HOMOGENIZATION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

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1 HOMOGENIZATION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Silvia Jiménez B.Sc. in Mathematics, Universidad de Costa Rica, Costa Rica, 00 M.S. in Mathematics, Louisiana State University, USA, 006 August 00

2 Acknowledgements Caminante, son tus huellas el camino y nada más; Caminante, no hay camino, se hace camino al andar. Al andar se hace el camino, y al volver la vista atrás se ve la senda que nunca se ha de volver a pisar. Antonio Machado, Proverbios y Cantares XXIX. Before we commence, I would like to thank my advisor Robert Lipton for his time, his patience, the gift of a plant that never dies, his support, and his guidance during my years of graduate stu at Louisiana State University. His enthusiasm for mathematics, his motivation, and his advice helped me immensely. He is an amazing advisor and mathematician but more important than that, he is a wonderful human being. Thank you! The Graduate School, the Department of Mathematics at Louisiana State University, and Board of Regents have supported me several times for travel to conferences, for which I am grateful. Further, I have been supported by NSF Grant DMS and AFOSR Grant FA I would particularly like to thank Prof. Stephen Shipman, more than a professor he has become a friend; Prof. Susanne Brenner, for her guidance and infinite energy; and my AWM Mentor Prof. Irina Mitrea, for her invaluable help and advice. Moreover, I want to thank all the members of my final exam committee, the members of the LSU SIAM Chapter, the AWM Chapter, and the LSU Judo Club, you all have had a very positive impact in my academic and personal life. To all the friends that I made here, the ones that are still here and the ones that alrea left, thank you very much. It was a great opportunity for me to attend LSU and I have thoroughly enoyed it. I dedicate this dissertation to my family: my parents William Jiménez Solís and Xenia Bolaños Rodríguez, my brothers William and José, my aunts Marcela, Vera, adira and olanda, my uncle Mario, my cousins Sofía, Alonso, and Ignacio, and especially to my grandma Berta. Thank you all so much for the visits, phone calls, chats, s, support, and love. ii

3 Table of Contents Acknowledgements Abstract ii v Chapter Introduction Chapter Basic Ideas in Homogenization Theory Motivation and Examples Dimensional Example of Homogenization Homogenization in R n Some Special Cases with Closed Form Expressions for the Homogenized Operator b The Hashin Structure (96) The Mortola-Steffé Structure and the Checkerboard Structure 0 Chapter 3 Dirichlet Boundary Value Problem Description of the Problem Microgeometries Considered Notation and Preliminary Results Properties of A Dirichlet BVP and Homogenization Theorem Properties of b Properties of p Chapter 4 Higher Order Integrability of the Homogenized Solution Statement of the Theorem on the Higher Order Integrability of the Homogenized Solution Proof of Higher Order Integrability of the Homogenized Solution... 6 Chapter 5 Corrector Theorem Statement of the Corrector Theorem Some Properties of Correctors Proof of the Corrector Theorem iii

4 Chapter 6 Lower Bounds on Field Concentrations Statement of the Lower Bound on the Amplification of the Macroscopic Field by the Microstructure oung Measures Proof of Lower Bound on the Amplification of the Macroscopic Field by the Microstructure Chapter 7 Nonlinear Neutral Inclusions Double Coated Nonlinear Neutral Inclusions Statement of the Problem and Result Calculations Bibliography Vita iv

5 Abstract This dissertation is concerned with properties of local fields inside composites made from two materials with different power law behavior. This simple constitutive model is frequently used to describe several phenomena ranging from plasticity to optical nonlinearities in dielectric media. We provide the corrector theory for the strong approximation of fields inside composites made from two power-law materials with different exponents. The correctors are used to develop bounds on the local singularity strength for gradient fields inside microstructured media. The bounds are multiscale in nature and can be used to measure the amplification of applied macroscopic fields by the microstructure. These results are shown to hold for finely mixed periodic dispersions of inclusions and for layers. v

6 Chapter Introduction We consider heterogeneous materials that have inhomogeneities on length scales that are much larger than the atomic scale but which are essentially homogeneous at macroscopic length scales. Heterogeneous materials such as fiber reinforced composites and polycrystalline metals and dielectrics appear in many physical contexts. The determination of the macroscopic effective properties for problems in heat transfer, elasticity, and electro magnetics is an important problem. It is also equally important to understand the behavior of the local fields, such as higher moments of fields inside heterogeneous media. The presence of large local fields either electric or mechanical often precede the onset of material failure [KM86]. Heterogeneities can amplify the applied load and generate local fields with very high intensities. The goal of the analysis presented in this research is to develop tools for quantifying the effect of load transfer between length scales inside heterogeneous media. In this thesis, we provide methods for quantitatively measuring the excursions of local fields generated by applied loads. These local quantities are extremely useful for understanding the evolution of nonlinear phenomena such as plasticity or damage. The research developed in this thesis investigates the properties of local fields inside mixtures of two nonlinear power law materials. This simple constitutive model is frequently used to describe several phenomena ranging from plasticity to optical nonlinearities in dielectric media. The main achievement of this thesis is that it develops the corrector theory necessary for the stu of local fields inside mixtures of two power law materials. Further the corrector theory is applied to deliver new multiscale tools to bound the local singularity strength inside micro-structured media in terms of the macroscopic applied fields. The thesis is organized as follows. In the next Chapter we provide background and motivate the theory of homogenization for a simple class of examples. In Chapter 3 we introduce the equilibrium problem for two phase nonlinear power law materials. Here the materials are assumed to have two different power law exponents and represent strongly nonlinear materials. In this thesis we consider two common two phase power law microstructures. The first is given by a periodic dispersion of particles embedded in a matrix and the second given by layered microstructures. In Chapter 4 we establish higher order integrability properties for the gradient of the solution of the equilibrium problem inside the material with the larger exponent. This is used in Chapter 5 where we develop the corrector theory necessary for the stu of local fields inside mixtures of two power law materials. In Chapter 6 we provide lower bounds on the local field strength inside microstructured media in terms of the macro-

7 scopic applied fields. We conclude in Chapter 7 where we will introduce special neutrally conducting microstructures made with power law materials.

8 Chapter Basic Ideas in Homogenization Theory The theory of homogenization or averaging of partial differential equations dates back to the late sixties [Spa67], it has been very rapidly developed during the last two decades, and it is now established as a distinct discipline within mathematics. Homogenization theory is concerned with the derivation of equations for averages of solutions of equations with rapidly varying coefficients. This problem arises in obtaining macroscopic, or homogenized, or effective equations for systems with a fine microscopic structure. The goal is to represent a complex, rapidly-varying medium with a slowly-varying medium in which the fine scale structure is averaged out in an appropriate way.. Motivation and Examples Suppose we would like to know the stationary temperature distribution in an homogeneous bo R 3 with an internal heat source f, heat conductivity A and zero temperature on the boundary. The model to describe this problem is given by the following boundary value problem: Find u W,p 0 (), < p <, such that div (A ( u)) = f on, (.) where is a bounded open subset of R n, f is a given function on, and A : R n R n satisfies suitable continuity and monotonicity conditions that allows the existence and uniqueness of the solution of (.). Now, suppose that we would like to be able to model the case when the underlying material is heterogeneous. Then we replace A in (.) with a map A : R n R n to obtain { div (A (x, u)) = f on u W,p (.) 0 (). Since (.) depends on x, this is much more difficult to handle than (.). An interesting special case is a two-phase composite where one material is periodically distributed in the other. In this case, the underlying periodic inclusions are often microscopic 3

