Homogenization of Nonlinear Partial Differential Equations
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 Jim´enez B.Sc. in Mathematics, Universidad de Costa Rica, Costa Rica, 2002 M.S. in Mathematics, Louisiana State University, USA, 2006 August 2010
Acknowledgements “Caminante, son tus huellas el camino y nada m´as; Caminante, no hay camino, se hace camino al andar. Al andar se hace el camino, y al volver la vista atr´as 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 study 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-0807265 and AFOSR Grant FA9550-08-1-0095. 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 already left, thank you very much. It was a great opportunity for me to attend LSU and I have thoroughly enjoyed it. I dedicate this dissertation to my family: my parents William Jim´enez Sol´ıs and Xenia Bola˜ nos Rodr´ıguez, my brothers William and Jos´e, my aunts Marcela, Vera, Yadira and Yolanda, my uncle Mario, my cousins Sof´ıa, Alonso, and Ignacio, and especially to my grandma Berta. Thank you all so much for the visits, phone calls, chats, emails, support, and love.
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Table of Contents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 2 Basic Ideas in Homogenization Theory . . . . . . . . . . . . . . . . . . . . 2.1 Motivation and Examples . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 1-Dimensional Example of Homogenization . . . . . . . . . . . 2.1.2 Homogenization in Rn . . . . . . . . . . . . . . . . . . . . . . 2.2 Some Special Cases with Closed Form Expressions for the Homogenized Operator b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Hashin Structure (1962) . . . . . . . . . . . . . . . . . . . 2.2.2 The Mortola-Steff´e Structure and the Checkerboard Structure
3 3 5 8 9 9 10
Chapter 3 Dirichlet Boundary Value Problem . . . . . . . 3.1 Description of the Problem . . . . . . . . . 3.2 Microgeometries Considered . . . . . . . . 3.3 Notation and Preliminary Results . . . . . 3.3.1 Properties of A . . . . . . . . . . . 3.3.2 Dirichlet BVP and Homogenization 3.3.3 Properties of b . . . . . . . . . . . 3.3.4 Properties of p . . . . . . . . . . .
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12 12 13 14 14 19 21 24
Chapter 4 Higher Order Integrability of the Homogenized Solution . . . . . . . . . . 4.1 Statement of the Theorem on the Higher Order Integrability of the Homogenized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Proof of Higher Order Integrability of the Homogenized Solution . . .
25
Chapter 5 Corrector Theorem . . . . . . . . . . . . . 5.1 Statement of the Corrector Theorem 5.2 Some Properties of Correctors . . . . 5.3 Proof of the Corrector Theorem . . .
31 31 32 51
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Chapter 6 Lower Bounds on Field Concentrations . . . . . . . . . . . . . . . . . . . 6.1 Statement of the Lower Bound on the Amplification of the Macroscopic Field by the Microstructure . . . . . . . . . . . . . . . . . . . . . . . 6.2 Young Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Proof of Lower Bound on the Amplification of the Macroscopic Field by the Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 7 Nonlinear Neutral Inclusions . . . . . . . . . . . 7.1 Double Coated Nonlinear Neutral Inclusions 7.1.1 Statement of the Problem and Result 7.2 Calculations . . . . . . . . . . . . . . . . . .
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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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.
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Chapter 1 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 study 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 study 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 macro1
scopic applied fields. We conclude in Chapter 7 where we will introduce special neutrally conducting microstructures made with power law materials.
2
Chapter 2 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.
2.1
Motivation and Examples
Suppose we would like to know the stationary temperature distribution in an homogeneous body Ω ⊂ R3 with an internal heat source f , heat conductivity A and zero temperature on the boundary ∂Ω. The model to describe this problem is given by the following boundary value problem: Find u ∈ W01,p (Ω), 1 < p < ∞, such that −div (A (∇u)) = f on Ω,
(2.1)
where Ω is a bounded open subset of Rn , f is a given function on Ω, and A : Rn → Rn satisfies suitable continuity and monotonicity conditions that allows the existence and uniqueness of the solution of (2.1). Now, suppose that we would like to be able to model the case when the underlying material is heterogeneous. Then we replace A in (2.1) with a map A : Ω × Rn → Rn to obtain ( −div (A (x, ∇u)) = f on Ω (2.2) u ∈ W01,p (Ω). Since (2.2) depends on x, this is much more difficult to handle than (2.1). 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
with respect to Ω. By periodicity, we can divide Ω into periodic cells, and we call the representative unit cell by Y (the microstructure of a given periodic material can be described ¢ ¡ by several different period cells). This is described by maps of the form Aǫ (x, ξ) = A xǫ , ξ , where A(·, ξ) is assumed to be Y -periodic and ǫ is the fineness of the periodic structure. Equation (2.2) becomes ( ¡ ¡ ¢¢ −div (Aǫ (x, ∇uǫ )) = −div A xǫ , ∇uǫ = f on Ω (2.3) uǫ ∈ W01,p (Ω). The function uǫ can be interpreted as the electric potential, magnetic potential, or the temperature and Aǫ describes the physical properties of the different materials constituting the body (they are the dielectric coefficients, the magnetic permeability and the thermic conductivity coefficients, respectively). Let {ǫk }∞ k=0 be a sequence of positive real numbers such that ǫk → 0 as k → ∞. In this way we get a sequence of problems, one for each value of k. The smaller ǫk gets, the finer the microstructure becomes. It is natural to ask ourselves if there exists some type of convergence of the solutions uǫk . If we assume convergence in an appropriate sense, that is uǫk → u, as k → ∞, we could also ask if u satisfies an equation of a similar type as the one uǫk satisfies ( −div (b (x, ∇u)) = f, on Ω u ∈ W01,p (Ω), 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 Ω (2.4) u ∈ W01,p (Ω). The “homogenized” b represents the physical parameters of a homogeneous body, whose behavior is equivalent, from a “macroscopic” point of view, to the behavior of the material with the given periodic microstructure, described by (2.3). 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 ½Z ³ ¾ Z ´ x Eǫ = min g , ∇u(x) dx − f udx , ǫ u∈W01,p (Ω) Ω Ω 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
called Γ-convergence (Introduced by DeGiorgi [DG75]). From the theory of Γ-convergence it follows that Eǫ → Ehom , as ǫ → 0, where ½Z ¾ Z Ehom = min ghom (∇u(x)) dx − f udx . 1,p u∈W0 (Ω)
Ω
Ω
Here the homogenized energy density function ghom is given by Z 1 ghom (ξ) = g(x, ξ + ∇u)dx, min 1,p |Y | u∈Wper (Y ) Y 1,p where Wper (Y ) is the set of all functions u ∈ W 1,p (Y ) which are Y -periodic and have mean value zero. Again we note that the limit problem does not depend on x, that is, ghom 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.
2.1.1
1-Dimensional Example of Homogenization
Let Ω = (0, 1), f ∈ H −1 (Ω), and A ∈ L∞ (Ω) be a measurable and periodic function with period 1 satisfying 0 < β1 ≤ A(x) ≤ β2 < ∞, for a.e. x ∈ R. (2.5) Remark 2.1. For example, consider a periodic mixture of two materials. Let χ1 be the the characteristic function of material 1 and χ2 be the the characteristic function of material 2, both periodic of periodicity 1. Let A(y) = β1 χ1 (y) + β2 χ2 (y) defined in Ω = (0, 1) and extend it to all R by periodicity, and call this extension by A as well. Note that A ∈ L∞ (Ω), because kAkL∞ (Ω) = β2 and clearly it satisfies (2.5) for all x ∈ R. ³ ´ We define Aǫk (x) = A ǫxk . The weak form of (2.3) becomes Z
1
Aǫk (x) ∂x uǫk (x)∂x φ(x)dx =
0
uǫk ∈
W01,2 (0, 1),
Z
0
1
f (x)φ(x)dx for every φ ∈ W01,2 (0, 1),
(2.6)
and (2.3) becomes ( −∂x (Aǫk (x)∂x uǫk (x)) = f (x) uǫk ∈ W01,2 (0, 1). 5
in (0, 1),
(2.7)
By a standard result in the existence theory of partial differential equations (using LaxMilgram Lemma [Eva98]), there exists a unique solution of these problems for each k. By choosing φ = uǫk in (2.6) and taking (2.5) into account, we obtain by H¨older’s inequality that Z 1 2 β1 k∂x uǫk kL2 (0,1) ≤ Aǫk (x) |∂x uǫk (x)|2 dx Z0 1 f (x)uǫk (x)dx = 0
≤ kf kH −1 (Ω) kuǫk kW 1,2 (Ω) . 0
Recall that
kuǫk k2W 1,2 (Ω) = kuǫk k2L2 (Ω) + k∂x uǫk k2L2 (Ω) . 0
The Poincar´e inequality for functions with zero boundary values states that there is a constant C only depending on Ω = (0, 1) such that kuǫk kL2 (Ω) ≤ C k∂x uǫk kL2 (Ω) . This implies that
kuǫk k2W 1,2 (Ω) ≤ C, 0
(2.8)
where C is a constant independent of k. Since W01,2 (Ω) is reflexive, there exists a subsequence, still denoted by {uǫk }, such that uǫk ⇀ u∗ in W01,2 (Ω).
(2.9)
Since W01,2 (Ω) is compactly embedded in L2 (Ω), we have by the Rellich Embedding Theorem that uǫk → u∗ in L2 (Ω).
In general, however, we only have that
∂x uǫk ⇀ ∂x u∗ in L2 (Ω). Since A is 1-periodic, we have that {Aǫk } converges weakly* L∞ (Ω), as k → ∞, to its arithmetic mean hAi, i.e., Z 1 ∗ A(y)dy in L∞ (Ω). (2.10) Aǫk ⇀ hAi = 0
From (2.6), (2.9), and (2.10), it could be reasonable to assume that, in the limit, we have Z 1 Z 1 f (x)φ(x)dx for every φ ∈ W01,2 (0, 1), hAi ∂x u∗ (x)∂φ(x)dx = 0 0 u∗ ∈ W01,2 (0, 1).
However, this is not true in general, since Aǫk ∂x uǫk is the product of two weakly converging sequences. This is the main difficulty in the limit process. To obtain the correct answer we 6
proceed in the following way: first we note that, according to (2.10) and (2.8), {Aǫk ∂x uǫk } is bounded in L2 (Ω) and that (2.6) implies that −∂x (Aǫk (x)∂x uǫk ) = f . Hence there is a constant C independent of k such that kAǫk ∂x uǫk kW 1,2 (Ω) ≤ C. As before, since W 1,2 (Ω) is reflexive, there exists a subsequence, still denoted {Aǫk ∂x uǫk } and a M 0 ∈ L2 (Ω) such that Since
n
1 Aǫ k
Aǫk ∂x uǫk → M 0 in L2 (Ω). o ® converges to A1 weakly* in L∞ (Ω) by periodicity (and hence weakly in
L2 (Ω)), we have
∂x uǫk =
µ
1 Aǫk
¶
¿ À 1 (Aǫk ∂x uǫk ) ⇀ M 0 , in L2 (Ω). A
(2.11)
Thus, by (2.9) and (2.11), we see that
®−1 1 M 0 = 1 ® ∂x u∗ = A−1 ∂ x u∗ . A
Now, by passing to the limit in (2.6) we obtain that Z 1 Z 1 b∂x du∗ ∂x φdx = f (x)φ(x)dx for every φ ∈ W01,2 (0, 1), 0 0 u∗ ∈ W01,2 (0, 1).
where the homogenized operator is given by b =
1
−1
= hA−1 i , the harmonic mean of A;
h i ¿ À 1 1 1 ≤ ≤ , β2 A β1 we conclude that the homogenized equation has a unique solution and thus that the whole sequence {uǫk } converges. and since
1 A
Remark 2.2. For the example given in Remark 2.1, we obtain M 0 = hθ ∂x u∗ , where ¡ ¢−1 hθ = θ1 β1−1 + θ2 β2−1
and
θ1 =
Z
0
1
χ1 (y)dy, and θ2 = 1 − θ1 .
Remark 2.3. The corresponding homogenization problem for the one-dimensional Poisson equation Z Z A (x) |∂ u |p−2 ∂ u ∂ φdx = f (x)φ(x)dx for every φ ∈ W01,p (Ω), ǫk x ǫk x ǫk x Ω Ω uǫk ∈ W01,p (Ω) D 1 E1−p . gives the homogenized operator b = A 1−p 7
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 L2 (Ω) converges in the sense of distributions to the product of their limits.
2.1.2
Homogenization in Rn
Assume that A satisfies suitable structure conditions. Remark 2.4. A common assumption is that A(x, ξ) satisfies the conditions |A(x, ξ1 ) − A(x, ξ2 )| ≤ c1 λ(x) (1 + |ξ1 | + |ξ2 |)p−1−α |ξ1 − ξ2 |α ,
(A(x, ξ1 ) − A(x, ξ2 ), ξ1 − ξ2 ) ≥ c2 λ(x) (1 + |ξ1 | + |ξ2 |)p−β |ξ1 − ξ2 |β ,
for constants c1 ,c2 > 0, where α and β satisfy 0 ≤ α ≤ min(1, p − 1) and max(p, 2) ≤ β < ∞ (see, for example, [DMD90, FM87]). For example, these conditions are satisfied by the p-Poisson operator A(x, ξ) = λ(x) |ξ|p−2 ξ, ³ ³ ¡ √ ¢2−p ´ ¡ √ ¢2−p ´ where c1 , c2 can be chosen as c1 = max p − 1, 2 2 , and c2 = min p − 1, 2 2 . We have the following homogenization theorem.
Theorem 2.5. Let 1 < p < ∞ and q its dual conjugate. The solutions uǫk of ( −div (Aǫk (x, ∇uǫk )) = f on Ω, uǫk ∈ W01,p (Ω) satisfy
uǫk ⇀ u in W01,p (Ω),
(2.12)
and Aǫk (x, ∇uǫk ) ⇀ b(∇u) in Lq (Ω, Rn ),
as k → ∞, where u is the solution of the homogenized equation ( −div (b (∇u)) = f on Ω, u ∈ W01,p (Ω), where the homogenized operator b : Rn → Rn is defined by Z 1 b(ξ) = A(x, ξ + ∇ω ξ (x))dx, |Y | Y
and where ω ξ is the solution of the local problem on Y Z ¡ ¡ ¢ ¢ 1,p A x, ξ + ∇ω ξ , ∇φ dx = 0 for every φ ∈ Wper (Y ), Y ξ 1,p ω ∈ Wper (Y ).
(2.13)
(2.14)
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
2.2
Some Special Cases with Closed Form Expressions for the Homogenized Operator b
The homogenized operator b in (2.13) depends on the solution of a cell problem (2.14), 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 (2.3) with p = 2 and A(x, ξ) = λ(x)ξ (linear), with ξ ∈ R2 . In Chapter 7 of this thesis, we obtain similar results in an example that deals with nonlinear materials.
2.2.1
The Hashin Structure (1962)
We study a three-phase composite consisting of three isotropic materials (coated sphere assemblage), let us call them materials 1, 2, and 3, with conductivity λ(x)I = [σ1 χΩ1 (x) + σ2 χΩ2 (x) + σ3 χΩ3 (x)] I where χΩi is the characteristic function for the set Ωi and I is the unit matrix. Let the unit cell geometry be described by ½ ¾ 1 Ω1 = {x : |x| ≤ r1 } , Ω2 = x : r1 ≤ |x| ≤ r2 < , 2 ¾ ½ 1 Ω3 = x : |xi | < ∧ |x| ≥ r2 , i = 1, 2 . 2 In order to compute the homogenized coefficients (2.13), we need to solve the cell problem (2.14) ¡ ¢ −div λ(y)∇φξ (y) = 0 on Y , (2.15)
where φξ (y) = ξ · y + ω ξ (y) and ω ξ (y) is Y -periodic. In the case ξ = e~1 =[1 0]T , we look for a solution of C1 x ³1 , ´ K2 φe~1 (x) = x1 C2 + |x| , 2 x , 1
the type x ∈ Ω1 ,
x ∈ Ω2 ,
(2.16)
x ∈ Ω3 .
It is easily seen that (2.16) satisfies (2.15) on Y . By physical reasons, the solution φξ (x) as well as the flux λ(x)∂n φξ must be continuous over the boundaries Ω1 ∩ Ω2 and Ω2 ∩ Ω3 . This gives four equations to solve for the three unknowns C1 , C2 , and K2 . In order to get a consistent solution, we get that σ3 must be ´´ ³ ³ µ ¶ σ 1 + σ1 + m σ1 − 1 1 σ2 2 σ2 K2 ´ , ³ (2.17) σ3 = σ2 C2 − 2 = r2 1 + σ1 − m σ1 − 1 σ2
9
1
σ2
r2
where m1 = r12 , the volume fraction of material 1 in material 2. Since we now know the 2 solution ω e1 (y) = φe1 (y) − e1 · y of the cell problem, we can compute the homogenized coefficients Z b(e~1 ) = λ(x)(e~1 + ∇ω e~1 )dx = [σ3 0]T Y
and similarly
b(e~2 ) =
Z
Y
λ(x)(e~2 + ∇ω e~2 )dx = [0 σ3 ]T .
This means that we can put the coated disk consisting of material 1 coated by material 2 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 σ3 . For more details see [Mil02].
2.2.2
The Mortola-Steff´ e Structure and the Checkerboard Structure
Let Y = (0, 1)2 and divide it into four equal parts µ ¶ µ ¶ µ ¶ ¶ µ 1 1 1 1 Y1 = 0, , 1 , Y2 = ,1 × ,1 , × 2 2 2 2 ¶ µ ¶ µ ¶ µ ¶ µ 1 1 1 1 , 1 × 0, × 0, , Y4 = . Y3 = 0, 2 2 2 2
We study a four-phase composite consisting of four isotropic materials, let us call them materials 1, 2, 3, and 4, with conductivity λ(x)I = [αχY1 (x) + βχY2 (x) + γχY3 (x) + δχY4 (x)] I,
where χYi (x) is the characteristic function for the set Yi and I is the unit matrix. In 1985, Mortola and Steff´e [MS85] conjectured that the homogenized conductivity coefficients of this structure are
(λij ) = where
µ
λ11 0
¶ 0 , λ22
λ11 =
sµ
αβγ + αβδ + αγδ + βγδ α+β+γ+δ
¶
(α + γ) (β + δ) , (α + β) (γ + δ)
λ22 =
sµ
αβγ + αβδ + αγδ + βγδ α+β+γ+δ
¶
(α + β) (γ + δ) . (α + γ) (β + δ)
This conjecture was proven by Milton in 2000. 10
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 p λ11 = λ22 = αβ, the geometric mean. This was proved already in 1970 by Dykhne, but Schulgasser (1977) showed that this was a corollary of Keller’s phase interchange identity from 1963.
11
Chapter 3 Dirichlet Boundary Value Problem In this chapter, we study boundary value problems associated with fields inside composites made from two materials with different power law behavior.
3.1
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 χ1 and χ2 , where χ1 (y) = 1 in material one and is zero outside and χ2 (y) = 1 − χ1 (y). The constitutive law for the heterogeneous medium is described by A : Rn × Rn → Rn , A (y, ξ) = σ(y) |ξ|p(y)−2 ξ
= σ1 χ1 (y) |ξ|p1 −2 ξ + σ2 χ2 (y) |ξ|p2 −2 ξ;
(3.1)
where σ(y) = χ1 (y) σ1 + χ2 (y) σ2 , and p(y) = χ1 (y) p1 + χ2 (y) p2 , periodic in y, with unit period cell Y = (0, 1)n . This simple constitutive model is used in the mathematical description of many physical phenomena including plasticity [PCS97, PCW99, Suq93, Idi08], nonlinear dielectrics [GNP01, GK03, LK98, TW94a, TW94b], and fluid flow [Ruˇz00, AR06]. We study the problem of periodic homogenization associated with the solutions uǫ to the problems ³ ³x ´´ −div A , ∇uǫ = f on Ω, uǫ ∈ W01,p1 (Ω), (3.2) ǫ where Ω is a bounded open subset of Rn , 2 ≤ p1 ≤ p2 , f ∈ W −1,q2 (Ω), and 1/p1 + 1/q2 = 1. The differential operator appearing on the left hand side of (3.2) is commonly referred to as the pǫ (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 uǫ to the problems (3.2). It was proved in [ZKO94] that {uǫ }ǫ>0 converges weakly in W 1,p1 (Ω) to the solution u of the homogenized problem −div (b (∇u)) = f on Ω, u ∈ W01,p1 (Ω), 12
(3.3)
where the monotone map b : Rn → Rn (independent of f and Ω) can be obtained by solving an auxiliary problem for the operator (3.2) on a periodicity cell. The notion of homogenization is intimately tied to the Γ-convergence of a suitable family energy functionals Iǫ 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 Iǫ and the solutions uǫ are their minimizers. The homogenized solution is precisely the minimizer of the Γ-limit of the sequence {Iǫ }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 pǫ (x)-Laplacian boundary value problems for smooth exponential functions pǫ (x) uniformly converging to a limit function p0 (x) [AAPP08]. There the convergence of the family of solutions for these homogenization problems is expressed in the topology of Lp0 (·) (Ω) [AAPP08].
3.2
Microgeometries Considered
We carry out this investigation for two nonlinear power-law materials periodically distributed inside a domain Ω. Here Ω is an open bounded subset of Rn , 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 Y = (0, 1)n of the microstructure. Let F be an open subset of Y of material one, with smooth boundary ∂F , such that F ⊂ Y . The function χ1 (y) = 1 inside F and 0 outside and χ2 (y) = 1 − χ1 (y). We extend χ1 (y) and χ2 (y) by periodicity to Rn and the ǫ-periodic mixture inside Ω is described by the oscillatory characteristic functions χǫ1 (x) = χ1 (x/ǫ) and χǫ2 (x) = χ2 (x/ǫ). Here we will consider the case where F is given by a simply connected inclusion embedded inside a host material (see Figure 3.1). A distribution of such inclusions is commonly referred to as a periodic dispersion of inclusions.
F Y
Figure 3.1: 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 R1 , sandwiched between layers of material two, denoted 13
by R2 . The interior boundary of R1 is denoted by Γ. Here χ1 (y) = 1 for y ∈ R1 and 0 in R2 , and χ2 (y) = 1 − χ1 (y) (see Figure 3.2). R2 R1 R2
Γ Figure 3.2: Unit cell: Layered material
3.3
Notation and Preliminary Results
On the unit cell Y , the constitutive law for the nonlinear material is given by (3.1) with exponents p1 and p2 satisfying 2 ≤ p1 ≤ p2 . Their H¨older conjugates (or dual conjugates) 1,pi (Y ) are denoted by q2 = p1 /(p1 − 1) and q1 = p2 /(p2 − 1) respectively. For i = 1, 2, Wper 1,pi denotes the set of all functions u ∈ W (Y ) with mean value zero that have the same trace 1,pi on the opposite faces of Y . Each function u ∈ Wper (Y ) can be extended by periodicity to a 1,pi n function of Wloc (R ). The Euclidean norm and the scalar product in Rn are denoted by |·| and (·, ·), respectively. If D ⊂ Rn , |D| denotes the Lebesgue measure and χD (x) denotes its characteristic function. The constitutive law for the ǫ-periodic composite is described by Aǫ (x, ξ) = A (x/ǫ, ξ), for every ǫ > 0, for every x ∈ Ω, and for every ξ ∈ Rn . 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.
3.3.1
Properties of A
The function A, defined in (3.1), satisfies the following properties: 1. For all ξ ∈ Rn , A(·, ξ) is Y -periodic and Lebesgue measurable. 2. |A(y, 0)| = 0 for all y ∈ Rn . 3. Continuity: for almost every y ∈ Rn and for every ξi ∈ Rn (i = 1, 2) we have £ |A(y, ξ1 ) − A(y, ξ2 )| ≤ C χ1 (y) |ξ1 − ξ2 | (1 + |ξ1 | + |ξ2 |)p1 −2 ¤ + χ2 (y) |ξ1 − ξ2 | (1 + |ξ1 | + |ξ2 |)p2 −2 .
(3.4)
4. Monotonicity: for almost every y ∈ Rn and for every ξi ∈ Rn (i = 1, 2) we have (A(y, ξ1 ) − A(y, ξ2 ), ξ1 − ξ2 ) ≥ C (χ1 (y) |ξ1 − ξ2 |p1 + χ2 (y) |ξ1 − ξ2 |p2 ) .
14
(3.5)
Proof of (3.4): Continuity of A Proof. By (3.1), we have |A(y, ξ1 ) − A(y, ξ2 )| ¯ ¯ = ¯σ1 χ1 (y) |ξ1 |p1 −2 ξ1 + σ2 χ2 (y) |ξ1 |p2 −2 ξ1 − σ1 χ1 (y) |ξ2 |p1 −2 ξ2 − σ2 χ2 (y) |ξ2 |p2 −2 ξ2 ¯ ¯ £ ¤¯ ¯ £ ¤¯ ≤ ¯σ1 χ1 (y) |ξ1 |p1 −2 ξ1 − |ξ2 |p1 −2 ξ2 ¯ + ¯σ2 χ2 (y) |ξ1 |p2 −2 ξ1 − |ξ2 |p2 −2 ξ2 ¯ ¯ ¯¢ ¯ ¯ ¡ ≤ C χ1 (y) ¯|ξ1 |p1 −2 ξ1 − |ξ2 |p1 −2 ξ2 ¯ + χ2 (y) ¯|ξ1 |p2 −2 ξ1 − |ξ2 |p2 −2 ξ2 ¯ Let us study the expression ¯ ¯ pi −2 ¯|ξ1 | ξ1 − |ξ2 |pi −2 ξ2 ¯ , for i = 1, 2.
Observe that
¯2 ¯ pi −2 ¯|ξ1 | ξ1 − |ξ2 |pi −2 ξ2 ¯
= |ξ1 |2(pi −1) + |ξ2 |2(pi −1) − 2 |ξ1 |pi −2 |ξ2 |pi −2 ξ1 · ξ2
= |ξ1 |2(pi −1) + |ξ2 |2(pi −1) − 2 |ξ1 |pi −1 |ξ2 |pi −1 + 2 |ξ1 |pi −1 |ξ2 |pi −1 − 2 |ξ1 |pi −2 |ξ2 |pi −2 ξ1 · ξ2 ¡ ¢2 = |ξ1 |pi −1 − |ξ2 |pi −1 + 2 |ξ1 |pi −2 |ξ2 |pi −2 (|ξ1 | |ξ2 | − ξ1 · ξ2 ) ¡ ¢2 ¡ ¢ = |ξ1 |pi −1 − |ξ2 |pi −1 + |ξ1 |pi −2 |ξ2 |pi −2 2 (|ξ1 | |ξ2 | − ξ1 · ξ2 )
By Remark 3.1, we have
¡ ¢2 ≤ |ξ1 |pi −1 − |ξ2 |pi −1 + |ξ1 |pi −2 |ξ2 |pi −2 |ξ1 − ξ2 |2
By Remark 3.2, we have
¯ ¯2 ¡ ¢2 ≤ ¯|ξ1 |pi −1 − |ξ2 |pi −1 ¯ + |ξ1 |pi −2 + |ξ2 |pi −2 |ξ1 − ξ2 |2
And by Remark 3.3, we have
¡ ¢2 ¡ ¢2 ≤ (pi − 1)2 |ξ1 |pi −2 + |ξ2 |pi −2 ||ξ1 | − |ξ2 ||2 + |ξ1 |pi −2 + |ξ2 |pi −2 |ξ1 − ξ2 |2 ¡ ¢2 ¡ ¢2 ≤ (pi − 1)2 |ξ1 |pi −2 + |ξ2 |pi −2 |ξ1 − ξ2 |2 + |ξ1 |pi −2 + |ξ2 |pi −2 |ξ1 − ξ2 |2 ¢2 ¡ ≤ C |ξ1 |pi −2 + |ξ2 |pi −2 |ξ1 − ξ2 |2 . ¤2 £ ≤ C (1 + |ξ1 | + |ξ2 |)pi −2 |ξ1 − ξ2 |2 .
Taking square root on both sides, we obtain ¯ ¯ pi −2 ¯|ξ1 | ξ1 − |ξ2 |pi −2 ξ2 ¯ ≤ C (1 + |ξ1 | + |ξ2 |)pi −2 |ξ1 − ξ2 | ,
which proves (3.4).
15
Remark 3.1. Observe that 0 ≤ (|ξ1 | − |ξ2 |)2 ⇔ 0 ≤ |ξ1 |2 − 2 |ξ1 | |ξ2 | + |ξ2 |2 ⇔ 2 |ξ1 | |ξ2 | ≤ |ξ1 |2 + |ξ2 |2
⇔ 2 |ξ1 | |ξ2 | − 2ξ1 · ξ2 ≤ |ξ1 |2 − 2ξ1 · ξ2 + |ξ2 |2
⇔ 2 (|ξ1 | |ξ2 | − ξ1 · ξ2 ) ≤ |ξ1 − ξ2 |2 . Remark 3.2. For i = 1, 2, we have ¡ ¢ |ξi |pi −2 ≤ |ξ1 |pi −2 + |ξ2 |pi −2 .
Remark 3.3. Consider the function f : R+ −→ R+ defined by f (x) = xpi −1 , where pi ≥ 2. Since f is a convex function, it satisfies ( f ′ (|ξ1 |) (|ξ2 | − |ξ1 |) ≤ f (|ξ2 |) − f (|ξ1 |) , f ′ (|ξ2 |) (|ξ1 | − |ξ2 |) ≤ f (|ξ1 |) − f (|ξ2 |) ; or equivalently, ( (pi − 1) |ξ1 |pi −2 (|ξ2 | − |ξ1 |) ≤ |ξ2 |pi −1 − |ξ1 |pi −1 , (pi − 1) |ξ2 |pi −2 (|ξ1 | − |ξ2 |) ≤ |ξ1 |pi −1 − |ξ2 |pi −1 . Then (pi − 1) |ξ2 |pi −2 (|ξ1 | − |ξ2 |) ≤ |ξ1 |pi −1 − |ξ2 |pi −1 ≤ (pi − 1) |ξ1 |pi −2 (|ξ1 | − |ξ2 |) . Therefore we have ¯ pi −1 ¯ ¡ ¢ ¯|ξ1 | − |ξ2 |pi −1 ¯ ≤ (pi − 1) |ξ1 |pi −2 + |ξ2 |pi −2 ||ξ1 | − |ξ2 || .
Proof of (3.5): Monotonicity of A Proof. By (3.1), we have
(A(y, ξ1 ) − A(y, ξ2 ), ξ1 − ξ2 ) = (A(y, ξ1 ), ξ1 ) − (A(y, ξ1 ), ξ2 ) − (A(y, ξ2 ), ξ1 ) + (A(y, ξ2 ), ξ2 )
= σ1 χ1 (y) |ξ1 |p1 + σ2 χ2 (y) |ξ1 |p2 − σ1 χ1 (y) |ξ1 |p1 −2 ξ1 · ξ2 − σ2 χ2 (y) |ξ1 |p2 −2 ξ1 · ξ2
− σ1 χ1 (y) |ξ2 |p1 −2 ξ1 · ξ2 − σ2 χ2 (y) |ξ2 |p2 −2 ξ1 · ξ2 + σ1 χ1 (y) |ξ2 |p1 + σ2 χ2 (y) |ξ2 |p2 ¤ £ = σ1 χ1 (y) |ξ1 |p1 − |ξ1 |p1 −2 ξ1 · ξ2 − |ξ2 |p1 −2 ξ1 · ξ2 + |ξ2 |p1 £ ¤ + σ2 χ2 (y) |ξ1 |p2 − |ξ1 |p2 −2 ξ1 · ξ2 − |ξ2 |p2 −2 ξ1 · ξ2 + |ξ2 |p2 £ ¡ ¢¤ = σ1 χ1 (y) |ξ1 |p1 + |ξ2 |p1 − ξ1 · ξ2 |ξ1 |p1 −2 − |ξ2 |p1 −2 £ ¡ ¢¤ + σ2 χ2 (y) |ξ1 |p2 + |ξ2 |p2 − ξ1 · ξ2 |ξ1 |p2 −2 − |ξ2 |p2 −2 16
Let us study the expression
• If |ξ1 | = |ξ2 |:
¡ ¢ |ξ1 |pi + |ξ2 |pi − ξ1 · ξ2 |ξ1 |pi −2 + |ξ2 |pi −2 , for i = 1, 2. ¢ ¡ |ξ1 |pi + |ξ1 |pi − ξ1 · ξ2 |ξ1 |pi −2 + |ξ1 |pi −2 = |ξ1 |pi + |ξ1 |pi − 2 |ξ1 |pi −2 ξ1 · ξ2 £ ¤ = |ξ1 |pi −2 |ξ1 |2 + |ξ1 |2 − 2ξ1 · ξ2 £ ¤ = |ξ1 |pi −2 |ξ1 |2 − 2ξ1 · ξ2 + |ξ2 |2
= |ξ1 |pi −2 |ξ1 − ξ2 |2 . Since
¯ ¯ ¯ ¯ 1 ¯¯ ξ1 ξ2 ¯¯ 1 ¯¯ ξ1 ξ2 ¯¯ − − = ≤ 1, 2 ¯ |ξ1 | |ξ1 | ¯ 2 ¯ |ξ1 | |ξ2 | ¯
then
|ξ1 | ≥
1 |ξ1 − ξ2 | . 2
Therefore ¡ ¢ |ξ1 |pi + |ξ2 |pi − ξ1 · ξ2 |ξ1 |pi −2 + |ξ2 |pi −2 = |ξ1 |pi −2 |ξ1 − ξ2 |2 µ ¶pi −2 1 ≥ |ξ1 − ξ2 |pi −2 |ξ1 − ξ2 |2 2 = 22−pi |ξ1 − ξ2 |pi . • If |ξ1 | > |ξ2 | > 0, we can write
ξ2 = βξ1 + γω,
where ω 6= ~0 is a vector orthogonal to ξ1 , and β, γ ∈ R with |β| < 1. Since
ξ1 · ξ2 = ξ1 · (βξ1 + γω) = β |ξ1 |2 ,
(3.6)
we obtain ¢ ¡ ¢ ¡ |ξ1 |pi + |ξ2 |pi − ξ1 · ξ2 |ξ1 |pi −2 + |ξ2 |pi −2 = |ξ1 |pi + |ξ2 |pi − β |ξ1 |2 |ξ1 |pi −2 + |ξ2 |pi −2 . – For β ≤ 0:
¡ ¢ |ξ1 |pi + |ξ2 |pi − β |ξ1 |2 |ξ1 |pi −2 + |ξ2 |pi −2 ≥ |ξ1 |pi + |ξ2 |pi ≥ 21−pi |ξ1 − ξ2 |pi .
17
– For 0 < β < 14 : ¡ ¢ |ξ1 |pi + |ξ2 |pi − β |ξ1 |2 |ξ1 |pi −2 + |ξ2 |pi −2 = |ξ1 |pi + |ξ2 |pi − β |ξ1 |pi − β |ξ2 |pi −2 |ξ1 |2 ≥ |ξ1 |pi − 2β |ξ1 |pi = |ξ1 |pi (1 − 2β) 1 > |ξ1 |pi 2 1 1 = |ξ1 |pi + |ξ1 |pi 4 4 |ξ2 |pi |ξ1 |pi > + 4 4 −(pi +1) ≥2 |ξ1 − ξ2 |pi . – For
1 4
≤ β < 1:
¡ ¢ |ξ1 |pi + |ξ2 |pi − |ξ1 |pi −2 + |ξ2 |pi −2 ξ1 · ξ2
= |ξ1 |pi − |ξ1 |pi −2 ξ1 · ξ2 + |ξ2 |pi − |ξ2 |pi −2 ξ1 · ξ2 ¡ ¢ ¡ ¢ = |ξ1 |pi −2 |ξ1 |2 − ξ1 · ξ2 + |ξ2 |pi −2 |ξ2 |2 − ξ1 · ξ2 ¡ ¢ ¡ ¢ ≥ |ξ2 |pi −2 |ξ1 |2 − ξ1 · ξ2 + |ξ2 |pi −2 |ξ2 |2 − ξ1 · ξ2 ¡ ¢ = |ξ2 |pi −2 |ξ1 |2 − 2ξ1 · ξ2 + |ξ2 |2 = |ξ2 |pi −2 |ξ1 − ξ2 |2 .
By (3.6), we have
Therefore
obtaining this way Then
1 |ξ1 | ≤ ≤ 4. |ξ2 | β |ξ1 − ξ2 | |ξ1 | ≤ + 1 ≤ 5; |ξ2 | |ξ2 | |ξ2 | ≥ 5−1 |ξ1 − ξ2 | .
¡ ¢ |ξ1 |pi + |ξ2 |pi − |ξ1 |pi −2 + |ξ2 |pi −2 ξ1 · ξ2 = |ξ2 |pi −2 |ξ1 − ξ2 |2
≥ 52−pi |ξ1 − ξ2 |pi −2 |ξ1 − ξ2 |2 = 52−pi |ξ1 − ξ2 |pi .
© ª Taking C = min 52−pi , 2−(pi +1) , we have proved ¡ ¢ |ξ1 |pi + |ξ2 |pi − ξ1 · ξ2 |ξ1 |pi −2 + |ξ2 |pi −2 ≥ C |ξ1 − ξ2 |pi ,
for pi ≥ 2. Therefore
(A(y, ξ1 ) − A(y, ξ2 ), ξ1 − ξ2 ) ≥ C (χ1 (y) |ξ1 − ξ2 |p1 + χ2 (y) |ξ1 − ξ2 |p2 ) . A different proof for the monotonicity and continuity of A can be found in [Bys05]. 18
3.3.2
Dirichlet BVP and Homogenization Theorem
We shall consider the following Dirichlet boundary value problem ( −div (Aǫ (x, ∇uǫ )) = f on Ω, uǫ ∈ W01,p1 (Ω);
(3.7)
where f ∈ W −1,q2 (Ω). The following homogenization result holds. Theorem 3.4 (Homogenization Theorem). As ǫ → 0, the solutions uǫ of (3.7) converge weakly to u in W 1,p1 (Ω), where u is the solution of ( −div (b (∇u)) = f on Ω, (3.8) u ∈ W01,p1 (Ω); and the function b : Rn → Rn (independent of f and Ω) is defined for all ξ ∈ Rn by Z b(ξ) = A(y, p(y, ξ))dy,
(3.9)
Y
where p : Rn × Rn → Rn is defined by p(y, ξ) = ξ + ∇υξ (y), where υξ is the solution to the cell problem: Z (A(y, ξ + ∇υ ), ∇w) dy = 0, for every w ∈ W 1,p1 (Y ), ξ per Y 1,p1 υξ ∈ Wper (Y ).
(3.10)
(3.11)
For a proof of Theorem 3.4, see Chapter 15 of [ZKO94].
Lemma 3.5. The following a priori bound is satisfied µZ ¶ Z p1 p2 ǫ ǫ χ1 (x) |∇uǫ (x)| dx + χ2 (x) |∇uǫ (x)| dx ≤ C < ∞. sup ǫ>0
Ω
Ω
Proof. The weak formulation for (3.7) is given by Z Z (Aǫ (x, ∇uǫ (x)), ∇ϕ(x))dx = f (x)ϕ(x)dx, Ω
Ω
for all ϕ ∈ W01,p1 (Ω). In particular, taking ϕ = uǫ above, we obtain Z Z (Aǫ (x, ∇uǫ (x)), ∇uǫ (x))dx = f (x)uǫ (x)dx. Ω
Ω
19
(3.12)
The last equation can be rewritten as Z Z Z p1 p2 ǫ ǫ σ1 χ1 (x) |∇uǫ | dx + σ2 χ2 (x) |∇uǫ | dx = f (x)uǫ (x)dx. Ω
Ω
(3.13)
Ω
Applying H¨older’s inequality to the right hand side of (3.13), we obtain Z Z Z ǫ f (x)uǫ (x)dx = χ1 (x)f (x)uǫ (x)dx + χǫ1 (x)f (x)uǫ (x)dx Ω
Ω
≤
Ω
µZ
q2
Ω
+
|f (x)| dx
µZ
Ω
¶ q1 µZ 2
Ω
q1
|f (x)| dx
χǫ1 (x) |uǫ (x)|p1
¶ q1 µZ 1
Ω
dx p2
¶ p1
χǫ2 (x) |uǫ (x)| dx
1
¶ p1
2
.
(3.14)
If we combine (3.13) and (3.14), and we use the fact that f ∈ W −1,q2 (Ω), we get Z Z p1 ǫ σ1 χ1 (x) |∇uǫ (x)| dx + σ2 χǫ2 (x) |∇uǫ (x)|p2 dx Ω Ω " µZ ¶ p1 µZ ¶ p1 # 1 2 χǫ1 (x) |uǫ (x)|p1 dx ≤C + χǫ2 (x) |uǫ (x)|p2 dx Ω
Ω
Poincar´e’s inequality (see [Neˇc67]) gives ≤C
" µZ
Ω
χǫ1 (x) |∇uǫ |p1 dx
¶ p1
1
+
µZ
Ω
χǫ2 (x) |∇uǫ |p2 dx
¶ p1 # 2
and applying Young’s inequality, we obtain δ p1 ≤C p1 ·
Z
Ω
χǫ1 (x) |∇uǫ |p1
δ −q2 δ p2 + dx + q2 p2
Z
Ω
χǫ2 (x) |∇uǫ |p2
¸ δ −q1 . dx + q1
By rearranging the terms in the inequality, one gets µ ¶Z ¶Z µ δ p1 δ p2 p1 ǫ σ1 − C χ1 (x) |∇uǫ | dx + σ2 − C χǫ2 (x) |∇uǫ |p2 dx p1 p 2 Ω Ω δ −q2 δ −q1 + . ≤ q2 q1 n o δ p1 δ p2 Therefore, by taking δ small enough so that min σ1 − C p1 , σ2 − C p2 is positive, one obtains Z Z p1 ǫ χ1 (x) |∇uǫ | dx + χǫ2 (x) |∇uǫ |p2 dx ≤ C, Ω
Ω
where C does not depend on ǫ.
20
3.3.3
Properties of b
The function b, defined in (3.9), satisfies the following properties 1. Continuity: for every ξ1 , ξ2 ∈ Rn , we have h p1 −2 1 |b(ξ1 ) − b(ξ2 )| ≤ C |ξ1 − ξ2 | p1 −1 (1 + |ξ1 |p1 + |ξ2 |p1 + |ξ1 |p2 + |ξ2 |p2 ) p1 −1 p2 −2 i 1 + |ξ1 − ξ2 | p2 −1 (1 + |ξ1 |p1 + |ξ2 |p1 + |ξ1 |p2 + |ξ2 |p2 ) p2 −1
(3.15)
2. Monotonicity:for every ξ1 , ξ2 ∈ Rn , we have (b(ξ1 ) − b(ξ2 ), ξ1 − ξ2 ) (3.16) µZ ¶ Z ≥C χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )|p1 dy + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )|p2 dy Y
Y
≥ 0.
