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(8) Some Developments of Homogenization Theory and Rothe’s Method. Licentiate Dissertation by Johan Dasht Department of Mathematics, Luleå University of Technology, 2005..

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(10) Some Developments of Homogenization Theory and Rothe’s Method. Licentiate Dissertation by Johan Dasht Department of Mathematics, Luleå University of Technology, 2005.. Abstract: This thesis is devoted to homogenization theory and some generalizations of Rothe’s method to non-cylindrical domains. It consists of two introductory chapters and four papers. Chapters 1 and 2 serve as a self-contained overview to the theory of homogenization. In the introduction we present the idea behind homogenization theory and in the second chapter some further results in homogenization theory are presented and some of the homogenization results from chapter 1 are proved. Paper A deals with a numerical study of stochastic homogenization and paper B deals with some generalizations of Rothe’s method to non-cylindrical domains. Paper C is devoted to numerical analysis of the convergence in homogenization of composites and finally the degeneracy in stochastic homogenization is considered in the paper D..

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(12) Preface This thesis consists of two chapters in homogenization theory and four papers, where the present author has made contributions. In the first chapter we present some of the main ideas in the theory of homogenization as a self-contained introduction to the subject. In the second chapter we consider dierent tools and theorems in homogenization (both under periodic and stochastic settings) to study the asymptotic behavior of the solutions of partial dierential equations. The references for chapter 1 and chapter 2 are presented in a unified way after chapter 2. Chapter 1 J. Dasht, Introduction to Homogenization Theory, Department of Mathematics, Luleå University of Technology, Sweden, 2005. Chapter 2 J. Dasht, Some Results in Homogenization Theory, Department of Mathematics, Luleå University of Technology, Sweden, 2005. Here we shortly present the content of the four papers included in this thesis. In paper A we present and discuss some new aspects of stochastic homogenization. In particular, we compare some dierent methods for computing the homogenized (or eective) conductivity of composite materials. In paper B we introduce and present a new generalization of Rothe’s method to non-cylindrical domains. Two methods to handle this problem are presented and applied. In papers C and D we present a numerical study of the convergence in homogenization of composites and the degeneracy in stochastic homogenization, respectively. A J. Dasht, J. Byström and P. Wall, A Numerical Study of Stochastic Homogenization, Journal of Analysis and Applications, Vol. 2, No. 3, 159-171, 2004. B J. Dasht, J. Engström, A. Kufner and L.-E. Persson, Rothe’s method for parabolic equations on non-cylindrical domains, Department of mathematics, Submitted, 2005. C J. Dasht, J. Engström and P. Wall, Numerical Analysis of the Convergence in Homogenization of Composites, Proceedings of the International Conference on Composites Engineering ICCE/9 (Ed: David Hui), San Diego, 2002. D J. Dasht, J. Engström and L.-E. Persson, Degeneracy in Stochastic Homogenization, Proceedings of the International Conference on Composites Engineering ICCE/10 (Ed: David Hui), New Orleans, 2003..

(13) Acknowledgments: I want to express my gratitude to my main supervisor Professor Lars-Erik Persson for all his support and encouragement during my time as Ph.D. student, it has meant a lot to me. I also want to thank my second supervisor Associate Professor Peter Wall and my co-authors Licentiate Jonas Engström, Dr. Johan Byström, Professor Lars-Erik Persson, Professor Alois Kufner and Associate Professor Peter Wall and all the colleagues at the Department of Mathematics. Moreover, I am grateful to the Swedish Research Council for their financial support and the Wallenberg Foundations for travel and conference grants..

(14) Chapter 1 J. Dasht, Introduction to Homogenization Theory, Department of Mathematics, Luleå University of Technology, Sweden, 2005..

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(16) Introduction to Homogenization Theory. In this chapter we present and give an introduction to the mathematical theory of homogenization. The theory of homogenization with start in the late sixties has been developed during the last decades and it is now a well established discipline in mathematics. It has applications in dierent fields, e.g. composite engineering, material science, geophysics, fluid mechanics, elasticity etc. Composite materials contain two or more finely mixed constituents. They have in general ’better’ properties than the properties of their individual constituents. Two separate scales characterize these materials, the microscopic one, describing the heterogeneities, and the macroscopic one, describing the general behavior of the composite. The aim of homogenization in this case is to give the macroscopic properties of the composite by taking into account the properties of the microscopic structure.. 1. A model problem in homogenization. To obtain a picture of the field of homogenization and the idea behind it, we start by model problem. Consider a homogeneous body occupying T (a bounded subset of R2 ) with the thermal conductivity described by the constant matrix D. If i represents the heat source and zero temperature on the boundary CT of the body, then the temperature x({) at the point { 5 T satisfies the following Dirichlet problem: ; ? div(DGx({)) = i ({) in T, (1) = x = 0 on CT, where Gx denotes the gradient of x. This is a classical elliptic boundary value problem and if i is su!ciently smooth, it admits a unique solution x.. 1.

(17) Q1. Q2. Figure 1: T consisting of two dierent constituents, T1 and T2 . Let us now consider a two-phase material, i.e. a body containing two constituents, one occupying T1 and the second T2 , with T1 _T2 = > and T = T1 ^ T2 ^ (CT1 _ CT2 ), see Figure 1. Suppose that the thermal conductivity of the body occupying T1 is D1 and that of the body occupying T2 is D2 . Then the corresponding equation to (1) is ; div (D({)Gx({)) = i({) in T1 ^ T2 , A A A A A A A A ? x = 0 on CT, (2) A A x = x on CT _ CT , A 1 2 1 2 A A A A A = D1 Gx1 · q1 = D2 Gx2 · q2 , with x({) = xl ({) on Tl and ql is the outward normal unit vector to CTl for l = 1> 2 and q1 = q2 . Note that from (2), it follows that the gradient of x is discontinuous, and in general the flux t({) = DGx is not dierentiable. Hence one can not expect solutions of class F 2 . The question is in which function space one can have a solution if we reformulate this problem. Let K be an appropriate Sobolev space taking into account the boundary conditions on x. Then we should consider equation (2) in the weak sense: ; ? div(D({)Gx({)) = i ({) in T, (3) = x 5 K, which should be understood in the following sense (where (·> ·) denotes the. 2.