9 with respect to. By periodicity, we can divide into periodic cells, and we call the representative unit cell by (the microstructure of a given periodic material can be described by several different period cells). This is described by maps of the form A ǫ (x,ξ) = A ( x,ξ), ǫ where A(,ξ) is assumed to be -periodic and ǫ is the fineness of the periodic structure. Equation (.) becomes { div (A ǫ (x, u ǫ )) = div ( A ( x u ǫ W,p 0 (). ǫ, u ǫ)) = f on (.3) The function u ǫ can be interpreted as the electric potential, magnetic potential, or the temperature and A ǫ describes the physical properties of the different materials constituting the bo (they are the dielectric coefficients, the magnetic permeability and the thermic conductivity coefficients, respectively). Let {ǫ k } k=0 be a sequence of positive real numbers such that ǫ k 0 as k. In this way we get a sequence of problems, one for each value of k. The smaller ǫ k gets, the finer the microstructure becomes. It is natural to ask ourselves if there exists some type of convergence of the solutions u ǫk. If we assume convergence in an appropriate sense, that is u ǫk u, as k, we could also ask if u satisfies an equation of a similar type as the one u ǫk satisfies { div (b (x, u)) = f, on u W,p 0 (), and if this is the case, how to find b. For large values of k, the material behaves like a homogeneous material from a macroscopic point of view, even though the material is strongly heterogeneous at a microscopic level. This makes it reasonable to assume that b should be independent of x, which means that u satisfies a homogenized equation of the form { div (b ( u)) = f, on u W,p (.4) 0 (). The homogenized b represents the physical parameters of a homogeneous bo, whose behavior is equivalent, from a macroscopic point of view, to the behavior of the material with the given periodic microstructure, described by (.3). Homogenization Theory deals with the questions mentioned above. Another approach to answer those questions is by using the fact that the state of the material u can be often found as the solution of a minimization problem of the form E ǫ = min u W,p 0 () { g ( x ǫ, u(x) ) } fu, where the local energy density function g(,ξ) is periodic and is assumed to satisfy the so called natural growth conditions. The convergence of this type of integral functionals is 4

10 called Γ-convergence (Introduced by DeGiorgi [DG75]). From the theory of Γ-convergence it follows that E ǫ E hom, as ǫ 0, where { } E hom = g hom ( u(x)) fu. min u W,p 0 () Here the homogenized energy density function g hom is given by g hom (ξ) = min g(x,ξ u), u Wper(,p ) where W,p per( ) is the set of all functions u W,p ( ) which are -periodic and have mean value zero. Again we note that the limit problem does not depend on x, that is, g hom is the energy density function of a homogeneous material. To demonstrate some of the techniques and difficulties encountered in the homogenization procedure, we consider homogenization of the one dimensional Poisson equation. This simple example reveals the main difficulty... -Dimensional Example of Homogenization Let = (0, ), f H (), and A L () be a measurable and periodic function with period satisfying 0 < β A(x) β <, for a.e. x R. (.5) Remark.. For example, consider a periodic mixture of two materials. Let χ be the the characteristic function of material and χ be the the characteristic function of material, both periodic of periodicity. Let A(y) = β χ (y) β χ (y) defined in = (0, ) and extend it to all R by periodicity, and call this extension by A as well. Note that A L (), because A L () = β and clearly it satisfies (.5) for all x R. ( x We define A ǫk (x) = A ǫ k ). The weak form of (.3) becomes A ǫk (x) x u ǫk (x) x φ(x) = 0 u ǫk W, 0 (0, ), 0 f(x)φ(x) for every φ W, 0 (0, ), and (.3) becomes { x (A ǫk (x) x u ǫk (x)) = f(x) in (0, ), u ǫk W, 0 (0, ). (.6) (.7) 5

11 By a standard result in the existence theory of partial differential equations (using Lax- Milgram Lemma [Eva98]), there exists a unique solution of these problems for each k. By choosing φ = u ǫk in (.6) and taking (.5) into account, we obtain by Hölder s inequality that β x u ǫk L (0,) A ǫk (x) x u ǫk (x) = 0 0 f(x)u ǫk (x) f H () u ǫ k W, 0 (). Recall that u ǫk W, 0 () = u ǫ k L () xu ǫk L (). The Poincaré inequality for functions with zero boundary values states that there is a constant C only depending on = (0, ) such that This implies that u ǫk L () C xu ǫk L (). u ǫk W, 0 () C, (.8) where C is a constant independent of k. Since W, 0 () is reflexive, there exists a subsequence, still denoted by {u ǫk }, such that u ǫk u in W, 0 (). (.9) Since W, 0 () is compactly embedded in L (), we have by the Rellich Embedding Theorem that u ǫk u in L (). In general, however, we only have that x u ǫk x u in L (). Since A is -periodic, we have that {A ǫk } converges weakly* L (), as k, to its arithmetic mean A, i.e., A ǫk A = 0 A(y) in L (). (.0) From (.6), (.9), and (.0), it could be reasonable to assume that, in the limit, we have A x u (x) φ(x) = f(x)φ(x) for every φ W, 0 (0, ), 0 0 u W, 0 (0, ). However, this is not true in general, since A ǫk x u ǫk is the product of two weakly converging sequences. This is the main difficulty in the limit process. To obtain the correct answer we 6

12 proceed in the following way: first we note that, according to (.0) and (.8), {A ǫk x u ǫk } is bounded in L () and that (.6) implies that x (A ǫk (x) x u ǫk ) = f. Hence there is a constant C independent of k such that A ǫk x u ǫk W, () C. As before, since W, () is reflexive, there exists a subsequence, still denoted {A ǫk x u ǫk } and a M 0 L () such that Since { } A ǫk L ()), we have A ǫk x u ǫk M 0 in L (). converges to A weakly* in L () by periodicity (and hence weakly in ( x u ǫk = A ǫk Thus, by (.9) and (.), we see that ) (A ǫk x u ǫk ) M 0, in L (). (.) A M 0 = A x u = A x u. Now, by passing to the limit in (.6) we obtain that b x du x φ = f(x)φ(x) for every φ W, 0 (0, ), 0 0 u W, 0 (0, ). where the homogenized operator is given by b = A = A, the harmonic mean of A; and since, β A β we conclude that the homogenized equation has a unique solution and thus that the whole sequence {u ǫk } converges. Remark.. For the example given in Remark., we obtain M 0 = h θ x u, where and θ = h θ = ( θ β θ β 0 ) χ (y), and θ = θ. Remark.3. The corresponding homogenization problem for the one-dimensional Poisson equation A ǫk (x) x u ǫk p x u ǫk x φ = f(x)φ(x) for every φ W,p 0 (), u ǫk W,p 0 () gives the homogenized operator b = A p p. 7

13 In higher dimensions, the problem of passing to the limit is rather delicate and requires the introduction of new techniques. One of the main tools to overcome this difficulty is the Compensated Compactness method introduced by Murat and Tartar [MT97]. This method shows that under some additional assumptions, the product of two weakly convergent sequences in L () converges in the sense of distributions to the product of their limits... Homogenization in R n Assume that A satisfies suitable structure conditions. Remark.4. A common assumption is that A(x,ξ) satisfies the conditions A(x,ξ ) A(x,ξ ) c λ(x) ( ξ ξ ) p α ξ ξ α, (A(x,ξ ) A(x,ξ ),ξ ξ ) c λ(x) ( ξ ξ ) p β ξ ξ β, for constants c,c > 0, where α and β satisfy 0 α min(,p ) and max(p, ) β < (see, for example, [DMD90, FM87]). For example, these conditions are satisfied by the p-poisson operator A(x,ξ) = λ(x) ξ p ξ, ( where c, c can be chosen as c = max p, ( ) ) ( p, and c = min p, ( ) ) p. We have the following homogenization theorem. Theorem.5. Let < p < and q its dual conugate. The solutions u ǫk of { div (A ǫk (x, u ǫk )) = f on, u ǫk W,p 0 () satisfy and u ǫk u in W,p 0 (), (.) A ǫk (x, u ǫk ) b( u) in L q (, R n ), as k, where u is the solution of the homogenized equation { div (b ( u)) = f on, u W,p 0 (), where the homogenized operator b : R n R n is defined by b(ξ) = A(x,ξ ω ξ (x)), (.3) and where ω ξ is the solution of the local problem on ( ( ) A x,ξ ω ξ, φ ) = 0 for every φ Wper(,p ), ω ξ W,p per( ). (.4) A common technique to prove this theorem is Tartar s method of oscillating test functions related to the notion of compensated compactness mentioned above. Another technique is the two-scale convergence method. For a proof see [FM87]. 8