Proof of Continuity of b (3.15) Proof. By (3.9), (3.4), we have |b(ξ1 ) − b(ξ2 )| ¯Z ¯ Z ¯ ¯ = ¯¯ A(y, p(y, ξ1 ))dy − A(y, p(y, ξ2 ))dy ¯¯ Y ZY |A(y, p(y, ξ1 )) − A(y, p(y, ξ2 ))| dy ≤ Y ·Z ≤C χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| (1 + |p(y, ξ1 )| + |p(y, ξ2 )|)p1 −2 dy ¸ Z Y p2 −2 + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| (1 + |p(y, ξ1 )| + |p(y, ξ2 )|) dy Y
Applying H¨older’s inequality in both integrals, we obtain ≤C
"µZ
Y
× +
p1
µZ
Y
µZ
Y
×
χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy
χ1 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)
χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy
Y
1
q2 (p1 −2)
p2
µZ
¶ p1
dy
¶ q1
dy
¶ q1 #
¶ p1
2
2
q1 (p2 −2)
χ2 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)
21
1
By (3.5), we have "µZ ≤C
Y
× +
χ1 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)q2 (p1 −2) dy
Y
¶ q1
Y
Y
q2 (p1 −2)
Y
χ1 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)
dy
¶ q1
(A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), p(y, ξ1 ) − p(y, ξ2 )) dy Y µZ ¶1# Y
= C (b(ξ1 ) − b(ξ2 ), ξ1 − ξ2 ) + (b(ξ1 ) − b(ξ2 ), ξ1 − ξ2 )
1 p1
1 p2
µZ Y
1
¶ p1
2
q2 (p1 −2)
Y
µZ
¶ p1
q1
χ2 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)q1 (p2 −2) dy
By (3.11), we get "
2
2
µZ ×
¶ p1
q1
χ2 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)q1 (p2 −2) dy
(A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), p(y, ξ1 ) − p(y, ξ2 )) dy
µZ
1
2
χ2 (y) (A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), p(y, ξ1 ) − p(y, ξ2 )) dy Y µZ ¶1#
"µZ ×
+
µZ
µZ ×
≤C
χ1 (y) (A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), p(y, ξ1 ) − p(y, ξ2 )) dy
¶ p1
χ1 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)
χ2 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)q1 (p2 −2) dy
Applying the Cauchy-Schwarz inequality and H¨older’s inequality we have “ ”“ ” µZ ¶ p 1−1 p1p−1 1 1 1 1 ≤ C |b(ξ1 ) − b(ξ2 )| p1 |ξ1 − ξ2 | p1 χ1 (y)1p1 −1 Y
×
µZ
Y
χ1 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|) 1
1
+ |b(ξ1 ) − b(ξ2 )| p2 |ξ1 − ξ2 | p2 ×
µZ
Y
dy
µZ
p1 (p1 −2)(p1 −1) (p1 −1)(p1 −2)
χ2 (y)1p2 −1
Y
χ2 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|) 22
¶
“
1 p2 −1
dy
”“
p2 (p2 −2)(p2 −1) (p2 −1)(p2 −2)
¶ (p1p−1)(p 1 −2) (p −1) 1
p2 −1 p2
dy
1
”
¶ (p2p−1)(p 2 −2) (p −1) 2
2
¶ q1
2
¶ q1 # 1
"
≤ C |b(ξ1 ) − b(ξ2 )| + |b(ξ1 ) − b(ξ2 )|
1 p1
1 p2
|ξ1 − ξ2 |
|ξ1 − ξ2 |
1 p1
1 p2
1 p1
θ1 1 p2
θ2
µZ
Y
µZ
Y
χ1 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)p1 dy
χ2 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)p2 dy
¶ p1p−2 1
¶ p2p−2 # 2
Lemma 5.3 delivers · 1 p1 −2 1 1 ≤ C |b(ξ1 ) − b(ξ2 )| p1 |ξ1 − ξ2 | p1 θ1p1 (1 + |ξ1 |p1 θ1 + |ξ2 |p1 θ1 + |ξ1 |p2 θ2 + |ξ2 |p2 θ2 ) p1 ¸ 1 p2 −2 1 1 p2 p1 p1 p2 p2 p2 p2 p2 + |b(ξ1 ) − b(ξ2 )| |ξ1 − ξ2 | θ2 (1 + |ξ1 | θ1 + |ξ2 | θ1 + |ξ1 | θ2 + |ξ2 | θ2 ) Applying Young’s inequality we obtain · p1 δ |b(ξ1 ) − b(ξ2 )| δ p2 |b(ξ1 ) − b(ξ2 )| + ≤C p1 p2 1
1
1 p2 −1
1 p2 −1
p1 −2
δ −q2 |ξ1 − ξ2 | p1 −1 θ1p1 −1 (1 + |ξ1 |p1 θ1 + |ξ2 |p1 θ1 + |ξ1 |p2 θ2 + |ξ2 |p2 θ2 ) p1 −1 + q2 +
δ
−q1
|ξ1 − ξ2 |
θ2
p1
p1
p2
p2
(1 + |ξ1 | θ1 + |ξ2 | θ1 + |ξ1 | θ2 + |ξ2 | θ2 ) q1
p2 −2 p2 −1
Therefore ¶¸ · µ p1 δ p2 δ |b(ξ1 ) − b(ξ2 )| + 1−C p p2 1 1 p1 −2 1 p1 −1 p1 p1 p2 p2 −q2 p1 −1 p1 −1 δ |ξ − ξ | θ (1 + |ξ | θ + |ξ | θ + |ξ | θ + |ξ | θ ) 1 2 1 1 2 1 1 2 2 2 1 ≤C q2 1 p2 −2 1 p2 −1 p1 p1 p2 p2 −q1 p2 −1 p2 −1 θ2 δ |ξ1 − ξ2 | (1 + |ξ1 | θ1 + |ξ2 | θ1 + |ξ1 | θ2 + |ξ2 | θ2 ) + q1
Taking δ small enough, we obtain
|b(ξ1 ) − b(ξ2 )| · 1 p1 −2 1 ≤ C |ξ1 − ξ2 | p1 −1 θ1p1 −1 (1 + |ξ1 |p1 θ1 + |ξ2 |p1 θ1 + |ξ1 |p2 θ2 + |ξ2 |p2 θ2 ) p1 −1 ¸ 1 p2 −2 1 p2 −1 p1 p1 p2 p2 p2 −1 p2 −1 (1 + |ξ1 | θ1 + |ξ2 | θ1 + |ξ1 | θ2 + |ξ2 | θ2 ) + |ξ1 − ξ2 | . θ2
23
Proof of (3.16): Monotonicity of b Proof. Using (3.11) and (3.5), we have (b(ξ2 ) − b(ξ1 ), ξ2 − ξ1 ) µZ ¶ Z = A(y, p(y, ξ2 ))dy − A(y, p(y, ξ1 ))dy, ξ2 − ξ1 Y Y Z = (A(y, p(y, ξ2 )) − A(y, p(y, ξ1 )), ξ2 − ξ1 ) dy ZY = (A(y, p(y, ξ2 )) − A(y, p(y, ξ1 )), p(y, ξ2 ) − p(y, ξ1 )) dy Y µZ ≥C χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )|p1 dy ¶ Z Y p2 + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy ≥ 0. Y
3.3.4
Properties of p
1,p1 (Rn ), Since the solution υξ of (3.11) can be extended by periodicity to a function of Wloc ′ then (3.11) is equivalent to −div(A(y, ξ + ∇υξ (y))) = 0 over D (Rn ), i.e., ′
−div (A(y, p(y, ξ))) = 0 in D (Rn ) for every ξ ∈ Rn . Moreover, by (3.11), we have Z Z (A(y, p(y, ξ)), p(y, ξ)) dy = (A(y, p(y, ξ)), ξ) dy = (b(ξ), ξ) . Y
(3.17)
(3.18)
Y
For ǫ > 0, define pǫ : Rn × Rn → Rn by ³x ´ ³x´ pǫ (x, ξ) = p , ξ = ξ + ∇υξ . ǫ ǫ
(3.19)
where υξ is the unique solution of (3.11). The functions p and pǫ are easily seen to satisfy the following properties p(·, ξ) is Y -periodic and pǫ (x, ξ) is ǫ-periodic in x. Z p(y, ξ)dy = ξ.
(3.20) (3.21)
Y
A
³·
ǫ
pǫ (·, ξ) ⇀ ξ in Lp1 (Ω; Rn ) as ǫ → 0.
(3.22)
p(y, 0) = 0 for almost every y. ´ , pǫ (·, ξ) ⇀ b(ξ) in Lq2 (Ω; Rn ), as ǫ → 0.
(3.23)
24
(3.24)
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 W01,p2 (Ω). In the following chapters we will apply these facts to establish strong approximations for the sequences {χǫi ∇uǫ }ǫ>0 in Lp2 (Ω, Rn ). 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.
4.1
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 4.1. Given a periodic dispersion of inclusions or a layered material then the solution u of (3.8) belongs to W01,p2 (Ω). Before we can prove this theorem we need some definitions. Definition 4.2. Functions f (x, ξ) depending on two variables x, ξ ∈ Rn will be referred to as Lagrangians. Definition 4.3. If the Lagrangian f (x, ξ), ξ ∈ Rn , x ∈ Ω ⊂ Rn satisfies −c0 + c1 |ξ|p ≤ f (x, ξ) ≤ c2 |ξ|p + c0
(4.1)
with c0 ≥ 0,c1 ,c2 > 0, and p > 1, then it is called standard. A much wider class of Lagrangians which includes the standard ones, is specified by the estimate −c0 + c1 |ξ|p1 ≤ f (x, ξ) ≤ c2 |ξ|p2 + c0 (4.2) 25
with p2 ≥ p1 > 1. These are called nonstandard Lagrangians. Definition 4.4. The conjugate of a nonstandard Lagrangian f , denoted g = f ∗ , is defined by g(x, ξ) = sup {ξ · η − f (x, η)} , (4.3) η∈Rn
and satisfies the estimate −c0 + c∗1 |ξ|q1 ≤ g(x, ξ) ≤ c∗2 |ξ|q2 + c0 .
4.2
(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 σ1 σ2 f˜(x, ξ) = q(x) |ξ|p(x) = χ1 (x) |ξ|p1 + χ2 (x) |ξ|p2 , p1 p2
(4.5)
with
σ2 σ1 χ1 (x) + χ2 (x); p1 p2 where ξ ∈ Rn and x ∈ Ω ⊂ Rn . Here ∇ξ f˜(x, ξ) = A (x, ξ), where A(x, ξ) is given by (3.1). We consider the rescaled Lagrangian ³x ´ σ σ2 1 f˜ǫ (x, ξ) = f˜ , ξ = χǫ1 (x) |ξ|p1 + χǫ2 (x) |ξ|p2 , (4.6) ǫ p1 p2 q(x) =
where χǫi (x) = χi (x/ǫ), i = 1, 2, ξ ∈ Rn , and x ∈ Ω ⊂ Rn . The Dirichlet problem given by (3.7) is associated with the variational problem given by ½Z ¾ ǫ ˜ fǫ (x, ∇u)dx − hf, ui , (4.7) E1 (f ) = inf 1,p u∈W0
1 (Ω)
Ω
with f ∈ W −1,q2 (Ω). Here (3.7) is the Euler equation for (4.7). However, we also consider ½Z ¾ ǫ f˜ǫ (x, ∇u)dx − hf, ui , (4.8) E2 (f ) = inf 1,p u∈W0
2 (Ω)
Ω
with f ∈ W −1,q2 (Ω) (See [Zhi92]). Here h·, ·i is the duality pairing between W01,p1 (Ω) and W −1,q2 (Ω). From [ZKO94], we have lim Eiǫ = Ei , for i = 1, 2, where ǫ→0
Ei =
inf 1,p
u∈W0
i (Ω)
½Z
Ω
¾ ˆ ˜ fi (∇u(x))dx − hf, ui . 26
(4.9)
In (4.9), f˜ˆi (ξ) is given by fˆ˜i (ξ) =
inf 1,p
i
v in Wper (Y )
and satisfies
Z
Y
f˜(y, ξ + ∇v(y))dy
(4.10)
−c0 + c1 |ξ|p1 ≤ fˆ˜i (ξ) ≤ c2 |ξ|p2 + c0 .
(4.11)
In general, (see [Zhi95]) Lavrentiev phenomenon can occur and E1 < E2 . 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 E1 = E2 , where fˆ˜ = fˆ˜1 = fˆ˜2 and c2 + c1 |ξ|p2 ≤ fˆ˜(ξ). Moreover, ∇ξ fˆ˜(ξ) = b(ξ), where b is the homogenized operator (3.9). Proof. Theorem 4.5 has been proved for dispersed periodic media in Chapter 14 of [ZKO94]. We prove Theorem 4.5 for layers following the steps outlined in [ZKO94]. We first show that fˆ˜ = fˆ˜1 = fˆ˜2 holds for layered media. Then we show that the homogenized Lagrangian fˆ˜ satisfies the standard estimate given by −c0 + c1 |ξ|p2 ≤ fˆ˜(ξ) ≤ c2 |ξ|p2 + c0
(4.12)
with c0 ≥ 0, and c1 ,c2 > 0 . We introduce the space of functions W∗1,p2 (R2 ) that belong to W 1,p2 (R2 ) and are periodic on ∂R2 ∩ ∂Y .
Lemma 4.6. Any function v ∈ W∗1,p2 (R2 ) can be extended to R1 in such a way that the 1,p2 extension v˜(y) belongs to Wper (Y ) and v˜(y) = v(y) on R2 . Proof. Let ϕ to be the solution of ∆p 2 ϕ = 0 ϕ takes periodic boundary values on opposite faces of ∂Y ∩ ∂R1 ϕ¯¯ = v¯¯ 1
, on R1 , on Γ
2
Here the subscript 1 indicates the trace on the R1 side of Γ and 2 indicates the trace on the R2 side of Γ. For a proof of existence of the solution ϕ see [Eva82] or [Lew77]. The extension v˜ is given by ( v , in R2 . v˜ = ϕ , on R1 .
27
ˆ˜ ˆ˜ p1 ˜ˆ ˜ˆ It is clear Z that f1 ≤ f2 . To prove f1 = f2 , it suffices to show that for every v ∈ Wper (Y ) 1,p2 f˜(y, ξ + ∇v(y))dy < ∞ there exists a sequence vǫ ∈ Wper (Y ) such that satisfying Y
lim ǫ→0
Z
Y
f˜(y, ξ + ∇vǫ (y))dy =
Z
Y
f˜(y, ξ + ∇v(y))dy.
For v as above, let v˜ be as in Lemma 4.6 and set z = v − v˜. It is clear that z ∈ W 1,p1 (R1 ), is periodic on opposite faces of ∂Y ∩ ∂R1 , zero on Γ and we write Z Z Z ˜ f (y, ξ + ∇v(y))dy = f2 (ξ + ∇v(y))dy + f1 (ξ + ∇˜ v (y) + ∇z(y))dy, R2
Y
R1
where f1 (ξ) = σp11 |ξ|p1 and f2 (ξ) = σp22 |ξ|p2 ; i.e, f1 and f2 are standard Lagrangians satisfying (4.1) with exponents p1 and p2 respectively. We can choose a sequence {zǫ }ǫ>0 ∈ C∞ 0 (R1 ) such that zǫ vanishes in R2 and zǫ → z in W 1,p1 (R1 ). 1,p2 Define vǫ ∈ Wper (Y ) by ( vǫ =
v v˜ + zǫ
in R2 , in R1 .
1,p1 Since vǫ → v in Wper (Y ), we see that Z lim f˜(y, ξ + ∇vǫ (y))dy ǫ→0 Y µZ ¶ Z = lim f2 (ξ + ∇v(y))dy + f1 (ξ + ∇˜ v (y) + ∇zǫ (y))dy ǫ→0 R2 R1 Z Z = f2 (ξ + ∇v(y))dy + f1 (ξ + ∇˜ v (y) + ∇z(y))dy R2 R1 Z = f˜(y, ξ + ∇v(y))dy. Y
Therefore fˆ˜ = fˆ˜1 = fˆ˜2 for layered media. We establish (4.12) by introducing the convex conjugate of fˆ˜. We denote the convex dual of fˆ˜i (ξ) by gˆ˜i (ξ) and the convex dual of f˜ by g˜. It is easily verified (see [Zhi92]) that Z ˆ g˜i (ξ) = infq g˜(y, ξ + w(y))dy (4.13) w in Sol i (Y )
and
Y
−c0 + c∗1 |ξ|q1 ≤ gˆ˜i (ξ) ≤ c∗2 |ξ|q2 + c0 .
(4.14)
Here Solqi (Y ) are the solenoidal vector fields belonging to Lqi (Y, Rn ) and having mean value zero Solqi (Y ) = {w ∈ Lqi (Y ; Rn ) : div w = 0, w · n anti-periodic} . 28
We will show that gˆ˜ = gˆ˜1 = gˆ˜2 satisfies gˆ˜(ξ) ≤ c2 |ξ|q1 + c1 , and apply duality to recover
fˆ˜(ξ) ≥ c∗2 |ξ|p2 + c∗1 . To get the upper bound on gˆ˜ we use the following lemma.
Lemma 4.7. There exists τ with div τ = 0 in Y , such that τ · n is anti-periodic on the boundary of Y , τ = −ξ in R1 , and Z |τ (y)|q1 dy ≤ C |ξ|q1 . Y
Proof. Let the function ϕ ∈ W∗1,p2 (R2 ) be the solution of p−2 · n is anti-periodic on ∂R2 ∩ ∂Y ; ∇ϕ|∇ϕ| ∆p2 ϕ = 0 in R2 ; ∇ϕ |∇ϕ|p2 −2 · n¯¯ = −ξ · n¯¯ ; on Γ, 2
1
where the subscript 1 indicates the trace on the R1 side of Γ and 2 indicates the trace on the R Z 2 side of Γ. The Z Neumann problem given above is the stationarity condition for the energy |∇φ|p2 dx − φξ · n dS when minimized over all φ ∈ W∗1,p2 (R2 ). The solution of the R2
Γ
Neumann problem is unique up to a constant. Here the anti-periodic boundary condition on ∇ϕ|∇ϕ|p−2 · n is the natural boundary condition for the problem. Now we define τ according to ( −ξ; in R1 τ= p2 −2 ∇ϕ |∇ϕ| ; in R2
and it follows that |ξ|q1 ; q1 |τ | = h¡ ¢ i q1 ¢ ¡ ∇ϕ |∇ϕ|p2 −2 2 2 = |∇ϕ|p2 −1 q1 = |∇ϕ|p2 ;
in R1 in R2 .
Then, for ψ ∈ W∗1,p2 (R2 ) we have Z |∇ϕ|p2 −2 ∇ϕ · ∇ψdy R2 Z Z p2 −2 = ψ |∇ϕ| ∇ϕ · ndS + ψ |∇ϕ|p2 −2 ∇ϕ · ndS Γ ∂R2 ∩∂Y Z = − ψξ · ndS. Γ
29
(4.15)
(4.16)
Set ψ = ϕ in (4.16) and an application of H¨older’s inequality gives Z Z p2 |∇ϕ| dx = − ϕξ · ndS R2 ZΓ =− ∇(ϕξ)dx R2 Z =− (∇ϕ) · ξdx R2
≤ Then
Z
µZ
p2
R2
|∇ϕ| dx p2
R2
|∇ϕ| dx ≤
¶1/p2 µZ
R2
Z
R2
q1
|ξ| dx
|ξ|q1 dx.
¶1/q1
.
(4.17)
Therefore, using (4.15) and (4.17), we have Z |τ (y)|q1 dy Y Z Z q1 = |τ (y)| dy + |τ (y)|q1 dy ZR1 Z R2 = |ξ|q1 dy + |∇ϕ(y)|p2 dy R1
R2
q1
≤ C |ξ| .
Taking gˆ˜ to be the conjugate of fˆ˜, and choosing τ in Solq1 (Y ) as in Lemma 4.7, we obtain Z ˆ g˜(ξ) = inf g˜(y, ξ + τ )dy τ in Solq1 (Y ) Y Z ≤ g˜(y, ξ + τ )dy Y Z Z ≤ g˜(y, 0)dy + g˜(y, ξ + τ )dy R1 R2 Z ≤ c1 + c2 |ξ + τ |q1 dy R2
≤ c1 + c2 |ξ|q1 ,
and the left hand inequality in (4.12) follows from duality. This concludes the proof of Theorem 4.5. Collecting results we now prove Theorem 4.1. Indeed the minimizer of E1 is precisely the solution u of (3.8). Theorem 4.5 establishes the coercivity of E1 over W01,p2 (Ω), thus the solution u lies in W01,p2 (Ω). 30
Chapter 5 Corrector Theorem In this chapter, we develop new strong convergence results that capture the asymptotic behavior of the gradients ∇uǫ , as ǫ tends to 0. Our approach delivers strong approximations for the gradients inside each phase χǫi ∇uǫ , i = 1, 2. Homogenization theory relates the average behavior seen at large length scales to the underlying heterogeneous structure. It allows one to approximate {∇uǫ }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 ∇uǫ decays to 0. We present a new corrector result that delivers an approximation to ∇uǫ 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 W01,p2 (Ω) (See Chapter 4). With this higher order integrability in hand, we provide an algorithm for building correctors and establish strong approximations for the sequences {χi ǫ∇uǫ }ǫ>0 in Lp2 (Ω, Rn ), see Theorem 5.2. When the host phase has a lower power-law exponent than the included phase one can only conclude that the homogenized solution lies in W01,p1 (Ω) and the techniques developed here do not apply.
5.1
Statement of the Corrector Theorem
We now describe the family of correctors that provide a strong approximation of the sequence {χǫi ∇uǫ }ǫ>0 in the Lpi (Ω, Rn ) norm. We denote the rescaled period cell with side length ǫ > 0 by Yǫ and write Yǫi = ǫi + Yǫ , where i ∈ Zn . In what follows it is convenient to define the index set Iǫ = {i ∈ Zn : Yǫi ⊂ Ω}. For ϕ ∈ Lp2 (Ω; Rn ), we define the local average operator Mǫ associated with the partition Yǫi , i ∈ Iǫ by X Z 1 χYǫi (x) i ϕ(y)dy; if x ∈ ∪i∈Iǫ Yǫi , |Y | i Yǫ ǫ Mǫ (ϕ)(x) = i∈ Iǫ (5.1) 0; i if x ∈ Ω \ ∪i∈Iǫ Yǫ . 31
Remark 5.1. The family Mǫ has the following properties 1. For i = 1, 2, kMǫ (ϕ) − ϕkLpi (Ω;Rn ) → 0 as ǫ → 0. For a proof, see, for instance Chapter 8 of [Zaa58]. 2. Mǫ (ϕ) → ϕ a.e. on Ω. For a proof, see, for instance Chapter 8 of [Zaa58]. 3. From Jensen’s inequality, we have kMǫ (ϕ)kLpi (Ω;Rn ) ≤ kϕkLpi (Ω;Rn ) , for every ϕ ∈ Lp2 (Ω; Rn ) and i = 1, 2. The strong approximation to the sequence {χǫi ∇uǫ }ǫ>0 is given by the following corrector theorem. Theorem 5.2 (Corrector Theorem). Let f ∈ W −1,q2 (Ω), let uǫ 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 = 1, 2, we have Z (5.2) |χǫi (x)pǫ (x, Mǫ (∇u)(x)) − χǫi (x)∇uǫ (x)|pi dx → 0, as ǫ → 0. Ω
Before we can give the proof of this theorem, we need the results from the following section.
5.2
Some Properties of Correctors
In this section, we state and prove a priori bounds and convergence properties for the sequences pǫ defined in (3.19), ∇uǫ , and Aǫ (x, pǫ (x, ∇uǫ )) that are used in the proof of Theorem 5.2. 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 ∈ Rn , we have Z Z p1 χ1 (y) |p(y, ξ)| dy + χ2 (y) |p(y, ξ)|p2 dy ≤ C (1 + |ξ|p1 θ1 + |ξ|p2 θ2 ) , Y
and by a change of variables, we obtain Z Z p1 ǫ χ1 (x) |pǫ (x, ξ)| dx + χǫ2 (x) |pǫ (x, ξ)|p2 dx ≤ C (1 + |ξ|p1 θ1 + |ξ|p2 θ2 ) |Yǫ | . Yǫ
(5.3)
Y
Yǫ
Proof. Let ξ ∈ Rn . By (3.5), we have that (A(y, p(y, ξ)), p(y, ξ)) ≥ C (χ1 (y) |p(y, ξ)|p1 + χ2 (y) |p(y, ξ)|p2 ) .
32
(5.4)
Integrating both sides over Y and using (3.11), we get Z Z p1 χ1 (y) |p(y, ξ)| dy + χ2 (y) |p(y, ξ)|p2 dy Y Y Z ≤C (A(y, p(y, ξ)), p(y, ξ)) dy Y Z =C (A(y, p(y, ξ)), ξ) dy Y
By Cauchy-Schwarz Inequality and (3.4), we have ≤C
Z
|A(y, p(y, ξ))| |ξ| dy
Y
·Z
χ1 (y) |p(y, ξ)| (1 + |p(y, ξ)|)p1 −2 |ξ| dy ¸ Z Y p2 −2 + χ2 (y) |p(y, ξ)| (1 + |p(y, ξ)|) |ξ| dy Y ·Z ¸ Z p1 −1 p2 −1 ≤C χ1 (y)(1 + |p(y, ξ)|) |ξ| dy + χ2 (y)(1 + |p(y, ξ)|) |ξ| dy
≤C
Y
Y
Using Young’s Inequality, we obtain
δ ≤C δ
+
q1
Z
q2
p1
Y
Z
Y
χ1 (y)(1 + |p(y, ξ)|) dy q2
δ
q1
Y
+
p2
χ2 (y)(1 + |p(y, ξ)|) dy
Z
−p1
δ
−p2
Z
Y
+
χ1 (y) |ξ|p1 dy p1 p2
χ2 (y) |ξ| dy p2
· Z = C δ q2 χ1 (y)(1 + |p(y, ξ)|)p1 dy + δ −p1 |ξ|p1 θ1 ¸ Z Y p2 q1 −p2 p2 +δ χ2 (y)(1 + |p(y, ξ)|) dy + δ |ξ| θ2 Y · Z q2 q2 ≤ C δ θ1 + δ χ1 (y) |p(y, ξ)|p1 dy + δ −p1 |ξ|p1 θ1 + δ q1 θ2 Y ¸ Z p2 p2 q1 −p2 +δ χ2 (y) |p(y, ξ)| dy + δ |ξ| θ2 Y £ ¡ ¢ ≤ C (δ q2 θ1 + δ q1 θ2 ) + δ −p1 |ξ|p1 θ1 + δ −p2 |ξ|p2 θ2 µZ ¶¸ Z p1 p2 q1 q2 +(δ + δ ) χ1 (y) |p(y, ξ)| dy + χ2 (y) |p(y, ξ)| dy . Y
Y
33
Doing some algebraic manipulations, we obtain µZ ¶ Z p1 p2 q1 q2 (1 − C(δ + δ )) χ1 (y) |p(y, ξ)| dy + χ2 (y) |p(y, ξ)| dy Y Y h ¡ ¢i ≤ C (δ q2 θ1 + δ q1 θ2 ) + δ −p1 |ξ|p1 θ1 + δ −p2 |ξ|p2 θ2 On choosing an appropiate δ, we finally obtain (5.3).
Lemma 5.4. For every ξ1 , ξ2 ∈ Rn , we have Z Z p1 χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )|p2 dy Y Y · 1 p1 −2 p1 ≤ C (1 + |ξ1 |p1 θ1 + |ξ1 |p2 θ2 + |ξ2 |p1 θ1 + |ξ2 |p2 θ2 ) p1 −1 |ξ1 − ξ2 | p1 −1 θ1p1 −1 ¸ 1 p2 p2 −2 p −1 p1 p2 p1 p2 2 + (1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) p2 −1 |ξ1 − ξ2 | p2 −1 θ2 , and by doing a change a variables, we obtain Z Z p1 ǫ χ1 (x) |pǫ (x, ξ1 ) − pǫ (x, ξ2 )| dx + χǫ2 (x) |pǫ (x, ξ1 ) − pǫ (x, ξ2 )|p2 dx Yǫ Yǫ · 1 p1 p1 −2 p1 −1 p1 p2 p1 p2 p1 −1 p1 −1 |ξ1 − ξ2 | θ1 ≤ C (1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) ¸ 1 p2 p2 −2 p2 −1 p1 p2 p1 p2 p2 −1 p2 −1 |ξ1 − ξ2 | θ2 + (1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) |Yǫ | Proof. By (3.5), we have (A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), p(y, ξ1 ) − p(y, ξ2 )) ≥ C (χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )|p1 + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )|p2 ) . Integrating over Y and using (3.11) and the Cauchy-Schwarz inequality, we get Z Z p1 χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )|p2 dy Y Y Z ≤C (A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), p(y, ξ1 ) − p(y, ξ2 )) dy Y Z =C (A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), ξ1 − ξ2 ) dy Y Z ≤C |A(y, p(y, ξ1 )) − A(y, p(y, ξ2 ))| |ξ1 − ξ2 | dy Y
Using (3.4), we have ·Z ≤C χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| (1 + |p(y, ξ1 )| + |p(y, ξ2 )|)p1 −2 |ξ1 − ξ2 | dy ¸ Z Y p2 −2 + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| (1 + |p(y, ξ1 )| + |p(y, ξ2 )|) |ξ1 − ξ2 | dy Y
34
(5.5)
(5.6)
p1 Using H¨older’s inequality in the first expression with r1 = , r2 = p1 , and r3 = p1 ; and p1 − 2 p2 in the second expression with s1 = , s2 = p2 , and s3 = p2 , we obtain p2 − 2 " µZ ¶ p1 −2 ≤C
Y
× +
µZ
Y
µZ
Y
× ≤C
+
Y
χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy Z
χ1 (y)dy +
Y
µZ
Y
µZ
χ2 (y)dy +
Z
Y
2
1
χ2 (y) |p(y, ξ1 )| dy +
χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )|p2 dy
p2
Y
¶ p1
¶ p1
2
¶ p1
1
2
¶ p1 µZ
p2
Y
µZ
χ1 (y) |ξ1 − ξ2 | dy
Y
Z
χ1 (y) |p(y, ξ1 )| dy +
χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )|p1 dy
p1
¶ p2p−2
p1
Y
Y
×
1
χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy
p2
" µZ ×
¶ p1 µZ
p1
χ2 (y) (1 + |p(y, ξ1 )| + |p(y, ξ2 )|)p2 dy
µZ
p1
χ1 (y) (1 + |p(y, ξ1 )| + |p(y, ξ2 )|)p1 dy
χ2 (y) |ξ1 − ξ2 | dy p1
Y
¶ p1 #
χ1 (y) |p(y, ξ2 )| dy
2
¶ p1p−2 1
1
|ξ1 − ξ2 | θ1p1 Z
Y
p2
χ2 (y) |p(y, ξ2 )| dy #
¶ p2p−2 2
1
|ξ1 − ξ2 | θ2p2
Use Lemma 5.3 to get h p1 −2 ≤ C (1 + |ξ1 |p1 θ1 + |ξ1 |p2 θ2 + |ξ2 |p1 θ1 + |ξ2 |p2 θ2 ) p1 µZ ¶ p1 1 1 p1 p1 × |ξ1 − ξ2 | θ1 χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy Y
p2 −2
p1
+ (1 + |ξ1 | θ1 + |ξ1 |p2 θ2 + |ξ2 |p1 θ1 + |ξ2 |p2 θ2 ) p2 µZ ¶ p1 # 1 2 p2 p2 × |ξ1 − ξ2 | θ2 χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy Y
By Young’s Inequality, we obtain q2 (p1 −2)q2 p1 p2 p1 p2 q2 p 1 −q2 p1 δ (1 + |ξ | θ + |ξ | θ + |ξ | θ + |ξ | θ ) |ξ − ξ | θ 1 1 1 2 2 1 2 2 1 2 1 ≤C q2 Z Z p1 p2 p1 χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )|p2 dy δ χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy δ Y Y + + p1 p2 35
+
δ
−q1
p1
p2
p1
p2
(1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) q1
(p2 −2)q1 p2
q1
q1 p2
|ξ1 − ξ2 | θ2 .
Straightforward algebraic manipulations deliver µZ ¶ Z p1 p2 kδ χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy Y Y q2 (p1 −2)q2 p1 p2 p1 p2 q2 p 1 −q2 p1 δ (1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) |ξ1 − ξ2 | θ1 ≤C q2 q1 (p2 −2)q1 p q p1 p2 p1 p2 1 −q1 δ (1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) p2 |ξ1 − ξ2 | θ2 2 + q1 1 p1 −2 p1 p1 −1 p1 p2 p1 p2 −q2 p1 −1 p1 −1 δ (1 + |ξ | θ + |ξ | θ + |ξ | θ + |ξ | θ ) |ξ − ξ | θ 1 1 1 2 2 1 2 2 1 2 1 =C q2 1 p2 p2 −2 p2 −1 p1 p2 p1 p2 −q1 p2 −1 p2 −1 |ξ1 − ξ2 | θ2 δ (1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) , + q1
n³ ´ ³ ´o p p where kδ = min 1 − cδp11 , 1 − cδp22 . The result (5.5) follows on choosing δ small enough so that kδ is positive. Lemma 5.5. Let ϕ be such that ½Z ¾ Z p1 p2 ǫ ǫ χ1 (x) |ϕ(x)| dx + χ2 (x) |ϕ(x)| dx < ∞, sup ǫ>0
Ω
Ω
and let Ψ be a simple function of the form Ψ(x) =
m X
ηj χΩj (x),
(5.7)
j=0
with ηj ∈ Rn \ {0}, Ωj ⊂⊂ Ω, |∂Ωj | = 0, Ωj ∩ Ωk = ∅ for j 6= k and j, k = 1, ..., m; and set m [ η0 = 0 and Ω0 = Ω \ Ωj . Then j=1
µZ
χǫ1 (x) |pǫ (x, Mǫ ϕ(x)) − pǫ (x, Ψ(x))|p1 dx lim sup ǫ→0 Ω ¶ Z p2 ǫ + χ2 (x) |pǫ (x, Mǫ ϕ(x)) − pǫ (x, Ψ(x))| dx Ω ( 2 ·µ Z Z X p1 ǫ ≤ lim sup C |Ω| + χ1 (x) |ϕ(x)| dx + χǫ2 (x) |ϕ(x)|p2 dx ǫ→0
i=1
Ω
Ω
36
+
Z
χǫ1 (x) |Ψ(x)|p1
×
µZ
Ω
Ω
dx +
Z
Ω
χǫi (x) |ϕ(x)
χǫ2 (x) |Ψ(x)|p2 pi
− Ψ(x)| dx
¶ p 1−1 i
dx #)
−2 ¶ ppi −1 i
(5.8)
Proof. Let Ψ of the form (5.7). For every ǫ > 0, let us denote by [ Yǫi for i ∈ Iǫ ; Ωǫ = and for j = 0, 1, 2, ..., m, we set
© ª Iǫj = i ∈ Iǫ : Yǫi ⊆ Ωj ,
and Furthermore,
© ª Jǫj = i ∈ Iǫ : Yǫi ∩ Ωj 6= ∅, Yǫi \ Ωj 6= ∅ . Eǫj =
[
Yǫi ,
(5.9)
[
Yǫi ;
(5.10)
i∈Iǫj
and Fǫj =
i∈Jǫj
with
¯ j¯ ¯Fǫ ¯ → 0, as ǫ → 0.
Set
ξǫi
1 = i |Yǫ |
Z
ϕ(x)dx.