(18) Figure 2: T1 periodically distributed in T2 . scalar product on Rq ) Z ; Z A A (D({)Gx> Gy) g{ = ? T. A A =. i y g{, ; y 5 K, T. (4). x 5 K.. Consider a two-phase material but assume this time that one material is distributed periodically in the other (i.e. a periodic composite material), see Figure 2. This is a realistic assumption for a large class of applications. This periodicity can be represented by a small parameter %. In mathematical models of microscopically heterogeneous materials, various local characteristics are usually described by maps of the form D% ({) = D({@%). Thus we assume that D% has this form and D is a given periodic function of period \ , hence it follows from (4) that ; Z Z A A (D% Gx% > Gy) g{ = i y g{, ; y 5 K, ? T T (5) A A = x% 5 K. Observe that two scales characterize problem (5), the macroscopic variable { and the microscopic one {@%, describing the micro oscillations. The rapid oscillation of D% makes this problem di!cult to treat, in particular from the numerical point of view. Letting % $ 0 is the mathematical homogenization of problem (5) since this makes the heterogeneities smaller and smaller, which means that we homogenize the mixture. Answering the following questions are some of the 3.

(19) Figure 3: A periodic two-phase composite. important part of this introduction and the mathematical theory of homogenization: Does the temperature x% converge to some limit function x0 ? If that is true, does x0 solve some limit boundary value problem? Let us now describe the main idea behind the theory of homogenization. Assume that D satisfies suitable conditions such that ; ? div (DGx) = i in T, =. x 5 Z01>2 (T) ,. has a unique solution. We note that Z01>2 (T) is the usual Sobolev space, i.e. x 5 O2 (T), Gx 5 O2 (T; Rq ) and x vanishes at the boundary. Consider again a periodic two phase composite material, where one material is periodically distributed in the other. Furthermore, assume that the underlying periodic inclusion are microscopic with respect to the overall body T. This periodicity can be used to divide T into periodic cells \ , see Figure 3. We emphasize that the periodicity of the microstructure of a given material can be described in dierent ways, i.e. with dierent periodic cells. Mathematically this is described by the map of the form D% ({) = D ({@%), where D (·) is \ -periodic and % is a sequence of positive numbers converging to zero, with other words % is the parameter that varies the frequencies of D= Hence we obtain ; ? div (D% Gx% ) = i in T, (6) = 1>2 x% 5 Z0 (T) , 4.

(20) Figure 4: The microstructure of a composite for dierent %. and we have a sequence of problems. The microstructure becomes finer and finer as % tends to zero and one natural question is, what happens to the sequence of solutions x% , i.e. will they converge in some sense to a function x0 and if so, is x0 the solution of the homogenized equation ; ? div (D0 Gx0 ) = i in T, (7) = x0 5 Z01>2 (T) . Another central question in homogenization is to find a way to obtain the homogenized matrix D0 . We note that even though the material is strongly heterogeneous, in a macroscopically level it appears as a homogeneous material, see Figure 4. The homogenized matrix D0 describes the eective property e.g. the eective heat conductivity. The convergence of partial dierential operators of the above type is an important special case of J-convergence of monotone operators, see [7], [14], [25] and [42]. The questions discussed in this section are some of the main ideas in the theory of homogenization. In the next section we explore these questions in more detail with a one dimensional problem.. 2. An example in one dimension. In this section we have a simple and concrete example in one dimension. Let T =]g1 > g2 [ be an interval in R, i 5 O2 (T), % a sequence of positive real numbers converging to zero and let d% ({) = d ({@%), where d : R $ R is a. 5.

(21) measurable \ -periodic function satisfying 0 ? d ({) ? 4 a.e. on R,. (8). where \ is the unit cube. Consider the Dirichlet boundary value problem ; µ ¶ g gx % A A d% ({) ({) = i in T, ? g{ g{ (9) A A = x% 5 Z01>2 (T) . or equivalently ¶ ; Z µ Z gx% gy A A d% > g{ = ? g{ g{ T A A = x% 5 Z01>2 (T) .. iy g{, ; y 5 Z01>2 (T) , T. (10). For every % A 0, equation (9), is an example of the stationary heat problem in a one dimensional %\ -periodic medium. By standard existence theory for every fixed % there exists a unique solution x% 5 Z01>2 (T) of problem (10). We shall construct a second order homogenized problem such that x% converges to x0 , where x0 is the solution to ; µ ¶ g gx 0 A A d0 ({) = i in T, ? g{ g{ (11) A A = x0 5 Z01>2 (T) , We start by proving that x% is bounded in Z01>2 (T) to obtain a convergent subsequence. Take y = x% in (10). By using Hölder and the Poincaré inequality we obtain that ° ° µ ¶2 Z ° gx% °2 gx% ° ° ° d% g{ = g{ °O2 (T;Rq ) g{ T Z. Thus kx% kZ 1>2 (T) 0. ° ° ° gx% ° ° i x% g{ n ki kO2 (T) ° . ° g{ ° 2 T O (T;Rq ). ° ° ° gx% ° Fn ° F° ki kO2 (T) = F1 ki kO2 (T) , ° g{ ° 2 O (T;Rq ) 6.

(22) where F1 is a positive constant that only depend on T. Hence the sequence x% is bounded in Z01>2 (T). Since Z01>2 (T) is a reflexive Banach space, there exists a subsequence, still denoted by x% and there exists x0 5 Z01>2 (T), such that x% - x0 in Z01>2 (T) . (12) W. Since d is a periodic function, we have Rd% - M (d) in O" (T) and hence weakly in O2 (T), where M (d) = 1@|\ | \ d g| is the average of d over the cell \ . Moreover, since d% 5 O" (T) and M (d) is a constant, we have that d% > M (d) 5 O2 (T). Next, define gx% ({) , (13) % ({) = d% ({) g{ which by (9) satisfies g % = i in T. (14) g{ Also ÃZ ¯ ! 12 ¯ ° ° ¯ gx% ¯2 ° gx% ° ¯d% ¯ ° k % kO2 (T) = ° ° g{ ° 2 F2 ki kO2 (T) . ¯ g{ ¯ g{ T O (T) Hence % is bounded in O2 (T). Now by using equation (14), we have ³ ´ 12 ³ ´ 12 2 2 2 2 k % kZ 1>2 (T) = k % kO2 (T) + ki kO2 (T) F2 ki kO2 (T) + ki kO2 (T) F3 ,. where F3 is a positive constant. Thus % is bounded in Z 1>2 (T), and hence % has a convergent subsequence, still denoted by % , which is weakly convergent in Z 1>2 (T). The injection Z 1>2 (T) /$ O2 (T) is compact (Rellich imbedding Theorem), hence the weak convergence of % implies that the sequence % converges strongly in O2 (T), i.e. % $ 0 in O2 (T). Note that assumption (8) implies that 1@d% is bounded in O" (T) and M (1@d) 6= 0, therefore W 1@d% - M (1@d) in O" (T), and hence weakly in O2 (T). Consequently µ ¶ 1 1 % - M in O2 (T) . (15) d% d 0 But (12), (13) and (15) imply that µ ¶ gx0 1 =M . g{ d 0 7. (16).