14 . Some Special Cases with Closed Form Expressions for the Homogenized Operator b The homogenized operator b in (.3) depends on the solution of a cell problem (.4), which means that the effective properties of a composite depend in a complicated way on the microstructure. We describe two special cases when we can get closed form expressions for b. We consider (.3) with p = and A(x,ξ) = λ(x)ξ (linear), with ξ R. In Chapter 7 of this thesis, we obtain similar results in an example that deals with nonlinear materials... The Hashin Structure (96) We stu a three-phase composite consisting of three isotropic materials (coated sphere assemblage), let us call them materials,, and 3, with conductivity λ(x)i = [σ χ (x) σ χ (x) σ 3 χ 3 (x)] I where χ i is the characteristic function for the set i and I is the unit matrix. Let the unit cell geometry be described by { = {x : x r }, = x : r x r < }, 3 = { x : x i < } x r,i =,. In order to compute the homogenized coefficients (.3), we need to solve the cell problem (.4) div ( λ(y) φ ξ (y) ) = 0 on, (.5) where φ ξ (y) = ξ y ω ξ (y) and ω ξ (y) is -periodic. In the case ξ = e =[ 0] T, we look for a solution of the type C x, ) x, φ e (x) = x (C K, x x, x, x 3. (.6) It is easily seen that (.6) satisfies (.5) on. By physical reasons, the solution φ ξ (x) as well as the flux λ(x) n φ ξ must be continuous over the boundaries and 3. This gives four equations to solve for the three unknowns C, C, and K. In order to get a consistent solution, we get that σ 3 must be ( σ 3 = σ C K ) = r ( )) σ ( σ σ σ m σ ( ), (.7) σ σ σ m σ 9

15 where m = r, the volume fraction of material in material. Since we now know the r solution ω e (y) = φ e (y) e y of the cell problem, we can compute the homogenized coefficients b( e ) = λ(x)( e ω e ) = [σ 3 0] T and similarly b( e ) = λ(x)( e ω e ) = [0 σ 3 ] T. This means that we can put the coated disk consisting of material coated by material into the homogeneous isotropic material 3 without changing the effective properties (neutral inclusions). By filling the whole cell with such homothetically coated disks, we get an isotropic two-phase composite with conductivity σ 3. For more details see [Mil0]... The Mortola-Steffé Structure and the Checkerboard Structure Let = (0, ) and divide it into four equal parts ( = 0, ) ( ) ( ) ( ),, =,,, ( 3 = 0, ) ( 0, ) ( ) (, 4 =, 0, ). We stu a four-phase composite consisting of four isotropic materials, let us call them materials,, 3, and 4, with conductivity λ(x)i = [αχ (x) βχ (x) γχ 3 (x) δχ 4 (x)]i, where χ i (x) is the characteristic function for the set i and I is the unit matrix. In 985, Mortola and Steffé [MS85] conectured that the homogenized conductivity coefficients of this structure are (λ i ) = ( λ 0 0 λ ), where λ = (αβγ αβδ αγδ βγδ α β γ δ ) (α γ) (β δ) (α β) (γ δ), λ = (αβγ αβδ αγδ βγδ α β γ δ This conecture was proven by Milton in 000. ) (α β) (γ δ) (α γ) (β δ). 0

16 If we let δ = α and γ = β, we get the so called checkerboard structure. We immediately see that the homogenized conductivity coefficients for the checkerboard structure are λ = λ = αβ, the geometric mean. This was proved alrea in 970 by Dykhne, but Schulgasser (977) showed that this was a corollary of Keller s phase interchange identity from 963.

17 Chapter 3 Dirichlet Boundary Value Problem In this chapter, we stu boundary value problems associated with fields inside composites made from two materials with different power law behavior. 3. Description of the Problem The geometry of the composite is specified by the indicator function of the sets occupied by each of the materials. The indicator function of material one and two are denoted by χ and χ, where χ (y) = in material one and is zero outside and χ (y) = χ (y). The constitutive law for the heterogeneous medium is described by A : R n R n R n, A (y,ξ) = σ(y) ξ p(y) ξ = σ χ (y) ξ p ξ σ χ (y) ξ ξ; (3.) where σ(y) = χ (y)σ χ (y)σ, and p(y) = χ (y)p χ (y), periodic in y, with unit period cell = (0, ) n. This simple constitutive model is used in the mathematical description of many physical phenomena including plasticity [PCS97, PCW99, Suq93, Idi08], nonlinear dielectrics [GNP0, GK03, LK98, TW94a, TW94b], and fluid flow [Ruž00, AR06]. We stu the problem of periodic homogenization associated with the solutions u ǫ to the problems ( x )) div A( ǫ, u ǫ = f on, u ǫ W,p 0 (), (3.) where is a bounded open subset of R n, p, f W,q (), and /p /q =. The differential operator appearing on the left hand side of (3.) is commonly referred to as the p ǫ (x)-laplacian. For the case at hand, the exponents p(x) and coefficients σ(x) are taken to be simple functions. Because the level sets associated with these functions can be quite general and irregular they are referred to as rough exponents and coefficients. In this context all solutions are understood in the usual weak sense [ZKO94]. One of the basic problems in homogenization theory is to understand the asymptotic behavior as ǫ 0, of the solutions u ǫ to the problems (3.). It was proved in [ZKO94] that {u ǫ } ǫ>0 converges weakly in W,p () to the solution u of the homogenized problem div (b ( u)) = f on, u W,p 0 (), (3.3)

18 where the monotone map b : R n R n (independent of f and ) can be obtained by solving an auxiliary problem for the operator (3.) on a periodicity cell. The notion of homogenization is intimately tied to the Γ-convergence of a suitable family energy functionals I ǫ as ǫ 0 [DM93], [ZKO94]. Here the connection is natural in that the family of boundary value problems (3.3) correspond to the Euler equations of the associated energy functional I ǫ and the solutions u ǫ are their minimizers. The homogenized solution is precisely the minimizer of the Γ-limit of the sequence {I ǫ } ǫ>0. The connections between Γ limits and homogenization for the power-law materials studied here can be found in [ZKO94]. The explicit formula for the Γ-limit of the associated energy functionals for layered materials was obtained recently in [PS06]. The earlier work of [DMD90] provides the corrector theory for homogenization of monotone operators that in our case applies to composite materials made from constituents having the same power-law growth but with rough coefficients σ(x). The corrector theory for monotone operators with uniform power law growth is developed further in [EP04] where it is used to extend multiscale finite element methods to nonlinear equations for stationary random media. Recent work considers the homogenization of p ǫ (x)-laplacian boundary value problems for smooth exponential functions p ǫ (x) uniformly converging to a limit function p 0 (x) [AAPP08]. There the convergence of the family of solutions for these homogenization problems is expressed in the topology of L p 0( ) () [AAPP08]. 3. Microgeometries Considered We carry out this investigation for two nonlinear power-law materials periodically distributed inside a domain. Here is an open bounded subset of R n, which represents a sample of the material. The length scale of the microstructure relative to the domain is denoted by ǫ. The periodic mixture is described as follows. We introduce the unit period cell = (0, ) n of the microstructure. Let F be an open subset of of material one, with smooth boundary F, such that F. The function χ (y) = inside F and 0 outside and χ (y) = χ (y). We extend χ (y) and χ (y) by periodicity to R n and the ǫ-periodic mixture inside is described by the oscillatory characteristic functions χ ǫ (x) = χ (x/ǫ) and χ ǫ (x) = χ (x/ǫ). Here we will consider the case where F is given by a simply connected inclusion embedded inside a host material (see Figure 3.). A distribution of such inclusions is commonly referred to as a periodic dispersion of inclusions. F Figure 3.: Unit cell: Dispersed Microstructure We also consider layered materials. For this case, the representative unit cell consists of a layer of material one, denoted by R, sandwiched between layers of material two, denoted 3