Yǫi
For ǫ sufficiently small Ωj is contained in Ωǫ , for all j 6= 0. From (5.7), we have Z Z p1 ǫ χ1 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, Ψ)| dx + χǫ2 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, Ψ)|p2 dx Ω ¯ Ã m Ω !¯p1 Z ¯ ¯ X ¯ ¯ = χǫ1 (x) ¯pǫ (x, Mǫ ϕ) − pǫ x, ηj χΩj ¯ dx ¯ ¯ Ω j=0 ¯ !¯p2 Ã Z m ¯ ¯ X ¯ ¯ + χǫ2 (x) ¯pǫ (x, Mǫ ϕ) − pǫ x, ηj χΩj ¯ dx ¯ ¯ Ω j=0
or equivalently, m Z X χǫ1 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, ηj )|p1 dx = j=0 Ωj m Z X
+
j=0
Ωj
χǫ2 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, ηj )|p2 dx 37
(5.11) (5.12)
Using the fact that Ωj ⊂ Eǫj ∪ Fǫj and (5.1), we have ≤
m Z X
Eǫj
j=0 m Z X
+
Fǫj
j=0
+
m Z X
Eǫj
j=0
+
χǫ1 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, ηj )|p1 dx
m Z X
Fǫj
j=0
χǫ1 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, ηj )|p1 dx χǫ2 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, ηj )|p2 dx χǫ2 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, ηj )|p2 dx
¯ p1 ¯ Ã ! ¯ ¯ X ¯ ¯ i ǫ χYǫi (x)ξǫ − pǫ (x, ηj )¯ dx χ1 (x) ¯pǫ x, = j ¯ ¯ j=0 Eǫ i∈Iǫ ¯ ¯ p1 Ã ! m Z ¯ ¯ X X ¯ ¯ χǫ1 (x) ¯pǫ x, + χYǫi (x)ξǫi − pǫ (x, ηj )¯ dx j ¯ ¯ j=0 Fǫ i∈Iǫ ¯ ¯ p2 ! Ã m Z ¯ ¯ X X ¯ ¯ ǫ i χ2 (x) ¯pǫ x, + χYǫi (x)ξǫ − pǫ (x, ηj )¯ dx j ¯ ¯ j=0 Eǫ i∈Iǫ ¯ ¯ Ã ! m Z ¯ ¯ p2 X X ¯ ¯ ǫ i χ2 (x) ¯pǫ x, + χYǫi (x)ξǫ − pǫ (x, ηj )¯ dx ¯ ¯ Fǫj m Z X
j=0
i∈Iǫ
Using (5.9), (5.10) and reorganizing the sums, we get
=
m X j=0
+
m X j=0
+
m X j=0
+
m X j=0
XZ i∈Iǫj
Yǫi
XZ
i∈Jǫj
i∈Iǫj
¯ ¡ ¯p ¢ χǫ1 (x) ¯pǫ x, ξǫi − pǫ (x, ηj )¯ 1 dx
Yǫi
XZ
Yǫi
XZ
i∈Jǫj
Yǫi
¯ ¡ ¯p ¢ χǫ1 (x) ¯pǫ x, ξǫi − pǫ (x, ηj )¯ 1 dx
¯ ¯ ¡ ¢ p χǫ2 (x) ¯pǫ x, ξǫi − pǫ (x, ηj )¯ 2 dx
¯ ¯ ¡ ¢ p χǫ2 (x) ¯pǫ x, ξǫi − pǫ (x, ηj )¯ 2 dx
38
m X µZ X = j=0
+
Yǫi
i∈Iǫj
Z
Yǫi
χǫ2 (x) ¯pǫ
¯ ¡ ¯p ¢ χǫ1 (x) ¯pǫ x, ξǫi − pǫ (x, ηj )¯ 1 dx
¯ ¡
m X µZ X + j=0
+
Z
Yǫi
x, ξǫi
Yǫi
i∈Jǫj
χǫ2 (x) ¯pǫ
¯ ¡
¢
¯p − pǫ (x, ηj )¯ 2 dx
¶¸
¯p − pǫ (x, ηj )¯ 2 dx
¶¸
¯ ¡ ¯p ¢ χǫ1 (x) ¯pǫ x, ξǫi − pǫ (x, ηj )¯ 1 dx
x, ξǫi
¢
Using Lemma 5.4, we obtain
≤C
m X · X ¡ j=0
i∈Iǫj
¯ ¯p ¯ ¯p ¢ p1 −2 1 + ¯ξǫi ¯ 1 θ1 + ¯ξǫi ¯ 2 θ2 + |ηj |p1 θ1 + |ηj |p2 θ2 p1 −1
1 ¯ ¯ ¯ p1 ¯ × ¯ξǫi − ηj ¯ p1 −1 θ1p1 −1 ¯Yǫi ¯ ¸¾ ¯ i ¯ p1 ¯ i ¯ p2 ¯ p2 p 1−1 ¯ i ¯ ¯ i ¡ ¢ pp2 −2 p1 p2 −1 2 p −1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Yǫ + 1 + ξǫ θ1 + ξǫ θ2 + |ηj | θ1 + |ηj | θ2 2 ξǫ − ηj 2 θ2 m X · X ¯ ¯p ¯ ¯p ¡ ¢ p1 −2 +C 1 + ¯ξǫi ¯ 1 θ1 + ¯ξǫi ¯ 2 θ2 + |ηj |p1 θ1 + |ηj |p2 θ2 p1 −1 j
j=0
i∈Jǫ
1 ¯ ¯ ¯ p1 ¯ × ¯ξǫi − ηj ¯ p1 −1 θ1p1 −1 ¯Yǫi ¯ ¸¾ ¯ p2 p 1−1 ¯ i ¯ ¯ i ¯ i ¯ p1 ¯ i ¯ p2 ¡ ¢ pp2 −2 p p 1 2 −1 2 p −1 ¯ Yǫ ¯ + 1 + ¯ξǫ ¯ θ1 + ¯ξǫ ¯ θ2 + |ηj | θ1 + |ηj | θ2 2 ¯ξǫ − ηj ¯ 2 θ2
≤C
m XZ X j=0 i∈Iǫj
· ¡
Yǫi
1 ¯ p1 ¯ ¯p ¯ ¯p ¢ p1 −2 ¯ 1 + ¯ξǫi ¯ 1 θ1 + ¯ξǫi ¯ 2 θ2 + |ηj |p1 θ1 + |ηj |p2 θ2 p1 −1 ¯ξǫi − ηj ¯ p1 −1 θ1p1 −1
¸ −2 ¯ ¯ p2 p 1−1 ¯ i ¯ p1 ¯ i ¯p2 ¡ ¢ pp2 −1 p1 p2 i 2 p2 −1 ¯ ¯ ¯ ¯ ¯ ¯ 2 ξǫ − ηj θ2 dx + 1 + ξǫ θ1 + ξǫ θ2 + |ηj | θ1 + |ηj | θ2
+C
m XZ X j=0 i∈Jǫj
Yǫi
· ¡
1 ¯ p1 ¯ ¯p ¯ ¯p ¢ p1 −2 ¯ 1 + ¯ξǫi ¯ 1 θ1 + ¯ξǫi ¯ 2 θ2 + |ηj |p1 θ1 + |ηj |p2 θ2 p1 −1 ¯ξǫi − ηj ¯ p1 −1 θ1p1 −1
¸ −2 ¯ ¯ p2 p 1−1 ¯ i ¯ p1 ¯ i ¯p2 ¡ ¢ pp2 −1 p1 p2 i 2 p2 −1 ¯ ¯ ¯ ¯ ¯ ¯ 2 ξǫ − ηj θ2 dx + 1 + ξǫ θ1 + ξǫ θ2 + |ηj | θ1 + |ηj | θ2
39
Using (5.9), (5.10) again ¯ ¯ p1 ¯ ¯ p2 pp1 −2 ¯X ¯ ¯X ¯ 1 −1 1 ¯ ¯ ¯ ¯ p p i i 1 2 1 + ¯¯ =C θ1p1 −1 χYǫi ξǫ ¯¯ θ1 + ¯¯ χYǫi ξǫ ¯¯ θ2 + |ηj | θ1 + |ηj | θ2 j ¯i∈I j ¯ ¯i∈I j ¯ j=0 Eǫ
m Z X
ǫ
ǫ
¯ ¯ p2 ¯ ¯ p1 ¯ p2 −1 1 ¯X ¯ p1 −1 Z ¯X ¯ ¯ ¯ ¯ i i ¯ dx + θ2p2 −1 χYǫi ξǫ − ηj ¯¯ × ¯¯ χYǫi ξǫ − ηj ¯¯ ¯ j Eǫ ¯ ¯ ¯i∈I j ¯ i∈Iǫj ǫ ¯ ¯ p1 ¯ ¯ p2 pp2 −2 ¯ ¯ ¯ ¯ 2 −1 ¯X ¯ ¯X ¯ p p i i 1 2 × 1 + ¯¯ χYǫi ξǫ ¯¯ θ1 + ¯¯ dx χYǫi ξǫ ¯¯ θ2 + |ηj | θ1 + |ηj | θ2 ¯i∈I j ¯ ¯i∈I j ¯ ǫ ǫ ¯ ¯ p1 ¯ ¯p2 ¯X ¯ ¯X ¯ m Z X ¯ ¯ ¯ ¯ i i ¯ ¯ ¯ 1 + +C χYǫi ξǫ ¯ θ1 + ¯ χYǫi ξǫ ¯¯ θ2 + |ηj |p1 θ1 ¯ Fǫj ¯ j ¯ ¯ j ¯ j=0
i∈Jǫ
i∈Jǫ
¯ ¯ ¯ p1 ¯ p2 ¯X ¯ p1 −1 ¯ p2 −1 1 Z ¯X 1 p1 −2 ¯ ¯ ¯ ¯ p −1 p2 i ¯ i ξ − ηj ¯ dx + θ2p2 −1 χYǫi ξǫi − ηj ¯¯ χ + |ηj | θ2 ) p1 −1 θ1 1 ¯¯ Y ǫ ǫ ¯ ¯ Fǫj ¯ ¯i∈J j ¯ ¯ i∈Jǫj ǫ −2 ¯ ¯ p1 ¯ ¯p2 pp2 −1 ¯X ¯ ¯X ¯ 2 ¯ ¯ ¯ ¯ p1 p2 i¯ i¯ ¯ ¯ × 1+¯ χYǫi ξǫ ¯ θ1 + ¯ χYǫi ξǫ ¯ θ2 + |ηj | θ1 + |ηj | θ2 dx ¯i∈J j ¯ ¯i∈J j ¯ ǫ
ǫ
By H¨older’s Inequality, we get
≤C
m X j=0
−2 ¯ ¯ p1 ¯ ¯ p2 pp1 −1 ¯X ¯ ¯X ¯ 1 ¯ ¯ ¯ ¯ p1 p2 i i ¯ 1 + ¯ iξ ¯ iξ ¯ χ θ + χ 1 Y Y ǫ ǫ ǫ ǫ ¯ ¯ ¯ ¯ θ2 + |ηj | θ1 + |ηj | θ2 dx j Eǫ ¯i∈I j ¯ ¯i∈I j ¯
Z
ǫ
ǫ
¯ ¯ ¯ p2 1 ¯p1 1 p1 −1 p2 −1 ¯X ¯X ¯ ¯ Z ¯ ¯ ¯ ¯ i i θ2 ¯¯ + θ1 ¯¯ χYǫi ξǫ − ηj ¯¯ dx χYǫi ξǫ − ηj ¯¯ dx Eǫj Eǫj ¯i∈I j ¯i∈I j ¯ ¯ ǫ ǫ −2 ¯ p2 ¯ ¯ p1 ¯ pp2 −1 ¯ ¯X ¯ ¯X 2 Z ¯ ¯ ¯ ¯ p1 p2 i i ¯ 1 + ¯ iξ ¯ iξ ¯ χ θ + χ × 1 Y Y ǫ ǫ ǫ ǫ ¯ θ2 + |ηj | θ1 + |ηj | θ2 dx ¯ ¯ ¯ j Eǫ ¯ ¯i∈I j ¯ ¯i∈I j ǫ ǫ ¯ ¯ p1 ¯X ¯ Z m X ¯ ¯ i 1 + ¯ +C χYǫi ξǫ ¯¯ θ1 + |ηj |p1 θ1 ¯ Fǫj ¯i∈J j ¯ j=0 Z ×
ǫ
40
−2 ¯ p1 1 ¯ p2 ¯ ¯ pp1 −1 p1 −1 ¯ ¯ ¯X ¯X 1 Z ¯ ¯ ¯ ¯ p χYǫi ξǫi − ηj ¯¯ dx χYǫi ξǫi ¯¯ θ2 + |ηj | 2 θ2 dx θ1 ¯¯ + ¯¯ j Fǫ ¯ ¯ ¯i∈J j ¯i∈J j ǫ ǫ ¯ ¯ p2 1 p2 −1 ¯X ¯ Z ¯ ¯ θ2 ¯¯ + χYǫi ξǫi − ηj ¯¯ dx j Fǫ ¯i∈J j ¯ ǫ −2 ¯ ¯p1 ¯ ¯p2 pp2 −1 ¯X ¯ ¯X ¯ 2 Z ¯ ¯ ¯ ¯ p1 p2 i¯ i¯ ¯ 1 + ¯ dx i i × χ ξ θ + χ ξ 1 Yǫ ǫ ¯ Yǫ ǫ ¯ θ2 + |ηj | θ1 + |ηj | θ2 ¯ ¯ Fǫj ¯i∈J j ¯ ¯i∈J j ¯ ǫ
ǫ
Reorganizing and using (5.12), we have
¯ ¯p1 ¯ ¯ p2 ¯X ¯ ¯ Z ¯X ¯ ¯ ¯ ¯ i i ¯ ¯ iξ ¯ iξ ¯ ≤C χ θ dx + χ 1 Y Y ǫ ǫ ǫ ǫ ¯ ¯ ¯ ¯ θ2 dx Eǫj ¯ Eǫj ¯ ¯ ¯ j j j=0 i∈Iǫ i∈Iǫ ¯ ¯ p1 1 p1 −1 ¯ ¯ Z ¯ j¯ ¯ j ¯¢ p1 −2 ¯X ¯ p1 p i 2 + |ηj | θ1 ¯Eǫ ¯ + |ηj | θ2 ¯Eǫ ¯ p1 −1 × θ1 ¯¯ χYǫi ξǫ − ηj ¯¯ dx Eǫj ¯i∈I j ¯ ǫ ¯ ¯ p1 ¯ ¯ p2 ¯ ¯ Z ¯X Z ¯X ¯ j¯ ¯ ¯ ¯ ¯ i i ¯ ¯ iξ ¯ iξ ¯ + ¯Eǫ ¯ + χ θ dx + χ 1 Y Y ǫ ǫ ǫ ǫ ¯ ¯ ¯ ¯ θ2 dx Eǫj ¯ Eǫj ¯ ¯ ¯ j j i∈Iǫ i∈Iǫ ¯ ¯ p2 1 p2 −1 ¯ ¯ Z ¯X ¯ ¯ j ¯¢ p2 −2 ¯ j¯ p p1 i 2 p −1 θ2 ¯¯ χYǫi ξǫ − ηj ¯¯ dx + |ηj | θ1 ¯Eǫ ¯ + |ηj | θ2 ¯Eǫ ¯ 2 × j Eǫ ¯i∈I j ¯ ǫ ¯ ¯ p1 ¯ ¯ p2 ¯ ¯ Z ¯X Z ¯X m X ¯ ¯ ¯ j¯ ¯ ¯ i i ¯ ¯ ¯ ¯ ¯ Fǫ + χYǫi ξǫ ¯ θ1 dx + +C χYǫi ξǫ ¯¯ θ2 dx ¯ ¯ Fǫj ¯ Fǫj ¯ ¯ ¯ j=0 i∈Jǫj i∈Jǫj ¯ p1 1 ¯ p1 −1 ¯ ¯X Z p −2 ¯ ¯ ¯ ¯ ¯ ¯ ¢ 1 χYǫi ξǫi − ηj ¯¯ dx + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p1 −1 × θ1 ¯¯ j Fǫ ¯ ¯i∈J j ǫ ¯ ¯ p1 ¯ ¯ p2 ¯ ¯ Z ¯X Z ¯X ¯ ¯ j¯ ¯ ¯ ¯ i¯ i¯ ¯ ¯ ¯ ¯ iξ iξ + Fǫ + χ θ dx + χ θ2 dx 1 Y Y ǫ ǫ¯ ǫ ǫ¯ ¯ ¯ Fǫj ¯ Fǫj ¯ ¯ ¯ j j i∈Jǫ i∈Jǫ ¯ ¯ p2 1 p2 −1 ¯ ¯ Z p2 −2 X ¯ ¯ j¯ ¯ ¯ ¯ ¢ p1 p θ2 ¯¯ + |ηj | θ1 ¯Fǫ ¯ + |ηj | 2 θ2 ¯Fǫj ¯ p2 −1 × χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯ m X
Z ¯ ¯ ¯Eǫj ¯ +
ǫ
41
≤C
m X j=0
Z ¯ ¯ X ¯Eǫj ¯ + i∈Iǫj
Yǫj
X ¯ i ¯ p1 ¯ξǫ ¯ θ1 dx + i∈Iǫj
Z
¯ i ¯ p2 ¯ξǫ ¯ θ2 dx
Yǫj
p1 −2 XZ ¯ ¯ ¯ ¯ ¢ p1 p2 j¯ j ¯ p1 −1 ¯ ¯ × + |ηj | θ1 Eǫ + |ηj | θ2 Eǫ i∈Iǫj
¯ ¯ X + ¯Eǫj ¯ + i∈Iǫj
Z
Yǫj
X ¯ i ¯ p1 ¯ξǫ ¯ θ1 dx + i∈Iǫj
¯ ¯¢ p2 −2 ¯ ¯ + |ηj |p1 θ1 ¯Eǫj ¯ + |ηj |p2 θ2 ¯Eǫj ¯ p2 −1
Z
Yǫj
Yǫj
p 1−1 1 ¯ i ¯ p1 ¯ ¯ θ1 ξǫ − ηj dx
¯ i ¯ p2 ¯ξǫ ¯ θ2 dx
XZ × i∈Iǫj
p 1−1 2 ¯ p2 ¯ i θ2 ¯ξǫ − ηj ¯ dx j
Yǫ
¯ ¯ p1 ¯ ¯ p2 ¯X ¯ ¯ Z ¯X ¯ ¯ j¯ ¯ ¯ ¯ i i ¯ ¯ ¯Fǫ ¯ + +C χYǫi ξǫ ¯¯ θ1 dx + χYǫi ξǫ ¯¯ θ2 dx ¯ ¯ Fǫj ¯ Fǫj ¯ ¯ ¯ j=0 i∈Jǫj i∈Jǫj ¯ ¯ p1 1 p1 −1 ¯ ¯ Z p −2 X ¯ ¯ ¯ j ¯¢ 1 ¯ j¯ p2 p1 i p1 −1 ¯ ¯ ¯ ¯ ¯ ¯ θ1 ¯ × χYǫi ξǫ − ηj ¯ dx + |ηj | θ1 Fǫ + |ηj | θ2 Fǫ Fǫj ¯i∈J j ¯ ǫ ¯ ¯ p1 ¯ ¯ p2 ¯ ¯ Z ¯X Z ¯X ¯ ¯ j¯ ¯ ¯ ¯ i¯ i¯ ¯ ¯ ¯ ¯ + Fǫ + χYǫi ξǫ ¯ θ1 dx + χYǫi ξǫ ¯ θ2 dx ¯ ¯ Fǫj ¯ Fǫj ¯ ¯ ¯ i∈Jǫj i∈Jǫj ¯ ¯ p2 1 p2 −1 ¯ ¯ Z p2 −2 X ¯ ¯ ¯ ¯ ¯ ¯ ¢ θ2 ¯¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1 × χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯ m X
Z
ǫ
By (5.1) and (5.9), we have m X ¯ ¯ ¯ ¯ p2 X ¯ ¯ X ¯ j ¯ ¯ i ¯ p1 ¯Yǫj ¯ ¯ξǫi ¯ θ2 ¯Yǫ ¯ ¯ξǫ ¯ θ1 + ¯Eǫj ¯ + ≤C j=0
i∈Iǫj
i∈Iǫj
¯ ¯ ¯ ¯¢ + |ηj |p1 θ1 ¯Eǫj ¯ + |ηj |p2 θ2 ¯Eǫj ¯
p1 −2 p1 −1
p 1−1 1 X¯ ¯ ¯ ¯ p1 i j ¯Yǫ ¯ θ1 ¯ξǫ − ηj ¯ × i∈Iǫj
X ¯ ¯ ¯ ¯ p2 ¯ ¯ X ¯ j ¯ ¯ i ¯ p1 ¯Yǫj ¯ ¯ξǫi ¯ θ2 ¯Yǫ ¯ ¯ξǫ ¯ θ1 + + ¯Eǫj ¯ + i∈Iǫj
i∈Iǫj
¯ ¯ ¯ ¯¢ p2 −2 + |ηj |p1 θ1 ¯Eǫj ¯ + |ηj |p2 θ2 ¯Eǫj ¯ p2 −1
p 1−1 2 X¯ ¯ ¯ ¯ p2 j¯ i ¯ ¯ ¯ Yǫ θ2 ξǫ − ηj × i∈Iǫj
42
Z m ·µ X ¯ j¯ ¯Fǫ ¯ + +C
p1
|Mǫ ϕ| θ1 dx +
Fǫj
j=0
¯ ¯¢ p1 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p1 −1 µ
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
=C
m X j=0
¯ ¯ X ¯Eǫj ¯ + i∈Iǫj
Z
Yǫj
¯ ¯ p1 1 p1 −1 ¯X ¯ ¯ ¯ i θ1 ¯¯ χYǫi ξǫ − ηj ¯¯ dx Fǫj ¯i∈J j ¯
i∈Iǫj
Z
Yǫj
¯ ¯
p χǫ1 (x) ¯ξǫi ¯ 1
¯ ¯
Fǫj
j=0
Z
ǫ
XZ
dx +
i∈Iǫj
dx +
XZ
Yǫj
i∈Iǫj
p1
¯ ¯ + ¯Fǫj ¯ +
Z
i∈Iǫj
|Mǫ ϕ| θ1 dx +
Fǫj
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
Yǫj
Z
Fǫj
¯ ¯p χǫ2 (x) ¯ξǫi ¯ 2 dx
χǫ1 (x) ¯ξǫi ¯
p 1−1 1
¯p − ηj ¯ 1 dx
¯ ¯p χǫ2 (x) ¯ξǫi ¯ 2 dx
XZ ×
p 1−1 2 ¯p2 ¯ i ǫ ¯ ¯ χ2 (x) ξǫ − ηj dx j
Yǫ
|Mǫ ϕ|p2 θ2 dx
¯ ¯ p1 1 p1 −1 ¯X ¯ ¯ ¯ θ1 ¯¯ χYǫi ξǫi − ηj ¯¯ dx j Fǫ ¯i∈J j ¯
Z p1 −2 ¯ ¯ ¯ ¯ ¢ p2 p1 j ¯ p1 −1 j¯ ¯ ¯ + |ηj | θ1 Fǫ + |ηj | θ2 Fǫ × µ
Yǫj
i∈Iǫj
XZ ×
¯ ¯ ¯ ¯¢ p2 −2 + |ηj |p1 θ1 ¯Eǫj ¯ + |ηj |p2 θ2 ¯Eǫj ¯ p2 −1
Z m ·µ X ¯ j¯ ¯Fǫ ¯ + +C
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ θ2 ¯¯ × χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯
p χǫ1 (x) ¯ξǫi ¯ 1
¯ ¯¢ p1 −2 ¯ ¯ + |ηj | θ1 ¯Eǫj ¯ + |ηj |p2 θ2 ¯Eǫj ¯ p1 −1 ¯ ¯ X + ¯Eǫj ¯ +
ǫ
|Mǫ ϕ|p2 θ2 dx
p1
|Mǫ ϕ|p2 θ2 dx
Fǫj
Z ×
¯ ¯¢ p2 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
Z
ǫ
|Mǫ ϕ|p2 θ2 dx
¯ ¯ ¯ ¯¢ p2 −2 + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ θ2 ¯¯ × χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z
43
ǫ
¯ ¯p1 ¯ ¯ p2 ¯X ¯ ¯X ¯ Z ¯ ¯ ¯ ¯ ǫ ǫ i i χ1 (x) ¯¯ χ2 (x) ¯¯ =C χYǫi (x)ξǫ ¯¯ dx + χYǫi (x)ξǫ ¯¯ dx Eǫj Eǫj ¯i∈I j ¯ ¯i∈I j ¯ j=0 ǫ ǫ ¯ ¯ p 1−1 p1 ¯ ¯ 1 Z ¯ j ¯¢ p1 −2 ¯ j¯ ¯X ¯ p p1 ǫ i 2 χ1 (x) ¯¯ + |ηj | θ1 ¯Eǫ ¯ + |ηj | θ2 ¯Eǫ ¯ p1 −1 × χYǫi (x)ξǫ − ηj ¯¯ dx Eǫj ¯i∈I j ¯ ¯ ¯p1 ¯ǫ ¯ p2 ¯X ¯ ¯X ¯ Z Z ¯ ¯ j¯ ¯ ¯ ¯ ǫ ǫ i i χ1 (x) ¯¯ χ2 (x) ¯¯ χYǫi (x)ξǫ ¯¯ dx + + ¯Eǫ ¯ + χYǫi (x)ξǫ ¯¯ dx Eǫj Eǫj ¯i∈I j ¯ ¯i∈I j ¯ ǫ ǫ ¯ ¯ p2 1 p2 −1 ¯ ¯ Z ¯ j ¯¢ p2 −2 ¯ j¯ ¯X ¯ p p1 ǫ i 2 p −1 χ2 (x) ¯¯ + |ηj | θ1 ¯Eǫ ¯ + |ηj | θ2 ¯Eǫ ¯ 2 × χYǫi (x)ξǫ − ηj ¯¯ dx j Eǫ ¯i∈I j ¯ ǫ ·µ Z Z m X ¯ ¯ ¯Fǫj ¯ + |Mǫ ϕ|p2 θ2 dx +C |Mǫ ϕ|p1 θ1 dx + m X
¯ ¯ ¯Eǫj ¯ +
Z
Fǫj
Fǫj
j=0
¯ ¯¢ ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ µ
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
|Mǫ ϕ|p1 θ1 dx +
¯ ¯ p1 1 p1 −1 ¯X ¯ ¯ ¯ i ¯ ¯ θ1 ¯ χYǫi ξǫ − ηj ¯ dx Fǫj ¯i∈J j ¯
Z ×
p1 −2 p1 −1
Z
Fǫj
|Mǫ ϕ|p2 θ2 dx
¯ ¯ ¯ ¯¢ p2 −2 + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
By (5.1) and (5.7), we get Z m ·µ X ¯ j¯ ¯ ¯ ≤C Eǫ +
Eǫj
j=0
+
Z
j
χǫ1 (x) |ηj |p1
µEǫ Z ¯ j¯ ¯ ¯ + Eǫ +
Eǫj
+
Z
Eǫj
+C
χǫ1 (x) |Mǫ ϕ|p1
dx +
j=0
Eǫj
¯ j¯ ¯Fǫ ¯ +
Z
Fǫj
Z
Eǫj
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ i θ2 ¯¯ × χYǫi ξǫ − ηj ¯¯ dx j Fǫ ¯i∈J j ¯
dx +
χǫ2 (x) |ηj |p2
χǫ1 (x) |Mǫ ϕ|p1 dx +
χǫ1 (x) |ηj |p1 dx +
m ·µ X
Z
ǫ
Z
Eǫj
Z
Z
Eǫj
dx
χǫ2 (x) |Mǫ ϕ|p2 dx
−2 ¶ pp1 −1 1
×
µZ
Ω
χǫ1 (x) |Mǫ ϕ
p1
− Ψ| dx
¶ p 1−1 1
χǫ2 (x) |Mǫ ϕ|p2 dx
χǫ2 (x) |ηj |p2 dx
|Mǫ ϕ|p1 θ1 dx +
ǫ
Z
Fǫj
−2 ¶ pp2 −1 2
×
µZ
Ω
|Mǫ ϕ|p2 θ2 dx
44
χǫ2 (x) |Mǫ ϕ − Ψ|p2 dx
¶ p 1−1 # 2
¯ ¯¢ p1 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p1 −1 µ
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
¯ ¯ p1 1 p1 −1 ¯X ¯ ¯ ¯ i θ1 ¯¯ χYǫi ξǫ − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z ×
|Mǫ ϕ|p2 θ2 dx
¯ ¯¢ p2 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
≤C +
m X j=0
Z
Ωj
+
+
× Ã
"Ã
|Ωj | +
µZ
Ω
Ωj
×
χǫ1 (x) |Mǫ ϕ
Ωj
χǫ1 (x) |ηj |p1 dx + χǫ1 (x) |Mǫ ϕ
|Ωj | +
Z
Z
Z
Ωj
Ω
Z
χǫ2 (x) |ηj |p2 dx
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
¶ p 1−1 1
dx
p2
p1
|Mǫ ϕ| θ1 dx +
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
Ωj
χǫ2 (x) |Mǫ ϕ − ϕ + ϕ|p2 dx
1
Z
− ϕ + ϕ| dx + χǫ2 (x) |ηj |p2
Z
−2 ! pp1 −1
p1
¯ ¯¢ p1 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p1 −1 µ
ǫ
− ϕ + ϕ| dx +
− ϕ + ϕ − Ψ| dx
Fǫj
Z
p1
− ϕ + ϕ − Ψ| dx
Ωj
Z m ·µ X ¯ j¯ ¯Fǫ ¯ + +C
¯ p2 1 ¯ p2 −1 ¯ ¯X ¯ ¯ χYǫi ξǫi − ηj ¯¯ dx θ2 ¯¯ × Fǫj ¯ ¯i∈J j
p1
dx +
χǫ2 (x) |Mǫ ϕ
j=0
Ωj
χǫ1 (x) |Mǫ ϕ
χǫ1 (x) |ηj |p1
µZ
Z
ǫ
Ωj
!
χǫ2 (x) |Mǫ ϕ − ϕ + ϕ|p2 dx
p2 −2 p2 −1
¶ p 1−1 # 2
Z
Fǫj
|Mǫ ϕ|p2 θ2 dx
¯ p1 1 ¯ p1 −1 ¯ ¯X ¯ ¯ i χYǫi ξǫ − ηj ¯¯ dx θ1 ¯¯ Fǫj ¯ ¯i∈J j
Z ×
ǫ
|Mǫ ϕ|p2 θ2 dx
¯ ¯¢ p2 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ θ2 ¯¯ × χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z
45
ǫ
≤C
"Ã
m X j=0
+
χǫ2 (x) |Mǫ ϕ
Z
χǫ2 (x) |ηj |p2
Ωj
+
+
Ωj
Z
Ωj
+
|Ωj | +
Z
Ã
|Ωj | +
Z
Ωj
× +C
Z
Ωj
Ω
− ϕ| dx +
dx
×
dx +
Z
Ωj
µZ
χǫ1 (x) |Mǫ ϕ
Ωj
Ω
χǫ1 (x) |ηj |p1 Z
χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx +
m ·µ X
¯ j¯ ¯Fǫ ¯ +
j=0
Z
Ω
|Mǫ ϕ| θ1 dx +
¯ ¯¢ p1 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p1 −1 µ
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
Z
Ωj
χǫ1 (x) |ϕ|p1 dx Z
dx +
Ωj
p1
− ϕ| dx +
Z
Ωj
Fǫj
+
χǫ2 (x) |Mǫ ϕ
m Z X
χǫ2 (x) |ηj |p2
Ωj
j=0
+
j=0
+
j=0
Z
Ω
Ωj
−2 ! pp1 −1 1
dx
χǫ1 (x) |ϕ − Ψ|p1 dx
×
− Ψ| dx
¶ p 1−1
−2 ! pp2 −1 2
dx
¶ p 1−1 # 2
ǫ
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ θ2 ¯¯ × χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z
m Z X
ǫ
j=0
µZ
Ω
Ωj
χǫ2 (x) |ϕ|p2
Ωj
χǫ1 (x) |ϕ|p1 dx
dx +
m Z X j=0
χǫ1 (x) |Mǫ ϕ − ϕ|p1 dx
¶ p 1−1 1
46
1
χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx
¯ ¯ p1 1 p1 −1 ¯X ¯ ¯ ¯ i θ1 ¯¯ χYǫi ξǫ − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z ×
j=0
− ϕ| dx +
Ωj
p1
|Mǫ ϕ|p2 θ2 dx
Ωj
p2
Ω
χǫ2 (x) |ηj |p2
Reorganizing the sums, we obtain "Ã m m Z m Z X X X p1 ǫ χ1 (x) |Mǫ ϕ − ϕ| dx + ≤C |Ωj | + m Z X
χǫ1 (x) |ϕ
Z
χǫ1 (x) |ϕ|p1 dx +
dx +
Z
Z
|Mǫ ϕ|p2 θ2 dx
¯ ¯ ¯ ¯¢ p2 −2 + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
j=0
χǫ1 (x) |ηj |p1 dx
χǫ2 (x) |ϕ − Ψ|p2 dx
p1
Fǫj
Ωj
χǫ2 (x) |ϕ|p2
− ϕ| dx + 1
Z
Z
p2
! pp1 −2 −1
p1
χǫ1 (x) |Mǫ ϕ − ϕ|p1 dx +
χǫ2 (x) |ϕ|p2
µZ
χǫ1 (x) |Mǫ ϕ
Ωj
χǫ1 (x) |ηj |p1 dx
+ +
à m X j=0 m XZ
Ωj
j=0
+
|Ωj | +
m Z X j=0
×
Ωj
µZ
Ω
m Z X
Ωj
j=0
χǫ2 (x) |Mǫ ϕ
p2
− ϕ| dx +
χǫ2 (x) |ηj |p2 dx
−2 ! pp2 −1
χǫ2 (x) |Mǫ ϕ − ϕ| dx +
m Z X j=0
Z
χǫ2 (x) |ϕ|p2
Ωj
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
dx +
Ω
p2
|Mǫ ϕ| θ1 dx +
Z
Fǫj
m Z X j=0
Ωj
χǫ1 (x) |ηj |p1 dx
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
¶ p 1−1 # 2
|Mǫ ϕ|p2 θ2 dx
¯ ¯ p1 1 p1 −1 ¯ ¯ ¯X ¯ θ1 ¯¯ χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z p1 −2 ¯ ¯ ¯ ¯ ¢ p1 p2 j¯ j ¯ p1 −1 ¯ ¯ + |ηj | θ1 Fǫ + |ηj | θ2 Fǫ × µ
Ωj
j=0
χǫ1 (x) |ϕ|p1 dx
χǫ2 (x) |ϕ − Ψ| dx
p1
Fǫj
j=0
− ϕ| dx +
m Z X
2
p2
Z m ·µ X ¯ j¯ ¯Fǫ ¯ + +C
p1
χǫ1 (x) |Mǫ ϕ
ǫ
|Mǫ ϕ|p2 θ2 dx
¯ ¯ p2 1 p2 −1 ¯ ¯ p2 −2 X ¯ ¯ ¯ ¯ ¯ ¯ ¢ θ2 ¯¯ χYǫi ξǫi − ηj ¯¯ dx + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1 × Fǫj ¯i∈J j ¯ ǫ ·µ Z Z Z p1 p1 ǫ ǫ ≤C |Ω| + χ1 (x) |Mǫ ϕ − ϕ| dx + χ1 (x) |ϕ| dx + χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx
Ω
Z
Ω
Ω
¯ ¯ p1 ¯ ¯p2 ! p1 −2 Z m m p1 −1 ¯X ¯ ¯X ¯ ¯ ¯ ¯ ¯ + χǫ2 (x) |ϕ|p2 dx + χǫ1 (x) ¯ χΩj (x)ηj ¯ dx + χǫ2 (x) ¯ χΩj (x)ηj ¯ dx ¯ ¯ ¯ ¯ Ω Ω Ω j=0 j=0 µZ ¶ p 1−1 Z 1 p1 p1 ǫ ǫ × χ1 (x) |Mǫ ϕ − ϕ| dx + χ1 (x) |ϕ − Ψ| dx Ω Ω µ Z Z Z p1 p1 ǫ ǫ + |Ω| + χ1 (x) |Mǫ ϕ − ϕ| dx + χ1 (x) |ϕ| dx + χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx Z
Z
Ω
Ω
Ω
¯ m ¯ p1 ¯ m ¯p2 ! p2 −2 Z p2 −1 ¯X ¯ ¯X ¯ ¯ ¯ ¯ ¯ + χǫ2 (x) |ϕ|p2 dx + χǫ1 (x) ¯ χΩj (x)ηj ¯ dx + χǫ2 (x) ¯ χΩj (x)ηj ¯ dx ¯ ¯ ¯ ¯ Ω Ω Ω j=0 j=0 µZ ¶ p 1−1 # Z 2 p2 p2 ǫ ǫ × χ2 (x) |Mǫ ϕ − ϕ| dx + χ2 (x) |ϕ − Ψ| dx Z
Z
Ω
Ω
Z m ·µ X ¯ j¯ ¯Fǫ ¯ + +C j=0
Fǫj
p1
|Mǫ ϕ| θ1 dx +
Z
Fǫj
|Mǫ ϕ|p2 θ2 dx
47
¯ ¯¢ ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ µ
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
|Mǫ ϕ|p1 θ1 dx +
¯ ¯ p1 1 p1 −1 ¯X ¯ ¯ ¯ i θ1 ¯¯ χYǫi ξǫ − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z ×
p1 −2 p1 −1
Z
Fǫj
ǫ
|Mǫ ϕ|p2 θ2 dx
¯ ¯¢ p2 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ i θ2 ¯¯ × χYǫi ξǫ − ηj ¯¯ dx j Fǫ ¯i∈J j ¯
Z
ǫ
By (5.7), we have ·µ Z Z Z p1 p1 ǫ ǫ =C |Ω| + χ1 (x) |Mǫ ϕ − ϕ| dx + χ1 (x) |ϕ| dx + χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx Ω
+
Z
Ω
Ω
χǫ2 (x) |ϕ|p2 dx +
µZ
Z
Ω
χǫ1 (x) |Ψ|p1 dx + Z
p1
χǫ1 (x) |Mǫ ϕ
Ω
Z
Ω
χǫ2 (x) |Ψ|p2 dx
χǫ1 (x) |ϕ
−2 ¶ pp1 −1 1
¶ p 1−1 1
p1
× − ϕ| dx + − Ψ| dx Ω Ω µ Z Z Z p1 p1 ǫ ǫ + |Ω| + χ1 (x) |Mǫ ϕ − ϕ| dx + χ1 (x) |ϕ| dx + χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx Ω
+
Ω
Z
χǫ2 (x) |ϕ| dx +
×
µZ
p2
Ω
+C
Ω
Z
Ω
p1
χǫ1 (x) |Ψ| dx +
χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx +
m ·µ X j=0
¯ j¯ ¯Fǫ ¯ +
Z
Fǫj
Ω
Z
Z
Ω
p1
Ω
¯ ¯ + ¯Fǫj ¯ +
Z
|Mǫ ϕ| θ1 dx +
Fǫj
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
χǫ2 (x) |Ψ| dx
χǫ2 (x) |ϕ − Ψ|p2 dx Z
Fǫj
−2 ¶ pp2 −1 2
¶ p 1−1 # 2
|Mǫ ϕ|p2 θ2 dx
¯ ¯ p1 1 p1 −1 ¯ ¯ ¯X ¯ θ1 ¯¯ χYǫi ξǫi − ηj ¯¯ dx j Fǫ ¯i∈J j ¯
Z p −2 ¯ ¯ ¯ ¯ ¢ 1 p1 p2 j¯ j ¯ p1 −1 ¯ ¯ + |ηj | θ1 Fǫ + |ηj | θ2 Fǫ × µ
p2
ǫ
|Mǫ ϕ|p2 θ2 dx
¯ ¯¢ p2 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ i θ2 ¯¯ × χYǫi ξǫ − ηj ¯¯ dx j Fǫ ¯i∈J j ¯
Z
ǫ
Since |∂Ωj | = 0 for j 6= 0, then we have that |Fǫj | → 0 as ǫ → 0 for every j = 0, 1, 2, ..., m. Also, by Property 1 of Mǫ in Remark 5.1, we have Z χǫi (x) |Mǫ ϕ − ϕ|pi dx → 0, Ω
48
as ǫ → 0, for i = 1, 2. Therefore, taking lim sup as ǫ → 0 above, we obtain (5.8). Lemma 5.6. If the microstructure is dispersed or layered, we have that ½Z ¾ pi ǫ χi (x) |pǫ (x, Mǫ ∇u)| dx ≤ C < ∞, for i = 1, 2. sup ǫ>0
Ω
Proof. Using (5.1), we have Z Z p1 ǫ χ1 (x) |pǫ (x, Mǫ ∇u)| dx + χǫ2 (x) |pǫ (x, Mǫ ∇u)|p2 dx Ω Ω Z Z p ǫ 1 = χ1 (x) |pǫ (x, Mǫ ∇u)| dx + χǫ2 (x) |pǫ (x, Mǫ ∇u)|p2 dx Ωǫ Ωǫ XZ XZ ¯ ¯ ¯ ¯p p 1 χǫ1 (x) ¯pǫ (x, ξǫi )¯ dx + = χǫ2 (x) ¯pǫ (x, ξǫi )¯ 2 dx Yǫi
i∈Iǫ
=
X ·Z
Yǫi
i∈Iǫ
Yǫi
i∈Iǫ
p χǫ1 (x) ¯pǫ (x, ξǫi )¯ 1
¯
¯
dx +
Z
Yǫi
¯ ¯p χǫ2 (x) ¯pǫ (x, ξǫi )¯ 2
dx
¸
By Lemma 5.3, Jensen’s inequality, and Theorem 4.1 we get X¡ ¯ ¯p ¯ ¯p ¢ ¯ ¯ ≤C 1 + ¯ξǫi ¯ 1 θ1 + ¯ξǫi ¯ 2 θ2 ¯Yǫi ¯ i∈Iǫ
X ¡¯ ¯ ¯ ¯p1 ¯ ¯ ¯ ¯p2 ¯ ¯¢ ¯Yǫi ¯ + ¯ξǫi ¯ θ1 ¯Yǫi ¯ + ¯ξǫi ¯ θ2 ¯Yǫi ¯ =C i∈I
´ ³ ǫ ≤ C |Ω| + kMǫ ∇ukpL1p1 (Ω) + kMǫ ∇ukpL2p2 (Ω) ´ ³ ≤ C |Ω| + k∇ukpL1p1 (Ω) + k∇ukpL2p2 (Ω) < ∞ (uniformly with respect to ǫ).
Z
Lemma 5.7. For all j = 0, ..., m, we have that |(Aǫ (x, pǫ (x, ηj )) , ∇uǫ )| dx as well as Ωj Z |(Aǫ (x, ∇uǫ ) , pǫ (x, ηj ))| dx are uniformly bounded with respect to ǫ. Ωj
Proof. Using H¨older’s Inequality, (3.4), and (3.12), we obtain Z Z |Aǫ (x, pǫ (x, ηj ))| |∇uǫ | dx |(Aǫ (x, pǫ (x, ηj )) , ∇uǫ )| dx ≤ Ωj
Ωj
à Z ≤C
Ωj
χǫ1 (x) (1 + |pǫ (x, ηj )|)p1 dx
! q1
2
≤ C, where C does not depend on ǫ. 49
+
ÃZ
Ωj
! q1 1 χǫ2 (x) (1 + |pǫ (x, ηj )|)p2 dx
The proof of the uniform boundedness of
Z
Ωj
manner.
|(Aǫ (x, ∇uǫ ) , pǫ (x, ηj ))| dx follows in the same
Lemma 5.8. As ǫ → 0, up to a subsequence, (Aǫ (·, pǫ (·, ηj )) , ∇uǫ (·)) converges weakly to a function gj ∈ L1 (Ωj ; R), for all j = 0, ..., m. In a similar way, up to a subsequence, (Aǫ (·, ∇uǫ (·)) , pǫ (·, ηj )) converges weakly to a function hj ∈ L1 (Ωj ; R), for all j = 0, ..., m. Proof. We prove the first statement of the lemma, the second statement follows in a similar way. The lemma follows from the Dunford-Pettis theorem (see [Dac89]). To apply this theorem we establish the following conditions given by: Z |(Aǫ (x, pǫ (x, ηj )) , ∇uǫ )| dx is uniformly bounded with respect to ǫ, and 1. Ωj
2. For all j = 0, ..., m, (Aǫ (·, pǫ (·, ηj )) , ∇uǫ ) is equiintegrable. The first condition is proved in Lemma 5.7. For the second condition, we have that χǫ1 (·) |Aǫ (·, pǫ (·, ηj ))|q2 and χǫ2 (·) |Aǫ (·, pǫ (·, ηj ))|q1 are equiintegrable (see for example Theorem 1.5 of [Dac89]). By (3.12), for any E ⊂ Ω, we have (µZ ( ¶ p1 )) i pi ǫ χi (x) |∇uǫ | dx max sup ≤ C. i=1,2
ǫ>0
E
1/q
1/q
Let α > 0 arbitrary and choose α1 > 0 and α2 > 0 such that α1 2 + α2 1 < α/C. For α1 and α2 , there exist λ(α1 ) > 0 and λ(α2 ) > 0 such that for every E ⊂ Ω with |E| < min {λ(α1 ), λ(α2 )}, Z Z q2 ǫ χ1 (x) |Aǫ (x, pǫ (x, ηj ))| dx < α1 , and χǫ2 (x) |Aǫ (x, pǫ (x, ηj ))|q1 dx < α2 . E
E
Take λ = λ(α) = min {λ(α1 ), λ(α2 )}. Then, for all E ⊂ Ω with |E| < λ(α), we have Z Z |(Aǫ (x, pǫ (x, ηj )) , ∇uǫ )| dx ≤ |Aǫ (x, pǫ (x, ηj ))| |∇uǫ | dx E
E
≤
µZ
E
+ ≤
χǫ1 (x) |Aǫ
µZ
(x, pǫ (x, ηj ))| dx
χǫ2 (x) |Aǫ
E 1/q2 C(α1
+
q2
1/q α2 1 )
¶ q1 µZ 2
E
q1
(x, pǫ (x, ηj ))| dx
χǫ1 (x) |∇uǫ |p1
¶ q1 µZ 1
E
< α,
χǫ2 (x) |∇uǫ |p2
for every α > 0, and so (Aǫ (·, pǫ (·, ηj )) , ∇uǫ ) is equiintegrable.
50
dx
¶ p1
dx
1
¶ p1
2
5.3
Proof of the Corrector Theorem
We are now in the position to give the proof of Theorem 5.2. Proof. Let uǫ ∈ W01,p1 (Ω) the solutions of (3.7). By (3.5), we have that Z [χǫ1 (x) |pǫ (x, Mǫ ∇u(x)) − ∇uǫ (x)|p1 + χǫ2 (x) |pǫ (x, Mǫ ∇u(x)) − ∇uǫ (x)|p2 ] dx Ω Z ≤C (Aǫ (x, pǫ (x, Mǫ ∇u(x))) − Aǫ (x, ∇uǫ (x)) , pǫ (x, Mǫ ∇u(x)) − ∇uǫ (x)) dx Ω
To prove Theorem 5.2, we show that Z (Aǫ (x, pǫ (x, Mǫ ∇u)) − Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u) − ∇uǫ ) dx Ω Z Z = (Aǫ (x, pǫ (x, Mǫ ∇u)) , pǫ (x, Mǫ ∇u)) dx − (Aǫ (x, pǫ (x, Mǫ ∇u)) , ∇uǫ ) dx Ω Ω Z Z − (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u)) dx + (Aǫ (x, ∇uǫ ) , ∇uǫ ) dx Ω
Ω
goes to 0, as ǫ → 0. This is done in four steps. In what follows, we use the following notation Z 1 i ∇u(x)dx. ξǫ = i |Yǫ | Yǫi Step 1 Let us prove that Z Z (Aǫ (x, pǫ (x, Mǫ ∇u)) , pǫ (x, Mǫ ∇u)) dx → (b(∇u), ∇u) dx, as ǫ → 0. Ω
Ω
Proof. From (3.18) and (5.1), we obtain Z (Aǫ (x, pǫ (x, Mǫ ∇u(x))) , pǫ (x, Mǫ ∇u(x))) dx Ω Z = (Aǫ (x, pǫ (x, Mǫ ∇u(x))) , pǫ (x, Mǫ ∇u(x))) dx Ωǫ X Z ³ ³ x ³ x ´´ ³ x ´´ A = ,p , ξǫi , p , ξǫi dx ǫ ǫ ǫ i i∈Iǫ Yǫ Z X ¡ ¡ ¡ ¢¢ ¡ ¢¢ A y, p y, ξǫi , p y, ξǫi dy = ǫn i∈Iǫ
=
XZ i∈Iǫ
=
Z
Ω
Ω
Y
¡ ¢ χYǫi (x) b(ξǫi ), ξǫi dx
(b(Mǫ ∇u(x)), Mǫ ∇u(x)) dx. 51
(5.13)
By (3.15) and the definition of q1 , we have Z
|b(Mǫ ∇u(x)) − b(∇u(x))|q1 dx Ω Z h p1 −2 1 ≤ C |Mǫ ∇u − ∇u| p1 −1 (1 + |Mǫ ∇u|p1 + |∇u|p1 + |Mǫ ∇u|p2 + |∇u|p2 ) p1 −1 Ω p2 −2 iq1 1 + |Mǫ ∇u − ∇u| p2 −1 (1 + |Mǫ ∇u|p1 + |∇u|p1 + |Mǫ ∇u|p2 + |∇u|p2 ) p2 −1 dx Z h p2 ≤C |Mǫ ∇u − ∇u| (p1 −1)(p2 −1) (1 + |Mǫ ∇u|p1 + |∇u|p1 Ω
p2
p2 (p1 −2)
+ |Mǫ ∇u|p2 + |∇u|p2 ) (p1 −1)(p2 −1) + |Mǫ ∇u − ∇u| (p2 −1)2 p1
p1
p2
p2
× (1 + |Mǫ ∇u| + |∇u| + |Mǫ ∇u| + |∇u| )
p2 (p2 −2) (p2 −1)2
¸
dx
By H¨older’s Inequality in the first integral with p = (p2 − 1)(p1 − 1) > 1 and in the second integral with p = (p2 − 1)2 > 1, we obtain ≤C
"µZ
Ω
× +
p2
µZ
Ω
µZ
Ω
×
|Mǫ ∇u(x) − ∇u(x)| dx p1
p2
p2
(1 + |Mǫ ∇u| + |∇u| + |Mǫ ∇u| + |∇u| )
|Mǫ ∇u(x) − ∇u(x)| dx
Ω
1 2 −1)(p1 −1)
p1
p2
µZ
¶ (p
¶
p2 (p1 −2) (p1 −1)(p2 −1)−1
dx
1 (p2 −1)2
(1 + |Mǫ ∇u|p1 + |∇u|p1 + |Mǫ ∇u|p2 + |∇u|p2 ) dx
¶ p2 (p2 −2) 2 (p2 −1)
¶ (p(p1 −1)(p 2 −1)−1 −1)(p −1) 1
2
By Theorem 4.1 and Jensen’s inequality, we have "µZ 1 ¶ (p −1)(p µZ ¶ 1 2# (p2 −1) 2 1 −1) |Mǫ ∇u − ∇u|p2 dx ≤C + |Mǫ ∇u − ∇u|p2 dx Ω
Ω
From Property 1 of Mǫ in Remark 5.1, we obtain that b(Mǫ ∇u) → b(∇u) in Lq1 (Ω; Rn ), as ǫ → 0. Now, (5.13) follows from (5.14) since Mǫ ∇u → ∇u in Lp2 (Ω; Rn ), so Z Z (Aǫ (x, pǫ (x, Mǫ ∇u)) , pǫ (x, Mǫ ∇u)) dx = (b(Mǫ ∇u(x)), Mǫ ∇u(x)) dx Ω Ω Z → (b(∇u(x)), ∇u(x)) dx, as ǫ → 0. Ω
52
(5.14)
Step 2 We now show that Z Z (Aǫ (x, pǫ (x, Mǫ ∇u)) , ∇uǫ ) dx → (b(∇u), ∇u) dx, as ǫ → 0. Ω
(5.15)
Ω
Proof. Let δ > 0. From Theorem 4.1 we have that ∇u ∈ Lp2 (Ω; Rn ) and there exists a simple function Ψ satisfying the assumptions of Lemma 5.5 such that k∇u − ΨkLp2 (Ω;Rn ) ≤ δ.
(5.16)
Let us write Z (Aǫ (x, pǫ (x, Mǫ ∇u(x))) , ∇uǫ (x)) dx Ω Z Z = (Aǫ (x, pǫ (x, Ψ)) , ∇uǫ ) dx + (Aǫ (x, pǫ (x, Mǫ ∇u)) − Aǫ (x, pǫ (x, Ψ)) , ∇uǫ ) dx. Ω
Ω
We first show that Z Z (Aǫ (x, pǫ (x, Ψ(x))) , ∇uǫ (x)) dx → (b(Ψ(x)), ∇u(x)) dx, as ǫ → 0. Ω
Ω
We have Z
Ω
(Aǫ (x, pǫ (x, Ψ(x))) , ∇uǫ (x)) dx =
m Z X
Ωj
j=0
(Aǫ (x, pǫ (x, ηj )) , ∇uǫ (x)) dx.
n q2 Now from Z (3.24), we have that Aǫ (·, pǫ (·, ηj )) ⇀ b(ηj ) ∈ L (Ωj ; R ), and by (3.17) we (Aǫ (x, pǫ (x, ηj )) , ∇ϕ) dx = 0, for ϕ ∈ W01,p1 (Ωj ). have that Ωj
Take ϕ = δuǫ , with δ ∈ C0∞ (Ωj ) to get Z Z (Aǫ (x, pǫ (x, ηj )) , (∇δ)uǫ ) dx + 0=
(Aǫ (x, pǫ (x, ηj )) , (∇uǫ )δ) dx.