(23) Figure 5: The set of all possible eigenvalues from Reuss-Voigt bounds. Since g % @g{ = i and % $ 0 = (1@M (1@d)) gx0 @g{, we get that g 0 @g{ = i . Thus x0 is the unique solution of the Dirichlet boundary value problem ; µ ¶ g gx 0 A 1 A = i in T, ? g{ M( d1 ) g{ (17) A A = x0 5 Z01>2 (T) . In this case, the homogenized coe!cient d0 is related to the harmonic mean, instead of the arithmetic mean of d despite what should be expected intuitively. Finally, by the uniqueness of the solution of (17), it follows that the whole sequence x% converges weakly in Z01>2 (T) to x0 .. 3. Bounds. Optimal bounds is one of the fields where the theory of homogenization can be applied successfully. The theory may be used to design material with desired properties. The idea of finding upper and lower bounds is a well studied area in homogenization. In this section we give a brief presentation of the field, for proofs and details of the results in this section we refer to [25], [26], [31], [34] and [38]. 8.

(24) Consider the linear problem ; ? div (D% Gx% ) = i in T =. x% = 0 on CT,. where D% = % L, L is the identity matrix, D% = D ({@%) and D is a \ -periodic matrix (\ is the unit cube). Assume now that : Rq $ R and for all %, 0 ? 1 2 ? 4 a.e. on T. One important question is whether we can characterize the corresponding homogenized matrix D0 , in the sense of finding all possible D0 . Before we study the subject presented above, we give some notation and preliminaries. A two-phase composite means that only takes two values, i.e. ({) = 1 "T1 + 2 "T2 , where Tl is the periodic extension of {{ 5 \ : ({) = l } and "Tl is the characteristic function of the set Tl . We define the volume fraction |T1 |@|T| = 5 [0> 1] and |T2 |@|T| = 1 and assume that 1 ? 2 . We start by our least sharp bounds, the Reuss-Voigt bounds of the homogenized matrix is kL D0 dL, where k is the harmonic mean of over a cell of periodicity, L is the identity matrix and d is the arithmetic mean of over a cell of periodicity. By the notation D E we mean that E D has positive eigenvalues and hence, if l are the eigenvalues of D0 , the Reuss-Voigt bounds can be rewritten as k l d. For a two-phase composite, we have 1. 1 l 1 + (1 ) 2 . + 13 2. In Figure 5 we have plotted the Reuss-Voigt bounds for an example in R2 . A sharper estimation is the Hashin-Shtrikman, see [21]. The HashinShtrikman give us the following bounds on the eigenvalues of the homogenized matrix D0 1X O l X , q 9.

(25) Figure 6: The set of all possible eigenvalues from Hashin-Shtrikman bounds. where. ® ( d)2 O = d. ® , q inf + ( d)2 (d inf )31. ® ( d)2. X = d ® . q sup + ( d)2 (d sup )31. Here h·i denotes the arithmetic mean. Then again for the case of two-phase composite, the Hashin-Shtrikman upper and lower bounds become (1 ) (2 1 )2 O = 1 + (1 ) 2 q1 + (2 1 ) and. (1 ) (2 1 )2 = q2 + (1 ) (1 2 ) In Figure 6, we plot an example in R2 , to see the sharper bounds of the Hashin-Shtrikman bounds. There is also a generalized Hashin-Shtrikman bounds, which gives even better estimates. Let the trace of a matrix D be denoted by wu (D), then the generalized Hashin-Shtrikman bounds are D E 1 q +(q31) inf 1 D E 1 wu (D0 inf L) 1 inf q X = 1 + (1 ) 2 . +(q31) inf . 10.

(26) Figure 7: The set of all possible eigenvalues from generalized HashinShtrikman bounds. and. D. 1 +(q31) sup . E. q 1 D E . 1 wu (sup L D0 ) sup +(q31) sup q 1. Also for this generalized Hashin-Shtrikman we give the bounds for the special case of two-phased composite q X 1 q 1 = + , wu (D0 1 L) 1 (1 ) (2 1 ) 1 (1 ) l=1 l q X 1 1 q (1 ) = . wu (2 L D0 ) 2 l (2 1 ) 2 l=1. In Figure 7, we plot an example above in R2 . We see in the Figure 7, when all three bounds are plotted in the same Figure that the classical and the generalized Hashin-Shtrikman bounds coincide in the isotropic case, i.e. when 1 = 2 . There are various generalizations of the results in this section. The results can for example be generalized without the assumption of periodicity or for example in the theory of reiterated homogenization problems, see [25]. For a further reading about bounds corresponding to nonlinear problem, see e.g. [29], [45], [51], [53] and [57]. 11.

(27) Figure 8: A three-layered sandwich structure.. 4. Applications. In this section we give a picture of some of the fields where theory of homogenization is used. For example some part of homogenization theory has been developed as an interplay between mathematics and composite material, fiber materials, materials with holes etc. But the theory can be used as an asymptotic method of analysis in many fields like geophysics, fluid mechanics, elasticity and finance mathematics. More specific homogenization has been applied to dierent kind of equations like, wave equation, Schrödinge equation and Maxwell’s equations etc. Below we consider two particular fields, composite materials and finance mathematics.. 4.1. Composite materials. A composite is a material containing two or more mixed components. They have in general ’better’ properties than the properties of their individual constituents, according to the performance one looks for. Examples of composites material are ceramics, fiberglass, concrete etc. Composite materials are widely used in industries and the everyday life. To emphasize the idea behind composite material, consider for example natural rubber (also known as polyisoprene) embedded in two layers of cotton fabric, making a three-layered sandwich, see Figure 8. This is the basic idea of making a good raincoats because, while the rubber made it waterproof, 12.