19 by R. The interior boundary of R is denoted by Γ. Here χ (y) = for y R and 0 in R, and χ (y) = χ (y) (see Figure 3.). R R R Figure 3.: Unit cell: Layered material Γ 3.3 Notation and Preliminary Results On the unit cell, the constitutive law for the nonlinear material is given by (3.) with exponents p and satisfying p. Their Hölder conugates (or dual conugates) are denoted by q = p /(p ) and q = /( ) respectively. For i =,, W,p i per ( ) denotes the set of all functions u W,p i ( ) with mean value zero that have the same trace on the opposite faces of. Each function u W,p i per ( ) can be extended by periodicity to a function of W,p i loc (Rn ). The Euclidean norm and the scalar product in R n are denoted by and (, ), respectively. If D R n, D denotes the Lebesgue measure and χ D (x) denotes its characteristic function. The constitutive law for the ǫ-periodic composite is described by A ǫ (x,ξ) = A (x/ǫ,ξ), for every ǫ > 0, for every x, and for every ξ R n. In the following, the letter C will represent a generic positive constant independent of ǫ, and it can take different values from line to line Properties of A The function A, defined in (3.), satisfies the following properties:. For all ξ R n, A(,ξ) is -periodic and Lebesgue measurable.. A(y, 0) = 0 for all y R n. 3. Continuity: for almost every y R n and for every ξ i R n (i =, ) we have A(y,ξ ) A(y,ξ ) C [ χ (y) ξ ξ ( ξ ξ ) p χ (y) ξ ξ ( ξ ξ ) ]. (3.4) 4. Monotonicity: for almost every y R n and for every ξ i R n (i =, ) we have (A(y,ξ ) A(y,ξ ),ξ ξ ) C (χ (y) ξ ξ p χ (y) ξ ξ ). (3.5) 4

20 Proof of (3.4): Continuity of A Proof. By (3.), we have A(y,ξ ) A(y,ξ ) = σ χ (y) ξ p ξ σ χ (y) ξ ξ σ χ (y) ξ p ξ σ χ (y) ξ ξ σ χ (y) [ ξ p ξ ξ p ξ ] σ χ (y) [ ξ ξ ξ ξ ] C ( χ (y) ξ p ξ ξ p ξ χ (y) ξ ξ ξ ξ ) Let us stu the expression ξ p i ξ ξ p i ξ, for i =,. Observe that ξ pi ξ ξ pi ξ = ξ (p i ) ξ (p i ) ξ p i ξ p i ξ ξ = ξ (p i ) ξ (p i ) ξ p i ξ p i ξ p i ξ p i ξ p i ξ p i ξ ξ = ( ξ p i ξ p i ) ξ p i ξ p i ( ξ ξ ξ ξ ) = ( ξ p i ξ p i ) ( ξ p i ξ p i ) ( ξ ξ ξ ξ ) By Remark 3., we have ( ξ p i ξ p i ) ξ p i ξ p i ξ ξ By Remark 3., we have ξ p i ξ p i ( ξ p i ξ p i ) ξ ξ And by Remark 3.3, we have (p i ) ( ξ p i ξ p i ) ξ ξ ( ξ p i ξ p i ) ξ ξ (p i ) ( ξ p i ξ p i ) ξ ξ ( ξ p i ξ p i ) ξ ξ C ( ξ p i ξ p i ) ξ ξ. C [ ( ξ ξ ) p i ] ξ ξ. Taking square root on both sides, we obtain ξ p i ξ ξ p i ξ C ( ξ ξ ) p i ξ ξ, which proves (3.4). 5

21 Remark 3.. Observe that 0 ( ξ ξ ) 0 ξ ξ ξ ξ Remark 3.. For i =,, we have ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ( ξ ξ ξ ξ ) ξ ξ. ξ i p i ( ξ p i ξ p i ). Remark 3.3. Consider the function f : R R defined by f(x) = x p i, where p i. Since f is a convex function, it satisfies { f ( ξ ) ( ξ ξ ) f ( ξ ) f ( ξ ), f ( ξ ) ( ξ ξ ) f ( ξ ) f ( ξ ) ; or equivalently, { (p i ) ξ p i ( ξ ξ ) ξ p i ξ p i, Then (p i ) ξ p i ( ξ ξ ) ξ p i ξ p i. (p i ) ξ p i ( ξ ξ ) ξ p i ξ p i (p i ) ξ p i ( ξ ξ ). Therefore we have ξ p i ξ p i (pi ) ( ξ p i ξ p i ) ξ ξ. Proof of (3.5): Monotonicity of A Proof. By (3.), we have (A(y,ξ ) A(y,ξ ),ξ ξ ) = (A(y,ξ ),ξ ) (A(y,ξ ),ξ ) (A(y,ξ ),ξ ) (A(y,ξ ),ξ ) = σ χ (y) ξ p σ χ (y) ξ σ χ (y) ξ p ξ ξ σ χ (y) ξ ξ ξ σ χ (y) ξ p ξ ξ σ χ (y) ξ ξ ξ σ χ (y) ξ p σ χ (y) ξ = σ χ (y) [ ξ p ξ p ξ ξ ξ p ξ ξ ξ p ] σ χ (y) [ ξ ξ ξ ξ ξ ξ ξ ξ ] = σ χ (y) [ ξ p ξ p ξ ξ ( ξ p ξ p )] σ χ (y) [ ξ ξ ξ ξ ( ξ ξ )] 6

22 Let us stu the expression ξ p i ξ p i ξ ξ ( ξ p i ξ p i ), for i =,. If ξ = ξ : ξ p i ξ p i ξ ξ ( ξ p i ξ p i ) = ξ p i ξ p i ξ p i ξ ξ = ξ p i [ ξ ξ ξ ξ ] = ξ p i [ ξ ξ ξ ξ ] = ξ p i ξ ξ. Since then ξ ξ ξ ξ = ξ ξ ξ ξ, ξ ξ ξ. Therefore ξ p i ξ p i ( ξ ξ ξ pi ξ p i ) = ξ pi ξ ξ ( ) pi ξ ξ pi ξ ξ If ξ > ξ > 0, we can write ξ = βξ γω, = p i ξ ξ p i. where ω 0 is a vector orthogonal to ξ, and β,γ R with β <. Since we obtain ξ ξ = ξ (βξ γω) = β ξ, (3.6) ξ p i ξ p i ξ ξ ( ξ p i ξ p i ) = ξ p i ξ p i β ξ ( ξ p i ξ p i ). For β 0: ξ p i ξ p i β ξ ( ξ p i ξ p i ) ξ p i ξ p i p i ξ ξ p i. 7