Ωj
Ωj
Taking the limit as ǫ → 0, and using the fact that uǫ ⇀ u in W01,p1 (Ω) and (3.24), we have by Lemma 5.8 that Z Z (Aǫ (x, pǫ (x, ηj )) , (∇uǫ )δ) dx gj δdx = lim ǫ→0 Ω Ωj j Z (Aǫ (x, pǫ (x, ηj )) , (∇δ)uǫ ) dx = − lim ǫ→0 Ω j Z =− (b(ηj ), (∇δ)u) dx Ωj Z (b(ηj ), (∇u)δ) dx. = Ωj
53
′
Then (Aǫ (·, pǫ (·, ηj )) , ∇uǫ (·)) ⇀ (b(ηj ), ∇u) in D (Ωj ), as ǫ → 0. Therefore, we may conclude that gj = (b(ηj ), ∇u), so n Z n Z X X (b(ηj ), ∇u) dx, as ǫ → 0. (Aǫ (x, pǫ (x, ηj )) , ∇uǫ ) dx → j=0
Ωj
j=0
Ωj
Thus, we get Z Z (Aǫ (x, pǫ (x, Ψ(x))) , ∇uǫ (x)) dx → (b(Ψ(x)), ∇u(x)) dx, as ǫ → 0. Ω
Ω
On the other hand, let us estimate Z (Aǫ (x, pǫ (x, Mǫ ∇u(x))) − Aǫ (x, pǫ (x, Ψ(x))) , ∇uǫ (x)) dx. Ω
By (3.4), we have ¯Z ¯ ¯ ¯ ¯ (Aǫ (x, pǫ (x, Mǫ ∇u)) − Aǫ (x, pǫ (x, Ψ)) , ∇uǫ ) dx¯ ¯ ¯ Ω Z ≤ |Aǫ (x, pǫ (x, Mǫ ∇u)) − Aǫ (x, pǫ (x, Ψ))| |∇uǫ | dx Ω Z ≤C χǫ1 (x) |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| (1 + |pǫ (x, Mǫ Du)| + |pǫ (x, Ψ)|)p1 −2 |∇uǫ | dx Ω Z +C χǫ2 (x) |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| (1 + |pǫ (x, Mǫ Du)| + |pǫ (x, Ψ)|)p2 −2 |∇uǫ | dx Ω
Applying H¨older’s inequality we obtain ¶ p1 µZ ¶ p1 µZ 1 1 p1 p1 ǫ ǫ χ1 (x) |∇uǫ | dx ≤C χ1 (x) |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx Ω
Ω
×
µZ
Ω
χǫ1 (x) (1 + |pǫ (x, Mǫ ∇u)| + |pǫ (x, Ψ)|) dx
+C
µZ
Ω
χǫ2 (x) |pǫ
×
µZ
χǫ2 (x) (1 + |pǫ (x, Mǫ ∇u)| + |pǫ (x, Ψ)|)p2 dx
≤C
Ω
µZ
Ω
p1
χǫ1 (x) |pǫ
p2
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx
p1
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx
¶ p1p−2 1
¶ p1 µZ 2
Ω
¶ p1 µZ 1
Ω
χǫ1 (x) |∇uǫ |p1
Ω
χǫ1 (x) (1 + |pǫ (x, Mǫ ∇u)|p1 + |pǫ (x, Ψ)|p1 ) dx
+C
µZ
Ω
χǫ2 (x) |pǫ
×
µZ
χǫ2 (x) (1
Ω
p2
¶ p1 µZ 2
Ω
p2
+ |pǫ (x, Mǫ ∇u)| + |pǫ (x, Ψ)| ) dx 54
¶ p1
2
2
×
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx
dx
¶ p2p−2
µZ
p2
χǫ2 (x) |∇uǫ |p2
dx
¶ p1p−2
¶ p1
1
1
χǫ2 (x) |∇uǫ |p2 ¶ p2p−2 2
dx
¶ p1
2
By (3.12), we get ≤C
µZ
χǫ1 (x) |pǫ
Ω
µ
× 1+ +C
µZ
Ω
Z
Ω
p1
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx
µ
× 1+
Z
Ω
1
Z
χǫ1 (x) |pǫ (x, Mǫ ∇u)|p1 dx +
χǫ2 (x) |pǫ
¶ p1
Ω
χǫ1 (x) |pǫ (x, Ψ)|p1 dx
p2
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx
χǫ2 (x) |pǫ
p2
(x, Mǫ ∇u)| dx +
Z
Ω
¶ p1
¶ p1p−2 1
2
p2
χǫ2 (x) |pǫ
(x, Ψ)| dx
¶ p2p−2 2
Using (5.6) and Lemma 5.3, we get ≤C +
"µZ
Ω
µZ
Ω
χǫ1 (x) |pǫ
χǫ2 (x) |pǫ
p1
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx p2
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx
¶ p1
1
¶ p1 # 2
Applying Lemma 5.5 and (5.16), we discover that ¯Z ¯ ¯ ¯ lim sup ¯¯ (Aǫ (x, pǫ (x, Mǫ ∇u)) − Aǫ (x, pǫ (x, Ψ)) , ∇uǫ ) dx¯¯ ǫ→0 Ω " µZ ¶1 ≤ lim sup C ǫ→0
+
µZ
χǫ2 (x) |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p2 dx
Ω
≤ lim sup C ǫ→0
+
" µZ
Ω
h
Ω
χǫ1 (x) |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p1 dx
q1
" µZ
Ω
χǫ1 (x) |∇u − Ψ|p1 dx
χǫ1 (x) |∇u − Ψ|p1 dx q2
≤ C (δ + δ )
1 p1
q1
q2
¶ p 1−1
+ (δ + δ )
1
1 p2
i
+
¶ p1 #
¶ p 1−1 1
µZ
Ω
,
p1
2
+
µZ
Ω
χǫ2 (x) |∇u − Ψ|p2 dx
χǫ2 (x) |∇u − Ψ|p2 dx
¶ p 1−1 # p11 2
¶ p 1−1 # p12 2
(5.17)
where C does not depend on δ. Since δ is arbitrary we conclude that the limit on the left hand side of (5.17) is equal to 0.
55
Finally, using (3.15), Theorem 4.1, and H¨older’s inequality, we obtain Z Z (b(∇u) − b(Ψ), ∇u) ≤ |b(∇u) − b(Ψ)| |∇u| dx Ω Ω " µZ ¶1# ¶ 1 µZ ≤C
≤C ≤C
Ω
µZ
|b(∇u) − b(Ψ)|q1 dx
½ZΩ h Ω
q1
|b(∇u) − b(Ψ)|q1 dx
Ω
p2
|∇u|p2 dx
¶ q1
1
p1 −2
1
|∇u − Ψ| p1 −1 (1 + |∇u|p1 + |∇u|p2 + |Ψ|p1 + |Ψ|p2 ) p1 −1 1 p2 −1
p1
p2
p1
p2
p2 −2 p2 −1
o q1
iq 1
1
dx (1 + |∇u| + |∇u| + |Ψ| + |Ψ| ) + |∇u − Ψ| ½Z · p2 p2 (p1 −2) ≤C |∇u − Ψ| (p1 −1)(p2 −1) (1 + |∇u|p1 + |∇u|p2 + |Ψ|p1 + |Ψ|p2 ) (p1 −1)(p2 −1) Ω
+ |∇u − Ψ|
p2 (p2 −1)2
p1
p2
p1
p2
(1 + |∇u| + |∇u| + |Ψ| + |Ψ| )
p2 (p2 −2) (p2 −1)2
¸
Applying H¨older’s inequality and Theorem 4.1 again, we obtain " µZ 1 ¶ (p −1)(p 1 2 −1) p2 |∇u(x) − Ψ(x)| dx ≤C
dx
¾ q1
1
Ω
× +
µZ
Ω
µZ
≤C
µZ
Ω
≤C δ
p1
Ω
q1 p1 −1
p1
p2
(1 + |∇u| + |∇u| + |Ψ| + |Ψ| )
|∇u(x) − Ψ(x)| dx
" µZ h
p2
p2
Ω
×
p1
¶
p2 (p1 −2) (p1 −1)(p2 −1)−1
1 (p2 −1)2
p2
p1
p2
(1 + |∇u| + |∇u| + |Ψ| + |Ψ| ) dx p2
|∇u(x) − Ψ(x)| dx +δ
q1 p2 −1
i q1
1
dx
¶ (p
1 1 −1)(p2 −1)
+
µZ
Ω
¶ (p2 −1)2 −1 2 (p2 −1)
¶ (p(p2 −1)(p 1 −2)−1 −1)(p −1) 1
2
q1
1
p2
|∇u(x) − Ψ(x)| dx
¶
1 (p2 −1)2
# q1
1
,
where C does not depend on δ. Now, since δ is arbitrarily small we conclude the proof of Step 2. Step 3 We will show that Z Z (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u)) dx → (b(∇u), ∇u) dx, as ǫ → 0. Ω
Ω
56
(5.18)
Proof. Let δ > 0. As in the proof of Step 2, assume Ψ is a simple function satisfying assumptions of Lemma 5.5 and such that k∇u − ΨkLp2 (Ω;Rn ) < δ. Let us write Z (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u)) dx Ω Z Z = (Aǫ (x, ∇uǫ ) , pǫ (x, Ψ)) dx + (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)) dx. Ω
Ω
We first show that Z Z (Aǫ (x, ∇uǫ (x)) , pǫ (x, Ψ(x))) dx → (b (∇u(x)) , Ψ(x)) dx. Ω
Ω
We start by writing Z
(Aǫ (x, ∇uǫ (x)) , pǫ (x, Ψ(x))) dx =
Ω
m Z X
(Aǫ (x, ∇uǫ (x)) , pǫ (x, ηj )) dx.
Ωj
j=0
From Lemma 5.8, up to a subsequence, (Aǫ (·, ∇uǫ (·)) , pǫ (·, ηj )) converges weakly to a function hj ∈ L1 (Ωj ; R), as ǫ → 0. By Theorem 3.4, we have Aǫ (·, ∇uǫ ) ⇀ b(∇u) ∈ Lq2 (Ω; Rn ) and −div (Aǫ (x, ∇uǫ )) = f = −div (b(∇u)) . Also, from (3.22), pǫ satisfies pǫ (·, ηj ) ⇀ ηj in Lp1 (Ωj , Rn ). ′ Arguing as in Step 2, we find that (Aǫ (x, ∇uǫ (x)) , pǫ (x, ηj )) ⇀ (b(∇u(x)), ηj ) in D (Ωj ), as ǫ → 0. Therefore, we may conclude that hj = (b(∇u), ηj ), and hence, n Z X j=0
Ωj
(Aǫ (x, ∇uǫ (x)) , pǫ (x, ηj )) dx →
n Z X j=0
Ωj
(b(∇u(x)), ηj ) dx, as ǫ → 0.
Thus, we get Z Z (Aǫ (x, ∇uǫ (x)) , pǫ (x, Ψ(x))) dx → (b(∇u(x)), Ψ(x)) dx, as ǫ → 0. Ω
Ω
Moreover, applying Cauchy-Schwarz inequality we have ¯Z ¯ ¯ ¯ ¯ (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)) dx¯ ¯ ¯ Ω Z ≤ |Aǫ (x, ∇uǫ (x))| |pǫ (x, Mǫ ∇u(x)) − pǫ (x, Ψ(x))| dx Ω
57
H¨older’s inequality delivers Z ≤ χǫ1 |Aǫ (x, ∇uǫ (x))| |pǫ (x, Mǫ ∇u(x)) − pǫ (x, Ψ(x))| dx Ω Z + χǫ2 |Aǫ (x, ∇uǫ (x))| |pǫ (x, Mǫ ∇u(x)) − pǫ (x, Ψ(x))| dx Ω
≤
µZ
Ω
+
q2
χǫ1 |Aǫ (x, ∇uǫ )|
µZ
χǫ2
Ω
¶ q1 µZ 2
Ω
q1
|Aǫ (x, ∇uǫ )|
≤C +
Ω
µZ
Ω
=C
Ω
+
µZ
+
" µZ
Ω
µZ
" µZ
Ω
+
µZ
Ω
(1 + |∇uǫ |)
χǫ1
χǫ1
χǫ1
χǫ2
p1
2
(1 + |∇uǫ |)
¶ q1 µZ 1
Ω
¶ q1 µZ 1
Ω
q2 (p1 −1)
(1 + |∇uǫ |)
χǫ1
χǫ1
|pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx
p1
|pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx
¶ q1 µZ 2
Ω
1
Ω
χǫ1
χǫ2 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p2 dx
1
¶ p1 i# 2
As in the proof of Step 2 we see that ¯Z ¯ ¯ ¯ lim sup ¯¯ (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)) dx¯¯ ǫ→0 Ω "µZ ¶1 +
µZ
Ω
Ω
χǫ1 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p1 dx p2
χǫ2 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx 58
¶ p1 # 2
p1
1
2
1
¶ p1 i# 2 p1
¶ p1
¶ p1
¶ p1 #
|pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx
|pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx
ǫ→0
¶ p1
χǫ2 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p2 dx p1
≤ C lim sup
2
p1
χǫ2 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p2 dx
¶ q1 µZ
¶ p1
χǫ2 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p2 dx
Ω
2
1
|pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx
¶ q1 µZ
¶ q1 µZ
¶ p1
p2
Ω
χǫ2 (1 + |∇uǫ |)q1 (p2 −1)
Ω
=C
1
q2 (p1 −1)
χǫ2 (1 + |∇uǫ |)p2
Ω
≤C
¶ q1 µZ
χǫ2 (1 + |∇uǫ |)q1 (p2 −1)
" µZ
χǫ1 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx
Ω
By (3.4) and (3.12) we get " µZ χǫ1
p1
¶ p1
1
¶ p1 # 2
By Lemma 5.5, we get i h 1 1 ≤ C (δ q2 + δ q1 ) p1 + (δ q2 + δ q1 ) p2 ,
where C does not depend on δ. Hence, proceeding as in Step 2, we find that ¯Z ¯ Z ¯ ¯ lim sup ¯¯ (Aǫ (x, ∇uǫ (x)) , pǫ (x, Mǫ ∇u(x))) dx − (b(∇u(x)), ∇u(x)) dx¯¯ ǫ→0 Ω ¯Z Z Ω ¯ = lim sup ¯¯ (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u)) dx − (Aǫ (x, ∇uǫ ) , pǫ (x, Ψ)) dx ǫ→0 Ω Ω Z Z + (Aǫ (x, ∇uǫ ) , pǫ (x, Ψ)) dx − (b(∇u), Ψ) dx Ω ¯ ZΩ Z ¯ + (b(∇u), Ψ) dx − (b(∇u), ∇u) dx¯¯ Ω Ω ¯Z ¯ Z ¯ ¯ ≤ lim sup ¯¯ (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u)) dx − (Aǫ (x, ∇uǫ ) , pǫ (x, Ψ)) dx¯¯ ǫ→0 ¯ΩZ ¯ Z Ω ¯ ¯ ¯ + lim sup ¯ (Aǫ (x, ∇uǫ ) , pǫ (x, Ψ)) dx − (b(∇u), Ψ) dx¯¯ ǫ→0 Ω ¯ZΩ ¯ Z ¯ ¯ ¯ + lim sup ¯ (b(∇u), Ψ) dx − (b(∇u), ∇u) dx¯¯ ǫ→0 Ω Ω ´ ³ 1 1 q1 p 2 q2 q1 p 1 q2 ≤ C (δ + δ ) + (δ + δ ) + 0 + δ kb(∇u)kLq2 (Ω,Rn ) ,
where C does not depend on δ. Now since δ is arbitrarily small, the proof of Step 3 is complete. Step 4 Finally, let us prove that Z Z (Aǫ (x, ∇uǫ (x)) , ∇uǫ (x)) dx → (b(∇u(x)), ∇u(x)) dx, as ǫ → 0. Ω
(5.19)
Ω
Proof. Since Z
Ω
and
(Aǫ (x, ∇uǫ ) , ∇uǫ ) dx = h−div (Aǫ (x, ∇uǫ )) , uǫ i = hf, uǫ i , Z
Ω
(b(∇u), ∇u) dx = h−div (b (∇u)) , ui = hf, ui ,
and uǫ ⇀ u in W 1,p1 (Ω), the result follows immediately. Finally, Theorem 5.2 follows from (5.13), (5.15), (5.18) and (5.19).
59
(5.20)
(5.21)
Chapter 6 Lower Bounds on Field Concentrations In composites, failure initiation is a multiscale phenomenon. A load applied at the structural scale is often amplified by the microstructure creating local zones of high field concentration. The regions containing high fields are often the first to suffer damage during service. Therefore it is of relevance to assess the load transfer between macroscopic and microscopic length scales. In this chapter, we bound the local singularity strength inside microstructured media in terms of the macroscopic applied fields. The strong approximations in Chapter 5 are used to develop new tools that provide lower bounds on the local gradient field intensity inside micro-structured media. These results provide a lower bound on the amplification of the macroscopic (average) gradient field by the microstructure. In [Lip06], similar lower bounds are established for field concentrations for mixtures of linear electrical conductors in the context of two scale convergence.
6.1
Statement of the Lower Bound on the Amplification of the Macroscopic Field by the Microstructure
We begin by presenting a general lower bound that holds for the composition of the sequence {χǫi ∇uǫ }ǫ>0 with any non-negative Carath´eodory function. Definition 6.1. A function ψ : Ω×Rn → R is a Carath´eodory function if ψ(x, ·) is continuous for almost every x ∈ Ω and if ψ(·, λ) is measurable in x for every λ ∈ Rn . The lower bound on the sequence obtained by the composition of ψ(x, ·) with χǫi (x)∇uǫ (x) is given by Theorem 6.2. For all Carath´eodory functions ψ ≥ 0 and measurable sets D ⊂ Ω, we have Z Z Z ψ (x, χi (y)p (y, ∇u(x))) dydx ≤ lim inf ψ (x, χǫi (x)∇uǫ (x)) dx, (i = 1, 2). D
Y
ǫ→0
60
D
If the sequence {ψ (x, χǫi (x)∇uǫ (x))}ǫ>0 is weakly convergent in L1 (Ω), then the inequality becomes an equality. Remark 6.3. As a direct consequence of Theorem 6.2, taking ψ(x, λ) = |λ|q with q ≥ 2, we display lower bounds on the Lq norm of the gradient fields inside each material that are given in terms of the correctors presented in Theorem 5.2 Z Z Z q χi (y) |p (y, ∇u(x))| dydx ≤ lim inf χǫi (x) |∇uǫ (x)|q dx, (6.1) D
ǫ→0
Y
D
for i = 1, 2. This result is still valid for q = ∞ if we have that χǫi (x)∇uǫ (x) belongs to L∞ (Ω; Rn ) (since ǫ χi (x)∇uǫ (x) belongs to Lpi (Ω; Rn ) by (3.12)) and if χi (y)p (y, ∇u(x)) belongs to L∞ (Y × Ω; Rn ) (since Z Z χi (y) |p (y, ∇u(x))|pi dydx D Y Z ≤C (1 + |∇u(x)|p1 + |∇u(x)|p2 ) dx < ∞, D
by Lemma 5.3 and Theorem 4.1).
It is clear from (6.1) that the Lq (Y × Ω; Rn ) integrability of p(y, ∇u(x)) provides a lower bound on the Lq (Ω; Rn ) integrability of ∇uǫ . Theorem 6.2 together with (6.1) provide explicit lower bounds on the gradient field inside each material. It relates the local excursions of the gradient inside each phase χǫi ∇uǫ to the average gradient ∇u through the multiscale quantity given by the corrector p(y, ∇u(x)).
6.2
Young Measures
We use the the results from Theorem 5.2 and Young Measures to study the behavior of gradients of solutions of the Dirichlet problem (3.7). These tools allow us to bound nonlinear quantities of these gradients from below in terms of the local solution p and the gradient of the homogenized solution u as stated in Section 6.1. Young Measures can be used as a tool to organize our ideas about oscillatory behavior and to deal in a consistent way with oscillations [Ped99]. Young Measures are a family of probability measures ν = {νx }x∈Ω associated with a sequence of functions f ǫ : Ω ⊂ Rn −→ Rn such that the supp(νx )⊂ Rn and they depend measurably on x ∈ Ω, which means that for any continuous function ϕ : Rn −→ R, the function Z ϕ(λ)dνx (λ) (6.2) ϕ(x) = Rn
is measurable. The fundamental property of this family of probability measures is that whenever {ϕ(f ǫ )}ǫ>0 converges weakly* in L∞ (Ω) (or more generally weak in some Lp (Ω)) the weak limit can be identified with the function ϕ in (6.2): Z Z Z ǫ lim ϕ(f )g(x)dx = ϕ(λ)dνx (λ)dx, (6.3) g(x) ǫ→0
Ω
Ω
1
for all g ∈ L (Ω).
61
Rn
6.3
Proof of Lower Bound on the Amplification of the Macroscopic Field by the Microstructure
The sequence {χǫi (x)∇uǫ (x)}ǫ>0 has a Young Measure ν i = {νxi }x∈Ω associated to it (see Theorem 6.2 and the discussion following in [Ped97]), for i = 1, 2. As a consequence of Theorem 5.2 proved in Chapter 5, we have ° ° ´ ³x ° ǫ ° , Mǫ (∇u)(x) − χǫi (x)∇uǫ (x)° p → 0, °χi (x)p ǫ L i (Ω;Rn ) as ǫ → 0 for i = 1, 2, which implies that the sequences ´o n ³x , Mǫ (∇u)(x) and {χǫi (x)∇uǫ (x)}ǫ>0 χǫi (x)p ǫ ǫ>0
share the same Young Measure ν i = {νxi }x∈Ω (see Lemma 6.3 of [Ped97]). The next Lemma identifies the Young measure ν i .
Lemma 6.4. For all φ ∈ C0 (Rn ) and for all ζ ∈ C0∞ (Rn ), we have Z Z Z Z i φ(λ)dνx (λ)dx = ζ(x) φ(χi (y)p(y, ∇u(x)))dydx. ζ(x) Ω
Rn
Ω
(6.4)
Y
(From discussion in Chapter 6 of [Ped97], this is enough to identify the Young measure ν i ) Proof. To prove (6.4), we will show that given φ ∈ C0 (Rn ) and ζ ∈ C0∞ (Rn ) Z Z Z ´´ ³ ³x ǫ , Mǫ (∇u) (x) dx = ζ(x) φ(χi (y)p(y, ∇u(x)))dydx. lim ζ(x)φ χi (x)p ǫ→0 Ω ǫ Ω Y
(6.5)
We consider the difference ¯Z ¯ Z Z ³ ³ ´ ³ ´´ ¯ ¯ ¯ ζ(x)φ χi x p x , Mǫ (∇u)(x) dx − ζ(x) φ (χi (y) p (y, ∇u(x))) dydx¯ ¯ ¯ ǫ ǫ Ω Ω Y
By (5.1), we have
¯ ¯ Z Z ¯X Z ¯ ³ ³ x ´ ³ x ´´ ¯ ¯ ≤¯ ζ(x)φ χi ζ(x) φ (χi (y) p (y, ∇u(x))) dydx¯ , ξǫi dx − p ¯ ¯ ǫ ǫ i Ωǫ Y i∈Iǫ Yǫ ¯ ¯Z Z Z ³ ³ x ´ ³ x ´´ ¯ ¯ , 0 dx − ζ(x) φ (χi (y) p (y, ∇u(x))) dydx¯¯ p + ¯¯ ζ(x)φ χi ǫ ǫ Ω\Ωǫ Y ǫ ¯ Ω\Ω ¯ Z Z ¯X Z ¯ ³ ³ x ´ ³ x ´´ ¯ ¯ i ζ(x)φ χi ≤¯ ζ(x) φ (χi (y) p (y, ∇u(x))) dydx¯ , ξǫ dx − p ¯ ¯ ǫ ǫ Yǫi Ωǫ Y i∈I ǫ
+ C |Ω \ Ωǫ | .
(6.6)
62
Note that the term C |Ω \ Ωǫ | goes to 0, as ǫ → 0. Now set xiǫ to be the center of Yǫi . On the first integral use the change of variables x = xiǫ + ǫy, where y belongs to Y , and since dx = ǫn dy, we get ¯ ¯ Z ¯ ¯X Z ³ ³ x ´ ³ x ´´ XZ ¯ ¯ i , ξǫ dx − p ζ(x) φ (χi (y) p (y, ∇u(x))) dydx¯ ζ(x)φ χi ¯ ¯ ¯ ǫ ǫ i i Y i∈Iǫ Yǫ i∈Iǫ Yǫ ¯ ¯X Z ¡ ¡ ¢¢ ¯ ζ(xiǫ + ǫy)φ χi (y) p y, ξǫi dy =¯ ǫn ¯ Y i∈Iǫ ¯ Z Z ¯ X ¯ ζ(x) φ (χi (y) p (y, ∇u(x))) dydx¯ − ¯ i Y Yǫ i∈Iǫ
Applying Taylor’s expansion for ζ, we obtain
¯ ¯X Z £ ¤ ¡ ¡ ¢¢ ¯ =¯ ǫn ζ(xiǫ ) + CO(ǫ) φ χi (y) p y, ξǫi dy ¯ Y i∈Iǫ ¯ Z Z ¯ X £ i ¤ ¯ − ζ(xǫ ) + CO(ǫ) φ (χi (y) p (y, ∇u(x))) dydx¯ ¯ i i∈Iǫ Yǫ Y ¯ ¯X Z ¡ ¡ ¢¢ ¯ ≤¯ ǫn ζ(xiǫ )φ χi (y) p y, ξǫi dy ¯ Y i∈Iǫ ¯ ¯ XZ Z ¯ − ζ(xiǫ )φ (χi (y) p (y, ∇u(x))) dydx¯ + CO(ǫ) ¯ i i∈Iǫ Yǫ Y ¯ ¯X Z Z ¡ ¡ ¢¢ ¯ =¯ ζ(xiǫ )φ χi (y) p y, ξǫi dydx ¯ i i∈Iǫ Yǫ Y ¯ ¯ XZ Z ¯ i − ζ(xǫ )φ (χi (y) p (y, ∇u(x))) dydx¯ + CO(ǫ) ¯ i i∈Iǫ Yǫ Y ¯ ¯ ¯X Z Z ¯ £ ¡ ¡ ¢¢ ¤ ¯ ¯ i i =¯ ζ(xǫ ) φ χi (y) p y, ξǫ − φ (χi (y) p (y, ∇u(x))) dydx¯ + CO(ǫ) ¯ ¯ Yǫi Y i∈Iǫ
63
Let us use a Taylor’s expansion for ζ again ¯ ¯X Z Z £ ¡ ¡ ¢¢ ¯ ≤¯ (ζ(x) + CO(ǫ)) φ χi (y) p y, ξǫi ¯ Yǫi Y i∈Iǫ
−φ (χi (y) p (y, ∇u(x)))] dydx| + CO(ǫ) ¯ ¯ ¯X Z Z ¯ £ ¡ ¡ ¢¢ ¤ ¯ ¯ ≤¯ ζ(x) φ χi (y) p y, ξǫi − φ (χi (y) p (y, ∇u(x))) dydx¯ + CO(ǫ) ¯ ¯ i Yǫ Y i∈I ¯Z ǫ ¯ Z ¯ ¯ = ¯¯ ζ(x) [φ (χi (y) p (y, Mǫ ∇u(x))) − φ (χi (y) p (y, ∇u(x)))] dydx¯¯ + CO(ǫ) Y ¯ZΩǫ ¯ Z ¯ ¯ ≤ ¯¯ |ζ(x)| |φ (χi (y) p (y, Mǫ ∇u(x))) − φ (χi (y) p (y, ∇u(x)))| dydx¯¯ + CO(ǫ) Ωǫ
Y
Because of the uniform Lipschitz continuity of φ, we get ¯Z ¯ ≤ C ¯¯
Ωǫ
|ζ(x)|
Z
Y
¯ ¯ |p (y, Mǫ ∇u(x)) − p (y, ∇u(x))| dydx¯¯ + CO(ǫ)
By H¨older’s inequality we have
¯Z " µZ ¶1/p1 ¯ ¯ p1 ≤C¯ χ1 (y) |p (y, Mǫ ∇u(x)) − p (y, ∇u(x))| dy |ζ(x)| ¯ Ωǫ Y µZ ¶1/p2 # ¯¯ ¯ + χ2 (y) |p (y, Mǫ ∇u(x)) − p (y, ∇u(x))|p2 dy dx¯ + CO(ǫ) ¯ Y
Using Lemma 5.4, we obtain ¯Z ¯ ≤ C ¯¯
Ωǫ
|ζ(x)| {[(1 + |Mǫ ∇u(x)|p1 θ1 + |Mǫ ∇u(x)|p2 θ2 + |∇u(x)|p1 θ1 p1
p1 −2
1
+ |∇u(x)|p2 θ2 ) p1 −1 |Mǫ ∇u(x) − ∇u(x)| p1 −1 θ1p1 −1 p2
1
+ |Mǫ ∇u(x) − ∇u(x)| p2 −1 θ2p2 −1 (1 + |Mǫ ∇u(x)|p1 θ1 + |Mǫ ∇u(x)|p2 θ2 p2 −2 i1/p1 p1 p2 p2 −1 + |∇u(x)| θ1 + |∇u(x)| θ2 ) + [(1 + |Mǫ ∇u(x)|p1 θ1 + |Mǫ ∇u(x)|p2 θ2 + |∇u(x)|p1 θ1 p1 −2
p1
1
+ |∇u(x)|p2 θ2 ) p1 −1 |Mǫ ∇u(x) − ∇u(x)| p1 −1 θ1p1 −1 p2
1
+ |Mǫ ∇u(x) − ∇u(x)| p2 −1 θ2p2 −1 (1 + |Mǫ ∇u(x)|p1 θ1 + |Mǫ ∇u(x)|p2 θ2 ¾ ¯ p2 −2 i1/p2 ¯ p1 p2 dx¯¯ + CO(ǫ) + |∇u(x)| θ1 + |∇u(x)| θ2 ) p2 −1 64
By H¨older’s inequality we get ≤C
(µZ
q2
Ωǫ
|ζ(x)| dx
¶1/q2 ·Z
Ωǫ
³
p1
|Mǫ ∇u(x) − ∇u(x)| p1 −1 p1 −2
× (1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) p1 −1 p2
+ |Mǫ ∇u(x) − ∇u(x)| p2 −1
i1/p1 p2 −2 ´ × (1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) p2 −1 dx µZ ¶1/q1 ·Z ³ p1 q1 + |ζ(x)| dx |Mǫ ∇u(x) − ∇u(x)| p1 −1 Ωǫ
Ωǫ
p1 −2
p1
× (1 + |Mǫ ∇u(x)| + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) p1 −1 p2
+ |Mǫ ∇u(x) − ∇u(x)| p2 −1 p1
p2
p1
p2
× (1 + |Mǫ ∇u(x)| + |Mǫ ∇u(x)| + |∇u(x)| + |∇u(x)| )
p2 −2 p2 −1
+ CO(ǫ).
´
dx
i1/p2 ¾
From the fact that ζ ∈ C0∞ (Rn ), we have ≤C
½·Z
Ωǫ
p1
|Mǫ ∇u(x) − ∇u(x)| p1 −1 p1 −2
× (1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) p1 −1 dx Z p2 + |Mǫ ∇u(x) − ∇u(x)| p2 −1 Ωǫ
p2 −2
× (1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) p2 −1 dx ·Z p1 + |Mǫ ∇u(x) − ∇u(x)| p1 −1
i1/p1
Ωǫ
p1 −2
× (1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) p1 −1 dx Z p2 + |Mǫ ∇u(x) − ∇u(x)| p2 −1 Ωǫ
p1
p2
p1
p2
× (1 + |Mǫ ∇u(x)| + |Mǫ ∇u(x)| + |∇u(x)| + |∇u(x)| ) + CO(ǫ)
65
p2 −2 p2 −1
dx
i1/p2 ¾
Applying H¨older’s Inequality again, we get ≤C
½·µZ
Ωǫ
(1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1
p2
+ |∇u(x)| ) dx) +
µZ
p1 −2 p1 −1
µZ
Ωǫ
p1
|Mǫ ∇u(x) − ∇u(x)| dx p2
Ωǫ
|Mǫ ∇u(x) − ∇u(x)| dx
¶ p 1−1 µZ 2
Ωǫ
¶ p 1−1 1
(1 + |Mǫ ∇u(x)|p1
p2 −2 i1/p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) dx) p2 −1 ·µZ + (1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1
Ωǫ
p2
+ |∇u(x)| ) dx) µZ
p1 −2 p1 −1
µZ
Ωǫ
p1
|Mǫ ∇u(x) − ∇u(x)| dx ¶ p 1−1 µZ
¶ p 1−1 1
dx
2
(1 + |Mǫ ∇u(x)|p1 Ωǫ Ωǫ ¾ p2 −2 i1/p2 p2 p1 p2 p2 −1 + CO(ǫ) + |Mǫ ∇u(x)| + |∇u(x)| + |∇u(x)| ) dx)
+
p2
|Mǫ ∇u(x) − ∇u(x)| dx
From Jensen’s inequality and Theorem 4.1, we obtain ≤C
("µZ
Ωǫ
+
µZ
+
" µZ
|Mǫ ∇u(x) − ∇u(x)| dx p2
Ωǫ
|Mǫ ∇u(x) − ∇u(x)| dx
µZ
Ωǫ
¶ p 1−1 1
¶ p 1−1 #1/p1
p1
Ωǫ
+
p1
2
|Mǫ ∇u(x) − ∇u(x)| dx
|Mǫ ∇u(x) − ∇u(x)|p2 dx
¶ p 1−1 1
dx
¶ p 1−1 #1/p2 2
+ CO(ǫ).
Finally, taking ǫ → 0 together with Property 1 of Mǫ in Remark 5.1, we obtain (6.5).
66
Therefore, from Proposition 4.4 of [Ped99] and Lemma 6.4 we have Z Z φ(λ)dνxi (λ)dx ζ(x) n Ω Z R Z = ζ(x) φ(χi (y)p(y, ∇u(x)))dydx Ω Y Z ´´ ³ ³x , Mǫ (∇u)(x) dx = lim ζ(x)φ χǫi (x)p ǫ→0 Ω ǫ Z ≤ lim ζ(x)φ (χǫi (x)∇uǫ (x)) dx, ǫ→0
Ω
for all φ ∈ C0 (Rn ) and for all ζ ∈ C0∞ (Rn ). The proof of Theorem 6.2 follows from Lemma 6.4 and Theorem 6.11 in [Ped97].
67
Chapter 7 Nonlinear Neutral Inclusions A neutral inclusion, when inserted in a matrix containing a uniform applied electric field, does not disturb the field outside the inclusion. The problem of finding neutral inclusions goes back to 1953 when Mansfield found that certain reinforced holes, which he called “neutral holes”, could be cut out of a uniformly stressed plate without disturbing the surrounding stress field in the plate [Man53]. The analogous problem of a “neutral elastic inhomogeneity” in which the introduction of the inhomogeneity into an elastic body (of a different material), does not disturb the original stress field in the uncut body, was first studied by Ru [Ru98]. The well known Hashin coated sphere is an example of a neutral coated inclusion [Has62]. Neutral spherical inclusions have been studied in [TR95],[LV96],[Lip97a], [Lip97b] and [LT99]. For more information on neutral coated inclusions see [Mil02]. In this chapter, we consider the problem of constructing neutral inclusions from nonlinear materials. We study the design of (double coated nonlinear) neutral inclusions that do not disturb the prescribed uniform applied electric field in the surrounding body.
7.1 7.1.1
Double Coated Nonlinear Neutral Inclusions Statement of the Problem and Result ~n
Middle
c
a
Core
x¯ b Outer
u = Ex1
Figure 7.1: Neutral Inclusion For a particular three phase coated sphere (see Figure 7.1), we apply the linear field ~ ~ = E e~1 , with e~1 = (1, 0) and ~x = (x1 , x2 )) as a boundary condition to E · ~x = Ex1 , (where E 68
the exterior boundary of the outer coating, to find that u = Ex1 solves 2−p1 σ 2−p2 ∆p1 u = 0 (nonlinear) in the core, ∆u = 0 (linear) in the middle coating, σ∆p2 u = 0 (nonlinear) in the outer coating,
(7.1)
where σ > 0 and ∆p represents the p-Laplacian, together with the usual interface conditions 1 (continuity of the electric potential and normal component of the current), for |E| = σ 2−p2 . Our calculations in Section 7.2 show that we can replace the three-phase coated disk with a disk composed only of linear material of conductivity one. One could continue to add coated disks of various sizes (see Figure 7.2) without disturbing the prescribed uniform applied electric field surrounding the inclusions. In fact, we can fill the space with these coated disks. The space filling configuration of triple coated disks can be solved explicitly 1 and the field inside it is precisely u = Ex1 when |E| = σ 2−p2 . This configuration of nonlinear materials dissipates energy the same as a linear material with thermal conductivity one. Middle
Core
Outer
Figure 7.2: Space Filling Configuration of Coated Disks
7.2
Calculations
Let x¯ be the center of the disk (see Figure 7.1). Inside the disk, we ask that σ1 ∆p1 u = 0 0 < |x − x¯| < a σ3 ∆u = 0 a < |x − x¯| < b σ2 ∆p2 u = 0 b < |x − x¯| < c,
(7.2)
where σ1 , σ2 , and σ3 are positive constants. x−¯ x . Have r = |x − x¯| and ~n = e~r = |x−¯ x| The solution u of (7.2) is such that
u continuous across |x − x¯| = a,
(7.3)
u continuous across |x − x¯| = b,
(7.4)
u = Ex1 at |x − x¯| = c,
(7.5)
69
and satisfies the transmission conditions and
σ1~n · |∇u|p1 −2 ∇u = σ3~n · ∇u, across |x − x¯| = a,
(7.6)
σ3~n · ∇u = σ2~n · |∇u|p2 −2 ∇u, across |x − x¯| = b.
(7.7)
We look for solution u of the form u = c1 r cos θ u = cr3 cos θ + c4 r cos θ u = c2 r cos θ
for 0 < r < a, for a < r < b, for b < r < c.
(7.8)
It is easily seen that (7.8) satisfies (7.2). In what follows, we explain how the unknowns c1 , c2 , c3 , and c4 are determined from (7.3), (7.4), (7.5), (7.6), and (7.7). First, we look at the conditions u must satisfy when r = a: By (7.3), we have c3 c1 a cos θ = cos θ + c4 a cos θ a c3 + c4 a ⇒ c1 a = a ⇒ c 1 a2 = c 3 + c 4 a2 ,
and from (7.6), we obtain
σ1 e~r · |∇u|p1 −2 ∇u = σ3 e~r · ∇u p1 −2
⇒ σ1 e~r · |c1 |
c1 (~ er cos θ − e~θ sin θ) = σ3
µ
−c3 cos θ + c4 cos θ a2
−σ3 c3 cos θ + σ3 c4 cos θ a2 = −σ3 c3 + a2 σ3 c4 .
(7.9)
¶
⇒ σ1 (sign(c1 )) |c1 |p1 −1 cos θ =
⇒ a2 σ1 (sign(c1 )) |c1 |p1 −1
(7.10)
Second, we look at the conditions u must satisfy when r = b: By (7.4), we have
and from (7.7), we obtain
c3 cos θ + c4 b cos θ = c2 b cos θ b c3 ⇒ + c4 b = c2 b b ⇒ c 3 + c 4 b2 = c 2 b2 ,
σ3 e~r · ∇u = σ2 e~r · |∇u|p2 −2 ∇u ¶ µ −c3 cos θ + c4 cos θ = σ2 (sign(c2 )) |c2 |p2 −1 cos θ ⇒ σ3 2 b ⇒ −σ3 c3 + b2 σ3 c4 = b2 σ2 (sign(c2 )) |c2 |p2 −1 . 70
(7.11)
(7.12)
Finally, we look at the conditions u must satisfy when r = c: By (7.5) we have c2 c cos θ = Ec cos θ ⇒ c2 = E.
(7.13)
Using (7.13), we can rewrite equations (7.11) and (7.12) in the following way: c3 + c4 b2 = Eb2 ,
(7.14)
−σ3 c3 + b2 σ3 c4 = b2 σ2 (sign(E)) |E|p2 −1 .
(7.15)
Now we use (7.9), (7.10), (7.14), and (7.15) to determine the unknowns c1 , c3 , and c4 (c2 = E by (7.13)). From (7.15), we have c 3 = b 2 c 4 − b2
σ2 (sign(E)) |E|p2 −1 , σ3
(7.16)
and (7.14) implies that c3 = Eb2 − c4 b2 .
(7.17)
If we combine (7.16) and (7.17), we obtain Eb2 − c4 b2 = b2 c4 − b2
σ2 (sign(E)) |E|p2 −1 σ3
σ2 (sign(E)) |E|p2 −1 σ3 ³ ´ E + (sign(E)) |E|p2 −1 σσ32 ⇒ c4 = , 2
⇒ 2c4 b2 = Eb2 + b2
(7.18)
and this way we find c4 . Evaluating c4 (given by (7.18)) in (7.17), we find c3 : 2
c3 = Eb − ⇒ c3 = ⇒ c3 =
E + (sign(E)) |E|p2 −1 2
³ ´ σ2 σ3
2Eb2 − Eb2 − (sign(E)) |E|p2 −1 2 ³ ´ p2 −1 σ2 2 Eb − (sign(E)) |E| b2 σ3 2
71
b2
³ ´ σ2 σ3
.
b2
(7.19)
Finally, to obtain c1 , we evaluate (7.18) and (7.19) in (7.9): ³ ´ ³ ´ E + (sign(E)) |E|p2 −1 σσ23 Eb2 − (sign(E)) |E|p2 −1 σσ23 b2 + a2 c 1 a2 = 2 2 ³ ´ E(a2 + b2 ) + (sign(E)) |E|p2 −1 σσ23 (a2 − b2 ) ⇒ c 1 a2 = 2 ³ ´³ ³ ¡ b ¢2 ´ ¡ b ¢2 ´ p2 −1 σ2 + (sign(E)) |E| E 1+ a 1− a σ3 . (7.20) ⇒ c1 = 2 Therefore the values of c1 , c2 , c3 , and c4 are given by (7.20), (7.13), (7.19), and (7.18), respectively. These values must also satisfy (7.10). Note that in (7.10), using (7.19) and (7.18), we have a2 σ1 (sign(c1 )) |c1 |p1 −1 = −σ3 c3 + a2 σ3 c4 ³ ´ ³ ´ Eb2 − (sign(E)) |E|p2 −1 σσ32 b2 E + (sign(E)) |E|p2 −1 σσ32 + a2 σ3 = −σ3 2 2 −σ3 Eb2 + (sign(E)) |E|p2 −1 σ2 b2 + σ3 Ea2 + (sign(E)) |E|p2 −1 σ2 a2 = 2 σ3 E(a2 − b2 ) + (sign(E)) |E|p2 −1 σ2 (a2 + b2 ) . = 2
Therefore, using (7.20) above, we get ´ ³ ´³ ¯ ³ ¡ b ¢2 ´ ¯¯p1 −1 ¯ E 1 + ¡ b ¢2 + (sign(E)) |E|p2 −1 σ2 1− a ¯ ¯ a σ3 ¯ σ1 (sign(c1 )) ¯¯ ¯ 2 ¯ ¯ ´ ´ ³ ³ ¡ ¢2 ¡ ¢2 + (sign(E)) |E|p2 −1 σ2 1 + ab σ3 E 1 − ab . = 2
(7.21)
We have that (7.21) is verified if σ3 = 1 σ2 = |E|2−p2 σ1 = |E|2−p1 2−p1
If we set σ = σ2 = |E|2−p2 , then σ1 = σ 2−p2 . This values of σ1 , σ2 , and σ3 correspond to 1 the coefficients in (7.1). We also obtain that |E| = σ 2−p2 . Here it can be checked that c1 = c2 = c4 = E and c3 = 0. Therefore u = Er cos θ is the solution to (7.1). 72
At r = c, we have u = Ec cos θ and σ2 e~r · |∇u|p2 −2 ∇u = |E|2−p2 |c2 |p2 −1 (sign(c2 )) cos θ = |E|2−p2 |E|p2 −1 (sign(E)) cos θ = |E| (sign(E)) cos θ = hE cos θ,
with h = 1 which does not depend on a, b, or c, which means that it is independent of scale.
73
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77
Vita Silvia Elena Jim´enez Bola˜ nos was born in July 1980, in Alajuela, Costa Rica. In August 2002 she finished her undergraduate studies in mathematics at the Universidad de Costa Rica, Costa Rica. In August 2004 she came to Louisiana State University to pursue graduate studies in mathematics and earned a master of science in mathematics in 2006. She is currently a candidate for the degree of Doctor of Philosophy in mathematics, which will be awarded in August 2010.