(28) y. x Figure 9: Oriented fiber are strong when pulled in { direction, while weak in | direction. the cotton layers made it comfortable to wear. This clever idea of Charles Macintosh (note that a raincoat in Britain is referred to as a "Macintosh", or just a "Mack") is a good example of composing materials to obtain better properties. Here the basic idea of composite is used: composites are maid to make a material that has the properties of both its components. In this case, he combined the water-resistance of polyisoprene and the comfort of cotton. Composites are usually made of two components, a fiber and matrix. The fiber is most often glass, but sometimes Kevlar, carbon fiber etc. The matrix is usually a thermoset like an epoxy resin, polydicyclopentadiene, or a polyamide. The fiber is embedded in the matrix in order to make the matrix stronger. Fiber-reinforced composites have the best properties of two worlds (i.e. materials). They are strong and light. They are often stronger than steel, but weight much less and hence composites can be used to make automobiles and aeroplane lighter, and thus much more fuel e!cient. In fiberglass, the fibers are not lined up in any particular direction. But by lining up all the fibers in the same direction, the composite become stronger. This has also the eect that, the composite is very strong in one direction and not as strong in the orthogonal direction, see Figure 9. If strength in more than one direction is needed, then pointing the fibers in more than one direction will give strength in several direction, this is usually done by using a woven fabric of the fibers to reinforce the composite. The woven fibers give a composite good strength in many directions, see Figure 10. Two separate scales characterize these materials, the microscopic one, describing the heterogeneities, and the macroscopic one, describing the general 13.

(29) y. x Figure 10: The woven fibres are in this case strong when pulled in { and | directions. behavior of the composite. The aim of homogenization is to give the macroscopic properties of the composite by taking into account the properties of the microscopic structure. In composites, the heterogeneities are very small compared with the global dimension of the sample. As the heterogeneities get smaller and smaller, the mixture becomes finer and finer, and the material appears at a first glance as a homogeneous material. One can assume that the heterogeneities are evenly distributed and from mathematical point of view, one can model this distribution by supposing that it is periodic distribution. The periodicity can be represented by a small parameter %. Since we are interested to know the global behavior of the composite material when the heterogeneities are small, by homogenization, we study the problem when % tends to zero, see previous sections. There are other examples in composites where the theory of homogenization may be applied, e.g. bounds, design of material property etc.. 4.2. Mathematical Finance. Mathematical Finance has produced a convergence of ideas between dierent applied fields and the demand for sophisticated ideas and creative solutions has increased. One field of Mathematical Finance is the theory of options pricing, This was initiated by Fisher Black and Myron Scholes 1973, see [9] and it has been under an intensive development since. An option is a contract that allows the holder to buy or sell a financial asset at a fixed price n in the future time W . There are two type of options, 14.

(30) calls and put. A call option gives the holder the right to buy an asset (e.g. a stock) by a certain price. A put option gives the holder right to sell an asset by a certain date for a certain price. The Black-Scholes (B-S) model is a model for pricing options, and it allowed a previously undreamed precision in pricing options and probably made the explosive growth in the markets for options and other derivatives. Myron Scholes and Robert Merton were awarded the Nobel prize for economics for their part in devising the Black-Scholes equations, their coinventor, Fischer Black (1938-95) was ineligible, having died. There are many dierent model nowadays based on the B-S model. In the original work the volatility was assumed to be constant. Roughly speaking, the volatility of a stock is a measure of how uncertain we are about future stock price movement. As volatility increases, the chance that the stock will do very well or very poorly increase. Same year after publication of the original paper of B-S model, Merton (see [33]) generalized the B-S model to account for a deterministic time dependent rather than constant volatility. In 1987, stochastic volatility models were studied by Hull & White [24], Scott [47] and Wiggins [59]. The underlying asset price is modelled as a stochastic process which is driven by a random volatility Itb r process. Although it has not been justified by mathematical or economical proof, it is believable that market volatility is random, at least because of the wild swings in the market prices. Here we consider one of the generalized Black-Scholes model of the price F ( > {) of a European option on a stock price whose price is modelled as the Itb r process b (w> [w ) gZw g[w = (w> [w ) gw + P which satisfies the PDE. ¡ ¢ 1 F = P2 ( > {) F {{ + u {F { F , 2 where = W w is the time to expiration of the contract, u is the constant spot b (W w> {) is a deterministic function called the interest rate, and P ( > {) = P volatility. If we now suppose the volatility ¡ is ¢random and rapidly varying in % time, so that it is given by ( > {) = % > { , where ( > {) = b (W w> {) for some stochastic process {b (w> {) > w 0} that stays positive and % A 0, then we define the associated stochastic option price F % ( > {) as the solution of the stochastic PDE 1 ³ ´ % F% = 2 > { F{{ + u ({F{% F % ) , 2 % 15.

(31) with initial condition F % (0> {) = ({ n)+ for a call option. Here there it may be an opportunity of using homogenization and asymptotic analysis to study the model when % tends to zero, this approximation will tell us how to deal with the risk from the randomness of the volatility. In the paper [49], it is shown that the above problem is connected to homogenization of the following equation ; x% ( > |) : [0> W ] × Rq $ R, A A A A ? P P 2x Cx% % % = ql>m=1 d%lm ( > |) C|C l C| + qm=1 e%m ( > |) Cx , C C|m m A A A A = x% (0> |) = k (|) , ¡ ¢ ¡ ¢ where the coe!cients d%lm = dlm % > | and e%m = em % > | are defined on the probability space (l> F> ). Under some technical conditions on d%lm and e%m a homogenization result is obtained for this equation. However such equations have been studied in some papers, see [49] and there are still open questions left, e.g. like having some periodic conditions on the volatility and what it would means in economic terms. It may very well be worth an investigation, and we believe that the result of the theory of homogenization could be a very useful tool to make a deeper investigation of these problems.. 16.

(32) Chapter 2 J. Dasht, Some Results in Homogenization Theory, Department of Mathematics, Luleå University of Technology, Sweden, 2005..

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(34) Some Results in Homogenization Theory. In this chapter we prove the convergence of the sequence of solutions x% of the model problem (6) mentioned in chapter 1 to the solution x0 of the homogenized problem (7) of chapter 1, by two dierent methods. Moreover, we will introduce the concept of two-scale convergence, homogenization of random operators and end up with homogenization of nonlinear monotone operators.. 1. Tartar’s method of oscillating test functions. In this section we prove that the sequence of solutions x% of the model problem (6), see chapter 1, converges weakly in Z01>2 (T) to the solution x0 of the homogenized problem (7),see chapter 1. Moreover, we present the precise description of the homogenized matrix D0 in terms of the so called cell problem. The method we use is known as the Tartar’s method of oscillating test functions in combination with a special case of compensated compactness (see [54] and [55]). More precisely, the problem is that the product of two weakly convergent sequences dier from the product of their weak limits. This type of problem is related to dierent compensated compactness results, for more information see e.g. [35], [36] and [56]. We will use the following result: Lemma 1 (Compensated compactness) Let x% be a sequence converging weakly to x in Z 1>2 (T), and let j% be a sequence in O2 (T; Rq ) converging weakly to j in O2 (T; Rq ). Moreover assume that div j% converges strongly to div j in Z 31>2 (T). Then Z Z (j% > Gx% ) * g{ $ (j> Gx) * g{ T. T. for every * 5 F0" (T). 1.