23 For 0 < β < 4 : ξ p i ξ p i β ξ ( ξ p i ξ p i ) = ξ p i ξ p i β ξ p i β ξ p i ξ ξ p i β ξ p i = ξ p i ( β) > ξ p i = 4 ξ p i 4 ξ p i > ξ p i ξ p i 4 4 (pi) ξ ξ p i. For 4 β < : ξ pi ξ pi ( ξ pi ξ pi ) ξ ξ By (3.6), we have Therefore obtaining this way Then = ξ p i ξ p i ξ ξ ξ p i ξ p i ξ ξ = ξ p i ( ξ ξ ξ ) ξ p i ( ξ ξ ξ ) ξ p i ( ξ ξ ξ ) ξ p i ( ξ ξ ξ ) = ξ p i ( ξ ξ ξ ξ ) = ξ p i ξ ξ. ξ ξ ξ ξ ξ β 4. ξ ξ 5; ξ 5 ξ ξ. ξ p i ξ p i ( ξ p i ξ p i ) ξ ξ = ξ p i ξ ξ Taking C = min { 5 p i, (p i) }, we have proved for p i. Therefore 5 p i ξ ξ p i ξ ξ = 5 p i ξ ξ p i. ξ p i ξ p i ξ ξ ( ξ p i ξ p i ) C ξ ξ p i, (A(y,ξ ) A(y,ξ ),ξ ξ ) C (χ (y) ξ ξ p χ (y) ξ ξ ). A different proof for the monotonicity and continuity of A can be found in [Bys05]. 8

24 3.3. Dirichlet BVP and Homogenization Theorem We shall consider the following Dirichlet boundary value problem { div (A ǫ (x, u ǫ )) = f on, u ǫ W,p 0 (); (3.7) where f W,q (). The following homogenization result holds. Theorem 3.4 (Homogenization Theorem). As ǫ 0, the solutions u ǫ of (3.7) converge weakly to u in W,p (), where u is the solution of { div (b ( u)) = f on, u W,p 0 (); (3.8) and the function b : R n R n (independent of f and ) is defined for all ξ R n by b(ξ) = A(y, p(y, ξ)), (3.9) where p : R n R n R n is defined by p(y,ξ) = ξ υ ξ (y), (3.0) where υ ξ is the solution to the cell problem: (A(y,ξ υ ξ ), w) = 0, for every w W,p per ( ), υ ξ W,p per ( ). (3.) For a proof of Theorem 3.4, see Chapter 5 of [ZKO94]. Lemma 3.5. The following a priori bound is satisfied ( ) sup χ ǫ (x) u ǫ (x) p χ ǫ (x) u ǫ (x) C <. (3.) ǫ>0 Proof. The weak formulation for (3.7) is given by (A ǫ (x, u ǫ (x)), ϕ(x)) = for all ϕ W,p 0 (). In particular, taking ϕ = u ǫ above, we obtain (A ǫ (x, u ǫ (x)), u ǫ (x)) = 9 f(x)ϕ(x), f(x)u ǫ (x).

25 The last equation can be rewritten as σ χ ǫ (x) u ǫ p σ χ ǫ (x) u ǫ = Applying Hölder s inequality to the right hand side of (3.3), we obtain f(x)u ǫ (x) = χ ǫ (x)f(x)u ǫ (x) χ ǫ (x)f(x)u ǫ (x) ( f(x) q ( f(x) q ) ( q ) ( q χ ǫ (x) u ǫ (x) p f(x)u ǫ (x). (3.3) ) p ) χ ǫ (x) u ǫ (x). (3.4) If we combine (3.3) and (3.4), and we use the fact that f W,q (), we get σ χ ǫ (x) u ǫ (x) p σ χ ǫ (x) u ǫ (x) [ ( ) ( ) ] C χ ǫ (x) u ǫ (x) p p χ ǫ (x) u ǫ (x) p Poincaré s inequality (see [Neč67]) gives [ ( C χ ǫ (x) u ǫ p ) p ( ) ] χ ǫ (x) u ǫ p and applying oung s inequality, we obtain [ δ p C p χ ǫ (x) u ǫ p δ q q δp χ ǫ (x) u ǫ δ q q ]. By rearranging the terms in the inequality, one gets ( ) σ C δp δ q q p δ q q. χ ǫ (x) u ǫ p ( σ C δ Therefore, by taking δ small enough so that min obtains χ ǫ (x) u ǫ p χ ǫ (x) u ǫ C, where C does not depend on ǫ. ) χ ǫ (x) u ǫ { σ C δp p,σ C δ } is positive, one 0

26 3.3.3 Properties of b The function b, defined in (3.9), satisfies the following properties. Continuity: for every ξ, ξ R n, we have [ p b(ξ ) b(ξ ) C ξ ξ ( ξ p ξ p ξ ξ ) p p (3.5) ] p ξ ξ ( ξ p ξ p ξ ξ ). Monotonicity:for every ξ, ξ R n, we have (b(ξ ) b(ξ ),ξ ξ ) (3.6) ( ) C χ (y) p(y,ξ ) p(y,ξ ) p χ (y) p(y,ξ ) p(y,ξ ) 0. Proof of Continuity of b (3.5) Proof. By (3.9), (3.4), we have b(ξ ) b(ξ ) = A(y,p(y,ξ )) A(y,p(y,ξ )) A(y,p(y,ξ )) A(y,p(y,ξ )) [ C χ (y) p(y,ξ ) p(y,ξ ) ( p(y,ξ ) p(y,ξ ) ) p ] χ (y) p(y,ξ ) p(y,ξ ) ( p(y,ξ ) p(y,ξ ) ) p Applying Hölder s inequality in both integrals, we obtain C [ ( ( ( ( ) χ (y) p(y,ξ ) p(y,ξ ) p p ) χ (y)( p(y,ξ ) p(y,ξ ) ) q (p ) q χ (y) p(y,ξ ) p(y,ξ ) ) ) ] χ (y)( p(y,ξ ) p(y,ξ ) ) q ( ) q

27 By (3.5), we have [ ( ) p C χ (y) (A(y,p(y,ξ )) A(y,p(y,ξ )),p(y,ξ ) p(y,ξ )) ( ( C ( [ ( ) χ (y)( p(y,ξ ) p(y,ξ ) ) q (p ) q χ (y) (A(y,p(y,ξ )) A(y,p(y,ξ )),p(y,ξ ) p(y,ξ )) ) ] χ (y)( p(y,ξ ) p(y,ξ ) ) q ( ) q ( ( ( (A(y,p(y,ξ )) A(y,p(y,ξ )),p(y,ξ ) p(y,ξ )) ) χ (y)( p(y,ξ ) p(y,ξ ) ) q (p ) q (A(y,p(y,ξ )) A(y,p(y,ξ )),p(y,ξ ) p(y,ξ )) ) ] χ (y)( p(y,ξ ) p(y,ξ ) ) q ( ) q By (3.), we get [ ( = C (b(ξ ) b(ξ ),ξ ξ ) p (b(ξ ) b(ξ ),ξ ξ ) ( ) p ) ) ) χ (y)( p(y,ξ ) p(y,ξ ) ) q (p ) q χ (y)( p(y,ξ ) p(y,ξ ) ) q ( ) Applying the Cauchy-Schwarz inequality and Hölder s inequality we have ( ) p C b(ξ ) b(ξ ) p ξ ξ p χ (y) p p p ( ) (p )(p ) χ (y)( p(y,ξ ) p(y,ξ ) ) p (p )(p ) p (p ) (p )(p ) b(ξ ) b(ξ ) ξ ξ ( ( ) p χ (y) p χ (y)( p(y,ξ ) p(y,ξ ) ) ( )( ) ( )( ) ) ( )( ) ( ) ) ] q