78
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 Jim´enez B.Sc. in Mathematics, Universidad de Costa Rica, Costa Rica, 2002 M.S. in Mathematics, Louisiana State University, USA, 2006 August 2010
Acknowledgements “Caminante, son tus huellas el camino y nada m´as; Caminante, no hay camino, se hace camino al andar. Al andar se hace el camino, y al volver la vista atr´as 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 study 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-0807265 and AFOSR Grant FA9550-08-1-0095. 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 already left, thank you very much. It was a great opportunity for me to attend LSU and I have thoroughly enjoyed it. I dedicate this dissertation to my family: my parents William Jim´enez Sol´ıs and Xenia Bola˜ nos Rodr´ıguez, my brothers William and Jos´e, my aunts Marcela, Vera, Yadira and Yolanda, my uncle Mario, my cousins Sof´ıa, Alonso, and Ignacio, and especially to my grandma Berta. Thank you all so much for the visits, phone calls, chats, emails, support, and love.
ii
Table of Contents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Chapter 2 Basic Ideas in Homogenization Theory . . . . . . . . . . . . . . . . . . . . 2.1 Motivation and Examples . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 1-Dimensional Example of Homogenization . . . . . . . . . . . 2.1.2 Homogenization in Rn . . . . . . . . . . . . . . . . . . . . . . 2.2 Some Special Cases with Closed Form Expressions for the Homogenized Operator b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Hashin Structure (1962) . . . . . . . . . . . . . . . . . . . 2.2.2 The Mortola-Steff´e Structure and the Checkerboard Structure
3 3 5 8 9 9 10
Chapter 3 Dirichlet Boundary Value Problem . . . . . . . 3.1 Description of the Problem . . . . . . . . . 3.2 Microgeometries Considered . . . . . . . . 3.3 Notation and Preliminary Results . . . . . 3.3.1 Properties of A . . . . . . . . . . . 3.3.2 Dirichlet BVP and Homogenization 3.3.3 Properties of b . . . . . . . . . . . 3.3.4 Properties of p . . . . . . . . . . .
. . . . . . . .
12 12 13 14 14 19 21 24
Chapter 4 Higher Order Integrability of the Homogenized Solution . . . . . . . . . . 4.1 Statement of the Theorem on the Higher Order Integrability of the Homogenized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Proof of Higher Order Integrability of the Homogenized Solution . . .
25
Chapter 5 Corrector Theorem . . . . . . . . . . . . . 5.1 Statement of the Corrector Theorem 5.2 Some Properties of Correctors . . . . 5.3 Proof of the Corrector Theorem . . .
31 31 32 51
iii
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Chapter 6 Lower Bounds on Field Concentrations . . . . . . . . . . . . . . . . . . . 6.1 Statement of the Lower Bound on the Amplification of the Macroscopic Field by the Microstructure . . . . . . . . . . . . . . . . . . . . . . . 6.2 Young Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Proof of Lower Bound on the Amplification of the Macroscopic Field by the Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . .
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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
Chapter 1 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 study 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 study 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 macro1
scopic applied fields. We conclude in Chapter 7 where we will introduce special neutrally conducting microstructures made with power law materials.
2
Chapter 2 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.
2.1
Motivation and Examples
Suppose we would like to know the stationary temperature distribution in an homogeneous body Ω ⊂ R3 with an internal heat source f , heat conductivity A and zero temperature on the boundary ∂Ω. The model to describe this problem is given by the following boundary value problem: Find u ∈ W01,p (Ω), 1 < p < ∞, such that −div (A (∇u)) = f on Ω,
(2.1)
where Ω is a bounded open subset of Rn , f is a given function on Ω, and A : Rn → Rn satisfies suitable continuity and monotonicity conditions that allows the existence and uniqueness of the solution of (2.1). Now, suppose that we would like to be able to model the case when the underlying material is heterogeneous. Then we replace A in (2.1) with a map A : Ω × Rn → Rn to obtain ( −div (A (x, ∇u)) = f on Ω (2.2) u ∈ W01,p (Ω). Since (2.2) depends on x, this is much more difficult to handle than (2.1). 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
with respect to Ω. By periodicity, we can divide Ω into periodic cells, and we call the representative unit cell by Y (the microstructure of a given periodic material can be described ¢ ¡ by several different period cells). This is described by maps of the form Aǫ (x, ξ) = A xǫ , ξ , where A(·, ξ) is assumed to be Y -periodic and ǫ is the fineness of the periodic structure. Equation (2.2) becomes ( ¡ ¡ ¢¢ −div (Aǫ (x, ∇uǫ )) = −div A xǫ , ∇uǫ = f on Ω (2.3) uǫ ∈ W01,p (Ω). The function uǫ can be interpreted as the electric potential, magnetic potential, or the temperature and Aǫ describes the physical properties of the different materials constituting the body (they are the dielectric coefficients, the magnetic permeability and the thermic conductivity coefficients, respectively). Let {ǫk }∞ k=0 be a sequence of positive real numbers such that ǫk → 0 as k → ∞. In this way we get a sequence of problems, one for each value of k. The smaller ǫk gets, the finer the microstructure becomes. It is natural to ask ourselves if there exists some type of convergence of the solutions uǫk . If we assume convergence in an appropriate sense, that is uǫk → u, as k → ∞, we could also ask if u satisfies an equation of a similar type as the one uǫk satisfies ( −div (b (x, ∇u)) = f, on Ω u ∈ W01,p (Ω), 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 Ω (2.4) u ∈ W01,p (Ω). The “homogenized” b represents the physical parameters of a homogeneous body, whose behavior is equivalent, from a “macroscopic” point of view, to the behavior of the material with the given periodic microstructure, described by (2.3). 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 ½Z ³ ¾ Z ´ x Eǫ = min g , ∇u(x) dx − f udx , ǫ u∈W01,p (Ω) Ω Ω 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
called Γ-convergence (Introduced by DeGiorgi [DG75]). From the theory of Γ-convergence it follows that Eǫ → Ehom , as ǫ → 0, where ½Z ¾ Z Ehom = min ghom (∇u(x)) dx − f udx . 1,p u∈W0 (Ω)
Ω
Ω
Here the homogenized energy density function ghom is given by Z 1 ghom (ξ) = g(x, ξ + ∇u)dx, min 1,p |Y | u∈Wper (Y ) Y 1,p where Wper (Y ) is the set of all functions u ∈ W 1,p (Y ) which are Y -periodic and have mean value zero. Again we note that the limit problem does not depend on x, that is, ghom 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.
2.1.1
1-Dimensional Example of Homogenization
Let Ω = (0, 1), f ∈ H −1 (Ω), and A ∈ L∞ (Ω) be a measurable and periodic function with period 1 satisfying 0 < β1 ≤ A(x) ≤ β2 < ∞, for a.e. x ∈ R. (2.5) Remark 2.1. For example, consider a periodic mixture of two materials. Let χ1 be the the characteristic function of material 1 and χ2 be the the characteristic function of material 2, both periodic of periodicity 1. Let A(y) = β1 χ1 (y) + β2 χ2 (y) defined in Ω = (0, 1) and extend it to all R by periodicity, and call this extension by A as well. Note that A ∈ L∞ (Ω), because kAkL∞ (Ω) = β2 and clearly it satisfies (2.5) for all x ∈ R. ³ ´ We define Aǫk (x) = A ǫxk . The weak form of (2.3) becomes Z
1
Aǫk (x) ∂x uǫk (x)∂x φ(x)dx =
0
uǫk ∈
W01,2 (0, 1),
Z
0
1
f (x)φ(x)dx for every φ ∈ W01,2 (0, 1),
(2.6)
and (2.3) becomes ( −∂x (Aǫk (x)∂x uǫk (x)) = f (x) uǫk ∈ W01,2 (0, 1). 5
in (0, 1),
(2.7)
By a standard result in the existence theory of partial differential equations (using LaxMilgram Lemma [Eva98]), there exists a unique solution of these problems for each k. By choosing φ = uǫk in (2.6) and taking (2.5) into account, we obtain by H¨older’s inequality that Z 1 2 β1 k∂x uǫk kL2 (0,1) ≤ Aǫk (x) |∂x uǫk (x)|2 dx Z0 1 f (x)uǫk (x)dx = 0
≤ kf kH −1 (Ω) kuǫk kW 1,2 (Ω) . 0
Recall that
kuǫk k2W 1,2 (Ω) = kuǫk k2L2 (Ω) + k∂x uǫk k2L2 (Ω) . 0
The Poincar´e inequality for functions with zero boundary values states that there is a constant C only depending on Ω = (0, 1) such that kuǫk kL2 (Ω) ≤ C k∂x uǫk kL2 (Ω) . This implies that
kuǫk k2W 1,2 (Ω) ≤ C, 0
(2.8)
where C is a constant independent of k. Since W01,2 (Ω) is reflexive, there exists a subsequence, still denoted by {uǫk }, such that uǫk ⇀ u∗ in W01,2 (Ω).
(2.9)
Since W01,2 (Ω) is compactly embedded in L2 (Ω), we have by the Rellich Embedding Theorem that uǫk → u∗ in L2 (Ω).
In general, however, we only have that
∂x uǫk ⇀ ∂x u∗ in L2 (Ω). Since A is 1-periodic, we have that {Aǫk } converges weakly* L∞ (Ω), as k → ∞, to its arithmetic mean hAi, i.e., Z 1 ∗ A(y)dy in L∞ (Ω). (2.10) Aǫk ⇀ hAi = 0
From (2.6), (2.9), and (2.10), it could be reasonable to assume that, in the limit, we have Z 1 Z 1 f (x)φ(x)dx for every φ ∈ W01,2 (0, 1), hAi ∂x u∗ (x)∂φ(x)dx = 0 0 u∗ ∈ W01,2 (0, 1).
However, this is not true in general, since Aǫk ∂x uǫk is the product of two weakly converging sequences. This is the main difficulty in the limit process. To obtain the correct answer we 6
proceed in the following way: first we note that, according to (2.10) and (2.8), {Aǫk ∂x uǫk } is bounded in L2 (Ω) and that (2.6) implies that −∂x (Aǫk (x)∂x uǫk ) = f . Hence there is a constant C independent of k such that kAǫk ∂x uǫk kW 1,2 (Ω) ≤ C. As before, since W 1,2 (Ω) is reflexive, there exists a subsequence, still denoted {Aǫk ∂x uǫk } and a M 0 ∈ L2 (Ω) such that Since
n
1 Aǫ k
Aǫk ∂x uǫk → M 0 in L2 (Ω). o ® converges to A1 weakly* in L∞ (Ω) by periodicity (and hence weakly in
L2 (Ω)), we have
∂x uǫk =
µ
1 Aǫk
¶
¿ À 1 (Aǫk ∂x uǫk ) ⇀ M 0 , in L2 (Ω). A
(2.11)
Thus, by (2.9) and (2.11), we see that
®−1 1 M 0 = 1 ® ∂x u∗ = A−1 ∂ x u∗ . A
Now, by passing to the limit in (2.6) we obtain that Z 1 Z 1 b∂x du∗ ∂x φdx = f (x)φ(x)dx for every φ ∈ W01,2 (0, 1), 0 0 u∗ ∈ W01,2 (0, 1).
where the homogenized operator is given by b =
1
−1
= hA−1 i , the harmonic mean of A;
h i ¿ À 1 1 1 ≤ ≤ , β2 A β1 we conclude that the homogenized equation has a unique solution and thus that the whole sequence {uǫk } converges. and since
1 A
Remark 2.2. For the example given in Remark 2.1, we obtain M 0 = hθ ∂x u∗ , where ¡ ¢−1 hθ = θ1 β1−1 + θ2 β2−1
and
θ1 =
Z
0
1
χ1 (y)dy, and θ2 = 1 − θ1 .
Remark 2.3. The corresponding homogenization problem for the one-dimensional Poisson equation Z Z A (x) |∂ u |p−2 ∂ u ∂ φdx = f (x)φ(x)dx for every φ ∈ W01,p (Ω), ǫk x ǫk x ǫk x Ω Ω uǫk ∈ W01,p (Ω) D 1 E1−p . gives the homogenized operator b = A 1−p 7
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 L2 (Ω) converges in the sense of distributions to the product of their limits.
2.1.2
Homogenization in Rn
Assume that A satisfies suitable structure conditions. Remark 2.4. A common assumption is that A(x, ξ) satisfies the conditions |A(x, ξ1 ) − A(x, ξ2 )| ≤ c1 λ(x) (1 + |ξ1 | + |ξ2 |)p−1−α |ξ1 − ξ2 |α ,
(A(x, ξ1 ) − A(x, ξ2 ), ξ1 − ξ2 ) ≥ c2 λ(x) (1 + |ξ1 | + |ξ2 |)p−β |ξ1 − ξ2 |β ,
for constants c1 ,c2 > 0, where α and β satisfy 0 ≤ α ≤ min(1, p − 1) and max(p, 2) ≤ β < ∞ (see, for example, [DMD90, FM87]). For example, these conditions are satisfied by the p-Poisson operator A(x, ξ) = λ(x) |ξ|p−2 ξ, ³ ³ ¡ √ ¢2−p ´ ¡ √ ¢2−p ´ where c1 , c2 can be chosen as c1 = max p − 1, 2 2 , and c2 = min p − 1, 2 2 . We have the following homogenization theorem.
Theorem 2.5. Let 1 < p < ∞ and q its dual conjugate. The solutions uǫk of ( −div (Aǫk (x, ∇uǫk )) = f on Ω, uǫk ∈ W01,p (Ω) satisfy
uǫk ⇀ u in W01,p (Ω),
(2.12)
and Aǫk (x, ∇uǫk ) ⇀ b(∇u) in Lq (Ω, Rn ),
as k → ∞, where u is the solution of the homogenized equation ( −div (b (∇u)) = f on Ω, u ∈ W01,p (Ω), where the homogenized operator b : Rn → Rn is defined by Z 1 b(ξ) = A(x, ξ + ∇ω ξ (x))dx, |Y | Y
and where ω ξ is the solution of the local problem on Y Z ¡ ¡ ¢ ¢ 1,p A x, ξ + ∇ω ξ , ∇φ dx = 0 for every φ ∈ Wper (Y ), Y ξ 1,p ω ∈ Wper (Y ).
(2.13)
(2.14)
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
2.2
Some Special Cases with Closed Form Expressions for the Homogenized Operator b
The homogenized operator b in (2.13) depends on the solution of a cell problem (2.14), 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 (2.3) with p = 2 and A(x, ξ) = λ(x)ξ (linear), with ξ ∈ R2 . In Chapter 7 of this thesis, we obtain similar results in an example that deals with nonlinear materials.
2.2.1
The Hashin Structure (1962)
We study a three-phase composite consisting of three isotropic materials (coated sphere assemblage), let us call them materials 1, 2, and 3, with conductivity λ(x)I = [σ1 χΩ1 (x) + σ2 χΩ2 (x) + σ3 χΩ3 (x)] I where χΩi is the characteristic function for the set Ωi and I is the unit matrix. Let the unit cell geometry be described by ½ ¾ 1 Ω1 = {x : |x| ≤ r1 } , Ω2 = x : r1 ≤ |x| ≤ r2 < , 2 ¾ ½ 1 Ω3 = x : |xi | < ∧ |x| ≥ r2 , i = 1, 2 . 2 In order to compute the homogenized coefficients (2.13), we need to solve the cell problem (2.14) ¡ ¢ −div λ(y)∇φξ (y) = 0 on Y , (2.15)
where φξ (y) = ξ · y + ω ξ (y) and ω ξ (y) is Y -periodic. In the case ξ = e~1 =[1 0]T , we look for a solution of C1 x ³1 , ´ K2 φe~1 (x) = x1 C2 + |x| , 2 x , 1
the type x ∈ Ω1 ,
x ∈ Ω2 ,
(2.16)
x ∈ Ω3 .
It is easily seen that (2.16) satisfies (2.15) on Y . By physical reasons, the solution φξ (x) as well as the flux λ(x)∂n φξ must be continuous over the boundaries Ω1 ∩ Ω2 and Ω2 ∩ Ω3 . This gives four equations to solve for the three unknowns C1 , C2 , and K2 . In order to get a consistent solution, we get that σ3 must be ´´ ³ ³ µ ¶ σ 1 + σ1 + m σ1 − 1 1 σ2 2 σ2 K2 ´ , ³ (2.17) σ3 = σ2 C2 − 2 = r2 1 + σ1 − m σ1 − 1 σ2
9
1
σ2
r2
where m1 = r12 , the volume fraction of material 1 in material 2. Since we now know the 2 solution ω e1 (y) = φe1 (y) − e1 · y of the cell problem, we can compute the homogenized coefficients Z b(e~1 ) = λ(x)(e~1 + ∇ω e~1 )dx = [σ3 0]T Y
and similarly
b(e~2 ) =
Z
Y
λ(x)(e~2 + ∇ω e~2 )dx = [0 σ3 ]T .
This means that we can put the coated disk consisting of material 1 coated by material 2 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 σ3 . For more details see [Mil02].
2.2.2
The Mortola-Steff´ e Structure and the Checkerboard Structure
Let Y = (0, 1)2 and divide it into four equal parts µ ¶ µ ¶ µ ¶ ¶ µ 1 1 1 1 Y1 = 0, , 1 , Y2 = ,1 × ,1 , × 2 2 2 2 ¶ µ ¶ µ ¶ µ ¶ µ 1 1 1 1 , 1 × 0, × 0, , Y4 = . Y3 = 0, 2 2 2 2
We study a four-phase composite consisting of four isotropic materials, let us call them materials 1, 2, 3, and 4, with conductivity λ(x)I = [αχY1 (x) + βχY2 (x) + γχY3 (x) + δχY4 (x)] I,
where χYi (x) is the characteristic function for the set Yi and I is the unit matrix. In 1985, Mortola and Steff´e [MS85] conjectured that the homogenized conductivity coefficients of this structure are
(λij ) = where
µ
λ11 0
¶ 0 , λ22
λ11 =
sµ
αβγ + αβδ + αγδ + βγδ α+β+γ+δ
¶
(α + γ) (β + δ) , (α + β) (γ + δ)
λ22 =
sµ
αβγ + αβδ + αγδ + βγδ α+β+γ+δ
¶
(α + β) (γ + δ) . (α + γ) (β + δ)
This conjecture was proven by Milton in 2000. 10
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 p λ11 = λ22 = αβ, the geometric mean. This was proved already in 1970 by Dykhne, but Schulgasser (1977) showed that this was a corollary of Keller’s phase interchange identity from 1963.
11
Chapter 3 Dirichlet Boundary Value Problem In this chapter, we study boundary value problems associated with fields inside composites made from two materials with different power law behavior.
3.1
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 χ1 and χ2 , where χ1 (y) = 1 in material one and is zero outside and χ2 (y) = 1 − χ1 (y). The constitutive law for the heterogeneous medium is described by A : Rn × Rn → Rn , A (y, ξ) = σ(y) |ξ|p(y)−2 ξ
= σ1 χ1 (y) |ξ|p1 −2 ξ + σ2 χ2 (y) |ξ|p2 −2 ξ;
(3.1)
where σ(y) = χ1 (y) σ1 + χ2 (y) σ2 , and p(y) = χ1 (y) p1 + χ2 (y) p2 , periodic in y, with unit period cell Y = (0, 1)n . This simple constitutive model is used in the mathematical description of many physical phenomena including plasticity [PCS97, PCW99, Suq93, Idi08], nonlinear dielectrics [GNP01, GK03, LK98, TW94a, TW94b], and fluid flow [Ruˇz00, AR06]. We study the problem of periodic homogenization associated with the solutions uǫ to the problems ³ ³x ´´ −div A , ∇uǫ = f on Ω, uǫ ∈ W01,p1 (Ω), (3.2) ǫ where Ω is a bounded open subset of Rn , 2 ≤ p1 ≤ p2 , f ∈ W −1,q2 (Ω), and 1/p1 + 1/q2 = 1. The differential operator appearing on the left hand side of (3.2) is commonly referred to as the pǫ (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 uǫ to the problems (3.2). It was proved in [ZKO94] that {uǫ }ǫ>0 converges weakly in W 1,p1 (Ω) to the solution u of the homogenized problem −div (b (∇u)) = f on Ω, u ∈ W01,p1 (Ω), 12
(3.3)
where the monotone map b : Rn → Rn (independent of f and Ω) can be obtained by solving an auxiliary problem for the operator (3.2) on a periodicity cell. The notion of homogenization is intimately tied to the Γ-convergence of a suitable family energy functionals Iǫ 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 Iǫ and the solutions uǫ are their minimizers. The homogenized solution is precisely the minimizer of the Γ-limit of the sequence {Iǫ }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 pǫ (x)-Laplacian boundary value problems for smooth exponential functions pǫ (x) uniformly converging to a limit function p0 (x) [AAPP08]. There the convergence of the family of solutions for these homogenization problems is expressed in the topology of Lp0 (·) (Ω) [AAPP08].
3.2
Microgeometries Considered
We carry out this investigation for two nonlinear power-law materials periodically distributed inside a domain Ω. Here Ω is an open bounded subset of Rn , 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 Y = (0, 1)n of the microstructure. Let F be an open subset of Y of material one, with smooth boundary ∂F , such that F ⊂ Y . The function χ1 (y) = 1 inside F and 0 outside and χ2 (y) = 1 − χ1 (y). We extend χ1 (y) and χ2 (y) by periodicity to Rn and the ǫ-periodic mixture inside Ω is described by the oscillatory characteristic functions χǫ1 (x) = χ1 (x/ǫ) and χǫ2 (x) = χ2 (x/ǫ). Here we will consider the case where F is given by a simply connected inclusion embedded inside a host material (see Figure 3.1). A distribution of such inclusions is commonly referred to as a periodic dispersion of inclusions.
F Y
Figure 3.1: 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 R1 , sandwiched between layers of material two, denoted 13
by R2 . The interior boundary of R1 is denoted by Γ. Here χ1 (y) = 1 for y ∈ R1 and 0 in R2 , and χ2 (y) = 1 − χ1 (y) (see Figure 3.2). R2 R1 R2
Γ Figure 3.2: Unit cell: Layered material
3.3
Notation and Preliminary Results
On the unit cell Y , the constitutive law for the nonlinear material is given by (3.1) with exponents p1 and p2 satisfying 2 ≤ p1 ≤ p2 . Their H¨older conjugates (or dual conjugates) 1,pi (Y ) are denoted by q2 = p1 /(p1 − 1) and q1 = p2 /(p2 − 1) respectively. For i = 1, 2, Wper 1,pi denotes the set of all functions u ∈ W (Y ) with mean value zero that have the same trace 1,pi on the opposite faces of Y . Each function u ∈ Wper (Y ) can be extended by periodicity to a 1,pi n function of Wloc (R ). The Euclidean norm and the scalar product in Rn are denoted by |·| and (·, ·), respectively. If D ⊂ Rn , |D| denotes the Lebesgue measure and χD (x) denotes its characteristic function. The constitutive law for the ǫ-periodic composite is described by Aǫ (x, ξ) = A (x/ǫ, ξ), for every ǫ > 0, for every x ∈ Ω, and for every ξ ∈ Rn . 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.
3.3.1
Properties of A
The function A, defined in (3.1), satisfies the following properties: 1. For all ξ ∈ Rn , A(·, ξ) is Y -periodic and Lebesgue measurable. 2. |A(y, 0)| = 0 for all y ∈ Rn . 3. Continuity: for almost every y ∈ Rn and for every ξi ∈ Rn (i = 1, 2) we have £ |A(y, ξ1 ) − A(y, ξ2 )| ≤ C χ1 (y) |ξ1 − ξ2 | (1 + |ξ1 | + |ξ2 |)p1 −2 ¤ + χ2 (y) |ξ1 − ξ2 | (1 + |ξ1 | + |ξ2 |)p2 −2 .
(3.4)
4. Monotonicity: for almost every y ∈ Rn and for every ξi ∈ Rn (i = 1, 2) we have (A(y, ξ1 ) − A(y, ξ2 ), ξ1 − ξ2 ) ≥ C (χ1 (y) |ξ1 − ξ2 |p1 + χ2 (y) |ξ1 − ξ2 |p2 ) .
14
(3.5)
Proof of (3.4): Continuity of A Proof. By (3.1), we have |A(y, ξ1 ) − A(y, ξ2 )| ¯ ¯ = ¯σ1 χ1 (y) |ξ1 |p1 −2 ξ1 + σ2 χ2 (y) |ξ1 |p2 −2 ξ1 − σ1 χ1 (y) |ξ2 |p1 −2 ξ2 − σ2 χ2 (y) |ξ2 |p2 −2 ξ2 ¯ ¯ £ ¤¯ ¯ £ ¤¯ ≤ ¯σ1 χ1 (y) |ξ1 |p1 −2 ξ1 − |ξ2 |p1 −2 ξ2 ¯ + ¯σ2 χ2 (y) |ξ1 |p2 −2 ξ1 − |ξ2 |p2 −2 ξ2 ¯ ¯ ¯¢ ¯ ¯ ¡ ≤ C χ1 (y) ¯|ξ1 |p1 −2 ξ1 − |ξ2 |p1 −2 ξ2 ¯ + χ2 (y) ¯|ξ1 |p2 −2 ξ1 − |ξ2 |p2 −2 ξ2 ¯ Let us study the expression ¯ ¯ pi −2 ¯|ξ1 | ξ1 − |ξ2 |pi −2 ξ2 ¯ , for i = 1, 2.
Observe that
¯2 ¯ pi −2 ¯|ξ1 | ξ1 − |ξ2 |pi −2 ξ2 ¯
= |ξ1 |2(pi −1) + |ξ2 |2(pi −1) − 2 |ξ1 |pi −2 |ξ2 |pi −2 ξ1 · ξ2
= |ξ1 |2(pi −1) + |ξ2 |2(pi −1) − 2 |ξ1 |pi −1 |ξ2 |pi −1 + 2 |ξ1 |pi −1 |ξ2 |pi −1 − 2 |ξ1 |pi −2 |ξ2 |pi −2 ξ1 · ξ2 ¡ ¢2 = |ξ1 |pi −1 − |ξ2 |pi −1 + 2 |ξ1 |pi −2 |ξ2 |pi −2 (|ξ1 | |ξ2 | − ξ1 · ξ2 ) ¡ ¢2 ¡ ¢ = |ξ1 |pi −1 − |ξ2 |pi −1 + |ξ1 |pi −2 |ξ2 |pi −2 2 (|ξ1 | |ξ2 | − ξ1 · ξ2 )
By Remark 3.1, we have
¡ ¢2 ≤ |ξ1 |pi −1 − |ξ2 |pi −1 + |ξ1 |pi −2 |ξ2 |pi −2 |ξ1 − ξ2 |2
By Remark 3.2, we have
¯ ¯2 ¡ ¢2 ≤ ¯|ξ1 |pi −1 − |ξ2 |pi −1 ¯ + |ξ1 |pi −2 + |ξ2 |pi −2 |ξ1 − ξ2 |2
And by Remark 3.3, we have
¡ ¢2 ¡ ¢2 ≤ (pi − 1)2 |ξ1 |pi −2 + |ξ2 |pi −2 ||ξ1 | − |ξ2 ||2 + |ξ1 |pi −2 + |ξ2 |pi −2 |ξ1 − ξ2 |2 ¡ ¢2 ¡ ¢2 ≤ (pi − 1)2 |ξ1 |pi −2 + |ξ2 |pi −2 |ξ1 − ξ2 |2 + |ξ1 |pi −2 + |ξ2 |pi −2 |ξ1 − ξ2 |2 ¢2 ¡ ≤ C |ξ1 |pi −2 + |ξ2 |pi −2 |ξ1 − ξ2 |2 . ¤2 £ ≤ C (1 + |ξ1 | + |ξ2 |)pi −2 |ξ1 − ξ2 |2 .
Taking square root on both sides, we obtain ¯ ¯ pi −2 ¯|ξ1 | ξ1 − |ξ2 |pi −2 ξ2 ¯ ≤ C (1 + |ξ1 | + |ξ2 |)pi −2 |ξ1 − ξ2 | ,
which proves (3.4).
15
Remark 3.1. Observe that 0 ≤ (|ξ1 | − |ξ2 |)2 ⇔ 0 ≤ |ξ1 |2 − 2 |ξ1 | |ξ2 | + |ξ2 |2 ⇔ 2 |ξ1 | |ξ2 | ≤ |ξ1 |2 + |ξ2 |2
⇔ 2 |ξ1 | |ξ2 | − 2ξ1 · ξ2 ≤ |ξ1 |2 − 2ξ1 · ξ2 + |ξ2 |2
⇔ 2 (|ξ1 | |ξ2 | − ξ1 · ξ2 ) ≤ |ξ1 − ξ2 |2 . Remark 3.2. For i = 1, 2, we have ¡ ¢ |ξi |pi −2 ≤ |ξ1 |pi −2 + |ξ2 |pi −2 .
Remark 3.3. Consider the function f : R+ −→ R+ defined by f (x) = xpi −1 , where pi ≥ 2. Since f is a convex function, it satisfies ( f ′ (|ξ1 |) (|ξ2 | − |ξ1 |) ≤ f (|ξ2 |) − f (|ξ1 |) , f ′ (|ξ2 |) (|ξ1 | − |ξ2 |) ≤ f (|ξ1 |) − f (|ξ2 |) ; or equivalently, ( (pi − 1) |ξ1 |pi −2 (|ξ2 | − |ξ1 |) ≤ |ξ2 |pi −1 − |ξ1 |pi −1 , (pi − 1) |ξ2 |pi −2 (|ξ1 | − |ξ2 |) ≤ |ξ1 |pi −1 − |ξ2 |pi −1 . Then (pi − 1) |ξ2 |pi −2 (|ξ1 | − |ξ2 |) ≤ |ξ1 |pi −1 − |ξ2 |pi −1 ≤ (pi − 1) |ξ1 |pi −2 (|ξ1 | − |ξ2 |) . Therefore we have ¯ pi −1 ¯ ¡ ¢ ¯|ξ1 | − |ξ2 |pi −1 ¯ ≤ (pi − 1) |ξ1 |pi −2 + |ξ2 |pi −2 ||ξ1 | − |ξ2 || .
Proof of (3.5): Monotonicity of A Proof. By (3.1), we have
(A(y, ξ1 ) − A(y, ξ2 ), ξ1 − ξ2 ) = (A(y, ξ1 ), ξ1 ) − (A(y, ξ1 ), ξ2 ) − (A(y, ξ2 ), ξ1 ) + (A(y, ξ2 ), ξ2 )
= σ1 χ1 (y) |ξ1 |p1 + σ2 χ2 (y) |ξ1 |p2 − σ1 χ1 (y) |ξ1 |p1 −2 ξ1 · ξ2 − σ2 χ2 (y) |ξ1 |p2 −2 ξ1 · ξ2
− σ1 χ1 (y) |ξ2 |p1 −2 ξ1 · ξ2 − σ2 χ2 (y) |ξ2 |p2 −2 ξ1 · ξ2 + σ1 χ1 (y) |ξ2 |p1 + σ2 χ2 (y) |ξ2 |p2 ¤ £ = σ1 χ1 (y) |ξ1 |p1 − |ξ1 |p1 −2 ξ1 · ξ2 − |ξ2 |p1 −2 ξ1 · ξ2 + |ξ2 |p1 £ ¤ + σ2 χ2 (y) |ξ1 |p2 − |ξ1 |p2 −2 ξ1 · ξ2 − |ξ2 |p2 −2 ξ1 · ξ2 + |ξ2 |p2 £ ¡ ¢¤ = σ1 χ1 (y) |ξ1 |p1 + |ξ2 |p1 − ξ1 · ξ2 |ξ1 |p1 −2 − |ξ2 |p1 −2 £ ¡ ¢¤ + σ2 χ2 (y) |ξ1 |p2 + |ξ2 |p2 − ξ1 · ξ2 |ξ1 |p2 −2 − |ξ2 |p2 −2 16
Let us study the expression
• If |ξ1 | = |ξ2 |:
¡ ¢ |ξ1 |pi + |ξ2 |pi − ξ1 · ξ2 |ξ1 |pi −2 + |ξ2 |pi −2 , for i = 1, 2. ¢ ¡ |ξ1 |pi + |ξ1 |pi − ξ1 · ξ2 |ξ1 |pi −2 + |ξ1 |pi −2 = |ξ1 |pi + |ξ1 |pi − 2 |ξ1 |pi −2 ξ1 · ξ2 £ ¤ = |ξ1 |pi −2 |ξ1 |2 + |ξ1 |2 − 2ξ1 · ξ2 £ ¤ = |ξ1 |pi −2 |ξ1 |2 − 2ξ1 · ξ2 + |ξ2 |2
= |ξ1 |pi −2 |ξ1 − ξ2 |2 . Since
¯ ¯ ¯ ¯ 1 ¯¯ ξ1 ξ2 ¯¯ 1 ¯¯ ξ1 ξ2 ¯¯ − − = ≤ 1, 2 ¯ |ξ1 | |ξ1 | ¯ 2 ¯ |ξ1 | |ξ2 | ¯
then
|ξ1 | ≥
1 |ξ1 − ξ2 | . 2
Therefore ¡ ¢ |ξ1 |pi + |ξ2 |pi − ξ1 · ξ2 |ξ1 |pi −2 + |ξ2 |pi −2 = |ξ1 |pi −2 |ξ1 − ξ2 |2 µ ¶pi −2 1 ≥ |ξ1 − ξ2 |pi −2 |ξ1 − ξ2 |2 2 = 22−pi |ξ1 − ξ2 |pi . • If |ξ1 | > |ξ2 | > 0, we can write
ξ2 = βξ1 + γω,
where ω 6= ~0 is a vector orthogonal to ξ1 , and β, γ ∈ R with |β| < 1. Since
ξ1 · ξ2 = ξ1 · (βξ1 + γω) = β |ξ1 |2 ,
(3.6)
we obtain ¢ ¡ ¢ ¡ |ξ1 |pi + |ξ2 |pi − ξ1 · ξ2 |ξ1 |pi −2 + |ξ2 |pi −2 = |ξ1 |pi + |ξ2 |pi − β |ξ1 |2 |ξ1 |pi −2 + |ξ2 |pi −2 . – For β ≤ 0:
¡ ¢ |ξ1 |pi + |ξ2 |pi − β |ξ1 |2 |ξ1 |pi −2 + |ξ2 |pi −2 ≥ |ξ1 |pi + |ξ2 |pi ≥ 21−pi |ξ1 − ξ2 |pi .
17
– For 0 < β < 14 : ¡ ¢ |ξ1 |pi + |ξ2 |pi − β |ξ1 |2 |ξ1 |pi −2 + |ξ2 |pi −2 = |ξ1 |pi + |ξ2 |pi − β |ξ1 |pi − β |ξ2 |pi −2 |ξ1 |2 ≥ |ξ1 |pi − 2β |ξ1 |pi = |ξ1 |pi (1 − 2β) 1 > |ξ1 |pi 2 1 1 = |ξ1 |pi + |ξ1 |pi 4 4 |ξ2 |pi |ξ1 |pi > + 4 4 −(pi +1) ≥2 |ξ1 − ξ2 |pi . – For
1 4
≤ β < 1:
¡ ¢ |ξ1 |pi + |ξ2 |pi − |ξ1 |pi −2 + |ξ2 |pi −2 ξ1 · ξ2
= |ξ1 |pi − |ξ1 |pi −2 ξ1 · ξ2 + |ξ2 |pi − |ξ2 |pi −2 ξ1 · ξ2 ¡ ¢ ¡ ¢ = |ξ1 |pi −2 |ξ1 |2 − ξ1 · ξ2 + |ξ2 |pi −2 |ξ2 |2 − ξ1 · ξ2 ¡ ¢ ¡ ¢ ≥ |ξ2 |pi −2 |ξ1 |2 − ξ1 · ξ2 + |ξ2 |pi −2 |ξ2 |2 − ξ1 · ξ2 ¡ ¢ = |ξ2 |pi −2 |ξ1 |2 − 2ξ1 · ξ2 + |ξ2 |2 = |ξ2 |pi −2 |ξ1 − ξ2 |2 .
By (3.6), we have
Therefore
obtaining this way Then
1 |ξ1 | ≤ ≤ 4. |ξ2 | β |ξ1 − ξ2 | |ξ1 | ≤ + 1 ≤ 5; |ξ2 | |ξ2 | |ξ2 | ≥ 5−1 |ξ1 − ξ2 | .
¡ ¢ |ξ1 |pi + |ξ2 |pi − |ξ1 |pi −2 + |ξ2 |pi −2 ξ1 · ξ2 = |ξ2 |pi −2 |ξ1 − ξ2 |2
≥ 52−pi |ξ1 − ξ2 |pi −2 |ξ1 − ξ2 |2 = 52−pi |ξ1 − ξ2 |pi .
© ª Taking C = min 52−pi , 2−(pi +1) , we have proved ¡ ¢ |ξ1 |pi + |ξ2 |pi − ξ1 · ξ2 |ξ1 |pi −2 + |ξ2 |pi −2 ≥ C |ξ1 − ξ2 |pi ,
for pi ≥ 2. Therefore
(A(y, ξ1 ) − A(y, ξ2 ), ξ1 − ξ2 ) ≥ C (χ1 (y) |ξ1 − ξ2 |p1 + χ2 (y) |ξ1 − ξ2 |p2 ) . A different proof for the monotonicity and continuity of A can be found in [Bys05]. 18
3.3.2
Dirichlet BVP and Homogenization Theorem
We shall consider the following Dirichlet boundary value problem ( −div (Aǫ (x, ∇uǫ )) = f on Ω, uǫ ∈ W01,p1 (Ω);
(3.7)
where f ∈ W −1,q2 (Ω). The following homogenization result holds. Theorem 3.4 (Homogenization Theorem). As ǫ → 0, the solutions uǫ of (3.7) converge weakly to u in W 1,p1 (Ω), where u is the solution of ( −div (b (∇u)) = f on Ω, (3.8) u ∈ W01,p1 (Ω); and the function b : Rn → Rn (independent of f and Ω) is defined for all ξ ∈ Rn by Z b(ξ) = A(y, p(y, ξ))dy,
(3.9)
Y
where p : Rn × Rn → Rn is defined by p(y, ξ) = ξ + ∇υξ (y), where υξ is the solution to the cell problem: Z (A(y, ξ + ∇υ ), ∇w) dy = 0, for every w ∈ W 1,p1 (Y ), ξ per Y 1,p1 υξ ∈ Wper (Y ).
(3.10)
(3.11)
For a proof of Theorem 3.4, see Chapter 15 of [ZKO94].
Lemma 3.5. The following a priori bound is satisfied µZ ¶ Z p1 p2 ǫ ǫ χ1 (x) |∇uǫ (x)| dx + χ2 (x) |∇uǫ (x)| dx ≤ C < ∞. sup ǫ>0
Ω
Ω
Proof. The weak formulation for (3.7) is given by Z Z (Aǫ (x, ∇uǫ (x)), ∇ϕ(x))dx = f (x)ϕ(x)dx, Ω
Ω
for all ϕ ∈ W01,p1 (Ω). In particular, taking ϕ = uǫ above, we obtain Z Z (Aǫ (x, ∇uǫ (x)), ∇uǫ (x))dx = f (x)uǫ (x)dx. Ω
Ω
19
(3.12)
The last equation can be rewritten as Z Z Z p1 p2 ǫ ǫ σ1 χ1 (x) |∇uǫ | dx + σ2 χ2 (x) |∇uǫ | dx = f (x)uǫ (x)dx. Ω
Ω
(3.13)
Ω
Applying H¨older’s inequality to the right hand side of (3.13), we obtain Z Z Z ǫ f (x)uǫ (x)dx = χ1 (x)f (x)uǫ (x)dx + χǫ1 (x)f (x)uǫ (x)dx Ω
Ω
≤
Ω
µZ
q2
Ω
+
|f (x)| dx
µZ
Ω
¶ q1 µZ 2
Ω
q1
|f (x)| dx
χǫ1 (x) |uǫ (x)|p1
¶ q1 µZ 1
Ω
dx p2
¶ p1
χǫ2 (x) |uǫ (x)| dx
1
¶ p1
2
.
(3.14)
If we combine (3.13) and (3.14), and we use the fact that f ∈ W −1,q2 (Ω), we get Z Z p1 ǫ σ1 χ1 (x) |∇uǫ (x)| dx + σ2 χǫ2 (x) |∇uǫ (x)|p2 dx Ω Ω " µZ ¶ p1 µZ ¶ p1 # 1 2 χǫ1 (x) |uǫ (x)|p1 dx ≤C + χǫ2 (x) |uǫ (x)|p2 dx Ω
Ω
Poincar´e’s inequality (see [Neˇc67]) gives ≤C
" µZ
Ω
χǫ1 (x) |∇uǫ |p1 dx
¶ p1
1
+
µZ
Ω
χǫ2 (x) |∇uǫ |p2 dx
¶ p1 # 2
and applying Young’s inequality, we obtain δ p1 ≤C p1 ·
Z
Ω
χǫ1 (x) |∇uǫ |p1
δ −q2 δ p2 + dx + q2 p2
Z
Ω
χǫ2 (x) |∇uǫ |p2
¸ δ −q1 . dx + q1
By rearranging the terms in the inequality, one gets µ ¶Z ¶Z µ δ p1 δ p2 p1 ǫ σ1 − C χ1 (x) |∇uǫ | dx + σ2 − C χǫ2 (x) |∇uǫ |p2 dx p1 p 2 Ω Ω δ −q2 δ −q1 + . ≤ q2 q1 n o δ p1 δ p2 Therefore, by taking δ small enough so that min σ1 − C p1 , σ2 − C p2 is positive, one obtains Z Z p1 ǫ χ1 (x) |∇uǫ | dx + χǫ2 (x) |∇uǫ |p2 dx ≤ C, Ω
Ω
where C does not depend on ǫ.
20
3.3.3
Properties of b
The function b, defined in (3.9), satisfies the following properties 1. Continuity: for every ξ1 , ξ2 ∈ Rn , we have h p1 −2 1 |b(ξ1 ) − b(ξ2 )| ≤ C |ξ1 − ξ2 | p1 −1 (1 + |ξ1 |p1 + |ξ2 |p1 + |ξ1 |p2 + |ξ2 |p2 ) p1 −1 p2 −2 i 1 + |ξ1 − ξ2 | p2 −1 (1 + |ξ1 |p1 + |ξ2 |p1 + |ξ1 |p2 + |ξ2 |p2 ) p2 −1
(3.15)
2. Monotonicity:for every ξ1 , ξ2 ∈ Rn , we have (b(ξ1 ) − b(ξ2 ), ξ1 − ξ2 ) (3.16) µZ ¶ Z ≥C χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )|p1 dy + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )|p2 dy Y
Y
≥ 0.