(35) Let D be a matrix such that D ({) = (dlm ({)), l> m = 1> = = = > q is \ -periodic (\ is the unit cube) and satisfies dlm 5 O" (Rq ) for every l> m = 1> = = = > q> dlm = dml on Rq for every l> m = 1> = = = > q,. (1) (2). and there exists a constant A 0 such that (D ({) > ) ||2 ,. (3). for d=h= { 5 Rq and for every 5 Rq . Consider the following Dirichlet boundary value problem on the bounded open subset T of Rq : ; ? div (D% Gx% ) = i in T, (4) = x% 5 Z01>2 (T) , where D% ({) = D ({@%), i 5 Z 31>2 (T) and % is a sequence of positive real numbers converging to zero. We shall now prove that as % $ 0, the solution x% of (4) converges to the solution x0 of the following homogenized problem: ; ? div (D0 Gx0 ) = i in T, (5) = x0 5 Z01>2 (T) , ¡ ¢ where the constant matrix D0 = d0lm is defined by (the summation convention over indices that appears twice is assumed): ¸ Z · Czn 0 dln = dln (|) + dlm (|) (|) g|, (6) C|m \ and zn is the unique solution of the local problem ; Z ¡ ¡ ¢ ¢ A A D (|) hn + Gz n > Gy g| = 0, ; y 5 Z#1>2 (\ ) , ? \. A A = zn 5 Z 1>2 (\ ) , #. (7). where Z#1>2 (\ ) is the subset of Z 1>2 (\ ) of all functions x with mean value zero which have the same trace on the opposite faces of \ . Before we prove the homogenization result below, we state the following lemma which will be used in the proof. 2.

(36) R Lemma 2 Let j 5 O2 (\ ; Rq ) such that \ (j> Gy) g| = 0 for every y 5 Z#1>s (\ ). Then j can be extended by periodicity to an element of O2orf (Rq ; Rq ), still denoted by j such that div j = 0 in D0 (Rq ). Theorem 3 Let x% and x0 be the solutions of (4) and (5), respectively. Then x% - x0 in Z01>2 (T) , D% Gx% - D0 Gx0. (8) q. 2. in O (T; R ) .. Proof. The weak formulation of (4) is ; Z A A (D% Gx% > Gy) = hi> yi , ; y 5 Z01>2 (T) , ? T. A A =. (9). (10). x% 5 Z01>2 (T) ,. where h·> ·i denote the canonical pairing over Z 31>2 (T)×Z01>2 (T). By taking y = x% in (10) and using (3), we obtain Z Z 2 2 |Gx% | g{ (D% Gx% > Gx% ) g{ kGx% kO2 (T;Rq ) = T. T. |hi> x% i| ki kZ 31>2 (T) kx% kZ 1>2 (T) . 0. By Poincaré inequality, kGxkO2 (T;Rq ) and kxkZ 1>2 (T) are equivalent norms on 0 Z01>2 (T), thus kx% kZ 1>2 (T) F1 , (11) 0. where F1 is independent of %. Now we define % 5 Rq by ³{´ Gx% ({) on T, % ({) = D %. (12). or for every l = 1> = = = > q. ( l ({))% = (dlm ({))%. Cx% ({) . C{m. Since (1) and (11) hold, we have k % kO2 (T;Rq ) F2 , 3. (13).

(37) where F2 is independent of %. Therefore, there exist xW 5 Z01>2 (T), W 5 O2 (T; Rq ) and two subsequences, still denoted by x% and % such that x% - xW in Z01>2 (T) , % - W in O2 (T; Rq ) .. (14) (15). Note that Rellich’s Theorem imply that x% $ xW. in O2 (T; Rq ) .. (16). If we write equation (10) in the form Z ( % > Gy) g{ = hi> yi , ; y 5 Z01>2 (T) , T. then we can pass to the limit for every fixed y 5 Z01>2 (T) to obtain Z ( W > Gy) g{ = hi> yi , ; y 5 Z01>2 (T) .. (17). T. Now set W ({) = D0 GxW ({). for d=h= { 5 T.. (18). Then (17) shows that xW 5 Z01>2 (T) satisfies the weak formulation of (5), and since this solution is unique, we may conclude that xW = x0 (one can prove that D0 satisfies the same ellipticity conditions as D, hence (5) has a unique solution). Therefore we only have to prove (18). Consider the local problem (7) and let zn 5 Z#1>2 (\ ) be the solution of 1>2 (7). We still denote its \ -periodic extension to Rq by zn 5 Zorf (Rq ). For n = 1> = = = > q we define ³ ´ ³ ´ n n { n { = (hn > {) + %z for d=h= { 5 Rq . z% ({) = {n + %z (19) % % By the periodicity property of this function we have z%n - {n in O2 (T) , Gz%n - hn in O2 (T; Rq ) Consequently, we have the following convergences ; n ? z% - {n in Z 1>2 (T) , =. (20) z%n $ {n. in O2 (T) . 4.

(38) By Lemma 2, z%n satisfies ¡ ¢ div D% Gz%n = 0 in D0 (Rq ) ,. which means that Z. Rq. and thus. Z. T. ¡ ¡. (21). ¢ D% Gz%n > G* g{ = 0, ; * 5 F0" (Rq ) ,. ¢ D% Gz%n > Gy g{ = 0, ; y 5 Z01>2 (T) .. (22). Let * 5 F0" (T). Set y = *z%n 5 Z01>2 (T) in (10) and y = *x% 5 Z01>2 (T) in (22). Then Z Z ¡ ¢ ¡ ¢. ® n D% Gx% > z% G* g{ + D% Gx% > *Gz%n g{ = i> *z%n , T. Z. T. T. ¡. ¢ D% Gz%n > x% G* g{ +. Z. T. ¡. ¢. (23). D% Gz%n > *Gx% g{ = 0.. Since dlm = dml for every l> m = 1> = = = > q, we have ¡ ¢ ¡ ¢ D% Gx% > Gz%n = D% Gz%n > Gx% , and thus. Z. T. ¡. D% Gx% >. ¡. Gz%n. ¢ ¢ * g{ =. Z. T. ¡. ¢ D% Gz%n > (Gx% ) * g{.. From (23) and (24) we obtain that Z Z ¡ ¢ ¡ ¢. ® n D% Gx% > z% G* g{ D% Gz%n > x% G* g{ = i> *z%n , T. T. for every * 5 F0" (T). Note that. ³ { ´ Czn ¡ ¢ % D% Gz%n l ({) = dlm ({) % C{m = dlm. ³{´ µ %. ¶ Czn ³ { ´ mn + , C|m % 5. (24). (25).