28 C [ b(ξ ) b(ξ ) p ξ ξ b(ξ ) b(ξ ) ξ ξ p p θ p θ Lemma 5.3 delivers [ C b(ξ ) b(ξ ) p ξ ξ b(ξ ) b(ξ ) ξ ξ p p θ p θ Applying oung s inequality we obtain C ( ( [ δ p b(ξ ) b(ξ ) p δp b(ξ ) b(ξ ) δ q ξ ξ δ q ξ ξ Therefore [ ( δ p C p δp p p θ p θ C δ q ξ ξ δ q ξ ξ ) p χ (y)( p(y,ξ ) p(y,ξ ) ) p p χ (y)( p(y,ξ ) p(y,ξ ) ) ) ] ( ξ p θ ξ p θ ξ θ ξ θ ) p p ( ξ p θ ξ p θ ξ θ ξ θ ) ( ξ p θ ξ p θ ξ θ ξ θ ) p p q ( ξ p θ ξ p θ ξ θ ξ θ ) p )] b(ξ ) b(ξ ) p p θ p θ Taking δ small enough, we obtain b(ξ ) b(ξ ) [ C ξ ξ ξ ξ p p θ p θ q ( ξ p θ ξ p θ ξ θ ξ θ ) p p q ( ξ p θ ξ p θ ξ θ ξ θ ) p q ( ξ p θ ξ p θ ξ θ ξ θ ) p p ( ξ p θ ξ p θ ξ θ ξ θ ) p ]. ] 3

29 Proof of (3.6): Monotonicity of b Proof. Using (3.) and (3.5), we have (b(ξ ) b(ξ ),ξ ξ ) ( ) = A(y,p(y,ξ )) A(y,p(y,ξ )),ξ ξ = (A(y,p(y,ξ )) A(y,p(y,ξ )),ξ ξ ) = (A(y,p(y,ξ )) A(y,p(y,ξ )),p(y,ξ ) p(y,ξ )) ( C χ (y) p(y,ξ ) p(y,ξ ) p ) χ (y) p(y,ξ ) p(y,ξ ) Properties of p Since the solution υ ξ of (3.) can be extended by periodicity to a function of W,p loc (Rn ), then (3.) is equivalent to div(a(y,ξ υ ξ (y))) = 0 over D (R n ), i.e., div (A(y,p(y,ξ))) = 0 in D (R n ) for every ξ R n. (3.7) Moreover, by (3.), we have (A(y,p(y,ξ)),p(y,ξ)) = For ǫ > 0, define p ǫ : R n R n R n by ( x ) ( x p ǫ (x,ξ) = p ǫ,ξ = ξ υ ξ ǫ (A(y,p(y,ξ)),ξ) = (b(ξ),ξ). (3.8) where υ ξ is the unique solution of (3.). The functions p and p ǫ are easily seen to satisfy the following properties ). (3.9) p(,ξ) is -periodic and p ǫ (x,ξ) is ǫ-periodic in x. (3.0) p(y, ξ) = ξ. (3.) p ǫ (,ξ) ξ in L p (; R n ) as ǫ 0. (3.) p(y, 0) = 0 for almost every y. (3.3) ( ) A ǫ,p ǫ(,ξ) b(ξ) in L q (; R n ), as ǫ 0. (3.4) 4

30 Chapter 4 Higher Order Integrability of the Homogenized Solution In this chapter, we display higher order integrability results for the field gradients inside dispersed microstructures and layered materials. For dispersions of inclusions, the included material is taken to have a lower power-law exponent than that of the host phase. For both of these cases it is shown that the homogenized solution lies in W, 0 (). In the following chapters we will apply these facts to establish strong approximations for the sequences {χ ǫ i u ǫ } ǫ>0 in L (, R n ). The approach taken here is variational and uses the homogenized Lagrangian associated with b(ξ) defined in (3.9). The integrability of the homogenized solution u of (3.8) is determined by the growth of the homogenized Lagrangian with respect to its argument. 4. Statement of the Theorem on the Higher Order Integrability of the Homogenized Solution We now state the higher order integrability properties of the homogenized solution for periodic dispersions of inclusions and layered microgeometries. Theorem 4.. Given a periodic dispersion of inclusions or a layered material then the solution u of (3.8) belongs to W, 0 (). Before we can prove this theorem we need some definitions. Definition 4.. Functions f(x,ξ) depending on two variables x,ξ R n will be referred to as Lagrangians. Definition 4.3. If the Lagrangian f(x,ξ), ξ R n, x R n satisfies c 0 c ξ p f(x,ξ) c ξ p c 0 (4.) with c 0 0,c,c > 0, and p >, then it is called standard. A much wider class of Lagrangians which includes the standard ones, is specified by the estimate c 0 c ξ p f(x,ξ) c ξ c 0 (4.) 5

31 with p >. These are called nonstandard Lagrangians. Definition 4.4. The conugate of a nonstandard Lagrangian f, denoted g = f, is defined by g(x,ξ) = sup {ξ η f(x,η)}, (4.3) η R n and satisfies the estimate c 0 c ξ q g(x,ξ) c ξ q c 0. (4.4) 4. Proof of Higher Order Integrability of the Homogenized Solution To proceed, we introduce the local Lagrangian associated with power-law composites. The Lagrangian corresponding to the problem studied here is given by with f(x,ξ) = q(x) ξ p(x) = σ p χ (x) ξ p σ χ (x) ξ, (4.5) q(x) = σ p χ (x) σ χ (x); where ξ R n and x R n. Here ξ f(x,ξ) = A (x,ξ), where A(x,ξ) is given by (3.). We consider the rescaled Lagrangian f ǫ (x,ξ) = f ( x ) ǫ,ξ = σ χ ǫ p (x) ξ p σ χ ǫ p (x) ξ, (4.6) where χ ǫ i(x) = χ i (x/ǫ), i =,, ξ R n, and x R n. The Dirichlet problem given by (3.7) is associated with the variational problem given by { } E(f) ǫ = f ǫ (x, u) f,u, (4.7) inf u W,p 0 () with f W,q (). Here (3.7) is the Euler equation for (4.7). However, we also consider { } E(f) ǫ = f ǫ (x, u) f,u, (4.8) inf u W, 0 () with f W,q () (See [Zhi9]). Here, is the duality pairing between W,p 0 () and W,q (). From [ZKO94], we have lim ǫ 0 E ǫ i = E i, for i =,, where E i = inf u W,p i 0 () { } ˆ f i ( u(x)) f,u. (4.9) 6

32 In (4.9), ˆ fi (ξ) is given by ˆ f i (ξ) = inf v in W,p i per ( ) f(y, ξ v(y)) (4.0) and satisfies c 0 c ξ p ˆ fi (ξ) c ξ c 0. (4.) In general, (see [Zhi95]) Lavrentiev phenomenon can occur and E < E. However, for periodic dispersed and layered microstructures, no Lavrentiev phenomenon occurs and we have the following Homogenization Theorem. Theorem 4.5. Homogenization Theorem for periodic dispersed and layered microstructures. For periodic dispersed and layered microstructures, the homogenized Dirichlet problems satisfy E = E, where ˆ f = ˆ f = ˆ f and c c ξ ˆ f(ξ). Moreover, ξ ˆ f(ξ) = b(ξ), where b is the homogenized operator (3.9). Proof. Theorem 4.5 has been proved for dispersed periodic media in Chapter 4 of [ZKO94]. We prove Theorem 4.5 for layers following the steps outlined in [ZKO94]. We first show that ˆ f = ˆ f = ˆ f holds for layered media. Then we show that the homogenized Lagrangian ˆ f satisfies the standard estimate given by c 0 c ξ ˆ f(ξ) c ξ c 0 (4.) with c 0 0, and c,c > 0. We introduce the space of functions W, (R ) that belong to W, (R ) and are periodic on R. Lemma 4.6. Any function v W, (R ) can be extended to R in such a way that the extension ṽ(y) belongs to W, per ( ) and ṽ(y) = v(y) on R. Proof. Let ϕ to be the solution of p ϕ = 0, on R ϕ takes periodic boundary values on opposite faces of R ϕ = v, on Γ Here the subscript indicates the trace on the R side of Γ and indicates the trace on the R side of Γ. For a proof of existence of the solution ϕ see [Eva8] or [Lew77]. The extension ṽ is given by { v, in R. ṽ = ϕ, on R. 7