Proof of Continuity of b (3.15) Proof. By (3.9), (3.4), we have |b(ξ1 ) − b(ξ2 )| ¯Z ¯ Z ¯ ¯ = ¯¯ A(y, p(y, ξ1 ))dy − A(y, p(y, ξ2 ))dy ¯¯ Y ZY |A(y, p(y, ξ1 )) − A(y, p(y, ξ2 ))| dy ≤ Y ·Z ≤C χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| (1 + |p(y, ξ1 )| + |p(y, ξ2 )|)p1 −2 dy ¸ Z Y p2 −2 + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| (1 + |p(y, ξ1 )| + |p(y, ξ2 )|) dy Y
Applying H¨older’s inequality in both integrals, we obtain ≤C
"µZ
Y
× +
p1
µZ
Y
µZ
Y
×
χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy
χ1 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)
χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy
Y
1
q2 (p1 −2)
p2
µZ
¶ p1
dy
¶ q1
dy
¶ q1 #
¶ p1
2
2
q1 (p2 −2)
χ2 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)
21
1
By (3.5), we have "µZ ≤C
Y
× +
χ1 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)q2 (p1 −2) dy
Y
¶ q1
Y
Y
q2 (p1 −2)
Y
χ1 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)
dy
¶ q1
(A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), p(y, ξ1 ) − p(y, ξ2 )) dy Y µZ ¶1# Y
= C (b(ξ1 ) − b(ξ2 ), ξ1 − ξ2 ) + (b(ξ1 ) − b(ξ2 ), ξ1 − ξ2 )
1 p1
1 p2
µZ Y
1
¶ p1
2
q2 (p1 −2)
Y
µZ
¶ p1
q1
χ2 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)q1 (p2 −2) dy
By (3.11), we get "
2
2
µZ ×
¶ p1
q1
χ2 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)q1 (p2 −2) dy
(A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), p(y, ξ1 ) − p(y, ξ2 )) dy
µZ
1
2
χ2 (y) (A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), p(y, ξ1 ) − p(y, ξ2 )) dy Y µZ ¶1#
"µZ ×
+
µZ
µZ ×
≤C
χ1 (y) (A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), p(y, ξ1 ) − p(y, ξ2 )) dy
¶ p1
χ1 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)
χ2 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)q1 (p2 −2) dy
Applying the Cauchy-Schwarz inequality and H¨older’s inequality we have “ ”“ ” µZ ¶ p 1−1 p1p−1 1 1 1 1 ≤ C |b(ξ1 ) − b(ξ2 )| p1 |ξ1 − ξ2 | p1 χ1 (y)1p1 −1 Y
×
µZ
Y
χ1 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|) 1
1
+ |b(ξ1 ) − b(ξ2 )| p2 |ξ1 − ξ2 | p2 ×
µZ
Y
dy
µZ
p1 (p1 −2)(p1 −1) (p1 −1)(p1 −2)
χ2 (y)1p2 −1
Y
χ2 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|) 22
¶
“
1 p2 −1
dy
”“
p2 (p2 −2)(p2 −1) (p2 −1)(p2 −2)
¶ (p1p−1)(p 1 −2) (p −1) 1
p2 −1 p2
dy
1
”
¶ (p2p−1)(p 2 −2) (p −1) 2
2
¶ q1
2
¶ q1 # 1
"
≤ C |b(ξ1 ) − b(ξ2 )| + |b(ξ1 ) − b(ξ2 )|
1 p1
1 p2
|ξ1 − ξ2 |
|ξ1 − ξ2 |
1 p1
1 p2
1 p1
θ1 1 p2
θ2
µZ
Y
µZ
Y
χ1 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)p1 dy
χ2 (y)(1 + |p(y, ξ1 )| + |p(y, ξ2 )|)p2 dy
¶ p1p−2 1
¶ p2p−2 # 2
Lemma 5.3 delivers · 1 p1 −2 1 1 ≤ C |b(ξ1 ) − b(ξ2 )| p1 |ξ1 − ξ2 | p1 θ1p1 (1 + |ξ1 |p1 θ1 + |ξ2 |p1 θ1 + |ξ1 |p2 θ2 + |ξ2 |p2 θ2 ) p1 ¸ 1 p2 −2 1 1 p2 p1 p1 p2 p2 p2 p2 p2 + |b(ξ1 ) − b(ξ2 )| |ξ1 − ξ2 | θ2 (1 + |ξ1 | θ1 + |ξ2 | θ1 + |ξ1 | θ2 + |ξ2 | θ2 ) Applying Young’s inequality we obtain · p1 δ |b(ξ1 ) − b(ξ2 )| δ p2 |b(ξ1 ) − b(ξ2 )| + ≤C p1 p2 1
1
1 p2 −1
1 p2 −1
p1 −2
δ −q2 |ξ1 − ξ2 | p1 −1 θ1p1 −1 (1 + |ξ1 |p1 θ1 + |ξ2 |p1 θ1 + |ξ1 |p2 θ2 + |ξ2 |p2 θ2 ) p1 −1 + q2 +
δ
−q1
|ξ1 − ξ2 |
θ2
p1
p1
p2
p2
(1 + |ξ1 | θ1 + |ξ2 | θ1 + |ξ1 | θ2 + |ξ2 | θ2 ) q1
p2 −2 p2 −1
Therefore ¶¸ · µ p1 δ p2 δ |b(ξ1 ) − b(ξ2 )| + 1−C p p2 1 1 p1 −2 1 p1 −1 p1 p1 p2 p2 −q2 p1 −1 p1 −1 δ |ξ − ξ | θ (1 + |ξ | θ + |ξ | θ + |ξ | θ + |ξ | θ ) 1 2 1 1 2 1 1 2 2 2 1 ≤C q2 1 p2 −2 1 p2 −1 p1 p1 p2 p2 −q1 p2 −1 p2 −1 θ2 δ |ξ1 − ξ2 | (1 + |ξ1 | θ1 + |ξ2 | θ1 + |ξ1 | θ2 + |ξ2 | θ2 ) + q1
Taking δ small enough, we obtain
|b(ξ1 ) − b(ξ2 )| · 1 p1 −2 1 ≤ C |ξ1 − ξ2 | p1 −1 θ1p1 −1 (1 + |ξ1 |p1 θ1 + |ξ2 |p1 θ1 + |ξ1 |p2 θ2 + |ξ2 |p2 θ2 ) p1 −1 ¸ 1 p2 −2 1 p2 −1 p1 p1 p2 p2 p2 −1 p2 −1 (1 + |ξ1 | θ1 + |ξ2 | θ1 + |ξ1 | θ2 + |ξ2 | θ2 ) + |ξ1 − ξ2 | . θ2
23
Proof of (3.16): Monotonicity of b Proof. Using (3.11) and (3.5), we have (b(ξ2 ) − b(ξ1 ), ξ2 − ξ1 ) µZ ¶ Z = A(y, p(y, ξ2 ))dy − A(y, p(y, ξ1 ))dy, ξ2 − ξ1 Y Y Z = (A(y, p(y, ξ2 )) − A(y, p(y, ξ1 )), ξ2 − ξ1 ) dy ZY = (A(y, p(y, ξ2 )) − A(y, p(y, ξ1 )), p(y, ξ2 ) − p(y, ξ1 )) dy Y µZ ≥C χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )|p1 dy ¶ Z Y p2 + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy ≥ 0. Y
3.3.4
Properties of p
1,p1 (Rn ), Since the solution υξ of (3.11) can be extended by periodicity to a function of Wloc ′ then (3.11) is equivalent to −div(A(y, ξ + ∇υξ (y))) = 0 over D (Rn ), i.e., ′
−div (A(y, p(y, ξ))) = 0 in D (Rn ) for every ξ ∈ Rn . Moreover, by (3.11), we have Z Z (A(y, p(y, ξ)), p(y, ξ)) dy = (A(y, p(y, ξ)), ξ) dy = (b(ξ), ξ) . Y
(3.17)
(3.18)
Y
For ǫ > 0, define pǫ : Rn × Rn → Rn by ³x ´ ³x´ pǫ (x, ξ) = p , ξ = ξ + ∇υξ . ǫ ǫ
(3.19)
where υξ is the unique solution of (3.11). The functions p and pǫ are easily seen to satisfy the following properties p(·, ξ) is Y -periodic and pǫ (x, ξ) is ǫ-periodic in x. Z p(y, ξ)dy = ξ.
(3.20) (3.21)
Y
A
³·
ǫ
pǫ (·, ξ) ⇀ ξ in Lp1 (Ω; Rn ) as ǫ → 0.
(3.22)
p(y, 0) = 0 for almost every y. ´ , pǫ (·, ξ) ⇀ b(ξ) in Lq2 (Ω; Rn ), as ǫ → 0.
(3.23)
24
(3.24)
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 W01,p2 (Ω). In the following chapters we will apply these facts to establish strong approximations for the sequences {χǫi ∇uǫ }ǫ>0 in Lp2 (Ω, Rn ). 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.
4.1
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 4.1. Given a periodic dispersion of inclusions or a layered material then the solution u of (3.8) belongs to W01,p2 (Ω). Before we can prove this theorem we need some definitions. Definition 4.2. Functions f (x, ξ) depending on two variables x, ξ ∈ Rn will be referred to as Lagrangians. Definition 4.3. If the Lagrangian f (x, ξ), ξ ∈ Rn , x ∈ Ω ⊂ Rn satisfies −c0 + c1 |ξ|p ≤ f (x, ξ) ≤ c2 |ξ|p + c0
(4.1)
with c0 ≥ 0,c1 ,c2 > 0, and p > 1, then it is called standard. A much wider class of Lagrangians which includes the standard ones, is specified by the estimate −c0 + c1 |ξ|p1 ≤ f (x, ξ) ≤ c2 |ξ|p2 + c0 (4.2) 25
with p2 ≥ p1 > 1. These are called nonstandard Lagrangians. Definition 4.4. The conjugate of a nonstandard Lagrangian f , denoted g = f ∗ , is defined by g(x, ξ) = sup {ξ · η − f (x, η)} , (4.3) η∈Rn
and satisfies the estimate −c0 + c∗1 |ξ|q1 ≤ g(x, ξ) ≤ c∗2 |ξ|q2 + c0 .
4.2
(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 σ1 σ2 f˜(x, ξ) = q(x) |ξ|p(x) = χ1 (x) |ξ|p1 + χ2 (x) |ξ|p2 , p1 p2
(4.5)
with
σ2 σ1 χ1 (x) + χ2 (x); p1 p2 where ξ ∈ Rn and x ∈ Ω ⊂ Rn . Here ∇ξ f˜(x, ξ) = A (x, ξ), where A(x, ξ) is given by (3.1). We consider the rescaled Lagrangian ³x ´ σ σ2 1 f˜ǫ (x, ξ) = f˜ , ξ = χǫ1 (x) |ξ|p1 + χǫ2 (x) |ξ|p2 , (4.6) ǫ p1 p2 q(x) =
where χǫi (x) = χi (x/ǫ), i = 1, 2, ξ ∈ Rn , and x ∈ Ω ⊂ Rn . The Dirichlet problem given by (3.7) is associated with the variational problem given by ½Z ¾ ǫ ˜ fǫ (x, ∇u)dx − hf, ui , (4.7) E1 (f ) = inf 1,p u∈W0
1 (Ω)
Ω
with f ∈ W −1,q2 (Ω). Here (3.7) is the Euler equation for (4.7). However, we also consider ½Z ¾ ǫ f˜ǫ (x, ∇u)dx − hf, ui , (4.8) E2 (f ) = inf 1,p u∈W0
2 (Ω)
Ω
with f ∈ W −1,q2 (Ω) (See [Zhi92]). Here h·, ·i is the duality pairing between W01,p1 (Ω) and W −1,q2 (Ω). From [ZKO94], we have lim Eiǫ = Ei , for i = 1, 2, where ǫ→0
Ei =
inf 1,p
u∈W0
i (Ω)
½Z
Ω
¾ ˆ ˜ fi (∇u(x))dx − hf, ui . 26
(4.9)
In (4.9), f˜ˆi (ξ) is given by fˆ˜i (ξ) =
inf 1,p
i
v in Wper (Y )
and satisfies
Z
Y
f˜(y, ξ + ∇v(y))dy
(4.10)
−c0 + c1 |ξ|p1 ≤ fˆ˜i (ξ) ≤ c2 |ξ|p2 + c0 .
(4.11)
In general, (see [Zhi95]) Lavrentiev phenomenon can occur and E1 < E2 . 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 E1 = E2 , where fˆ˜ = fˆ˜1 = fˆ˜2 and c2 + c1 |ξ|p2 ≤ fˆ˜(ξ). Moreover, ∇ξ fˆ˜(ξ) = b(ξ), where b is the homogenized operator (3.9). Proof. Theorem 4.5 has been proved for dispersed periodic media in Chapter 14 of [ZKO94]. We prove Theorem 4.5 for layers following the steps outlined in [ZKO94]. We first show that fˆ˜ = fˆ˜1 = fˆ˜2 holds for layered media. Then we show that the homogenized Lagrangian fˆ˜ satisfies the standard estimate given by −c0 + c1 |ξ|p2 ≤ fˆ˜(ξ) ≤ c2 |ξ|p2 + c0
(4.12)
with c0 ≥ 0, and c1 ,c2 > 0 . We introduce the space of functions W∗1,p2 (R2 ) that belong to W 1,p2 (R2 ) and are periodic on ∂R2 ∩ ∂Y .
Lemma 4.6. Any function v ∈ W∗1,p2 (R2 ) can be extended to R1 in such a way that the 1,p2 extension v˜(y) belongs to Wper (Y ) and v˜(y) = v(y) on R2 . Proof. Let ϕ to be the solution of ∆p 2 ϕ = 0 ϕ takes periodic boundary values on opposite faces of ∂Y ∩ ∂R1 ϕ¯¯ = v¯¯ 1
, on R1 , on Γ
2
Here the subscript 1 indicates the trace on the R1 side of Γ and 2 indicates the trace on the R2 side of Γ. For a proof of existence of the solution ϕ see [Eva82] or [Lew77]. The extension v˜ is given by ( v , in R2 . v˜ = ϕ , on R1 .
27
ˆ˜ ˆ˜ p1 ˜ˆ ˜ˆ It is clear Z that f1 ≤ f2 . To prove f1 = f2 , it suffices to show that for every v ∈ Wper (Y ) 1,p2 f˜(y, ξ + ∇v(y))dy < ∞ there exists a sequence vǫ ∈ Wper (Y ) such that satisfying Y
lim ǫ→0
Z
Y
f˜(y, ξ + ∇vǫ (y))dy =
Z
Y
f˜(y, ξ + ∇v(y))dy.
For v as above, let v˜ be as in Lemma 4.6 and set z = v − v˜. It is clear that z ∈ W 1,p1 (R1 ), is periodic on opposite faces of ∂Y ∩ ∂R1 , zero on Γ and we write Z Z Z ˜ f (y, ξ + ∇v(y))dy = f2 (ξ + ∇v(y))dy + f1 (ξ + ∇˜ v (y) + ∇z(y))dy, R2
Y
R1
where f1 (ξ) = σp11 |ξ|p1 and f2 (ξ) = σp22 |ξ|p2 ; i.e, f1 and f2 are standard Lagrangians satisfying (4.1) with exponents p1 and p2 respectively. We can choose a sequence {zǫ }ǫ>0 ∈ C∞ 0 (R1 ) such that zǫ vanishes in R2 and zǫ → z in W 1,p1 (R1 ). 1,p2 Define vǫ ∈ Wper (Y ) by ( vǫ =
v v˜ + zǫ
in R2 , in R1 .
1,p1 Since vǫ → v in Wper (Y ), we see that Z lim f˜(y, ξ + ∇vǫ (y))dy ǫ→0 Y µZ ¶ Z = lim f2 (ξ + ∇v(y))dy + f1 (ξ + ∇˜ v (y) + ∇zǫ (y))dy ǫ→0 R2 R1 Z Z = f2 (ξ + ∇v(y))dy + f1 (ξ + ∇˜ v (y) + ∇z(y))dy R2 R1 Z = f˜(y, ξ + ∇v(y))dy. Y
Therefore fˆ˜ = fˆ˜1 = fˆ˜2 for layered media. We establish (4.12) by introducing the convex conjugate of fˆ˜. We denote the convex dual of fˆ˜i (ξ) by gˆ˜i (ξ) and the convex dual of f˜ by g˜. It is easily verified (see [Zhi92]) that Z ˆ g˜i (ξ) = infq g˜(y, ξ + w(y))dy (4.13) w in Sol i (Y )
and
Y
−c0 + c∗1 |ξ|q1 ≤ gˆ˜i (ξ) ≤ c∗2 |ξ|q2 + c0 .
(4.14)
Here Solqi (Y ) are the solenoidal vector fields belonging to Lqi (Y, Rn ) and having mean value zero Solqi (Y ) = {w ∈ Lqi (Y ; Rn ) : div w = 0, w · n anti-periodic} . 28
We will show that gˆ˜ = gˆ˜1 = gˆ˜2 satisfies gˆ˜(ξ) ≤ c2 |ξ|q1 + c1 , and apply duality to recover
fˆ˜(ξ) ≥ c∗2 |ξ|p2 + c∗1 . To get the upper bound on gˆ˜ we use the following lemma.
Lemma 4.7. There exists τ with div τ = 0 in Y , such that τ · n is anti-periodic on the boundary of Y , τ = −ξ in R1 , and Z |τ (y)|q1 dy ≤ C |ξ|q1 . Y
Proof. Let the function ϕ ∈ W∗1,p2 (R2 ) be the solution of p−2 · n is anti-periodic on ∂R2 ∩ ∂Y ; ∇ϕ|∇ϕ| ∆p2 ϕ = 0 in R2 ; ∇ϕ |∇ϕ|p2 −2 · n¯¯ = −ξ · n¯¯ ; on Γ, 2
1
where the subscript 1 indicates the trace on the R1 side of Γ and 2 indicates the trace on the R Z 2 side of Γ. The Z Neumann problem given above is the stationarity condition for the energy |∇φ|p2 dx − φξ · n dS when minimized over all φ ∈ W∗1,p2 (R2 ). The solution of the R2
Γ
Neumann problem is unique up to a constant. Here the anti-periodic boundary condition on ∇ϕ|∇ϕ|p−2 · n is the natural boundary condition for the problem. Now we define τ according to ( −ξ; in R1 τ= p2 −2 ∇ϕ |∇ϕ| ; in R2
and it follows that |ξ|q1 ; q1 |τ | = h¡ ¢ i q1 ¢ ¡ ∇ϕ |∇ϕ|p2 −2 2 2 = |∇ϕ|p2 −1 q1 = |∇ϕ|p2 ;
in R1 in R2 .
Then, for ψ ∈ W∗1,p2 (R2 ) we have Z |∇ϕ|p2 −2 ∇ϕ · ∇ψdy R2 Z Z p2 −2 = ψ |∇ϕ| ∇ϕ · ndS + ψ |∇ϕ|p2 −2 ∇ϕ · ndS Γ ∂R2 ∩∂Y Z = − ψξ · ndS. Γ
29
(4.15)
(4.16)
Set ψ = ϕ in (4.16) and an application of H¨older’s inequality gives Z Z p2 |∇ϕ| dx = − ϕξ · ndS R2 ZΓ =− ∇(ϕξ)dx R2 Z =− (∇ϕ) · ξdx R2
≤ Then
Z
µZ
p2
R2
|∇ϕ| dx p2
R2
|∇ϕ| dx ≤
¶1/p2 µZ
R2
Z
R2
q1
|ξ| dx
|ξ|q1 dx.
¶1/q1
.
(4.17)
Therefore, using (4.15) and (4.17), we have Z |τ (y)|q1 dy Y Z Z q1 = |τ (y)| dy + |τ (y)|q1 dy ZR1 Z R2 = |ξ|q1 dy + |∇ϕ(y)|p2 dy R1
R2
q1
≤ C |ξ| .
Taking gˆ˜ to be the conjugate of fˆ˜, and choosing τ in Solq1 (Y ) as in Lemma 4.7, we obtain Z ˆ g˜(ξ) = inf g˜(y, ξ + τ )dy τ in Solq1 (Y ) Y Z ≤ g˜(y, ξ + τ )dy Y Z Z ≤ g˜(y, 0)dy + g˜(y, ξ + τ )dy R1 R2 Z ≤ c1 + c2 |ξ + τ |q1 dy R2
≤ c1 + c2 |ξ|q1 ,
and the left hand inequality in (4.12) follows from duality. This concludes the proof of Theorem 4.5. Collecting results we now prove Theorem 4.1. Indeed the minimizer of E1 is precisely the solution u of (3.8). Theorem 4.5 establishes the coercivity of E1 over W01,p2 (Ω), thus the solution u lies in W01,p2 (Ω). 30
Chapter 5 Corrector Theorem In this chapter, we develop new strong convergence results that capture the asymptotic behavior of the gradients ∇uǫ , as ǫ tends to 0. Our approach delivers strong approximations for the gradients inside each phase χǫi ∇uǫ , i = 1, 2. Homogenization theory relates the average behavior seen at large length scales to the underlying heterogeneous structure. It allows one to approximate {∇uǫ }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 ∇uǫ decays to 0. We present a new corrector result that delivers an approximation to ∇uǫ 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 W01,p2 (Ω) (See Chapter 4). With this higher order integrability in hand, we provide an algorithm for building correctors and establish strong approximations for the sequences {χi ǫ∇uǫ }ǫ>0 in Lp2 (Ω, Rn ), see Theorem 5.2. When the host phase has a lower power-law exponent than the included phase one can only conclude that the homogenized solution lies in W01,p1 (Ω) and the techniques developed here do not apply.
5.1
Statement of the Corrector Theorem
We now describe the family of correctors that provide a strong approximation of the sequence {χǫi ∇uǫ }ǫ>0 in the Lpi (Ω, Rn ) norm. We denote the rescaled period cell with side length ǫ > 0 by Yǫ and write Yǫi = ǫi + Yǫ , where i ∈ Zn . In what follows it is convenient to define the index set Iǫ = {i ∈ Zn : Yǫi ⊂ Ω}. For ϕ ∈ Lp2 (Ω; Rn ), we define the local average operator Mǫ associated with the partition Yǫi , i ∈ Iǫ by X Z 1 χYǫi (x) i ϕ(y)dy; if x ∈ ∪i∈Iǫ Yǫi , |Y | i Yǫ ǫ Mǫ (ϕ)(x) = i∈ Iǫ (5.1) 0; i if x ∈ Ω \ ∪i∈Iǫ Yǫ . 31
Remark 5.1. The family Mǫ has the following properties 1. For i = 1, 2, kMǫ (ϕ) − ϕkLpi (Ω;Rn ) → 0 as ǫ → 0. For a proof, see, for instance Chapter 8 of [Zaa58]. 2. Mǫ (ϕ) → ϕ a.e. on Ω. For a proof, see, for instance Chapter 8 of [Zaa58]. 3. From Jensen’s inequality, we have kMǫ (ϕ)kLpi (Ω;Rn ) ≤ kϕkLpi (Ω;Rn ) , for every ϕ ∈ Lp2 (Ω; Rn ) and i = 1, 2. The strong approximation to the sequence {χǫi ∇uǫ }ǫ>0 is given by the following corrector theorem. Theorem 5.2 (Corrector Theorem). Let f ∈ W −1,q2 (Ω), let uǫ 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 = 1, 2, we have Z (5.2) |χǫi (x)pǫ (x, Mǫ (∇u)(x)) − χǫi (x)∇uǫ (x)|pi dx → 0, as ǫ → 0. Ω
Before we can give the proof of this theorem, we need the results from the following section.
5.2
Some Properties of Correctors
In this section, we state and prove a priori bounds and convergence properties for the sequences pǫ defined in (3.19), ∇uǫ , and Aǫ (x, pǫ (x, ∇uǫ )) that are used in the proof of Theorem 5.2. 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 ∈ Rn , we have Z Z p1 χ1 (y) |p(y, ξ)| dy + χ2 (y) |p(y, ξ)|p2 dy ≤ C (1 + |ξ|p1 θ1 + |ξ|p2 θ2 ) , Y
and by a change of variables, we obtain Z Z p1 ǫ χ1 (x) |pǫ (x, ξ)| dx + χǫ2 (x) |pǫ (x, ξ)|p2 dx ≤ C (1 + |ξ|p1 θ1 + |ξ|p2 θ2 ) |Yǫ | . Yǫ
(5.3)
Y
Yǫ
Proof. Let ξ ∈ Rn . By (3.5), we have that (A(y, p(y, ξ)), p(y, ξ)) ≥ C (χ1 (y) |p(y, ξ)|p1 + χ2 (y) |p(y, ξ)|p2 ) .
32
(5.4)
Integrating both sides over Y and using (3.11), we get Z Z p1 χ1 (y) |p(y, ξ)| dy + χ2 (y) |p(y, ξ)|p2 dy Y Y Z ≤C (A(y, p(y, ξ)), p(y, ξ)) dy Y Z =C (A(y, p(y, ξ)), ξ) dy Y
By Cauchy-Schwarz Inequality and (3.4), we have ≤C
Z
|A(y, p(y, ξ))| |ξ| dy
Y
·Z
χ1 (y) |p(y, ξ)| (1 + |p(y, ξ)|)p1 −2 |ξ| dy ¸ Z Y p2 −2 + χ2 (y) |p(y, ξ)| (1 + |p(y, ξ)|) |ξ| dy Y ·Z ¸ Z p1 −1 p2 −1 ≤C χ1 (y)(1 + |p(y, ξ)|) |ξ| dy + χ2 (y)(1 + |p(y, ξ)|) |ξ| dy
≤C
Y
Y
Using Young’s Inequality, we obtain
δ ≤C δ
+
q1
Z
q2
p1
Y
Z
Y
χ1 (y)(1 + |p(y, ξ)|) dy q2
δ
q1
Y
+
p2
χ2 (y)(1 + |p(y, ξ)|) dy
Z
−p1
δ
−p2
Z
Y
+
χ1 (y) |ξ|p1 dy p1 p2
χ2 (y) |ξ| dy p2
· Z = C δ q2 χ1 (y)(1 + |p(y, ξ)|)p1 dy + δ −p1 |ξ|p1 θ1 ¸ Z Y p2 q1 −p2 p2 +δ χ2 (y)(1 + |p(y, ξ)|) dy + δ |ξ| θ2 Y · Z q2 q2 ≤ C δ θ1 + δ χ1 (y) |p(y, ξ)|p1 dy + δ −p1 |ξ|p1 θ1 + δ q1 θ2 Y ¸ Z p2 p2 q1 −p2 +δ χ2 (y) |p(y, ξ)| dy + δ |ξ| θ2 Y £ ¡ ¢ ≤ C (δ q2 θ1 + δ q1 θ2 ) + δ −p1 |ξ|p1 θ1 + δ −p2 |ξ|p2 θ2 µZ ¶¸ Z p1 p2 q1 q2 +(δ + δ ) χ1 (y) |p(y, ξ)| dy + χ2 (y) |p(y, ξ)| dy . Y
Y
33
Doing some algebraic manipulations, we obtain µZ ¶ Z p1 p2 q1 q2 (1 − C(δ + δ )) χ1 (y) |p(y, ξ)| dy + χ2 (y) |p(y, ξ)| dy Y Y h ¡ ¢i ≤ C (δ q2 θ1 + δ q1 θ2 ) + δ −p1 |ξ|p1 θ1 + δ −p2 |ξ|p2 θ2 On choosing an appropiate δ, we finally obtain (5.3).
Lemma 5.4. For every ξ1 , ξ2 ∈ Rn , we have Z Z p1 χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )|p2 dy Y Y · 1 p1 −2 p1 ≤ C (1 + |ξ1 |p1 θ1 + |ξ1 |p2 θ2 + |ξ2 |p1 θ1 + |ξ2 |p2 θ2 ) p1 −1 |ξ1 − ξ2 | p1 −1 θ1p1 −1 ¸ 1 p2 p2 −2 p −1 p1 p2 p1 p2 2 + (1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) p2 −1 |ξ1 − ξ2 | p2 −1 θ2 , and by doing a change a variables, we obtain Z Z p1 ǫ χ1 (x) |pǫ (x, ξ1 ) − pǫ (x, ξ2 )| dx + χǫ2 (x) |pǫ (x, ξ1 ) − pǫ (x, ξ2 )|p2 dx Yǫ Yǫ · 1 p1 p1 −2 p1 −1 p1 p2 p1 p2 p1 −1 p1 −1 |ξ1 − ξ2 | θ1 ≤ C (1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) ¸ 1 p2 p2 −2 p2 −1 p1 p2 p1 p2 p2 −1 p2 −1 |ξ1 − ξ2 | θ2 + (1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) |Yǫ | Proof. By (3.5), we have (A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), p(y, ξ1 ) − p(y, ξ2 )) ≥ C (χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )|p1 + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )|p2 ) . Integrating over Y and using (3.11) and the Cauchy-Schwarz inequality, we get Z Z p1 χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )|p2 dy Y Y Z ≤C (A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), p(y, ξ1 ) − p(y, ξ2 )) dy Y Z =C (A(y, p(y, ξ1 )) − A(y, p(y, ξ2 )), ξ1 − ξ2 ) dy Y Z ≤C |A(y, p(y, ξ1 )) − A(y, p(y, ξ2 ))| |ξ1 − ξ2 | dy Y
Using (3.4), we have ·Z ≤C χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| (1 + |p(y, ξ1 )| + |p(y, ξ2 )|)p1 −2 |ξ1 − ξ2 | dy ¸ Z Y p2 −2 + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| (1 + |p(y, ξ1 )| + |p(y, ξ2 )|) |ξ1 − ξ2 | dy Y
34
(5.5)
(5.6)
p1 Using H¨older’s inequality in the first expression with r1 = , r2 = p1 , and r3 = p1 ; and p1 − 2 p2 in the second expression with s1 = , s2 = p2 , and s3 = p2 , we obtain p2 − 2 " µZ ¶ p1 −2 ≤C
Y
× +
µZ
Y
µZ
Y
× ≤C
+
Y
χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy Z
χ1 (y)dy +
Y
µZ
Y
µZ
χ2 (y)dy +
Z
Y
2
1
χ2 (y) |p(y, ξ1 )| dy +
χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )|p2 dy
p2
Y
¶ p1
¶ p1
2
¶ p1
1
2
¶ p1 µZ
p2
Y
µZ
χ1 (y) |ξ1 − ξ2 | dy
Y
Z
χ1 (y) |p(y, ξ1 )| dy +
χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )|p1 dy
p1
¶ p2p−2
p1
Y
Y
×
1
χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy
p2
" µZ ×
¶ p1 µZ
p1
χ2 (y) (1 + |p(y, ξ1 )| + |p(y, ξ2 )|)p2 dy
µZ
p1
χ1 (y) (1 + |p(y, ξ1 )| + |p(y, ξ2 )|)p1 dy
χ2 (y) |ξ1 − ξ2 | dy p1
Y
¶ p1 #
χ1 (y) |p(y, ξ2 )| dy
2
¶ p1p−2 1
1
|ξ1 − ξ2 | θ1p1 Z
Y
p2
χ2 (y) |p(y, ξ2 )| dy #
¶ p2p−2 2
1
|ξ1 − ξ2 | θ2p2
Use Lemma 5.3 to get h p1 −2 ≤ C (1 + |ξ1 |p1 θ1 + |ξ1 |p2 θ2 + |ξ2 |p1 θ1 + |ξ2 |p2 θ2 ) p1 µZ ¶ p1 1 1 p1 p1 × |ξ1 − ξ2 | θ1 χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy Y
p2 −2
p1
+ (1 + |ξ1 | θ1 + |ξ1 |p2 θ2 + |ξ2 |p1 θ1 + |ξ2 |p2 θ2 ) p2 µZ ¶ p1 # 1 2 p2 p2 × |ξ1 − ξ2 | θ2 χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy Y
By Young’s Inequality, we obtain q2 (p1 −2)q2 p1 p2 p1 p2 q2 p 1 −q2 p1 δ (1 + |ξ | θ + |ξ | θ + |ξ | θ + |ξ | θ ) |ξ − ξ | θ 1 1 1 2 2 1 2 2 1 2 1 ≤C q2 Z Z p1 p2 p1 χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )|p2 dy δ χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy δ Y Y + + p1 p2 35
+
δ
−q1
p1
p2
p1
p2
(1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) q1
(p2 −2)q1 p2
q1
q1 p2
|ξ1 − ξ2 | θ2 .
Straightforward algebraic manipulations deliver µZ ¶ Z p1 p2 kδ χ1 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy + χ2 (y) |p(y, ξ1 ) − p(y, ξ2 )| dy Y Y q2 (p1 −2)q2 p1 p2 p1 p2 q2 p 1 −q2 p1 δ (1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) |ξ1 − ξ2 | θ1 ≤C q2 q1 (p2 −2)q1 p q p1 p2 p1 p2 1 −q1 δ (1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) p2 |ξ1 − ξ2 | θ2 2 + q1 1 p1 −2 p1 p1 −1 p1 p2 p1 p2 −q2 p1 −1 p1 −1 δ (1 + |ξ | θ + |ξ | θ + |ξ | θ + |ξ | θ ) |ξ − ξ | θ 1 1 1 2 2 1 2 2 1 2 1 =C q2 1 p2 p2 −2 p2 −1 p1 p2 p1 p2 −q1 p2 −1 p2 −1 |ξ1 − ξ2 | θ2 δ (1 + |ξ1 | θ1 + |ξ1 | θ2 + |ξ2 | θ1 + |ξ2 | θ2 ) , + q1
n³ ´ ³ ´o p p where kδ = min 1 − cδp11 , 1 − cδp22 . The result (5.5) follows on choosing δ small enough so that kδ is positive. Lemma 5.5. Let ϕ be such that ½Z ¾ Z p1 p2 ǫ ǫ χ1 (x) |ϕ(x)| dx + χ2 (x) |ϕ(x)| dx < ∞, sup ǫ>0
Ω
Ω
and let Ψ be a simple function of the form Ψ(x) =
m X
ηj χΩj (x),
(5.7)
j=0
with ηj ∈ Rn \ {0}, Ωj ⊂⊂ Ω, |∂Ωj | = 0, Ωj ∩ Ωk = ∅ for j 6= k and j, k = 1, ..., m; and set m [ η0 = 0 and Ω0 = Ω \ Ωj . Then j=1
µZ
χǫ1 (x) |pǫ (x, Mǫ ϕ(x)) − pǫ (x, Ψ(x))|p1 dx lim sup ǫ→0 Ω ¶ Z p2 ǫ + χ2 (x) |pǫ (x, Mǫ ϕ(x)) − pǫ (x, Ψ(x))| dx Ω ( 2 ·µ Z Z X p1 ǫ ≤ lim sup C |Ω| + χ1 (x) |ϕ(x)| dx + χǫ2 (x) |ϕ(x)|p2 dx ǫ→0
i=1
Ω
Ω
36
+
Z
χǫ1 (x) |Ψ(x)|p1
×
µZ
Ω
Ω
dx +
Z
Ω
χǫi (x) |ϕ(x)
χǫ2 (x) |Ψ(x)|p2 pi
− Ψ(x)| dx
¶ p 1−1 i
dx #)
−2 ¶ ppi −1 i
(5.8)
Proof. Let Ψ of the form (5.7). For every ǫ > 0, let us denote by [ Yǫi for i ∈ Iǫ ; Ωǫ = and for j = 0, 1, 2, ..., m, we set
© ª Iǫj = i ∈ Iǫ : Yǫi ⊆ Ωj ,
and Furthermore,
© ª Jǫj = i ∈ Iǫ : Yǫi ∩ Ωj 6= ∅, Yǫi \ Ωj 6= ∅ . Eǫj =
[
Yǫi ,
(5.9)
[
Yǫi ;
(5.10)
i∈Iǫj
and Fǫj =
i∈Jǫj
with
¯ j¯ ¯Fǫ ¯ → 0, as ǫ → 0.
Set
ξǫi
1 = i |Yǫ |
Z
ϕ(x)dx.