(39) for every l = 1> = = = > q. Hence ¸ Z · ¡ ¢ Cz%n n D% Gz% l ({) (|) g| = d0ln , dln (|) + dlm (|) C| m \. (26). weakly in O2 (T). Next, by Rellich’s Theorem, (14) and (20) we have the following convergences ; x% $ xW in O2 (T; Rq ) , A A A A ? z%n - {n in Z01>2 (T) , A A A A = n z% $ {n in O2 (T; Rq ) , for every * 5 F0" (T) and % = D% Gx% - W. in O2 (T; Rq ) .. Each term in (25) consists of a scalar product in O2 (T; Rq ) > of an element which converges strongly and another which converges weakly in O2 (T; Rq ), hence we are ready to pass to the limit to the (25) as % $ 0. Consequently Z Z ¡ ¢ n lim % > $ % G* g{ = [( W )l (Gl *) {n ] g{, %<0. lim. %<0. and hence. Z. T. Z. T. T. T. ¡. ¢. D% Gz%n > x% G*. g{ =. Z. T. [d0ln (Gl *) xW ] g{,. £ ¤ ( W )l (Gl *) {n d0ln (Gl *) xW g{ = hi> *{n i .. Moreover, by inserting y = *{n in (17), we obtain Z [( W )l Gl (* {n )] g{ = hi> *{n i , ; * 5 F0" (T) ,. (27). (28). T. and thus Z Z £¡ ¢ ¤ 0 ( W )l {n dln xW Gl * g{ = [( W )l Gl (* {n )] g{, ; * 5 F0" (T) . T. T. (29). 6.

(40) But by integration by part Z Z ¤ £ 0 ¤ £ 0 dln (Gl *) xW g{ = dln (Gl xW ) * g{, T. and hence by (30) in (29), for every n = 1> = = = > q we have Z Z £ 0 ¤ dln (Gl xW ) * g{ = ( W )n * g{, ; * 5 F0" (T) . T. (30). T. (31). T. Thus we have the desired result from (31), namely ( W )n = d0ln Gl xW. d=h= on T,. for every n = 1> = = = > q. This ends the proof of (18). Note that one can prove that D0 is symmetric, bounded and coercive, hence the homogenized equation has a unique solution. Since the homogenized operator is uniquely defined and we have uniqueness of the solution (5) we conclude that the convergences ; 1>2 ? x% - x0 in Z0 (T) , =. D% Gx% - D0 Gx0. in O2 (T; Rq ) ,. holds for the whole sequence, and not only for the extracted subsequence.. 2. The two-scale method. The concept of two-scale convergence was introduced by G. Nguetseng 1989 (see [39]) and developed further by G.R Allair (see [2] and [3]). It deals with convergence of integrals of the form x% ({) *({> {@%)g{. We start by the definition of the two-scale convergence and some theorems which will be useful in this context. We end up with proving the homogenization result of problem (4) using this method. For a more detailed and self-contained presentation of this subject, we recommend [40]. Let T be an open bounded subset of Rq , \ the unit cube in Rq , Fshu (\ ) the set of \ -periodic continuous functions defined on Rq and O2 (T; Fshu (\ )) the of functions i : T $ Fshu (\ ) which are measurable and satisfies R space 2 ||i ||Fshu (\ ) ? 4. Let % be a fix sequence of positive numbers converging T to 0. 7.

(41) Definition 4 (Two-scale convergence) A sequence x% 5 O2 (T) is said to two-scale converge to a limit x 5 O2 (T × \ ) if Z Z Z ³ {´ 1 x% ({) y {> x ({> |) y ({> |) g|g{ g{ $ % |\ | T \ T for every y 5 O2 (T; Fshu (\ )). We note that the two-scale limit is unique. Moreover, the term |\ | is the Lebesgue measure of \ and here |\ | = 1 and that the space O2 (T; [) means in general, the set of measurable functions x : { 5 T $ x ({) 5 [ such that ||x ({) ||[ 5 O2 (T). Before we deal with the homogenization results for (4) by the two-scale method, we state the following two theorems (for proof see, e.g. [40]) which we will need during the proof of the main theorem. Theorem 5 Let x% be a sequence in Z 1>2 (T) such that x% - x in Z 1>2 (T). Then x% two-scale converges to x, and there exists a subsequence %0 and x1 5 ¡ ¢ 1>2 O2 T; Zshu (\ ) such that Gx%0 two-scale converges to Gx + Gx1 .. Theorem 6 Let x% be a sequence in O2 (T) such that x% two-scale converges to x, then Z Z Z ³ {´ g{ = x% y {> x ({> |) y ({> |) g|g{, % T T \ ¡ ¡ ¢¢ for every y 5 O2shu \ ; F T . Now we are able to state and proof the main theorem in this section.. Theorem 7 The sequence of solutions x% of (4) converges weakly to x 5 Z01>2 (T) and the sequence Gx% two-scale converges to Gx + G| x¢1 ({> |), ¡ ({)1>2 1>2 2 where (x> x1 ) is the unique solution in Z0 (T) × O T; Zshu (\ ) of the homogenized equation Z Z Z (D (|) (Gx ({) + G| x1 ({> |)) > Gy ({) G| y1 ({> |)) g|g{ = i y g{, T. \. for every y 5. T. Z01>2. 2. (T) and y1 5 O. ¡. 1>2 T; Zshu. 8. ¢ (\ ) .. (32).