33 It is clear that ˆ f ˆ f. To prove ˆ f = ˆ f, it suffices to show that for every v W p satisfying f(y,ξ v(y)) < there exists a sequence v ǫ W, per ( ) such that f(y,ξ v ǫ (y)) = f(y,ξ v(y)). lim ǫ 0 per( ) For v as above, let ṽ be as in Lemma 4.6 and set z = v ṽ. It is clear that z W,p (R ), is periodic on opposite faces of R, zero on Γ and we write f(y,ξ v(y)) = f (ξ v(y)) R f (ξ ṽ(y) z(y)), R where f (ξ) = σ p ξ p and f (ξ) = σ ξ ; i.e, f and f are standard Lagrangians satisfying (4.) with exponents p and respectively. We can choose a sequence {z ǫ } ǫ>0 C 0 (R ) such that z ǫ vanishes in R and z ǫ z in W,p (R ). Define v ǫ W, per ( ) by v ǫ = Since v ǫ v in W,p per ( ), we see that { v in R, ṽ z ǫ in R. f(y,ξ v ǫ (y)) ( = lim f (ξ v(y)) ǫ 0 R f (ξ ṽ(y) z ǫ (y)) R ) = f (ξ v(y)) R f (ξ ṽ(y) z(y)) R = f(y,ξ v(y)). lim ǫ 0 Therefore ˆ f = ˆ f = ˆ f for layered media. We establish (4.) by introducing the convex conugate of ˆ f. We denote the convex dual of ˆ fi (ξ) by ˆ g i (ξ) and the convex dual of f by g. It is easily verified (see [Zhi9]) that ˆ g i (ξ) = inf g(y,ξ w(y)) (4.3) w in Sol q i( ) and c 0 c ξ q ˆ g i (ξ) c ξ q c 0. (4.4) Here Sol q i ( ) are the solenoidal vector fields belonging to L q i (, R n ) and having mean value zero Sol q i ( ) = {w L q i ( ; R n ) : divw = 0,w n anti-periodic}. 8

34 We will show that ˆ g = ˆ g = ˆ g satisfies ˆ g(ξ) c ξ q c, and apply duality to recover ˆ f(ξ) c ξ c. To get the upper bound on ˆ g we use the following lemma. Lemma 4.7. There exists τ with divτ = 0 in, such that τ n is anti-periodic on the boundary of, τ = ξ in R, and τ(y) q C ξ q. Proof. Let the function ϕ W, (R ) be the solution of ϕ ϕ p n is anti-periodic on R ; p ϕ = 0 in R ; ϕ ϕ n = ξ n ; on Γ, where the subscript indicates the trace on the R side of Γ and indicates the trace on the R side of Γ. The Neumann problem given above is the stationarity condition for the energy φ φξ nds when minimized over all φ W, (R ). The solution of the R Γ Neumann problem is unique up to a constant. Here the anti-periodic boundary condition on ϕ ϕ p n is the natural boundary condition for the problem. Now we define τ according to { ξ; in R τ = ϕ ϕ p ; in R and it follows that ξ q ; in R τ q = [ ( ϕ ϕ ) ] q = ( ϕ ) q = ϕ ; in R. (4.5) Then, for ψ W, (R ) we have ϕ p ϕ ψ (4.6) R = ψ ϕ p ϕ nds ψ ϕ p ϕ nds Γ R = ψξ nds. Γ 9

35 Set ψ = ϕ in (4.6) and an application of Hölder s inequality gives ϕ = ϕξ nds R Γ = (ϕξ) R = ( ϕ) ξ R ( ) /p ( ) /q ϕ ξ q. R R Then ϕ R ξ q. R (4.7) Therefore, using (4.5) and (4.7), we have τ(y) q = τ(y) q R τ(y) q R = ξ q R ϕ(y) R C ξ q. Taking ˆ g to be the conugate of ˆ f, and choosing τ in Sol q ( ) as in Lemma 4.7, we obtain ˆ g(ξ) = inf g(y,ξ τ) τ in Sol q ( ) g(y,ξ τ) g(y, 0) R g(y,ξ τ) R c c R ξ τ q c c ξ q, and the left hand inequality in (4.) follows from duality. This concludes the proof of Theorem 4.5. Collecting results we now prove Theorem 4.. Indeed the minimizer of E is precisely the solution u of (3.8). Theorem 4.5 establishes the coercivity of E over W, 0 (), thus the solution u lies in W, 0 (). 30

36 Chapter 5 Corrector Theorem In this chapter, we develop new strong convergence results that capture the asymptotic behavior of the gradients u ǫ, as ǫ tends to 0. Our approach delivers strong approximations for the gradients inside each phase χ ǫ i u ǫ, i =,. Homogenization theory relates the average behavior seen at large length scales to the underlying heterogeneous structure. It allows one to approximate { u ǫ } ǫ>0 in terms of u, where u is the solution of the homogenized problem (3.3). The homogenization result given in [ZKO94] shows that the average of the error incurred in this approximation of u ǫ decays to 0. We present a new corrector result that delivers an approximation to u ǫ up to an error that converges to zero strongly in the norm. The corrector results are presented for layered materials and for dispersions of inclusions embedded inside a host medium. For the dispersed microstructures the included material is taken to have the lower power law exponent than that of the host phase. For both of these cases it is shown that the homogenized solution lies in W, 0 () (See Chapter 4). With this higher order integrability in hand, we provide an algorithm for building correctors and establish strong approximations for the sequences {χ i ǫ u ǫ } ǫ>0 in L (,R n ), see Theorem 5.. When the host phase has a lower power-law exponent than the included phase one can only conclude that the homogenized solution lies in W,p 0 () and the techniques developed here do not apply. 5. Statement of the Corrector Theorem We now describe the family of correctors that provide a strong approximation of the sequence {χ ǫ i u ǫ } ǫ>0 in the L p i (, R n ) norm. We denote the rescaled period cell with side length ǫ > 0 by ǫ and write ǫ i = ǫi ǫ, where i Z n. In what follows it is convenient to define the index set I ǫ = {i Z n : ǫ i }. For ϕ L (; R n ), we define the local average operator M ǫ associated with the partition ǫ i, i I ǫ by χ i ǫ (x) ϕ(y); if x M ǫ (ϕ)(x) = i i Iǫ ǫ i, i I ǫ ǫ ǫ i (5.) 0; if x \ i Iǫ i ǫ. 3

