Suppose that w ε is in W 1,p ( Ω ε) and satisfies w ε W 1,p ( Ω ε) C. Then, there exist w in W 1,p (Ω) and ŵ in L p (Ω; W 1,p per (Y )) with M Y (ŵ) 0, such that, up to a subsequence, T ε (w ε ) w weakly in L p (Ω; W 1,p (Y )), T ε ( w ε ) w + y ŵ weakly in L p (Ω Y ). However, boundedness hypothesis is not readily satisfied. Therefore a more precise result is needed.

49 Neumann problem in perforated domains with Fredholm alternative Approximated domain Introduce the following notation: }. Ω ε = {x Ω dist(x, Ω) > ερ(y ), Ω ε = Ω ε Ω ε (dark blue). Ω ε Ω ε and is connected (as Ω is connected with Lipschitz boundary). If Y is a parallelotop, then ρ(y ) = 2 diam(y ). In general, ρ(y ) can be explicitly computed and is related to the number of b-parallelotops needed to cover Y.

50 Neumann problem in perforated domains with Fredholm alternative Suppose that hypothesis (H p ) holds. Then the set Ω ε is included in a single connected component of Ω ε, denoted Cε. But one should not think this is a fluke. It can happen even with an analytic boundary Ω: or a straight boundary: (The set C ε is in blue) Remark. All the connected components of Ω ε, others than C ε, lie near the boundary of Ω. Therefore, in some sense, C ε is the core connected component of Ω ε.

51 Neumann problem in perforated domains with Fredholm alternative We have now a convergence result for the unfolding in C ε. Theorem Let p be in ]1, + ] and suppose that hypothesis (H p ) holds. Assume that w ε in W 1,p loc (C ε ) satisfies w ε L p (C ε ) C and M Ω ε (w ε ) = 0. (1) Then, up to a subsequence (still denoted ε), there are two functions w in W 1,p (Ω) and ŵ in L p (Ω; W 1,p per (Y )), such that T ε (w ε ) w weakly in L p (Ω Y ) L p 1 Ωε loc (Ω; W 1,p (Y )), ( ) ( wε ) 1 C ε w + y ŵ weakly in L p (Ω Y ). T ε For p = +, the weak convergences above are replaced by weak- convergences.

52 Neumann problem in perforated domains with Fredholm alternative A candidate for a domain on which to consider an approximate problem appears now to be the core set C ε. For the space we will use W (C ε ). = {ϕ H 1 loc (C ε ), ϕ L 2 (C ε ), and M Ω ε (ϕ) = 0}, a Hilbert space for the norm ϕ W (C ε ) = ϕ L 2 (Cε ). As for the data, we make the following assumptions: f ε L 2 (Ω is bounded, f ε) ε vanishes outside Ω ε and (f M Ω ε ) = 0, ε Tε b (g ε ) L 2 (Ω S) and 1 ε g ε vanishes outside Ω ε. M S (T b ε (g ε )) L 2 (Ω) are bounded,

53 Neumann problem in perforated domains with Fredholm alternative Approximated problem Consider now the following problem: Find u ε in W (Cε ) such that for all ϕ W (Cε ), A ε (x) u ε (x) ϕ(x) dx = f ε (x)ϕ(x) dx+ g ε (x)ϕ(x) dσ. Ŝε C ε C ε There is a unique function u ε in W (C ε ), solution of this problem satisfying the a priori estimate u ε W (C ε ) C( f ε L 2 (Ω ε ) + T b ε (g ε ) L 2 ( S)).

54 Homogenization result Suppose that ε (g ε ) g strongly in L p (Ω S), 1 ε M S(Tε b (g ε )) G strongly in L p (Ω). T b One has the convergence T ε ((u ε ) 1 Ω ε ) u weakly in L 2 (Ω; H 1 (Y )), where u in V (Ω) is the unique solution of div (A 0 u) = f (x) + S ( ) Y G(x) G(x) dx div G in Ω, Ω A 0 u n = G n on Ω, M Ω (u) = 0.

55 Homogenization result where G(x) =. S Y M ( S(y M g)(x) M Y A(x, ) y χ 0 (x, ) ) in Ω, which belongs to L 2 (Ω). D. C, Alain Damlamian and Georges Griso, The periodic unfolding for a Fredholm alternative in perforated domains, to appear IMA Journal of Applied Math THANK YOU FOR YOUR ATTENTION

56 Homogenization result What { is essential in the proofs is the} following result concerning Ω ε = x Ω dist(x, Ω) > ερ(y ). Lemma (Poincaré-Wirtinger inequality for Ω ε ) Assume that Ω is a bounded domain with Lipschitz boundary. Then there exist δ 0 > 0 and a common Poincaré-Wirtinger constant C p for all the sets Ω ε for ε ]0, δ 0 ], i.e. φ W 1,p ( Ω ε ), φ M Ωε (φ) L p ( Ω ε) C p φ [L p ( Ω ε)], where C is independent of ε. The proof of the lemma is based on the equivalence of the uniform cone property and the Lipschitz boundary property.

57 Homogenization result I m fixing a hole where the rain gets in And stops my mind from wondering Where it will go John Lennon & Paul McCartney, Fixing A Hole, in Sgt. Peppers Lonely Hearts Club Band, The Beatles, ed. Northern Songs Ltd., Londres, 1967