(42) Proof. We proved in the previous section that ||x% ||Z 1>2 (T) ? f and by the 0 reflexivity of Z01>2 (T) we obtained a weakly convergent subsequence of x% , still denoted by %, i.e. x% - x in Z01>2 (T). By ¡ Theorem ¢(5), there exists 2 1>2 a subsequence, still denoted by %, and x1 5 O T; Zshu (\ ) such that Gx% two-scale converges to Gx ({) ¡+ G| x1 ({> |). ¢ Let z% = y ({) + %y1 ({> {@%), " " where y 5 F0 (T) and y1 5 G T> Fshu (\ ) . By using z% as test function in (10), which is possible since clearly z% 5 Z01>2 (T) by construction, we obtain Z ³ ³ { ´´ D% Gx% > Gy ({) + G| y1 {> g{+ % T Z Z ³ ³ { ´´ ³ { ´´ ³ D% Gx% > G{ y1 {> g{ = i y ({) + %y1 {> g{, % % % T T which can be rewritten as follows Z ³ h ³ { í´ w Gx% > D% Gy ({) + G| y1 {> g{+ % T Z ³ Z ³ { ´´ ³ ³ { ´´ w % Gx% > D% G{y1 {> i y ({) + %y1 {> g{ = g{, % % T T. (33). where we want to pass to limit as % $ 0. For the first term, this¡ is possible ¡ ¢¢ according to Theorem (6), since Gy ({) + G| y1 ({> {@%) 5 O2shu \ ; F T . Consequently Z ³ h ³ { í´ w lim g{ = Gx% > D% Gy ({) + G| y1 {> %<0 T % Z Z (D (|) [Gx ({) + G| x1 ({> |)] > Gy ({) + G| y1 ({> |)) g|g{. T. \. For the second therm in (33), by Hölder inequality and the fact that Gx% is bounded in O2 (T) (see previous section), we have Z ³ ³ { ´´ lim % Gx% > Dw% G{ y1 {> g{ = 0. %<0 % T To pass to¡the¢limit in the last term, notice that by definition of z% , we have y (·) + %y1 ·> %· - y in Z01>2 (T). Hence passing to the limit in the left hand side of (33), we obtain Z Z Z (D (|) [Gx ({) + G| x1 ({> |)] > Gy ({) + G| y1 ({> |)) g|g{ = iy g{. T. \. T. (34). 9.

(43) By using the relation x1 ({> |) =. q X. zl (|) x ({) ,. l=1. we see that the formulation (5) and (34) are equivalent. ¡ ¢ 1>2 By density (34) holds for each (y> y1 ) in Z01>2 (T) × O2 T; Zshu (\ ) . By proving that the solution (x> x1 ) of (34) is unique, then the proof above is true for the whole sequence and not only a subsequence. This is done by using Lax-Milgram Lemma. Let us define the Hilbert space K = Z01>2 (T) × ¢ ¡ 1>2 O2 T; Zshu (\ ) , with the norm defined by ||X ||2K = ||Gx||2O2 (T;Rq ) +||Gx1 ||2T×\ for X = (x> x1 ) 5 K, the bilinear form D0 Z Z D0 (X> Y ) = (D (|) [Gx ({) + G| x1 ({> |)] > Gy ({) + G| y1 ({> |)) g|g{, T. \. R and the functional I on K by hI> Y i = T I y g{. Then the homogenized problem (34) is: Find X in K such that D0 (X> Y ) = hI> Y i for each Y 5 K. The coerciveness and boundness of D0 is standard results, hence Lax-Milgram Lemma guarantees the existence and uniqueness of the solution X of D0 and the proof is done.. 3. Homogenization of random operators. In this section we introduce homogenization of random operators, in particular we study the homogenization of random operators for a Dirichlet boundary value problem.. 3.1. Preliminaries. We start by giving a brief presentation of the notations and preliminaries. Let (l> F> ) be a probability space, where the l is equipped with a -algebra F and a measure which is countably additive, non-negative and normalized with measure 1. A random variable i ($) is a measurable mapping i : l $ Rq . Now we define a dynamical system. Definition 8 An q-dimensional dynamical system is a family of mappings W{ : l $ l, { 5 Rq , which satisfy the following properties: 10.

(44) 1. W0 = L, L is the identity mapping and W{+| = W{ W| , {> | 5 Rq . 2. The map W{ : l $ l preserves the measure on l, i.e. for every { 5 Rq , and every X 5 F W{ X 5 F and (X ) = (W{ X ) . 3. For any measurable function i on l, the function i (W{ $) defined on Rq × l is measurable (Rq × l is endowed with the measure g{ × g, where g{ is the Lebesgue measure). We observe that W{ is invertible for every { 5 Rq , since from the first property we have W{ W3{ = W3{ W{ . Given a random variable i , the function ({> $) = i (W{ $) is a stationary random variable field. For any fixed $ 5 l the function $ ({) = i (W{ $) is called a realization of the field ({> $). The function i is said to be invariant if i ($) = i (W{ $) a.e. in l, for every { 5 Rq . A measurable set X is called invariant if W{ X = X for every { 5 Rq . A dynamical system is called ergodic if every invariant function is constant a.e. in l or equivalently if every invariant set X has either measure 1 or 0. A vector field i 5 O2orf (Rq ; Rq ) is called vortex-free in Rq if curl i = 0 in the weak sense, i.e. ¶ Z µ C! C! il im g{ = 0 for all ! 5 F0" (Rq ) . C{m C{l Rq A vector field i 5 O2 (l; Rq ) is said to be potential if almost all its realizations are potential vectors fields defined on Rq , we note that i is potential if and only if is is vortex-free. The subspace of O2 (l; Rq ) formed by potential 2 vectors is denoted by O2srw (l; Rq ) and Ysrw (l) is defined in the following way Z 2 2 q Ysrw (l) = {i 5 Osrw (l; R ) : i ($) g = 0}. Now we introduce the notation of mean value and an important theorem, Birkho ergodic theorem. Let i 5 O1orf (Rq ), the mean value of i is defined by Z ³{´ 1 M (i ) = lim i g{, %<0 |N| N % 11.

(45) for any Lebesgue measurable bounded set N. Assuming that the family i ({@%) is bounded in O2orf (Rq ), we define the mean value in terms of weak convergence by i ({@%) - M (i ) in O2orf (Rq ). Theorem 9 (Birkho ergodic theorem) Let i 5 O2 (l). Then for a.e. $ 5 l the realization I ({) = i (W{ $) possesses a mean value M (i ) in sense that ³{´ - M (i ) in O2orf (Rq ) . I % Moreover, the mean value M (i (W{ $)) is invariant as a function of $ and Z Z i ($) g = M (i (W{ $)) g. l. l. In particular, if the system is ergodic, then Z M (i (W{ $)) = i ($) g for d=h= $ 5 l. l. Proof. A proof of this can be found in [19].. 3.2. Homogenization of linear random operators. In this section we present some results on homogenization of random operators. Let i 5 Z 31>2 (T) with T Rq open and bounded and l be a probability space with an ergodic dynamical system W{ ({ 5 Rq ) defined on it. Consider the following Dirichlet boundary value problem ; ? div (D$% Gx$% ) = i , =. x$% 5 Z01>2 (T) , $ 5 l,. where D$ is a realization of a stationary random field. For more information about stochastic homogenization is referred to [25] and [42]. For more information concerning homogenization see, e.g. [7], [10], [14], [23], [25], [42] or [44]. Let D : l $ Rq be a function such that D ($) = (dlm ($)) is measurable and satisfies the following structure conditions: there exists constants 0 ? ? 4 such that (D ($) > ) ||2 , (35) 12.