Yǫi
For ǫ sufficiently small Ωj is contained in Ωǫ , for all j 6= 0. From (5.7), we have Z Z p1 ǫ χ1 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, Ψ)| dx + χǫ2 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, Ψ)|p2 dx Ω ¯ Ã m Ω !¯p1 Z ¯ ¯ X ¯ ¯ = χǫ1 (x) ¯pǫ (x, Mǫ ϕ) − pǫ x, ηj χΩj ¯ dx ¯ ¯ Ω j=0 ¯ !¯p2 Ã Z m ¯ ¯ X ¯ ¯ + χǫ2 (x) ¯pǫ (x, Mǫ ϕ) − pǫ x, ηj χΩj ¯ dx ¯ ¯ Ω j=0
or equivalently, m Z X χǫ1 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, ηj )|p1 dx = j=0 Ωj m Z X
+
j=0
Ωj
χǫ2 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, ηj )|p2 dx 37
(5.11) (5.12)
Using the fact that Ωj ⊂ Eǫj ∪ Fǫj and (5.1), we have ≤
m Z X
Eǫj
j=0 m Z X
+
Fǫj
j=0
+
m Z X
Eǫj
j=0
+
χǫ1 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, ηj )|p1 dx
m Z X
Fǫj
j=0
χǫ1 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, ηj )|p1 dx χǫ2 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, ηj )|p2 dx χǫ2 (x) |pǫ (x, Mǫ ϕ) − pǫ (x, ηj )|p2 dx
¯ p1 ¯ Ã ! ¯ ¯ X ¯ ¯ i ǫ χYǫi (x)ξǫ − pǫ (x, ηj )¯ dx χ1 (x) ¯pǫ x, = j ¯ ¯ j=0 Eǫ i∈Iǫ ¯ ¯ p1 Ã ! m Z ¯ ¯ X X ¯ ¯ χǫ1 (x) ¯pǫ x, + χYǫi (x)ξǫi − pǫ (x, ηj )¯ dx j ¯ ¯ j=0 Fǫ i∈Iǫ ¯ ¯ p2 ! Ã m Z ¯ ¯ X X ¯ ¯ ǫ i χ2 (x) ¯pǫ x, + χYǫi (x)ξǫ − pǫ (x, ηj )¯ dx j ¯ ¯ j=0 Eǫ i∈Iǫ ¯ ¯ Ã ! m Z ¯ ¯ p2 X X ¯ ¯ ǫ i χ2 (x) ¯pǫ x, + χYǫi (x)ξǫ − pǫ (x, ηj )¯ dx ¯ ¯ Fǫj m Z X
j=0
i∈Iǫ
Using (5.9), (5.10) and reorganizing the sums, we get
=
m X j=0
+
m X j=0
+
m X j=0
+
m X j=0
XZ i∈Iǫj
Yǫi
XZ
i∈Jǫj
i∈Iǫj
¯ ¡ ¯p ¢ χǫ1 (x) ¯pǫ x, ξǫi − pǫ (x, ηj )¯ 1 dx
Yǫi
XZ
Yǫi
XZ
i∈Jǫj
Yǫi
¯ ¡ ¯p ¢ χǫ1 (x) ¯pǫ x, ξǫi − pǫ (x, ηj )¯ 1 dx
¯ ¯ ¡ ¢ p χǫ2 (x) ¯pǫ x, ξǫi − pǫ (x, ηj )¯ 2 dx
¯ ¯ ¡ ¢ p χǫ2 (x) ¯pǫ x, ξǫi − pǫ (x, ηj )¯ 2 dx
38
m X µZ X = j=0
+
Yǫi
i∈Iǫj
Z
Yǫi
χǫ2 (x) ¯pǫ
¯ ¡ ¯p ¢ χǫ1 (x) ¯pǫ x, ξǫi − pǫ (x, ηj )¯ 1 dx
¯ ¡
m X µZ X + j=0
+
Z
Yǫi
x, ξǫi
Yǫi
i∈Jǫj
χǫ2 (x) ¯pǫ
¯ ¡
¢
¯p − pǫ (x, ηj )¯ 2 dx
¶¸
¯p − pǫ (x, ηj )¯ 2 dx
¶¸
¯ ¡ ¯p ¢ χǫ1 (x) ¯pǫ x, ξǫi − pǫ (x, ηj )¯ 1 dx
x, ξǫi
¢
Using Lemma 5.4, we obtain
≤C
m X · X ¡ j=0
i∈Iǫj
¯ ¯p ¯ ¯p ¢ p1 −2 1 + ¯ξǫi ¯ 1 θ1 + ¯ξǫi ¯ 2 θ2 + |ηj |p1 θ1 + |ηj |p2 θ2 p1 −1
1 ¯ ¯ ¯ p1 ¯ × ¯ξǫi − ηj ¯ p1 −1 θ1p1 −1 ¯Yǫi ¯ ¸¾ ¯ i ¯ p1 ¯ i ¯ p2 ¯ p2 p 1−1 ¯ i ¯ ¯ i ¡ ¢ pp2 −2 p1 p2 −1 2 p −1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Yǫ + 1 + ξǫ θ1 + ξǫ θ2 + |ηj | θ1 + |ηj | θ2 2 ξǫ − ηj 2 θ2 m X · X ¯ ¯p ¯ ¯p ¡ ¢ p1 −2 +C 1 + ¯ξǫi ¯ 1 θ1 + ¯ξǫi ¯ 2 θ2 + |ηj |p1 θ1 + |ηj |p2 θ2 p1 −1 j
j=0
i∈Jǫ
1 ¯ ¯ ¯ p1 ¯ × ¯ξǫi − ηj ¯ p1 −1 θ1p1 −1 ¯Yǫi ¯ ¸¾ ¯ p2 p 1−1 ¯ i ¯ ¯ i ¯ i ¯ p1 ¯ i ¯ p2 ¡ ¢ pp2 −2 p p 1 2 −1 2 p −1 ¯ Yǫ ¯ + 1 + ¯ξǫ ¯ θ1 + ¯ξǫ ¯ θ2 + |ηj | θ1 + |ηj | θ2 2 ¯ξǫ − ηj ¯ 2 θ2
≤C
m XZ X j=0 i∈Iǫj
· ¡
Yǫi
1 ¯ p1 ¯ ¯p ¯ ¯p ¢ p1 −2 ¯ 1 + ¯ξǫi ¯ 1 θ1 + ¯ξǫi ¯ 2 θ2 + |ηj |p1 θ1 + |ηj |p2 θ2 p1 −1 ¯ξǫi − ηj ¯ p1 −1 θ1p1 −1
¸ −2 ¯ ¯ p2 p 1−1 ¯ i ¯ p1 ¯ i ¯p2 ¡ ¢ pp2 −1 p1 p2 i 2 p2 −1 ¯ ¯ ¯ ¯ ¯ ¯ 2 ξǫ − ηj θ2 dx + 1 + ξǫ θ1 + ξǫ θ2 + |ηj | θ1 + |ηj | θ2
+C
m XZ X j=0 i∈Jǫj
Yǫi
· ¡
1 ¯ p1 ¯ ¯p ¯ ¯p ¢ p1 −2 ¯ 1 + ¯ξǫi ¯ 1 θ1 + ¯ξǫi ¯ 2 θ2 + |ηj |p1 θ1 + |ηj |p2 θ2 p1 −1 ¯ξǫi − ηj ¯ p1 −1 θ1p1 −1
¸ −2 ¯ ¯ p2 p 1−1 ¯ i ¯ p1 ¯ i ¯p2 ¡ ¢ pp2 −1 p1 p2 i 2 p2 −1 ¯ ¯ ¯ ¯ ¯ ¯ 2 ξǫ − ηj θ2 dx + 1 + ξǫ θ1 + ξǫ θ2 + |ηj | θ1 + |ηj | θ2
39
Using (5.9), (5.10) again ¯ ¯ p1 ¯ ¯ p2 pp1 −2 ¯X ¯ ¯X ¯ 1 −1 1 ¯ ¯ ¯ ¯ p p i i 1 2 1 + ¯¯ =C θ1p1 −1 χYǫi ξǫ ¯¯ θ1 + ¯¯ χYǫi ξǫ ¯¯ θ2 + |ηj | θ1 + |ηj | θ2 j ¯i∈I j ¯ ¯i∈I j ¯ j=0 Eǫ
m Z X
ǫ
ǫ
¯ ¯ p2 ¯ ¯ p1 ¯ p2 −1 1 ¯X ¯ p1 −1 Z ¯X ¯ ¯ ¯ ¯ i i ¯ dx + θ2p2 −1 χYǫi ξǫ − ηj ¯¯ × ¯¯ χYǫi ξǫ − ηj ¯¯ ¯ j Eǫ ¯ ¯ ¯i∈I j ¯ i∈Iǫj ǫ ¯ ¯ p1 ¯ ¯ p2 pp2 −2 ¯ ¯ ¯ ¯ 2 −1 ¯X ¯ ¯X ¯ p p i i 1 2 × 1 + ¯¯ χYǫi ξǫ ¯¯ θ1 + ¯¯ dx χYǫi ξǫ ¯¯ θ2 + |ηj | θ1 + |ηj | θ2 ¯i∈I j ¯ ¯i∈I j ¯ ǫ ǫ ¯ ¯ p1 ¯ ¯p2 ¯X ¯ ¯X ¯ m Z X ¯ ¯ ¯ ¯ i i ¯ ¯ ¯ 1 + +C χYǫi ξǫ ¯ θ1 + ¯ χYǫi ξǫ ¯¯ θ2 + |ηj |p1 θ1 ¯ Fǫj ¯ j ¯ ¯ j ¯ j=0
i∈Jǫ
i∈Jǫ
¯ ¯ ¯ p1 ¯ p2 ¯X ¯ p1 −1 ¯ p2 −1 1 Z ¯X 1 p1 −2 ¯ ¯ ¯ ¯ p −1 p2 i ¯ i ξ − ηj ¯ dx + θ2p2 −1 χYǫi ξǫi − ηj ¯¯ χ + |ηj | θ2 ) p1 −1 θ1 1 ¯¯ Y ǫ ǫ ¯ ¯ Fǫj ¯ ¯i∈J j ¯ ¯ i∈Jǫj ǫ −2 ¯ ¯ p1 ¯ ¯p2 pp2 −1 ¯X ¯ ¯X ¯ 2 ¯ ¯ ¯ ¯ p1 p2 i¯ i¯ ¯ ¯ × 1+¯ χYǫi ξǫ ¯ θ1 + ¯ χYǫi ξǫ ¯ θ2 + |ηj | θ1 + |ηj | θ2 dx ¯i∈J j ¯ ¯i∈J j ¯ ǫ
ǫ
By H¨older’s Inequality, we get
≤C
m X j=0
−2 ¯ ¯ p1 ¯ ¯ p2 pp1 −1 ¯X ¯ ¯X ¯ 1 ¯ ¯ ¯ ¯ p1 p2 i i ¯ 1 + ¯ iξ ¯ iξ ¯ χ θ + χ 1 Y Y ǫ ǫ ǫ ǫ ¯ ¯ ¯ ¯ θ2 + |ηj | θ1 + |ηj | θ2 dx j Eǫ ¯i∈I j ¯ ¯i∈I j ¯
Z
ǫ
ǫ
¯ ¯ ¯ p2 1 ¯p1 1 p1 −1 p2 −1 ¯X ¯X ¯ ¯ Z ¯ ¯ ¯ ¯ i i θ2 ¯¯ + θ1 ¯¯ χYǫi ξǫ − ηj ¯¯ dx χYǫi ξǫ − ηj ¯¯ dx Eǫj Eǫj ¯i∈I j ¯i∈I j ¯ ¯ ǫ ǫ −2 ¯ p2 ¯ ¯ p1 ¯ pp2 −1 ¯ ¯X ¯ ¯X 2 Z ¯ ¯ ¯ ¯ p1 p2 i i ¯ 1 + ¯ iξ ¯ iξ ¯ χ θ + χ × 1 Y Y ǫ ǫ ǫ ǫ ¯ θ2 + |ηj | θ1 + |ηj | θ2 dx ¯ ¯ ¯ j Eǫ ¯ ¯i∈I j ¯ ¯i∈I j ǫ ǫ ¯ ¯ p1 ¯X ¯ Z m X ¯ ¯ i 1 + ¯ +C χYǫi ξǫ ¯¯ θ1 + |ηj |p1 θ1 ¯ Fǫj ¯i∈J j ¯ j=0 Z ×
ǫ
40
−2 ¯ p1 1 ¯ p2 ¯ ¯ pp1 −1 p1 −1 ¯ ¯ ¯X ¯X 1 Z ¯ ¯ ¯ ¯ p χYǫi ξǫi − ηj ¯¯ dx χYǫi ξǫi ¯¯ θ2 + |ηj | 2 θ2 dx θ1 ¯¯ + ¯¯ j Fǫ ¯ ¯ ¯i∈J j ¯i∈J j ǫ ǫ ¯ ¯ p2 1 p2 −1 ¯X ¯ Z ¯ ¯ θ2 ¯¯ + χYǫi ξǫi − ηj ¯¯ dx j Fǫ ¯i∈J j ¯ ǫ −2 ¯ ¯p1 ¯ ¯p2 pp2 −1 ¯X ¯ ¯X ¯ 2 Z ¯ ¯ ¯ ¯ p1 p2 i¯ i¯ ¯ 1 + ¯ dx i i × χ ξ θ + χ ξ 1 Yǫ ǫ ¯ Yǫ ǫ ¯ θ2 + |ηj | θ1 + |ηj | θ2 ¯ ¯ Fǫj ¯i∈J j ¯ ¯i∈J j ¯ ǫ
ǫ
Reorganizing and using (5.12), we have
¯ ¯p1 ¯ ¯ p2 ¯X ¯ ¯ Z ¯X ¯ ¯ ¯ ¯ i i ¯ ¯ iξ ¯ iξ ¯ ≤C χ θ dx + χ 1 Y Y ǫ ǫ ǫ ǫ ¯ ¯ ¯ ¯ θ2 dx Eǫj ¯ Eǫj ¯ ¯ ¯ j j j=0 i∈Iǫ i∈Iǫ ¯ ¯ p1 1 p1 −1 ¯ ¯ Z ¯ j¯ ¯ j ¯¢ p1 −2 ¯X ¯ p1 p i 2 + |ηj | θ1 ¯Eǫ ¯ + |ηj | θ2 ¯Eǫ ¯ p1 −1 × θ1 ¯¯ χYǫi ξǫ − ηj ¯¯ dx Eǫj ¯i∈I j ¯ ǫ ¯ ¯ p1 ¯ ¯ p2 ¯ ¯ Z ¯X Z ¯X ¯ j¯ ¯ ¯ ¯ ¯ i i ¯ ¯ iξ ¯ iξ ¯ + ¯Eǫ ¯ + χ θ dx + χ 1 Y Y ǫ ǫ ǫ ǫ ¯ ¯ ¯ ¯ θ2 dx Eǫj ¯ Eǫj ¯ ¯ ¯ j j i∈Iǫ i∈Iǫ ¯ ¯ p2 1 p2 −1 ¯ ¯ Z ¯X ¯ ¯ j ¯¢ p2 −2 ¯ j¯ p p1 i 2 p −1 θ2 ¯¯ χYǫi ξǫ − ηj ¯¯ dx + |ηj | θ1 ¯Eǫ ¯ + |ηj | θ2 ¯Eǫ ¯ 2 × j Eǫ ¯i∈I j ¯ ǫ ¯ ¯ p1 ¯ ¯ p2 ¯ ¯ Z ¯X Z ¯X m X ¯ ¯ ¯ j¯ ¯ ¯ i i ¯ ¯ ¯ ¯ ¯ Fǫ + χYǫi ξǫ ¯ θ1 dx + +C χYǫi ξǫ ¯¯ θ2 dx ¯ ¯ Fǫj ¯ Fǫj ¯ ¯ ¯ j=0 i∈Jǫj i∈Jǫj ¯ p1 1 ¯ p1 −1 ¯ ¯X Z p −2 ¯ ¯ ¯ ¯ ¯ ¯ ¢ 1 χYǫi ξǫi − ηj ¯¯ dx + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p1 −1 × θ1 ¯¯ j Fǫ ¯ ¯i∈J j ǫ ¯ ¯ p1 ¯ ¯ p2 ¯ ¯ Z ¯X Z ¯X ¯ ¯ j¯ ¯ ¯ ¯ i¯ i¯ ¯ ¯ ¯ ¯ iξ iξ + Fǫ + χ θ dx + χ θ2 dx 1 Y Y ǫ ǫ¯ ǫ ǫ¯ ¯ ¯ Fǫj ¯ Fǫj ¯ ¯ ¯ j j i∈Jǫ i∈Jǫ ¯ ¯ p2 1 p2 −1 ¯ ¯ Z p2 −2 X ¯ ¯ j¯ ¯ ¯ ¯ ¢ p1 p θ2 ¯¯ + |ηj | θ1 ¯Fǫ ¯ + |ηj | 2 θ2 ¯Fǫj ¯ p2 −1 × χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯ m X
Z ¯ ¯ ¯Eǫj ¯ +
ǫ
41
≤C
m X j=0
Z ¯ ¯ X ¯Eǫj ¯ + i∈Iǫj
Yǫj
X ¯ i ¯ p1 ¯ξǫ ¯ θ1 dx + i∈Iǫj
Z
¯ i ¯ p2 ¯ξǫ ¯ θ2 dx
Yǫj
p1 −2 XZ ¯ ¯ ¯ ¯ ¢ p1 p2 j¯ j ¯ p1 −1 ¯ ¯ × + |ηj | θ1 Eǫ + |ηj | θ2 Eǫ i∈Iǫj
¯ ¯ X + ¯Eǫj ¯ + i∈Iǫj
Z
Yǫj
X ¯ i ¯ p1 ¯ξǫ ¯ θ1 dx + i∈Iǫj
¯ ¯¢ p2 −2 ¯ ¯ + |ηj |p1 θ1 ¯Eǫj ¯ + |ηj |p2 θ2 ¯Eǫj ¯ p2 −1
Z
Yǫj
Yǫj
p 1−1 1 ¯ i ¯ p1 ¯ ¯ θ1 ξǫ − ηj dx
¯ i ¯ p2 ¯ξǫ ¯ θ2 dx
XZ × i∈Iǫj
p 1−1 2 ¯ p2 ¯ i θ2 ¯ξǫ − ηj ¯ dx j
Yǫ
¯ ¯ p1 ¯ ¯ p2 ¯X ¯ ¯ Z ¯X ¯ ¯ j¯ ¯ ¯ ¯ i i ¯ ¯ ¯Fǫ ¯ + +C χYǫi ξǫ ¯¯ θ1 dx + χYǫi ξǫ ¯¯ θ2 dx ¯ ¯ Fǫj ¯ Fǫj ¯ ¯ ¯ j=0 i∈Jǫj i∈Jǫj ¯ ¯ p1 1 p1 −1 ¯ ¯ Z p −2 X ¯ ¯ ¯ j ¯¢ 1 ¯ j¯ p2 p1 i p1 −1 ¯ ¯ ¯ ¯ ¯ ¯ θ1 ¯ × χYǫi ξǫ − ηj ¯ dx + |ηj | θ1 Fǫ + |ηj | θ2 Fǫ Fǫj ¯i∈J j ¯ ǫ ¯ ¯ p1 ¯ ¯ p2 ¯ ¯ Z ¯X Z ¯X ¯ ¯ j¯ ¯ ¯ ¯ i¯ i¯ ¯ ¯ ¯ ¯ + Fǫ + χYǫi ξǫ ¯ θ1 dx + χYǫi ξǫ ¯ θ2 dx ¯ ¯ Fǫj ¯ Fǫj ¯ ¯ ¯ i∈Jǫj i∈Jǫj ¯ ¯ p2 1 p2 −1 ¯ ¯ Z p2 −2 X ¯ ¯ ¯ ¯ ¯ ¯ ¢ θ2 ¯¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1 × χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯ m X
Z
ǫ
By (5.1) and (5.9), we have m X ¯ ¯ ¯ ¯ p2 X ¯ ¯ X ¯ j ¯ ¯ i ¯ p1 ¯Yǫj ¯ ¯ξǫi ¯ θ2 ¯Yǫ ¯ ¯ξǫ ¯ θ1 + ¯Eǫj ¯ + ≤C j=0
i∈Iǫj
i∈Iǫj
¯ ¯ ¯ ¯¢ + |ηj |p1 θ1 ¯Eǫj ¯ + |ηj |p2 θ2 ¯Eǫj ¯
p1 −2 p1 −1
p 1−1 1 X¯ ¯ ¯ ¯ p1 i j ¯Yǫ ¯ θ1 ¯ξǫ − ηj ¯ × i∈Iǫj
X ¯ ¯ ¯ ¯ p2 ¯ ¯ X ¯ j ¯ ¯ i ¯ p1 ¯Yǫj ¯ ¯ξǫi ¯ θ2 ¯Yǫ ¯ ¯ξǫ ¯ θ1 + + ¯Eǫj ¯ + i∈Iǫj
i∈Iǫj
¯ ¯ ¯ ¯¢ p2 −2 + |ηj |p1 θ1 ¯Eǫj ¯ + |ηj |p2 θ2 ¯Eǫj ¯ p2 −1
p 1−1 2 X¯ ¯ ¯ ¯ p2 j¯ i ¯ ¯ ¯ Yǫ θ2 ξǫ − ηj × i∈Iǫj
42
Z m ·µ X ¯ j¯ ¯Fǫ ¯ + +C
p1
|Mǫ ϕ| θ1 dx +
Fǫj
j=0
¯ ¯¢ p1 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p1 −1 µ
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
=C
m X j=0
¯ ¯ X ¯Eǫj ¯ + i∈Iǫj
Z
Yǫj
¯ ¯ p1 1 p1 −1 ¯X ¯ ¯ ¯ i θ1 ¯¯ χYǫi ξǫ − ηj ¯¯ dx Fǫj ¯i∈J j ¯
i∈Iǫj
Z
Yǫj
¯ ¯
p χǫ1 (x) ¯ξǫi ¯ 1
¯ ¯
Fǫj
j=0
Z
ǫ
XZ
dx +
i∈Iǫj
dx +
XZ
Yǫj
i∈Iǫj
p1
¯ ¯ + ¯Fǫj ¯ +
Z
i∈Iǫj
|Mǫ ϕ| θ1 dx +
Fǫj
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
Yǫj
Z
Fǫj
¯ ¯p χǫ2 (x) ¯ξǫi ¯ 2 dx
χǫ1 (x) ¯ξǫi ¯
p 1−1 1
¯p − ηj ¯ 1 dx
¯ ¯p χǫ2 (x) ¯ξǫi ¯ 2 dx
XZ ×
p 1−1 2 ¯p2 ¯ i ǫ ¯ ¯ χ2 (x) ξǫ − ηj dx j
Yǫ
|Mǫ ϕ|p2 θ2 dx
¯ ¯ p1 1 p1 −1 ¯X ¯ ¯ ¯ θ1 ¯¯ χYǫi ξǫi − ηj ¯¯ dx j Fǫ ¯i∈J j ¯
Z p1 −2 ¯ ¯ ¯ ¯ ¢ p2 p1 j ¯ p1 −1 j¯ ¯ ¯ + |ηj | θ1 Fǫ + |ηj | θ2 Fǫ × µ
Yǫj
i∈Iǫj
XZ ×
¯ ¯ ¯ ¯¢ p2 −2 + |ηj |p1 θ1 ¯Eǫj ¯ + |ηj |p2 θ2 ¯Eǫj ¯ p2 −1
Z m ·µ X ¯ j¯ ¯Fǫ ¯ + +C
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ θ2 ¯¯ × χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯
p χǫ1 (x) ¯ξǫi ¯ 1
¯ ¯¢ p1 −2 ¯ ¯ + |ηj | θ1 ¯Eǫj ¯ + |ηj |p2 θ2 ¯Eǫj ¯ p1 −1 ¯ ¯ X + ¯Eǫj ¯ +
ǫ
|Mǫ ϕ|p2 θ2 dx
p1
|Mǫ ϕ|p2 θ2 dx
Fǫj
Z ×
¯ ¯¢ p2 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
Z
ǫ
|Mǫ ϕ|p2 θ2 dx
¯ ¯ ¯ ¯¢ p2 −2 + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ θ2 ¯¯ × χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z
43
ǫ
¯ ¯p1 ¯ ¯ p2 ¯X ¯ ¯X ¯ Z ¯ ¯ ¯ ¯ ǫ ǫ i i χ1 (x) ¯¯ χ2 (x) ¯¯ =C χYǫi (x)ξǫ ¯¯ dx + χYǫi (x)ξǫ ¯¯ dx Eǫj Eǫj ¯i∈I j ¯ ¯i∈I j ¯ j=0 ǫ ǫ ¯ ¯ p 1−1 p1 ¯ ¯ 1 Z ¯ j ¯¢ p1 −2 ¯ j¯ ¯X ¯ p p1 ǫ i 2 χ1 (x) ¯¯ + |ηj | θ1 ¯Eǫ ¯ + |ηj | θ2 ¯Eǫ ¯ p1 −1 × χYǫi (x)ξǫ − ηj ¯¯ dx Eǫj ¯i∈I j ¯ ¯ ¯p1 ¯ǫ ¯ p2 ¯X ¯ ¯X ¯ Z Z ¯ ¯ j¯ ¯ ¯ ¯ ǫ ǫ i i χ1 (x) ¯¯ χ2 (x) ¯¯ χYǫi (x)ξǫ ¯¯ dx + + ¯Eǫ ¯ + χYǫi (x)ξǫ ¯¯ dx Eǫj Eǫj ¯i∈I j ¯ ¯i∈I j ¯ ǫ ǫ ¯ ¯ p2 1 p2 −1 ¯ ¯ Z ¯ j ¯¢ p2 −2 ¯ j¯ ¯X ¯ p p1 ǫ i 2 p −1 χ2 (x) ¯¯ + |ηj | θ1 ¯Eǫ ¯ + |ηj | θ2 ¯Eǫ ¯ 2 × χYǫi (x)ξǫ − ηj ¯¯ dx j Eǫ ¯i∈I j ¯ ǫ ·µ Z Z m X ¯ ¯ ¯Fǫj ¯ + |Mǫ ϕ|p2 θ2 dx +C |Mǫ ϕ|p1 θ1 dx + m X
¯ ¯ ¯Eǫj ¯ +
Z
Fǫj
Fǫj
j=0
¯ ¯¢ ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ µ
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
|Mǫ ϕ|p1 θ1 dx +
¯ ¯ p1 1 p1 −1 ¯X ¯ ¯ ¯ i ¯ ¯ θ1 ¯ χYǫi ξǫ − ηj ¯ dx Fǫj ¯i∈J j ¯
Z ×
p1 −2 p1 −1
Z
Fǫj
|Mǫ ϕ|p2 θ2 dx
¯ ¯ ¯ ¯¢ p2 −2 + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
By (5.1) and (5.7), we get Z m ·µ X ¯ j¯ ¯ ¯ ≤C Eǫ +
Eǫj
j=0
+
Z
j
χǫ1 (x) |ηj |p1
µEǫ Z ¯ j¯ ¯ ¯ + Eǫ +
Eǫj
+
Z
Eǫj
+C
χǫ1 (x) |Mǫ ϕ|p1
dx +
j=0
Eǫj
¯ j¯ ¯Fǫ ¯ +
Z
Fǫj
Z
Eǫj
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ i θ2 ¯¯ × χYǫi ξǫ − ηj ¯¯ dx j Fǫ ¯i∈J j ¯
dx +
χǫ2 (x) |ηj |p2
χǫ1 (x) |Mǫ ϕ|p1 dx +
χǫ1 (x) |ηj |p1 dx +
m ·µ X
Z
ǫ
Z
Eǫj
Z
Z
Eǫj
dx
χǫ2 (x) |Mǫ ϕ|p2 dx
−2 ¶ pp1 −1 1
×
µZ
Ω
χǫ1 (x) |Mǫ ϕ
p1
− Ψ| dx
¶ p 1−1 1
χǫ2 (x) |Mǫ ϕ|p2 dx
χǫ2 (x) |ηj |p2 dx
|Mǫ ϕ|p1 θ1 dx +
ǫ
Z
Fǫj
−2 ¶ pp2 −1 2
×
µZ
Ω
|Mǫ ϕ|p2 θ2 dx
44
χǫ2 (x) |Mǫ ϕ − Ψ|p2 dx
¶ p 1−1 # 2
¯ ¯¢ p1 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p1 −1 µ
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
¯ ¯ p1 1 p1 −1 ¯X ¯ ¯ ¯ i θ1 ¯¯ χYǫi ξǫ − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z ×
|Mǫ ϕ|p2 θ2 dx
¯ ¯¢ p2 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
≤C +
m X j=0
Z
Ωj
+
+
× Ã
"Ã
|Ωj | +
µZ
Ω
Ωj
×
χǫ1 (x) |Mǫ ϕ
Ωj
χǫ1 (x) |ηj |p1 dx + χǫ1 (x) |Mǫ ϕ
|Ωj | +
Z
Z
Z
Ωj
Ω
Z
χǫ2 (x) |ηj |p2 dx
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
¶ p 1−1 1
dx
p2
p1
|Mǫ ϕ| θ1 dx +
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
Ωj
χǫ2 (x) |Mǫ ϕ − ϕ + ϕ|p2 dx
1
Z
− ϕ + ϕ| dx + χǫ2 (x) |ηj |p2
Z
−2 ! pp1 −1
p1
¯ ¯¢ p1 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p1 −1 µ
ǫ
− ϕ + ϕ| dx +
− ϕ + ϕ − Ψ| dx
Fǫj
Z
p1
− ϕ + ϕ − Ψ| dx
Ωj
Z m ·µ X ¯ j¯ ¯Fǫ ¯ + +C
¯ p2 1 ¯ p2 −1 ¯ ¯X ¯ ¯ χYǫi ξǫi − ηj ¯¯ dx θ2 ¯¯ × Fǫj ¯ ¯i∈J j
p1
dx +
χǫ2 (x) |Mǫ ϕ
j=0
Ωj
χǫ1 (x) |Mǫ ϕ
χǫ1 (x) |ηj |p1
µZ
Z
ǫ
Ωj
!
χǫ2 (x) |Mǫ ϕ − ϕ + ϕ|p2 dx
p2 −2 p2 −1
¶ p 1−1 # 2
Z
Fǫj
|Mǫ ϕ|p2 θ2 dx
¯ p1 1 ¯ p1 −1 ¯ ¯X ¯ ¯ i χYǫi ξǫ − ηj ¯¯ dx θ1 ¯¯ Fǫj ¯ ¯i∈J j
Z ×
ǫ
|Mǫ ϕ|p2 θ2 dx
¯ ¯¢ p2 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ θ2 ¯¯ × χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z
45
ǫ
≤C
"Ã
m X j=0
+
χǫ2 (x) |Mǫ ϕ
Z
χǫ2 (x) |ηj |p2
Ωj
+
+
Ωj
Z
Ωj
+
|Ωj | +
Z
Ã
|Ωj | +
Z
Ωj
× +C
Z
Ωj
Ω
− ϕ| dx +
dx
×
dx +
Z
Ωj
µZ
χǫ1 (x) |Mǫ ϕ
Ωj
Ω
χǫ1 (x) |ηj |p1 Z
χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx +
m ·µ X
¯ j¯ ¯Fǫ ¯ +
j=0
Z
Ω
|Mǫ ϕ| θ1 dx +
¯ ¯¢ p1 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p1 −1 µ
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
Z
Ωj
χǫ1 (x) |ϕ|p1 dx Z
dx +
Ωj
p1
− ϕ| dx +
Z
Ωj
Fǫj
+
χǫ2 (x) |Mǫ ϕ
m Z X
χǫ2 (x) |ηj |p2
Ωj
j=0
+
j=0
+
j=0
Z
Ω
Ωj
−2 ! pp1 −1 1
dx
χǫ1 (x) |ϕ − Ψ|p1 dx
×
− Ψ| dx
¶ p 1−1
−2 ! pp2 −1 2
dx
¶ p 1−1 # 2
ǫ
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ θ2 ¯¯ × χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z
m Z X
ǫ
j=0
µZ
Ω
Ωj
χǫ2 (x) |ϕ|p2
Ωj
χǫ1 (x) |ϕ|p1 dx
dx +
m Z X j=0
χǫ1 (x) |Mǫ ϕ − ϕ|p1 dx
¶ p 1−1 1
46
1
χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx
¯ ¯ p1 1 p1 −1 ¯X ¯ ¯ ¯ i θ1 ¯¯ χYǫi ξǫ − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z ×
j=0
− ϕ| dx +
Ωj
p1
|Mǫ ϕ|p2 θ2 dx
Ωj
p2
Ω
χǫ2 (x) |ηj |p2
Reorganizing the sums, we obtain "Ã m m Z m Z X X X p1 ǫ χ1 (x) |Mǫ ϕ − ϕ| dx + ≤C |Ωj | + m Z X
χǫ1 (x) |ϕ
Z
χǫ1 (x) |ϕ|p1 dx +
dx +
Z
Z
|Mǫ ϕ|p2 θ2 dx
¯ ¯ ¯ ¯¢ p2 −2 + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
j=0
χǫ1 (x) |ηj |p1 dx
χǫ2 (x) |ϕ − Ψ|p2 dx
p1
Fǫj
Ωj
χǫ2 (x) |ϕ|p2
− ϕ| dx + 1
Z
Z
p2
! pp1 −2 −1
p1
χǫ1 (x) |Mǫ ϕ − ϕ|p1 dx +
χǫ2 (x) |ϕ|p2
µZ
χǫ1 (x) |Mǫ ϕ
Ωj
χǫ1 (x) |ηj |p1 dx
+ +
à m X j=0 m XZ
Ωj
j=0
+
|Ωj | +
m Z X j=0
×
Ωj
µZ
Ω
m Z X
Ωj
j=0
χǫ2 (x) |Mǫ ϕ
p2
− ϕ| dx +
χǫ2 (x) |ηj |p2 dx
−2 ! pp2 −1
χǫ2 (x) |Mǫ ϕ − ϕ| dx +
m Z X j=0
Z
χǫ2 (x) |ϕ|p2
Ωj
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
dx +
Ω
p2
|Mǫ ϕ| θ1 dx +
Z
Fǫj
m Z X j=0
Ωj
χǫ1 (x) |ηj |p1 dx
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
¶ p 1−1 # 2
|Mǫ ϕ|p2 θ2 dx
¯ ¯ p1 1 p1 −1 ¯ ¯ ¯X ¯ θ1 ¯¯ χYǫi ξǫi − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z p1 −2 ¯ ¯ ¯ ¯ ¢ p1 p2 j¯ j ¯ p1 −1 ¯ ¯ + |ηj | θ1 Fǫ + |ηj | θ2 Fǫ × µ
Ωj
j=0
χǫ1 (x) |ϕ|p1 dx
χǫ2 (x) |ϕ − Ψ| dx
p1
Fǫj
j=0
− ϕ| dx +
m Z X
2
p2
Z m ·µ X ¯ j¯ ¯Fǫ ¯ + +C
p1
χǫ1 (x) |Mǫ ϕ
ǫ
|Mǫ ϕ|p2 θ2 dx
¯ ¯ p2 1 p2 −1 ¯ ¯ p2 −2 X ¯ ¯ ¯ ¯ ¯ ¯ ¢ θ2 ¯¯ χYǫi ξǫi − ηj ¯¯ dx + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1 × Fǫj ¯i∈J j ¯ ǫ ·µ Z Z Z p1 p1 ǫ ǫ ≤C |Ω| + χ1 (x) |Mǫ ϕ − ϕ| dx + χ1 (x) |ϕ| dx + χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx
Ω
Z
Ω
Ω
¯ ¯ p1 ¯ ¯p2 ! p1 −2 Z m m p1 −1 ¯X ¯ ¯X ¯ ¯ ¯ ¯ ¯ + χǫ2 (x) |ϕ|p2 dx + χǫ1 (x) ¯ χΩj (x)ηj ¯ dx + χǫ2 (x) ¯ χΩj (x)ηj ¯ dx ¯ ¯ ¯ ¯ Ω Ω Ω j=0 j=0 µZ ¶ p 1−1 Z 1 p1 p1 ǫ ǫ × χ1 (x) |Mǫ ϕ − ϕ| dx + χ1 (x) |ϕ − Ψ| dx Ω Ω µ Z Z Z p1 p1 ǫ ǫ + |Ω| + χ1 (x) |Mǫ ϕ − ϕ| dx + χ1 (x) |ϕ| dx + χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx Z
Z
Ω
Ω
Ω
¯ m ¯ p1 ¯ m ¯p2 ! p2 −2 Z p2 −1 ¯X ¯ ¯X ¯ ¯ ¯ ¯ ¯ + χǫ2 (x) |ϕ|p2 dx + χǫ1 (x) ¯ χΩj (x)ηj ¯ dx + χǫ2 (x) ¯ χΩj (x)ηj ¯ dx ¯ ¯ ¯ ¯ Ω Ω Ω j=0 j=0 µZ ¶ p 1−1 # Z 2 p2 p2 ǫ ǫ × χ2 (x) |Mǫ ϕ − ϕ| dx + χ2 (x) |ϕ − Ψ| dx Z
Z
Ω
Ω
Z m ·µ X ¯ j¯ ¯Fǫ ¯ + +C j=0
Fǫj
p1
|Mǫ ϕ| θ1 dx +
Z
Fǫj
|Mǫ ϕ|p2 θ2 dx
47
¯ ¯¢ ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ µ
¯ ¯ + ¯Fǫj ¯ +
Z
Fǫj
|Mǫ ϕ|p1 θ1 dx +
¯ ¯ p1 1 p1 −1 ¯X ¯ ¯ ¯ i θ1 ¯¯ χYǫi ξǫ − ηj ¯¯ dx Fǫj ¯i∈J j ¯
Z ×
p1 −2 p1 −1
Z
Fǫj
ǫ
|Mǫ ϕ|p2 θ2 dx
¯ ¯¢ p2 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ i θ2 ¯¯ × χYǫi ξǫ − ηj ¯¯ dx j Fǫ ¯i∈J j ¯
Z
ǫ
By (5.7), we have ·µ Z Z Z p1 p1 ǫ ǫ =C |Ω| + χ1 (x) |Mǫ ϕ − ϕ| dx + χ1 (x) |ϕ| dx + χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx Ω
+
Z
Ω
Ω
χǫ2 (x) |ϕ|p2 dx +
µZ
Z
Ω
χǫ1 (x) |Ψ|p1 dx + Z
p1
χǫ1 (x) |Mǫ ϕ
Ω
Z
Ω
χǫ2 (x) |Ψ|p2 dx
χǫ1 (x) |ϕ
−2 ¶ pp1 −1 1
¶ p 1−1 1
p1
× − ϕ| dx + − Ψ| dx Ω Ω µ Z Z Z p1 p1 ǫ ǫ + |Ω| + χ1 (x) |Mǫ ϕ − ϕ| dx + χ1 (x) |ϕ| dx + χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx Ω
+
Ω
Z
χǫ2 (x) |ϕ| dx +
×
µZ
p2
Ω
+C
Ω
Z
Ω
p1
χǫ1 (x) |Ψ| dx +
χǫ2 (x) |Mǫ ϕ − ϕ|p2 dx +
m ·µ X j=0
¯ j¯ ¯Fǫ ¯ +
Z
Fǫj
Ω
Z
Z
Ω
p1
Ω
¯ ¯ + ¯Fǫj ¯ +
Z
|Mǫ ϕ| θ1 dx +
Fǫj
|Mǫ ϕ|p1 θ1 dx +
Z
Fǫj
χǫ2 (x) |Ψ| dx
χǫ2 (x) |ϕ − Ψ|p2 dx Z
Fǫj
−2 ¶ pp2 −1 2
¶ p 1−1 # 2
|Mǫ ϕ|p2 θ2 dx
¯ ¯ p1 1 p1 −1 ¯ ¯ ¯X ¯ θ1 ¯¯ χYǫi ξǫi − ηj ¯¯ dx j Fǫ ¯i∈J j ¯
Z p −2 ¯ ¯ ¯ ¯ ¢ 1 p1 p2 j¯ j ¯ p1 −1 ¯ ¯ + |ηj | θ1 Fǫ + |ηj | θ2 Fǫ × µ
p2
ǫ
|Mǫ ϕ|p2 θ2 dx
¯ ¯¢ p2 −2 ¯ ¯ + |ηj |p1 θ1 ¯Fǫj ¯ + |ηj |p2 θ2 ¯Fǫj ¯ p2 −1
¯ ¯ p2 1 p2 −1 ¯X ¯ ¯ ¯ i θ2 ¯¯ × χYǫi ξǫ − ηj ¯¯ dx j Fǫ ¯i∈J j ¯
Z
ǫ
Since |∂Ωj | = 0 for j 6= 0, then we have that |Fǫj | → 0 as ǫ → 0 for every j = 0, 1, 2, ..., m. Also, by Property 1 of Mǫ in Remark 5.1, we have Z χǫi (x) |Mǫ ϕ − ϕ|pi dx → 0, Ω
48
as ǫ → 0, for i = 1, 2. Therefore, taking lim sup as ǫ → 0 above, we obtain (5.8). Lemma 5.6. If the microstructure is dispersed or layered, we have that ½Z ¾ pi ǫ χi (x) |pǫ (x, Mǫ ∇u)| dx ≤ C < ∞, for i = 1, 2. sup ǫ>0
Ω
Proof. Using (5.1), we have Z Z p1 ǫ χ1 (x) |pǫ (x, Mǫ ∇u)| dx + χǫ2 (x) |pǫ (x, Mǫ ∇u)|p2 dx Ω Ω Z Z p ǫ 1 = χ1 (x) |pǫ (x, Mǫ ∇u)| dx + χǫ2 (x) |pǫ (x, Mǫ ∇u)|p2 dx Ωǫ Ωǫ XZ XZ ¯ ¯ ¯ ¯p p 1 χǫ1 (x) ¯pǫ (x, ξǫi )¯ dx + = χǫ2 (x) ¯pǫ (x, ξǫi )¯ 2 dx Yǫi
i∈Iǫ
=
X ·Z
Yǫi
i∈Iǫ
Yǫi
i∈Iǫ
p χǫ1 (x) ¯pǫ (x, ξǫi )¯ 1
¯
¯
dx +
Z
Yǫi
¯ ¯p χǫ2 (x) ¯pǫ (x, ξǫi )¯ 2
dx
¸
By Lemma 5.3, Jensen’s inequality, and Theorem 4.1 we get X¡ ¯ ¯p ¯ ¯p ¢ ¯ ¯ ≤C 1 + ¯ξǫi ¯ 1 θ1 + ¯ξǫi ¯ 2 θ2 ¯Yǫi ¯ i∈Iǫ
X ¡¯ ¯ ¯ ¯p1 ¯ ¯ ¯ ¯p2 ¯ ¯¢ ¯Yǫi ¯ + ¯ξǫi ¯ θ1 ¯Yǫi ¯ + ¯ξǫi ¯ θ2 ¯Yǫi ¯ =C i∈I
´ ³ ǫ ≤ C |Ω| + kMǫ ∇ukpL1p1 (Ω) + kMǫ ∇ukpL2p2 (Ω) ´ ³ ≤ C |Ω| + k∇ukpL1p1 (Ω) + k∇ukpL2p2 (Ω) < ∞ (uniformly with respect to ǫ).
Z
Lemma 5.7. For all j = 0, ..., m, we have that |(Aǫ (x, pǫ (x, ηj )) , ∇uǫ )| dx as well as Ωj Z |(Aǫ (x, ∇uǫ ) , pǫ (x, ηj ))| dx are uniformly bounded with respect to ǫ. Ωj
Proof. Using H¨older’s Inequality, (3.4), and (3.12), we obtain Z Z |Aǫ (x, pǫ (x, ηj ))| |∇uǫ | dx |(Aǫ (x, pǫ (x, ηj )) , ∇uǫ )| dx ≤ Ωj
Ωj
à Z ≤C
Ωj
χǫ1 (x) (1 + |pǫ (x, ηj )|)p1 dx
! q1
2
≤ C, where C does not depend on ǫ. 49
+
ÃZ
Ωj
! q1 1 χǫ2 (x) (1 + |pǫ (x, ηj )|)p2 dx
The proof of the uniform boundedness of
Z
Ωj
manner.
|(Aǫ (x, ∇uǫ ) , pǫ (x, ηj ))| dx follows in the same
Lemma 5.8. As ǫ → 0, up to a subsequence, (Aǫ (·, pǫ (·, ηj )) , ∇uǫ (·)) converges weakly to a function gj ∈ L1 (Ωj ; R), for all j = 0, ..., m. In a similar way, up to a subsequence, (Aǫ (·, ∇uǫ (·)) , pǫ (·, ηj )) converges weakly to a function hj ∈ L1 (Ωj ; R), for all j = 0, ..., m. Proof. We prove the first statement of the lemma, the second statement follows in a similar way. The lemma follows from the Dunford-Pettis theorem (see [Dac89]). To apply this theorem we establish the following conditions given by: Z |(Aǫ (x, pǫ (x, ηj )) , ∇uǫ )| dx is uniformly bounded with respect to ǫ, and 1. Ωj
2. For all j = 0, ..., m, (Aǫ (·, pǫ (·, ηj )) , ∇uǫ ) is equiintegrable. The first condition is proved in Lemma 5.7. For the second condition, we have that χǫ1 (·) |Aǫ (·, pǫ (·, ηj ))|q2 and χǫ2 (·) |Aǫ (·, pǫ (·, ηj ))|q1 are equiintegrable (see for example Theorem 1.5 of [Dac89]). By (3.12), for any E ⊂ Ω, we have (µZ ( ¶ p1 )) i pi ǫ χi (x) |∇uǫ | dx max sup ≤ C. i=1,2
ǫ>0
E
1/q
1/q
Let α > 0 arbitrary and choose α1 > 0 and α2 > 0 such that α1 2 + α2 1 < α/C. For α1 and α2 , there exist λ(α1 ) > 0 and λ(α2 ) > 0 such that for every E ⊂ Ω with |E| < min {λ(α1 ), λ(α2 )}, Z Z q2 ǫ χ1 (x) |Aǫ (x, pǫ (x, ηj ))| dx < α1 , and χǫ2 (x) |Aǫ (x, pǫ (x, ηj ))|q1 dx < α2 . E
E
Take λ = λ(α) = min {λ(α1 ), λ(α2 )}. Then, for all E ⊂ Ω with |E| < λ(α), we have Z Z |(Aǫ (x, pǫ (x, ηj )) , ∇uǫ )| dx ≤ |Aǫ (x, pǫ (x, ηj ))| |∇uǫ | dx E
E
≤
µZ
E
+ ≤
χǫ1 (x) |Aǫ
µZ
(x, pǫ (x, ηj ))| dx
χǫ2 (x) |Aǫ
E 1/q2 C(α1
+
q2
1/q α2 1 )
¶ q1 µZ 2
E
q1
(x, pǫ (x, ηj ))| dx
χǫ1 (x) |∇uǫ |p1
¶ q1 µZ 1
E
< α,
χǫ2 (x) |∇uǫ |p2
for every α > 0, and so (Aǫ (·, pǫ (·, ηj )) , ∇uǫ ) is equiintegrable.
50
dx
¶ p1
dx
1
¶ p1
2
5.3
Proof of the Corrector Theorem
We are now in the position to give the proof of Theorem 5.2. Proof. Let uǫ ∈ W01,p1 (Ω) the solutions of (3.7). By (3.5), we have that Z [χǫ1 (x) |pǫ (x, Mǫ ∇u(x)) − ∇uǫ (x)|p1 + χǫ2 (x) |pǫ (x, Mǫ ∇u(x)) − ∇uǫ (x)|p2 ] dx Ω Z ≤C (Aǫ (x, pǫ (x, Mǫ ∇u(x))) − Aǫ (x, ∇uǫ (x)) , pǫ (x, Mǫ ∇u(x)) − ∇uǫ (x)) dx Ω
To prove Theorem 5.2, we show that Z (Aǫ (x, pǫ (x, Mǫ ∇u)) − Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u) − ∇uǫ ) dx Ω Z Z = (Aǫ (x, pǫ (x, Mǫ ∇u)) , pǫ (x, Mǫ ∇u)) dx − (Aǫ (x, pǫ (x, Mǫ ∇u)) , ∇uǫ ) dx Ω Ω Z Z − (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u)) dx + (Aǫ (x, ∇uǫ ) , ∇uǫ ) dx Ω
Ω
goes to 0, as ǫ → 0. This is done in four steps. In what follows, we use the following notation Z 1 i ∇u(x)dx. ξǫ = i |Yǫ | Yǫi Step 1 Let us prove that Z Z (Aǫ (x, pǫ (x, Mǫ ∇u)) , pǫ (x, Mǫ ∇u)) dx → (b(∇u), ∇u) dx, as ǫ → 0. Ω
Ω
Proof. From (3.18) and (5.1), we obtain Z (Aǫ (x, pǫ (x, Mǫ ∇u(x))) , pǫ (x, Mǫ ∇u(x))) dx Ω Z = (Aǫ (x, pǫ (x, Mǫ ∇u(x))) , pǫ (x, Mǫ ∇u(x))) dx Ωǫ X Z ³ ³ x ³ x ´´ ³ x ´´ A = ,p , ξǫi , p , ξǫi dx ǫ ǫ ǫ i i∈Iǫ Yǫ Z X ¡ ¡ ¡ ¢¢ ¡ ¢¢ A y, p y, ξǫi , p y, ξǫi dy = ǫn i∈Iǫ
=
XZ i∈Iǫ
=
Z
Ω
Ω
Y
¡ ¢ χYǫi (x) b(ξǫi ), ξǫi dx
(b(Mǫ ∇u(x)), Mǫ ∇u(x)) dx. 51
(5.13)
By (3.15) and the definition of q1 , we have Z
|b(Mǫ ∇u(x)) − b(∇u(x))|q1 dx Ω Z h p1 −2 1 ≤ C |Mǫ ∇u − ∇u| p1 −1 (1 + |Mǫ ∇u|p1 + |∇u|p1 + |Mǫ ∇u|p2 + |∇u|p2 ) p1 −1 Ω p2 −2 iq1 1 + |Mǫ ∇u − ∇u| p2 −1 (1 + |Mǫ ∇u|p1 + |∇u|p1 + |Mǫ ∇u|p2 + |∇u|p2 ) p2 −1 dx Z h p2 ≤C |Mǫ ∇u − ∇u| (p1 −1)(p2 −1) (1 + |Mǫ ∇u|p1 + |∇u|p1 Ω
p2
p2 (p1 −2)
+ |Mǫ ∇u|p2 + |∇u|p2 ) (p1 −1)(p2 −1) + |Mǫ ∇u − ∇u| (p2 −1)2 p1
p1
p2
p2
× (1 + |Mǫ ∇u| + |∇u| + |Mǫ ∇u| + |∇u| )
p2 (p2 −2) (p2 −1)2
¸
dx
By H¨older’s Inequality in the first integral with p = (p2 − 1)(p1 − 1) > 1 and in the second integral with p = (p2 − 1)2 > 1, we obtain ≤C
"µZ
Ω
× +
p2
µZ
Ω
µZ
Ω
×
|Mǫ ∇u(x) − ∇u(x)| dx p1
p2
p2
(1 + |Mǫ ∇u| + |∇u| + |Mǫ ∇u| + |∇u| )
|Mǫ ∇u(x) − ∇u(x)| dx
Ω
1 2 −1)(p1 −1)
p1
p2
µZ
¶ (p
¶
p2 (p1 −2) (p1 −1)(p2 −1)−1
dx
1 (p2 −1)2
(1 + |Mǫ ∇u|p1 + |∇u|p1 + |Mǫ ∇u|p2 + |∇u|p2 ) dx
¶ p2 (p2 −2) 2 (p2 −1)
¶ (p(p1 −1)(p 2 −1)−1 −1)(p −1) 1
2
By Theorem 4.1 and Jensen’s inequality, we have "µZ 1 ¶ (p −1)(p µZ ¶ 1 2# (p2 −1) 2 1 −1) |Mǫ ∇u − ∇u|p2 dx ≤C + |Mǫ ∇u − ∇u|p2 dx Ω
Ω
From Property 1 of Mǫ in Remark 5.1, we obtain that b(Mǫ ∇u) → b(∇u) in Lq1 (Ω; Rn ), as ǫ → 0. Now, (5.13) follows from (5.14) since Mǫ ∇u → ∇u in Lp2 (Ω; Rn ), so Z Z (Aǫ (x, pǫ (x, Mǫ ∇u)) , pǫ (x, Mǫ ∇u)) dx = (b(Mǫ ∇u(x)), Mǫ ∇u(x)) dx Ω Ω Z → (b(∇u(x)), ∇u(x)) dx, as ǫ → 0. Ω
52
(5.14)
Step 2 We now show that Z Z (Aǫ (x, pǫ (x, Mǫ ∇u)) , ∇uǫ ) dx → (b(∇u), ∇u) dx, as ǫ → 0. Ω
(5.15)
Ω
Proof. Let δ > 0. From Theorem 4.1 we have that ∇u ∈ Lp2 (Ω; Rn ) and there exists a simple function Ψ satisfying the assumptions of Lemma 5.5 such that k∇u − ΨkLp2 (Ω;Rn ) ≤ δ.
(5.16)
Let us write Z (Aǫ (x, pǫ (x, Mǫ ∇u(x))) , ∇uǫ (x)) dx Ω Z Z = (Aǫ (x, pǫ (x, Ψ)) , ∇uǫ ) dx + (Aǫ (x, pǫ (x, Mǫ ∇u)) − Aǫ (x, pǫ (x, Ψ)) , ∇uǫ ) dx. Ω
Ω
We first show that Z Z (Aǫ (x, pǫ (x, Ψ(x))) , ∇uǫ (x)) dx → (b(Ψ(x)), ∇u(x)) dx, as ǫ → 0. Ω
Ω
We have Z
Ω
(Aǫ (x, pǫ (x, Ψ(x))) , ∇uǫ (x)) dx =
m Z X
Ωj
j=0
(Aǫ (x, pǫ (x, ηj )) , ∇uǫ (x)) dx.
n q2 Now from Z (3.24), we have that Aǫ (·, pǫ (·, ηj )) ⇀ b(ηj ) ∈ L (Ωj ; R ), and by (3.17) we (Aǫ (x, pǫ (x, ηj )) , ∇ϕ) dx = 0, for ϕ ∈ W01,p1 (Ωj ). have that Ωj
Take ϕ = δuǫ , with δ ∈ C0∞ (Ωj ) to get Z Z (Aǫ (x, pǫ (x, ηj )) , (∇δ)uǫ ) dx + 0=
(Aǫ (x, pǫ (x, ηj )) , (∇uǫ )δ) dx.