37 Remark 5.. The family M ǫ has the following properties. For i =,, M ǫ (ϕ) ϕ L p i(;r n ) 0 as ǫ 0. For a proof, see, for instance Chapter 8 of [Zaa58].. M ǫ (ϕ) ϕ a.e. on. For a proof, see, for instance Chapter 8 of [Zaa58]. 3. From Jensen s inequality, we have for every ϕ L (; R n ) and i =,. M ǫ (ϕ) L p i(;r n ) ϕ L p i(;r n ), The strong approximation to the sequence {χ ǫ i u ǫ } ǫ>0 is given by the following corrector theorem. Theorem 5. (Corrector Theorem). Let f W,q (), let u ǫ be the solution to the problem (3.7), and let u be the solution to problem (3.8). Then, for periodic dispersions of inclusions and layered materials and i =,, we have χ ǫ i(x)p ǫ (x,m ǫ ( u)(x)) χ ǫ i(x) u ǫ (x) p i 0, as ǫ 0. (5.) Before we can give the proof of this theorem, we need the results from the following section. 5. Some Properties of Correctors In this section, we state and prove a priori bounds and convergence properties for the sequences p ǫ defined in (3.9), u ǫ, and A ǫ (x,p ǫ (x, u ǫ )) that are used in the proof of Theorem 5.. In the following, the letter C will represent a generic positive constant independent of ǫ, and it can take different values from line to line. Lemma 5.3. For every ξ R n, we have χ (y) p(y,ξ) p χ (y) p(y,ξ) C ( ξ p θ ξ θ ), (5.3) and by a change of variables, we obtain χ ǫ (x) p ǫ (x,ξ) p ǫ χ ǫ (x) p ǫ (x,ξ) C ( ξ p θ ξ θ ) ǫ. ǫ (5.4) Proof. Let ξ R n. By (3.5), we have that (A(y,p(y,ξ)),p(y,ξ)) C (χ (y) p(y,ξ) p χ (y) p(y,ξ) ). 3

38 Integrating both sides over and using (3.), we get χ (y) p(y,ξ) p χ (y) p(y,ξ) C (A(y,p(y,ξ)),p(y,ξ)) = C (A(y,p(y,ξ)),ξ) By Cauchy-Schwarz Inequality and (3.4), we have C A(y,p(y,ξ)) ξ [ C χ (y) p(y,ξ) ( p(y,ξ) ) p ξ ] χ (y) p(y,ξ) ( p(y,ξ) ) p ξ [ ] C χ (y)( p(y,ξ) ) p ξ χ (y)( p(y,ξ) ) p ξ Using oung s Inequality, we obtain C δ q δ q χ (y)( p(y,ξ) ) p δ p χ (y) ξ p q p χ (y)( p(y,ξ) ) δ q χ (y) ξ [ = C δ q χ (y)( p(y,ξ) ) p δ p ξ p θ ] δ q χ (y)( p(y,ξ) ) δ ξ θ [ C δ q θ δ q χ (y) p(y,ξ) p δ p ξ p θ δ q θ ] δ q χ (y) p(y,ξ) δ ξ θ C [ (δ q θ δ q θ ) ( δ p ξ p θ δ ξ ) θ ( )] (δ q δ q ) χ (y) p(y,ξ) p χ (y) p(y,ξ). 33

39 Doing some algebraic manipulations, we obtain ( ) ( C(δ q δ q )) χ (y) p(y,ξ) p χ (y) p(y,ξ) [ C (δ q θ δ q θ ) ( δ p ξ p θ δ ξ ) ] θ On choosing an appropiate δ, we finally obtain (5.3). Lemma 5.4. For every ξ,ξ R n, we have χ (y) p(y,ξ ) p(y,ξ ) p χ (y) p(y,ξ ) p(y,ξ ) [ C ( ξ p θ ξ θ ξ p θ ξ θ ) p p ξ ξ p ( ξ p θ ξ θ ξ p θ ξ θ ) ξ ξ p p θ p θ and by doing a change a variables, we obtain χ ǫ (x) p ǫ (x,ξ ) p ǫ (x,ξ ) p χ ǫ (x) p ǫ (x,ξ ) p ǫ (x,ξ ) ǫ [ ǫ C ( ξ p θ ξ θ ξ p θ ξ θ ) p p ξ ξ p p θ ( ξ p θ ξ θ ξ p θ ξ θ ) ξ ξ Proof. By (3.5), we have (A(y,p(y,ξ )) A(y,p(y,ξ )),p(y,ξ ) p(y,ξ )) p p θ C (χ (y) p(y,ξ ) p(y,ξ ) p χ (y) p(y,ξ ) p(y,ξ ) ). Integrating over and using (3.) and the Cauchy-Schwarz inequality, we get χ (y) p(y,ξ ) p(y,ξ ) p χ (y) p(y,ξ ) p(y,ξ ) C (A(y,p(y,ξ )) A(y,p(y,ξ )),p(y,ξ ) p(y,ξ )) = C (A(y,p(y,ξ )) A(y,p(y,ξ )),ξ ξ ) C A(y,p(y,ξ )) A(y,p(y,ξ )) ξ ξ Using (3.4), we have [ C χ (y) p(y,ξ ) p(y,ξ ) ( p(y,ξ ) p(y,ξ ) ) p ξ ξ ] χ (y) p(y,ξ ) p(y,ξ ) ( p(y,ξ ) p(y,ξ ) ) p ξ ξ 34 ], (5.5) ] ǫ (5.6)

40 Using Hölder s inequality in the first expression with r = p p, r = p, and r 3 = p ; and in the second expression with s =, s =, and s 3 =, we obtain C [ ( ( ( C ( [ ( ) p χ (y) ( p(y,ξ ) p(y,ξ ) ) p p ) ( χ (y) p(y,ξ ) p(y,ξ ) p p χ (y) ( p(y,ξ ) p(y,ξ ) ) ( ( ( ) ) χ (y) ξ ξ p p ) ( ) ] χ (y) p(y,ξ ) p(y,ξ ) p χ (y) ξ ξ p ) p χ (y) χ (y) p(y,ξ ) p χ (y) p(y,ξ ) p p χ (y) p(y,ξ ) p(y,ξ ) p χ (y) ) χ (y) p(y,ξ ) χ (y) p(y,ξ ) p(y,ξ ) ) p ξ ξ θ p ) χ (y) p(y,ξ ) p ] ξ ξ θ Use Lemma 5.3 to get C [( ξ p θ ξ θ ξ p θ ξ θ ) p p ( ) p ξ ξ θ χ (y) p(y,ξ ) p(y,ξ ) p p ( ξ p θ ξ θ ξ p θ ξ θ ) ( ) ] p ξ ξ θ χ (y) p(y,ξ ) p(y,ξ ) p By oung s Inequality, we obtain C δ q ( ξ p θ ξ θ ξ p θ ξ θ ) (p )q p δ p q χ (y) p(y,ξ ) p(y,ξ ) p δ p 35 ξ ξ q q p θ χ (y) p(y,ξ ) p(y,ξ )

41 δ q ( ξ p θ ξ θ ξ p θ ξ θ ) (p )q q ξ ξ q q p θ Straightforward algebraic manipulations deliver ( ) k δ χ (y) p(y,ξ ) p(y,ξ ) p χ (y) p(y,ξ ) p(y,ξ ) C δ q ( ξ p θ ξ θ ξ p θ ξ θ ) (p )q p ξ ξ q q p θ = C δ q ( ξ p θ ξ θ ξ p θ ξ θ ) (p )q q q. ξ ξ q q p θ δ q ( ξ p θ ξ θ ξ p θ ξ θ ) p p ξ ξ p δ q ( ξ p θ ξ θ ξ p θ ξ θ ) p ξ ξ {( ) ( where k δ = min cδp p, cδ )}. The result (5.5) follows on choosing δ small enough so that k δ is positive. Lemma 5.5. Let ϕ be such that { sup ǫ>0 χ ǫ (x) ϕ(x) p and let Ψ be a simple function of the form Ψ(x) = q q } χ ǫ (x) ϕ(x) <, p p θ p θ, m η χ (x), (5.7) with η R n \ {0},, = 0, k = for k and,k =,...,m; and set m η 0 = 0 and 0 = \. Then = ( lim sup ǫ 0 χ ǫ (x) p ǫ (x,m ǫ ϕ(x)) p ǫ (x, Ψ(x)) p ) χ ǫ (x) p ǫ (x,m ǫ ϕ(x)) p ǫ (x, Ψ(x)) { lim sup ǫ 0 C i= [( χ ǫ (x) ϕ(x) p χ ǫ (x) ϕ(x) 36

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