Ωj
Ωj
Taking the limit as ǫ → 0, and using the fact that uǫ ⇀ u in W01,p1 (Ω) and (3.24), we have by Lemma 5.8 that Z Z (Aǫ (x, pǫ (x, ηj )) , (∇uǫ )δ) dx gj δdx = lim ǫ→0 Ω Ωj j Z (Aǫ (x, pǫ (x, ηj )) , (∇δ)uǫ ) dx = − lim ǫ→0 Ω j Z =− (b(ηj ), (∇δ)u) dx Ωj Z (b(ηj ), (∇u)δ) dx. = Ωj
53
′
Then (Aǫ (·, pǫ (·, ηj )) , ∇uǫ (·)) ⇀ (b(ηj ), ∇u) in D (Ωj ), as ǫ → 0. Therefore, we may conclude that gj = (b(ηj ), ∇u), so n Z n Z X X (b(ηj ), ∇u) dx, as ǫ → 0. (Aǫ (x, pǫ (x, ηj )) , ∇uǫ ) dx → j=0
Ωj
j=0
Ωj
Thus, we get Z Z (Aǫ (x, pǫ (x, Ψ(x))) , ∇uǫ (x)) dx → (b(Ψ(x)), ∇u(x)) dx, as ǫ → 0. Ω
Ω
On the other hand, let us estimate Z (Aǫ (x, pǫ (x, Mǫ ∇u(x))) − Aǫ (x, pǫ (x, Ψ(x))) , ∇uǫ (x)) dx. Ω
By (3.4), we have ¯Z ¯ ¯ ¯ ¯ (Aǫ (x, pǫ (x, Mǫ ∇u)) − Aǫ (x, pǫ (x, Ψ)) , ∇uǫ ) dx¯ ¯ ¯ Ω Z ≤ |Aǫ (x, pǫ (x, Mǫ ∇u)) − Aǫ (x, pǫ (x, Ψ))| |∇uǫ | dx Ω Z ≤C χǫ1 (x) |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| (1 + |pǫ (x, Mǫ Du)| + |pǫ (x, Ψ)|)p1 −2 |∇uǫ | dx Ω Z +C χǫ2 (x) |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| (1 + |pǫ (x, Mǫ Du)| + |pǫ (x, Ψ)|)p2 −2 |∇uǫ | dx Ω
Applying H¨older’s inequality we obtain ¶ p1 µZ ¶ p1 µZ 1 1 p1 p1 ǫ ǫ χ1 (x) |∇uǫ | dx ≤C χ1 (x) |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx Ω
Ω
×
µZ
Ω
χǫ1 (x) (1 + |pǫ (x, Mǫ ∇u)| + |pǫ (x, Ψ)|) dx
+C
µZ
Ω
χǫ2 (x) |pǫ
×
µZ
χǫ2 (x) (1 + |pǫ (x, Mǫ ∇u)| + |pǫ (x, Ψ)|)p2 dx
≤C
Ω
µZ
Ω
p1
χǫ1 (x) |pǫ
p2
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx
p1
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx
¶ p1p−2 1
¶ p1 µZ 2
Ω
¶ p1 µZ 1
Ω
χǫ1 (x) |∇uǫ |p1
Ω
χǫ1 (x) (1 + |pǫ (x, Mǫ ∇u)|p1 + |pǫ (x, Ψ)|p1 ) dx
+C
µZ
Ω
χǫ2 (x) |pǫ
×
µZ
χǫ2 (x) (1
Ω
p2
¶ p1 µZ 2
Ω
p2
+ |pǫ (x, Mǫ ∇u)| + |pǫ (x, Ψ)| ) dx 54
¶ p1
2
2
×
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx
dx
¶ p2p−2
µZ
p2
χǫ2 (x) |∇uǫ |p2
dx
¶ p1p−2
¶ p1
1
1
χǫ2 (x) |∇uǫ |p2 ¶ p2p−2 2
dx
¶ p1
2
By (3.12), we get ≤C
µZ
χǫ1 (x) |pǫ
Ω
µ
× 1+ +C
µZ
Ω
Z
Ω
p1
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx
µ
× 1+
Z
Ω
1
Z
χǫ1 (x) |pǫ (x, Mǫ ∇u)|p1 dx +
χǫ2 (x) |pǫ
¶ p1
Ω
χǫ1 (x) |pǫ (x, Ψ)|p1 dx
p2
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx
χǫ2 (x) |pǫ
p2
(x, Mǫ ∇u)| dx +
Z
Ω
¶ p1
¶ p1p−2 1
2
p2
χǫ2 (x) |pǫ
(x, Ψ)| dx
¶ p2p−2 2
Using (5.6) and Lemma 5.3, we get ≤C +
"µZ
Ω
µZ
Ω
χǫ1 (x) |pǫ
χǫ2 (x) |pǫ
p1
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx p2
(x, Mǫ ∇u) − pǫ (x, Ψ)| dx
¶ p1
1
¶ p1 # 2
Applying Lemma 5.5 and (5.16), we discover that ¯Z ¯ ¯ ¯ lim sup ¯¯ (Aǫ (x, pǫ (x, Mǫ ∇u)) − Aǫ (x, pǫ (x, Ψ)) , ∇uǫ ) dx¯¯ ǫ→0 Ω " µZ ¶1 ≤ lim sup C ǫ→0
+
µZ
χǫ2 (x) |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p2 dx
Ω
≤ lim sup C ǫ→0
+
" µZ
Ω
h
Ω
χǫ1 (x) |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p1 dx
q1
" µZ
Ω
χǫ1 (x) |∇u − Ψ|p1 dx
χǫ1 (x) |∇u − Ψ|p1 dx q2
≤ C (δ + δ )
1 p1
q1
q2
¶ p 1−1
+ (δ + δ )
1
1 p2
i
+
¶ p1 #
¶ p 1−1 1
µZ
Ω
,
p1
2
+
µZ
Ω
χǫ2 (x) |∇u − Ψ|p2 dx
χǫ2 (x) |∇u − Ψ|p2 dx
¶ p 1−1 # p11 2
¶ p 1−1 # p12 2
(5.17)
where C does not depend on δ. Since δ is arbitrary we conclude that the limit on the left hand side of (5.17) is equal to 0.
55
Finally, using (3.15), Theorem 4.1, and H¨older’s inequality, we obtain Z Z (b(∇u) − b(Ψ), ∇u) ≤ |b(∇u) − b(Ψ)| |∇u| dx Ω Ω " µZ ¶1# ¶ 1 µZ ≤C
≤C ≤C
Ω
µZ
|b(∇u) − b(Ψ)|q1 dx
½ZΩ h Ω
q1
|b(∇u) − b(Ψ)|q1 dx
Ω
p2
|∇u|p2 dx
¶ q1
1
p1 −2
1
|∇u − Ψ| p1 −1 (1 + |∇u|p1 + |∇u|p2 + |Ψ|p1 + |Ψ|p2 ) p1 −1 1 p2 −1
p1
p2
p1
p2
p2 −2 p2 −1
o q1
iq 1
1
dx (1 + |∇u| + |∇u| + |Ψ| + |Ψ| ) + |∇u − Ψ| ½Z · p2 p2 (p1 −2) ≤C |∇u − Ψ| (p1 −1)(p2 −1) (1 + |∇u|p1 + |∇u|p2 + |Ψ|p1 + |Ψ|p2 ) (p1 −1)(p2 −1) Ω
+ |∇u − Ψ|
p2 (p2 −1)2
p1
p2
p1
p2
(1 + |∇u| + |∇u| + |Ψ| + |Ψ| )
p2 (p2 −2) (p2 −1)2
¸
Applying H¨older’s inequality and Theorem 4.1 again, we obtain " µZ 1 ¶ (p −1)(p 1 2 −1) p2 |∇u(x) − Ψ(x)| dx ≤C
dx
¾ q1
1
Ω
× +
µZ
Ω
µZ
≤C
µZ
Ω
≤C δ
p1
Ω
q1 p1 −1
p1
p2
(1 + |∇u| + |∇u| + |Ψ| + |Ψ| )
|∇u(x) − Ψ(x)| dx
" µZ h
p2
p2
Ω
×
p1
¶
p2 (p1 −2) (p1 −1)(p2 −1)−1
1 (p2 −1)2
p2
p1
p2
(1 + |∇u| + |∇u| + |Ψ| + |Ψ| ) dx p2
|∇u(x) − Ψ(x)| dx +δ
q1 p2 −1
i q1
1
dx
¶ (p
1 1 −1)(p2 −1)
+
µZ
Ω
¶ (p2 −1)2 −1 2 (p2 −1)
¶ (p(p2 −1)(p 1 −2)−1 −1)(p −1) 1
2
q1
1
p2
|∇u(x) − Ψ(x)| dx
¶
1 (p2 −1)2
# q1
1
,
where C does not depend on δ. Now, since δ is arbitrarily small we conclude the proof of Step 2. Step 3 We will show that Z Z (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u)) dx → (b(∇u), ∇u) dx, as ǫ → 0. Ω
Ω
56
(5.18)
Proof. Let δ > 0. As in the proof of Step 2, assume Ψ is a simple function satisfying assumptions of Lemma 5.5 and such that k∇u − ΨkLp2 (Ω;Rn ) < δ. Let us write Z (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u)) dx Ω Z Z = (Aǫ (x, ∇uǫ ) , pǫ (x, Ψ)) dx + (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)) dx. Ω
Ω
We first show that Z Z (Aǫ (x, ∇uǫ (x)) , pǫ (x, Ψ(x))) dx → (b (∇u(x)) , Ψ(x)) dx. Ω
Ω
We start by writing Z
(Aǫ (x, ∇uǫ (x)) , pǫ (x, Ψ(x))) dx =
Ω
m Z X
(Aǫ (x, ∇uǫ (x)) , pǫ (x, ηj )) dx.
Ωj
j=0
From Lemma 5.8, up to a subsequence, (Aǫ (·, ∇uǫ (·)) , pǫ (·, ηj )) converges weakly to a function hj ∈ L1 (Ωj ; R), as ǫ → 0. By Theorem 3.4, we have Aǫ (·, ∇uǫ ) ⇀ b(∇u) ∈ Lq2 (Ω; Rn ) and −div (Aǫ (x, ∇uǫ )) = f = −div (b(∇u)) . Also, from (3.22), pǫ satisfies pǫ (·, ηj ) ⇀ ηj in Lp1 (Ωj , Rn ). ′ Arguing as in Step 2, we find that (Aǫ (x, ∇uǫ (x)) , pǫ (x, ηj )) ⇀ (b(∇u(x)), ηj ) in D (Ωj ), as ǫ → 0. Therefore, we may conclude that hj = (b(∇u), ηj ), and hence, n Z X j=0
Ωj
(Aǫ (x, ∇uǫ (x)) , pǫ (x, ηj )) dx →
n Z X j=0
Ωj
(b(∇u(x)), ηj ) dx, as ǫ → 0.
Thus, we get Z Z (Aǫ (x, ∇uǫ (x)) , pǫ (x, Ψ(x))) dx → (b(∇u(x)), Ψ(x)) dx, as ǫ → 0. Ω
Ω
Moreover, applying Cauchy-Schwarz inequality we have ¯Z ¯ ¯ ¯ ¯ (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)) dx¯ ¯ ¯ Ω Z ≤ |Aǫ (x, ∇uǫ (x))| |pǫ (x, Mǫ ∇u(x)) − pǫ (x, Ψ(x))| dx Ω
57
H¨older’s inequality delivers Z ≤ χǫ1 |Aǫ (x, ∇uǫ (x))| |pǫ (x, Mǫ ∇u(x)) − pǫ (x, Ψ(x))| dx Ω Z + χǫ2 |Aǫ (x, ∇uǫ (x))| |pǫ (x, Mǫ ∇u(x)) − pǫ (x, Ψ(x))| dx Ω
≤
µZ
Ω
+
q2
χǫ1 |Aǫ (x, ∇uǫ )|
µZ
χǫ2
Ω
¶ q1 µZ 2
Ω
q1
|Aǫ (x, ∇uǫ )|
≤C +
Ω
µZ
Ω
=C
Ω
+
µZ
+
" µZ
Ω
µZ
" µZ
Ω
+
µZ
Ω
(1 + |∇uǫ |)
χǫ1
χǫ1
χǫ1
χǫ2
p1
2
(1 + |∇uǫ |)
¶ q1 µZ 1
Ω
¶ q1 µZ 1
Ω
q2 (p1 −1)
(1 + |∇uǫ |)
χǫ1
χǫ1
|pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx
p1
|pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx
¶ q1 µZ 2
Ω
1
Ω
χǫ1
χǫ2 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p2 dx
1
¶ p1 i# 2
As in the proof of Step 2 we see that ¯Z ¯ ¯ ¯ lim sup ¯¯ (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)) dx¯¯ ǫ→0 Ω "µZ ¶1 +
µZ
Ω
Ω
χǫ1 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p1 dx p2
χǫ2 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx 58
¶ p1 # 2
p1
1
2
1
¶ p1 i# 2 p1
¶ p1
¶ p1
¶ p1 #
|pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx
|pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx
ǫ→0
¶ p1
χǫ2 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p2 dx p1
≤ C lim sup
2
p1
χǫ2 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p2 dx
¶ q1 µZ
¶ p1
χǫ2 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)|p2 dx
Ω
2
1
|pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx
¶ q1 µZ
¶ q1 µZ
¶ p1
p2
Ω
χǫ2 (1 + |∇uǫ |)q1 (p2 −1)
Ω
=C
1
q2 (p1 −1)
χǫ2 (1 + |∇uǫ |)p2
Ω
≤C
¶ q1 µZ
χǫ2 (1 + |∇uǫ |)q1 (p2 −1)
" µZ
χǫ1 |pǫ (x, Mǫ ∇u) − pǫ (x, Ψ)| dx
Ω
By (3.4) and (3.12) we get " µZ χǫ1
p1
¶ p1
1
¶ p1 # 2
By Lemma 5.5, we get i h 1 1 ≤ C (δ q2 + δ q1 ) p1 + (δ q2 + δ q1 ) p2 ,
where C does not depend on δ. Hence, proceeding as in Step 2, we find that ¯Z ¯ Z ¯ ¯ lim sup ¯¯ (Aǫ (x, ∇uǫ (x)) , pǫ (x, Mǫ ∇u(x))) dx − (b(∇u(x)), ∇u(x)) dx¯¯ ǫ→0 Ω ¯Z Z Ω ¯ = lim sup ¯¯ (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u)) dx − (Aǫ (x, ∇uǫ ) , pǫ (x, Ψ)) dx ǫ→0 Ω Ω Z Z + (Aǫ (x, ∇uǫ ) , pǫ (x, Ψ)) dx − (b(∇u), Ψ) dx Ω ¯ ZΩ Z ¯ + (b(∇u), Ψ) dx − (b(∇u), ∇u) dx¯¯ Ω Ω ¯Z ¯ Z ¯ ¯ ≤ lim sup ¯¯ (Aǫ (x, ∇uǫ ) , pǫ (x, Mǫ ∇u)) dx − (Aǫ (x, ∇uǫ ) , pǫ (x, Ψ)) dx¯¯ ǫ→0 ¯ΩZ ¯ Z Ω ¯ ¯ ¯ + lim sup ¯ (Aǫ (x, ∇uǫ ) , pǫ (x, Ψ)) dx − (b(∇u), Ψ) dx¯¯ ǫ→0 Ω ¯ZΩ ¯ Z ¯ ¯ ¯ + lim sup ¯ (b(∇u), Ψ) dx − (b(∇u), ∇u) dx¯¯ ǫ→0 Ω Ω ´ ³ 1 1 q1 p 2 q2 q1 p 1 q2 ≤ C (δ + δ ) + (δ + δ ) + 0 + δ kb(∇u)kLq2 (Ω,Rn ) ,
where C does not depend on δ. Now since δ is arbitrarily small, the proof of Step 3 is complete. Step 4 Finally, let us prove that Z Z (Aǫ (x, ∇uǫ (x)) , ∇uǫ (x)) dx → (b(∇u(x)), ∇u(x)) dx, as ǫ → 0. Ω
(5.19)
Ω
Proof. Since Z
Ω
and
(Aǫ (x, ∇uǫ ) , ∇uǫ ) dx = h−div (Aǫ (x, ∇uǫ )) , uǫ i = hf, uǫ i , Z
Ω
(b(∇u), ∇u) dx = h−div (b (∇u)) , ui = hf, ui ,
and uǫ ⇀ u in W 1,p1 (Ω), the result follows immediately. Finally, Theorem 5.2 follows from (5.13), (5.15), (5.18) and (5.19).
59
(5.20)
(5.21)
Chapter 6 Lower Bounds on Field Concentrations In composites, failure initiation is a multiscale phenomenon. A load applied at the structural scale is often amplified by the microstructure creating local zones of high field concentration. The regions containing high fields are often the first to suffer damage during service. Therefore it is of relevance to assess the load transfer between macroscopic and microscopic length scales. In this chapter, we bound the local singularity strength inside microstructured media in terms of the macroscopic applied fields. The strong approximations in Chapter 5 are used to develop new tools that provide lower bounds on the local gradient field intensity inside micro-structured media. These results provide a lower bound on the amplification of the macroscopic (average) gradient field by the microstructure. In [Lip06], similar lower bounds are established for field concentrations for mixtures of linear electrical conductors in the context of two scale convergence.
6.1
Statement of the Lower Bound on the Amplification of the Macroscopic Field by the Microstructure
We begin by presenting a general lower bound that holds for the composition of the sequence {χǫi ∇uǫ }ǫ>0 with any non-negative Carath´eodory function. Definition 6.1. A function ψ : Ω×Rn → R is a Carath´eodory function if ψ(x, ·) is continuous for almost every x ∈ Ω and if ψ(·, λ) is measurable in x for every λ ∈ Rn . The lower bound on the sequence obtained by the composition of ψ(x, ·) with χǫi (x)∇uǫ (x) is given by Theorem 6.2. For all Carath´eodory functions ψ ≥ 0 and measurable sets D ⊂ Ω, we have Z Z Z ψ (x, χi (y)p (y, ∇u(x))) dydx ≤ lim inf ψ (x, χǫi (x)∇uǫ (x)) dx, (i = 1, 2). D
Y
ǫ→0
60
D
If the sequence {ψ (x, χǫi (x)∇uǫ (x))}ǫ>0 is weakly convergent in L1 (Ω), then the inequality becomes an equality. Remark 6.3. As a direct consequence of Theorem 6.2, taking ψ(x, λ) = |λ|q with q ≥ 2, we display lower bounds on the Lq norm of the gradient fields inside each material that are given in terms of the correctors presented in Theorem 5.2 Z Z Z q χi (y) |p (y, ∇u(x))| dydx ≤ lim inf χǫi (x) |∇uǫ (x)|q dx, (6.1) D
ǫ→0
Y
D
for i = 1, 2. This result is still valid for q = ∞ if we have that χǫi (x)∇uǫ (x) belongs to L∞ (Ω; Rn ) (since ǫ χi (x)∇uǫ (x) belongs to Lpi (Ω; Rn ) by (3.12)) and if χi (y)p (y, ∇u(x)) belongs to L∞ (Y × Ω; Rn ) (since Z Z χi (y) |p (y, ∇u(x))|pi dydx D Y Z ≤C (1 + |∇u(x)|p1 + |∇u(x)|p2 ) dx < ∞, D
by Lemma 5.3 and Theorem 4.1).
It is clear from (6.1) that the Lq (Y × Ω; Rn ) integrability of p(y, ∇u(x)) provides a lower bound on the Lq (Ω; Rn ) integrability of ∇uǫ . Theorem 6.2 together with (6.1) provide explicit lower bounds on the gradient field inside each material. It relates the local excursions of the gradient inside each phase χǫi ∇uǫ to the average gradient ∇u through the multiscale quantity given by the corrector p(y, ∇u(x)).
6.2
Young Measures
We use the the results from Theorem 5.2 and Young Measures to study the behavior of gradients of solutions of the Dirichlet problem (3.7). These tools allow us to bound nonlinear quantities of these gradients from below in terms of the local solution p and the gradient of the homogenized solution u as stated in Section 6.1. Young Measures can be used as a tool to organize our ideas about oscillatory behavior and to deal in a consistent way with oscillations [Ped99]. Young Measures are a family of probability measures ν = {νx }x∈Ω associated with a sequence of functions f ǫ : Ω ⊂ Rn −→ Rn such that the supp(νx )⊂ Rn and they depend measurably on x ∈ Ω, which means that for any continuous function ϕ : Rn −→ R, the function Z ϕ(λ)dνx (λ) (6.2) ϕ(x) = Rn
is measurable. The fundamental property of this family of probability measures is that whenever {ϕ(f ǫ )}ǫ>0 converges weakly* in L∞ (Ω) (or more generally weak in some Lp (Ω)) the weak limit can be identified with the function ϕ in (6.2): Z Z Z ǫ lim ϕ(f )g(x)dx = ϕ(λ)dνx (λ)dx, (6.3) g(x) ǫ→0
Ω
Ω
1
for all g ∈ L (Ω).
61
Rn
6.3
Proof of Lower Bound on the Amplification of the Macroscopic Field by the Microstructure
The sequence {χǫi (x)∇uǫ (x)}ǫ>0 has a Young Measure ν i = {νxi }x∈Ω associated to it (see Theorem 6.2 and the discussion following in [Ped97]), for i = 1, 2. As a consequence of Theorem 5.2 proved in Chapter 5, we have ° ° ´ ³x ° ǫ ° , Mǫ (∇u)(x) − χǫi (x)∇uǫ (x)° p → 0, °χi (x)p ǫ L i (Ω;Rn ) as ǫ → 0 for i = 1, 2, which implies that the sequences ´o n ³x , Mǫ (∇u)(x) and {χǫi (x)∇uǫ (x)}ǫ>0 χǫi (x)p ǫ ǫ>0
share the same Young Measure ν i = {νxi }x∈Ω (see Lemma 6.3 of [Ped97]). The next Lemma identifies the Young measure ν i .
Lemma 6.4. For all φ ∈ C0 (Rn ) and for all ζ ∈ C0∞ (Rn ), we have Z Z Z Z i φ(λ)dνx (λ)dx = ζ(x) φ(χi (y)p(y, ∇u(x)))dydx. ζ(x) Ω
Rn
Ω
(6.4)
Y
(From discussion in Chapter 6 of [Ped97], this is enough to identify the Young measure ν i ) Proof. To prove (6.4), we will show that given φ ∈ C0 (Rn ) and ζ ∈ C0∞ (Rn ) Z Z Z ´´ ³ ³x ǫ , Mǫ (∇u) (x) dx = ζ(x) φ(χi (y)p(y, ∇u(x)))dydx. lim ζ(x)φ χi (x)p ǫ→0 Ω ǫ Ω Y
(6.5)
We consider the difference ¯Z ¯ Z Z ³ ³ ´ ³ ´´ ¯ ¯ ¯ ζ(x)φ χi x p x , Mǫ (∇u)(x) dx − ζ(x) φ (χi (y) p (y, ∇u(x))) dydx¯ ¯ ¯ ǫ ǫ Ω Ω Y
By (5.1), we have
¯ ¯ Z Z ¯X Z ¯ ³ ³ x ´ ³ x ´´ ¯ ¯ ≤¯ ζ(x)φ χi ζ(x) φ (χi (y) p (y, ∇u(x))) dydx¯ , ξǫi dx − p ¯ ¯ ǫ ǫ i Ωǫ Y i∈Iǫ Yǫ ¯ ¯Z Z Z ³ ³ x ´ ³ x ´´ ¯ ¯ , 0 dx − ζ(x) φ (χi (y) p (y, ∇u(x))) dydx¯¯ p + ¯¯ ζ(x)φ χi ǫ ǫ Ω\Ωǫ Y ǫ ¯ Ω\Ω ¯ Z Z ¯X Z ¯ ³ ³ x ´ ³ x ´´ ¯ ¯ i ζ(x)φ χi ≤¯ ζ(x) φ (χi (y) p (y, ∇u(x))) dydx¯ , ξǫ dx − p ¯ ¯ ǫ ǫ Yǫi Ωǫ Y i∈I ǫ
+ C |Ω \ Ωǫ | .
(6.6)
62
Note that the term C |Ω \ Ωǫ | goes to 0, as ǫ → 0. Now set xiǫ to be the center of Yǫi . On the first integral use the change of variables x = xiǫ + ǫy, where y belongs to Y , and since dx = ǫn dy, we get ¯ ¯ Z ¯ ¯X Z ³ ³ x ´ ³ x ´´ XZ ¯ ¯ i , ξǫ dx − p ζ(x) φ (χi (y) p (y, ∇u(x))) dydx¯ ζ(x)φ χi ¯ ¯ ¯ ǫ ǫ i i Y i∈Iǫ Yǫ i∈Iǫ Yǫ ¯ ¯X Z ¡ ¡ ¢¢ ¯ ζ(xiǫ + ǫy)φ χi (y) p y, ξǫi dy =¯ ǫn ¯ Y i∈Iǫ ¯ Z Z ¯ X ¯ ζ(x) φ (χi (y) p (y, ∇u(x))) dydx¯ − ¯ i Y Yǫ i∈Iǫ
Applying Taylor’s expansion for ζ, we obtain
¯ ¯X Z £ ¤ ¡ ¡ ¢¢ ¯ =¯ ǫn ζ(xiǫ ) + CO(ǫ) φ χi (y) p y, ξǫi dy ¯ Y i∈Iǫ ¯ Z Z ¯ X £ i ¤ ¯ − ζ(xǫ ) + CO(ǫ) φ (χi (y) p (y, ∇u(x))) dydx¯ ¯ i i∈Iǫ Yǫ Y ¯ ¯X Z ¡ ¡ ¢¢ ¯ ≤¯ ǫn ζ(xiǫ )φ χi (y) p y, ξǫi dy ¯ Y i∈Iǫ ¯ ¯ XZ Z ¯ − ζ(xiǫ )φ (χi (y) p (y, ∇u(x))) dydx¯ + CO(ǫ) ¯ i i∈Iǫ Yǫ Y ¯ ¯X Z Z ¡ ¡ ¢¢ ¯ =¯ ζ(xiǫ )φ χi (y) p y, ξǫi dydx ¯ i i∈Iǫ Yǫ Y ¯ ¯ XZ Z ¯ i − ζ(xǫ )φ (χi (y) p (y, ∇u(x))) dydx¯ + CO(ǫ) ¯ i i∈Iǫ Yǫ Y ¯ ¯ ¯X Z Z ¯ £ ¡ ¡ ¢¢ ¤ ¯ ¯ i i =¯ ζ(xǫ ) φ χi (y) p y, ξǫ − φ (χi (y) p (y, ∇u(x))) dydx¯ + CO(ǫ) ¯ ¯ Yǫi Y i∈Iǫ
63
Let us use a Taylor’s expansion for ζ again ¯ ¯X Z Z £ ¡ ¡ ¢¢ ¯ ≤¯ (ζ(x) + CO(ǫ)) φ χi (y) p y, ξǫi ¯ Yǫi Y i∈Iǫ
−φ (χi (y) p (y, ∇u(x)))] dydx| + CO(ǫ) ¯ ¯ ¯X Z Z ¯ £ ¡ ¡ ¢¢ ¤ ¯ ¯ ≤¯ ζ(x) φ χi (y) p y, ξǫi − φ (χi (y) p (y, ∇u(x))) dydx¯ + CO(ǫ) ¯ ¯ i Yǫ Y i∈I ¯Z ǫ ¯ Z ¯ ¯ = ¯¯ ζ(x) [φ (χi (y) p (y, Mǫ ∇u(x))) − φ (χi (y) p (y, ∇u(x)))] dydx¯¯ + CO(ǫ) Y ¯ZΩǫ ¯ Z ¯ ¯ ≤ ¯¯ |ζ(x)| |φ (χi (y) p (y, Mǫ ∇u(x))) − φ (χi (y) p (y, ∇u(x)))| dydx¯¯ + CO(ǫ) Ωǫ
Y
Because of the uniform Lipschitz continuity of φ, we get ¯Z ¯ ≤ C ¯¯
Ωǫ
|ζ(x)|
Z
Y
¯ ¯ |p (y, Mǫ ∇u(x)) − p (y, ∇u(x))| dydx¯¯ + CO(ǫ)
By H¨older’s inequality we have
¯Z " µZ ¶1/p1 ¯ ¯ p1 ≤C¯ χ1 (y) |p (y, Mǫ ∇u(x)) − p (y, ∇u(x))| dy |ζ(x)| ¯ Ωǫ Y µZ ¶1/p2 # ¯¯ ¯ + χ2 (y) |p (y, Mǫ ∇u(x)) − p (y, ∇u(x))|p2 dy dx¯ + CO(ǫ) ¯ Y
Using Lemma 5.4, we obtain ¯Z ¯ ≤ C ¯¯
Ωǫ
|ζ(x)| {[(1 + |Mǫ ∇u(x)|p1 θ1 + |Mǫ ∇u(x)|p2 θ2 + |∇u(x)|p1 θ1 p1
p1 −2
1
+ |∇u(x)|p2 θ2 ) p1 −1 |Mǫ ∇u(x) − ∇u(x)| p1 −1 θ1p1 −1 p2
1
+ |Mǫ ∇u(x) − ∇u(x)| p2 −1 θ2p2 −1 (1 + |Mǫ ∇u(x)|p1 θ1 + |Mǫ ∇u(x)|p2 θ2 p2 −2 i1/p1 p1 p2 p2 −1 + |∇u(x)| θ1 + |∇u(x)| θ2 ) + [(1 + |Mǫ ∇u(x)|p1 θ1 + |Mǫ ∇u(x)|p2 θ2 + |∇u(x)|p1 θ1 p1 −2
p1
1
+ |∇u(x)|p2 θ2 ) p1 −1 |Mǫ ∇u(x) − ∇u(x)| p1 −1 θ1p1 −1 p2
1
+ |Mǫ ∇u(x) − ∇u(x)| p2 −1 θ2p2 −1 (1 + |Mǫ ∇u(x)|p1 θ1 + |Mǫ ∇u(x)|p2 θ2 ¾ ¯ p2 −2 i1/p2 ¯ p1 p2 dx¯¯ + CO(ǫ) + |∇u(x)| θ1 + |∇u(x)| θ2 ) p2 −1 64
By H¨older’s inequality we get ≤C
(µZ
q2
Ωǫ
|ζ(x)| dx
¶1/q2 ·Z
Ωǫ
³
p1
|Mǫ ∇u(x) − ∇u(x)| p1 −1 p1 −2
× (1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) p1 −1 p2
+ |Mǫ ∇u(x) − ∇u(x)| p2 −1
i1/p1 p2 −2 ´ × (1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) p2 −1 dx µZ ¶1/q1 ·Z ³ p1 q1 + |ζ(x)| dx |Mǫ ∇u(x) − ∇u(x)| p1 −1 Ωǫ
Ωǫ
p1 −2
p1
× (1 + |Mǫ ∇u(x)| + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) p1 −1 p2
+ |Mǫ ∇u(x) − ∇u(x)| p2 −1 p1
p2
p1
p2
× (1 + |Mǫ ∇u(x)| + |Mǫ ∇u(x)| + |∇u(x)| + |∇u(x)| )
p2 −2 p2 −1
+ CO(ǫ).
´
dx
i1/p2 ¾
From the fact that ζ ∈ C0∞ (Rn ), we have ≤C
½·Z
Ωǫ
p1
|Mǫ ∇u(x) − ∇u(x)| p1 −1 p1 −2
× (1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) p1 −1 dx Z p2 + |Mǫ ∇u(x) − ∇u(x)| p2 −1 Ωǫ
p2 −2
× (1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) p2 −1 dx ·Z p1 + |Mǫ ∇u(x) − ∇u(x)| p1 −1
i1/p1
Ωǫ
p1 −2
× (1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) p1 −1 dx Z p2 + |Mǫ ∇u(x) − ∇u(x)| p2 −1 Ωǫ
p1
p2
p1
p2
× (1 + |Mǫ ∇u(x)| + |Mǫ ∇u(x)| + |∇u(x)| + |∇u(x)| ) + CO(ǫ)
65
p2 −2 p2 −1
dx
i1/p2 ¾
Applying H¨older’s Inequality again, we get ≤C
½·µZ
Ωǫ
(1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1
p2
+ |∇u(x)| ) dx) +
µZ
p1 −2 p1 −1
µZ
Ωǫ
p1
|Mǫ ∇u(x) − ∇u(x)| dx p2
Ωǫ
|Mǫ ∇u(x) − ∇u(x)| dx
¶ p 1−1 µZ 2
Ωǫ
¶ p 1−1 1
(1 + |Mǫ ∇u(x)|p1
p2 −2 i1/p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1 + |∇u(x)|p2 ) dx) p2 −1 ·µZ + (1 + |Mǫ ∇u(x)|p1 + |Mǫ ∇u(x)|p2 + |∇u(x)|p1
Ωǫ
p2
+ |∇u(x)| ) dx) µZ
p1 −2 p1 −1
µZ
Ωǫ
p1
|Mǫ ∇u(x) − ∇u(x)| dx ¶ p 1−1 µZ
¶ p 1−1 1
dx
2
(1 + |Mǫ ∇u(x)|p1 Ωǫ Ωǫ ¾ p2 −2 i1/p2 p2 p1 p2 p2 −1 + CO(ǫ) + |Mǫ ∇u(x)| + |∇u(x)| + |∇u(x)| ) dx)
+
p2
|Mǫ ∇u(x) − ∇u(x)| dx
From Jensen’s inequality and Theorem 4.1, we obtain ≤C
("µZ
Ωǫ
+
µZ
+
" µZ
|Mǫ ∇u(x) − ∇u(x)| dx p2
Ωǫ
|Mǫ ∇u(x) − ∇u(x)| dx
µZ
Ωǫ
¶ p 1−1 1
¶ p 1−1 #1/p1
p1
Ωǫ
+
p1
2
|Mǫ ∇u(x) − ∇u(x)| dx
|Mǫ ∇u(x) − ∇u(x)|p2 dx
¶ p 1−1 1
dx
¶ p 1−1 #1/p2 2
+ CO(ǫ).
Finally, taking ǫ → 0 together with Property 1 of Mǫ in Remark 5.1, we obtain (6.5).
66
Therefore, from Proposition 4.4 of [Ped99] and Lemma 6.4 we have Z Z φ(λ)dνxi (λ)dx ζ(x) n Ω Z R Z = ζ(x) φ(χi (y)p(y, ∇u(x)))dydx Ω Y Z ´´ ³ ³x , Mǫ (∇u)(x) dx = lim ζ(x)φ χǫi (x)p ǫ→0 Ω ǫ Z ≤ lim ζ(x)φ (χǫi (x)∇uǫ (x)) dx, ǫ→0
Ω
for all φ ∈ C0 (Rn ) and for all ζ ∈ C0∞ (Rn ). The proof of Theorem 6.2 follows from Lemma 6.4 and Theorem 6.11 in [Ped97].
67
Chapter 7 Nonlinear Neutral Inclusions A neutral inclusion, when inserted in a matrix containing a uniform applied electric field, does not disturb the field outside the inclusion. The problem of finding neutral inclusions goes back to 1953 when Mansfield found that certain reinforced holes, which he called “neutral holes”, could be cut out of a uniformly stressed plate without disturbing the surrounding stress field in the plate [Man53]. The analogous problem of a “neutral elastic inhomogeneity” in which the introduction of the inhomogeneity into an elastic body (of a different material), does not disturb the original stress field in the uncut body, was first studied by Ru [Ru98]. The well known Hashin coated sphere is an example of a neutral coated inclusion [Has62]. Neutral spherical inclusions have been studied in [TR95],[LV96],[Lip97a], [Lip97b] and [LT99]. For more information on neutral coated inclusions see [Mil02]. In this chapter, we consider the problem of constructing neutral inclusions from nonlinear materials. We study the design of (double coated nonlinear) neutral inclusions that do not disturb the prescribed uniform applied electric field in the surrounding body.
7.1 7.1.1
Double Coated Nonlinear Neutral Inclusions Statement of the Problem and Result ~n
Middle
c
a
Core
x¯ b Outer
u = Ex1
Figure 7.1: Neutral Inclusion For a particular three phase coated sphere (see Figure 7.1), we apply the linear field ~ ~ = E e~1 , with e~1 = (1, 0) and ~x = (x1 , x2 )) as a boundary condition to E · ~x = Ex1 , (where E 68
the exterior boundary of the outer coating, to find that u = Ex1 solves 2−p1 σ 2−p2 ∆p1 u = 0 (nonlinear) in the core, ∆u = 0 (linear) in the middle coating, σ∆p2 u = 0 (nonlinear) in the outer coating,
(7.1)
where σ > 0 and ∆p represents the p-Laplacian, together with the usual interface conditions 1 (continuity of the electric potential and normal component of the current), for |E| = σ 2−p2 . Our calculations in Section 7.2 show that we can replace the three-phase coated disk with a disk composed only of linear material of conductivity one. One could continue to add coated disks of various sizes (see Figure 7.2) without disturbing the prescribed uniform applied electric field surrounding the inclusions. In fact, we can fill the space with these coated disks. The space filling configuration of triple coated disks can be solved explicitly 1 and the field inside it is precisely u = Ex1 when |E| = σ 2−p2 . This configuration of nonlinear materials dissipates energy the same as a linear material with thermal conductivity one. Middle
Core
Outer
Figure 7.2: Space Filling Configuration of Coated Disks
7.2
Calculations
Let x¯ be the center of the disk (see Figure 7.1). Inside the disk, we ask that σ1 ∆p1 u = 0 0 < |x − x¯| < a σ3 ∆u = 0 a < |x − x¯| < b σ2 ∆p2 u = 0 b < |x − x¯| < c,
(7.2)
where σ1 , σ2 , and σ3 are positive constants. x−¯ x . Have r = |x − x¯| and ~n = e~r = |x−¯ x| The solution u of (7.2) is such that
u continuous across |x − x¯| = a,
(7.3)
u continuous across |x − x¯| = b,
(7.4)
u = Ex1 at |x − x¯| = c,
(7.5)
69
and satisfies the transmission conditions and
σ1~n · |∇u|p1 −2 ∇u = σ3~n · ∇u, across |x − x¯| = a,
(7.6)
σ3~n · ∇u = σ2~n · |∇u|p2 −2 ∇u, across |x − x¯| = b.
(7.7)
We look for solution u of the form u = c1 r cos θ u = cr3 cos θ + c4 r cos θ u = c2 r cos θ
for 0 < r < a, for a < r < b, for b < r < c.
(7.8)
It is easily seen that (7.8) satisfies (7.2). In what follows, we explain how the unknowns c1 , c2 , c3 , and c4 are determined from (7.3), (7.4), (7.5), (7.6), and (7.7). First, we look at the conditions u must satisfy when r = a: By (7.3), we have c3 c1 a cos θ = cos θ + c4 a cos θ a c3 + c4 a ⇒ c1 a = a ⇒ c 1 a2 = c 3 + c 4 a2 ,
and from (7.6), we obtain
σ1 e~r · |∇u|p1 −2 ∇u = σ3 e~r · ∇u p1 −2
⇒ σ1 e~r · |c1 |
c1 (~ er cos θ − e~θ sin θ) = σ3
µ
−c3 cos θ + c4 cos θ a2
−σ3 c3 cos θ + σ3 c4 cos θ a2 = −σ3 c3 + a2 σ3 c4 .
(7.9)
¶
⇒ σ1 (sign(c1 )) |c1 |p1 −1 cos θ =
⇒ a2 σ1 (sign(c1 )) |c1 |p1 −1
(7.10)
Second, we look at the conditions u must satisfy when r = b: By (7.4), we have
and from (7.7), we obtain
c3 cos θ + c4 b cos θ = c2 b cos θ b c3 ⇒ + c4 b = c2 b b ⇒ c 3 + c 4 b2 = c 2 b2 ,
σ3 e~r · ∇u = σ2 e~r · |∇u|p2 −2 ∇u ¶ µ −c3 cos θ + c4 cos θ = σ2 (sign(c2 )) |c2 |p2 −1 cos θ ⇒ σ3 2 b ⇒ −σ3 c3 + b2 σ3 c4 = b2 σ2 (sign(c2 )) |c2 |p2 −1 . 70
(7.11)
(7.12)
Finally, we look at the conditions u must satisfy when r = c: By (7.5) we have c2 c cos θ = Ec cos θ ⇒ c2 = E.
(7.13)
Using (7.13), we can rewrite equations (7.11) and (7.12) in the following way: c3 + c4 b2 = Eb2 ,
(7.14)
−σ3 c3 + b2 σ3 c4 = b2 σ2 (sign(E)) |E|p2 −1 .
(7.15)
Now we use (7.9), (7.10), (7.14), and (7.15) to determine the unknowns c1 , c3 , and c4 (c2 = E by (7.13)). From (7.15), we have c 3 = b 2 c 4 − b2
σ2 (sign(E)) |E|p2 −1 , σ3
(7.16)
and (7.14) implies that c3 = Eb2 − c4 b2 .
(7.17)
If we combine (7.16) and (7.17), we obtain Eb2 − c4 b2 = b2 c4 − b2
σ2 (sign(E)) |E|p2 −1 σ3
σ2 (sign(E)) |E|p2 −1 σ3 ³ ´ E + (sign(E)) |E|p2 −1 σσ32 ⇒ c4 = , 2
⇒ 2c4 b2 = Eb2 + b2
(7.18)
and this way we find c4 . Evaluating c4 (given by (7.18)) in (7.17), we find c3 : 2
c3 = Eb − ⇒ c3 = ⇒ c3 =
E + (sign(E)) |E|p2 −1 2
³ ´ σ2 σ3
2Eb2 − Eb2 − (sign(E)) |E|p2 −1 2 ³ ´ p2 −1 σ2 2 Eb − (sign(E)) |E| b2 σ3 2
71
b2
³ ´ σ2 σ3
.
b2
(7.19)
Finally, to obtain c1 , we evaluate (7.18) and (7.19) in (7.9): ³ ´ ³ ´ E + (sign(E)) |E|p2 −1 σσ23 Eb2 − (sign(E)) |E|p2 −1 σσ23 b2 + a2 c 1 a2 = 2 2 ³ ´ E(a2 + b2 ) + (sign(E)) |E|p2 −1 σσ23 (a2 − b2 ) ⇒ c 1 a2 = 2 ³ ´³ ³ ¡ b ¢2 ´ ¡ b ¢2 ´ p2 −1 σ2 + (sign(E)) |E| E 1+ a 1− a σ3 . (7.20) ⇒ c1 = 2 Therefore the values of c1 , c2 , c3 , and c4 are given by (7.20), (7.13), (7.19), and (7.18), respectively. These values must also satisfy (7.10). Note that in (7.10), using (7.19) and (7.18), we have a2 σ1 (sign(c1 )) |c1 |p1 −1 = −σ3 c3 + a2 σ3 c4 ³ ´ ³ ´ Eb2 − (sign(E)) |E|p2 −1 σσ32 b2 E + (sign(E)) |E|p2 −1 σσ32 + a2 σ3 = −σ3 2 2 −σ3 Eb2 + (sign(E)) |E|p2 −1 σ2 b2 + σ3 Ea2 + (sign(E)) |E|p2 −1 σ2 a2 = 2 σ3 E(a2 − b2 ) + (sign(E)) |E|p2 −1 σ2 (a2 + b2 ) . = 2
Therefore, using (7.20) above, we get ´ ³ ´³ ¯ ³ ¡ b ¢2 ´ ¯¯p1 −1 ¯ E 1 + ¡ b ¢2 + (sign(E)) |E|p2 −1 σ2 1− a ¯ ¯ a σ3 ¯ σ1 (sign(c1 )) ¯¯ ¯ 2 ¯ ¯ ´ ´ ³ ³ ¡ ¢2 ¡ ¢2 + (sign(E)) |E|p2 −1 σ2 1 + ab σ3 E 1 − ab . = 2
(7.21)
We have that (7.21) is verified if σ3 = 1 σ2 = |E|2−p2 σ1 = |E|2−p1 2−p1
If we set σ = σ2 = |E|2−p2 , then σ1 = σ 2−p2 . This values of σ1 , σ2 , and σ3 correspond to 1 the coefficients in (7.1). We also obtain that |E| = σ 2−p2 . Here it can be checked that c1 = c2 = c4 = E and c3 = 0. Therefore u = Er cos θ is the solution to (7.1). 72
At r = c, we have u = Ec cos θ and σ2 e~r · |∇u|p2 −2 ∇u = |E|2−p2 |c2 |p2 −1 (sign(c2 )) cos θ = |E|2−p2 |E|p2 −1 (sign(E)) cos θ = |E| (sign(E)) cos θ = hE cos θ,
with h = 1 which does not depend on a, b, or c, which means that it is independent of scale.
73
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Vita Silvia Elena Jim´enez Bola˜ nos was born in July 1980, in Alajuela, Costa Rica. In August 2002 she finished her undergraduate studies in mathematics at the Universidad de Costa Rica, Costa Rica. In August 2004 she came to Louisiana State University to pursue graduate studies in mathematics and earned a master of science in mathematics in 2006. She is currently a candidate for the degree of Doctor of Philosophy in mathematics, which will be awarded in August 2010.
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