(46) dlm ($) 31 ,. (36). for a.e. $ 5 l and for every 5 Rq . 2 Let [ = Ysrw (l), 5 Rq , A : [ $ [ W be the operator defined by Z hA y> !i[×[ W = D ($) ( + y ($) > !) g for all ! 5 X, l. . and y 5 [ be the solution of the auxiliary problem. hA y> !i[×[W = 0 for all ! 5 X.. Denote by Y >$ ({) a realization of y , i.e. for every fixed $ 5 l we define 2 Y >$ ({) = y (W{ $), { 5 Rq . By definition of Ysrw (l) we observe that for 1>2 >$ 5 Zorf (Rq ) such that Y >$ ({) = a.e. $ 5 l there exists a function Q >$ GQ ({). Finally we define ³ ´ >$ { $ >$ ({) = (> {) + %Q . % % Then for a.e. $ 5 l ³ ´ ³ ´ >$ { >$ { G$ >$ ({) = + GQ = + Y . % % % Next by M (y q) = 0, Theorem (9) and the ergodicity of W{ we obtain that G$ >$ % ({) - . in O2orf (Rq ; Rq ) for d=h= $ 5 l=. Next consider the realization D (W{ $) ( + y (W{ $)) = D (W{ $) ( + Y >$ ({)) , where { 5 Rq . By the properties of D one can show that D ($) ( + y ($)) 5 O2 (l; Rq ) and by Theorem (9) we can pass to the limit and obtain ³ { ´´ ¡ ¢³ ¡ ¢ D W {% $ + Y >$ = D W {% $ (G$ >$ (37) % ({)) % Z D ($) ( + y ($)) g. -. l. in (R ; R ) for a.e. $ 5 l. We define the limit operator D0 : Rq $ Rq by the limit in (37) Z D0 = D ($) ( + y ($)) g (38) O2orf. q. q. l. for a.e. $ 5 l. Now we return to the main problem, the homogenization of the random monotone operator. 13.

(47) 3.3. The main problem. Let D = D($) = (dlm ($)) be a measurable matrix function satisfying (35) and (36) for every 5 RQ and for $ 5 l0 , where l0 is a measurable subset of l such that (l0 ) = 1. It can be shown that there exists a measurable subset l1 l0 such that (l1 ) = 1 and W{ $ 5 l0 for $ 5 l1 and for a.e. { 5 RQ . This implies that for $ 5 l1 the realizations D({> $) = D(W{$) have the following properties: D(·> $) is measurable and (D({> $)> ) ||2 , dlm ({> $) 31 , for every 5 RQ and a.e. { 5 RQ . Let i 5 Z 31>2 (T) with T Rq , open and bounded. Consider now the following Dirichlet boundary value problem ; ¡ ¡ ¢ ¢ ? div D {% > $ Gx$% = i , (39) = x% 5 Z01>2 (T) . By classical results in existence theory for boundary value problem defined by monotone operators (see e.g. [61]), for each % there exists a unique solution x$% 5 Z01>2 (T). Now we are in position to state the main theorem. Theorem 10 Let x$% be the solution of (39) and let D0 be given by (38). Then for almost every $ 5 l, we have x$% - x0 in Z01>2 (T) , ³{ ´ > $ Gx$% - D0 Gx0 in O2 (T; Rq ) , D % where x is the unique solution of the homogenized equation ½ div (D0 Gx0 ) = i , x0 5 Z01>2 (T) . Proof. For a proof of this result see e.g. [25], [27], [42] or [43]. We also refer to the articles [16] and [17] concerning related results in stochastic homogenization.. 14.

(48) 4. Homogenization of nonlinear monotone operators. Until now we have regarded linear problems, but the results can be generalized to nonlinear problems as well. Let us look at some results of homogenization of nonlinear monotone operators Ak : Z01>2 (T) $ Z 31>2 (T) of the form ³ ³{ ´´ k > Gx , A x = div D % where D : Rq $ Rq , and for every 5 Rq, D (·> ) is Lebesgue measurable and \ -periodic. Moreover, there exist constants 0 ? ? ? 4 such that (Strict monotonicity) (D ({> 1 ) D ({> 2 ) > 1 2 ) | 1 2 |2 , (Lipschitz-continuity) |D ({> 1 ) D ({> 2 )| | 1 2 | ,. and D ({> 0) = 0, for every 1 > 2 5 Rq and for d=h= { 5 Rq . Then the following Dirichlet boundary value problem ´´ ³ ³{ ; A > Gx% = i% in T, ? div D % (40) A = 1>2 x% 5 Z0 (T) , has a unique solution for every i% 5 Z 31>2 (T), where T is a bounded open subset of Rq . Moreover, let % be a sequence of positive real numbers converging to zero, i% converging strongly in Z 31>2 (T) to i and x% be the solution of (40), then x% - x0 in Z01>2 (T) , ³{ ´ > Gx% - D0 (Gx0 ) D %. (41) in O2 (T; Rq ) ,. where x0 is the unique solution of the homogenized problem ; ? div (D0 (Gx0 )) = i in T, =. x0 5 Z01>2 (T) .. The operator D0 : Rq $ Rq is defined for every 5 Rq by Z ¡ ¢ D0 () = D |> + G$ (|) g|, \. 15.

(49) where z is the unique solution of the local problem ; Z ¡ ¡ ¢ ¢ A A D |> + Gz (|) > Gy (|) g| = 0, ; y 5 Z#1>2 (\ ) , ? \. A A = z 5 Z 1>2 (\ ) . #. The results of this section were earlier proved in the linear case with two dierent tools, Tartar’s method of oscillating test functions and the two-scale method. Finally let us mention that other results in these two chapters can be generalized as well. For example for bounds concerning the s-Poisson equation see e.g. [30] and for bounds in the degenerated case see e.g. [13].. 4.1. Concluding remarks. One of the main research areas at the Department of Mathematics in Luleå University of Technology is the homogenization theory. It has been a fruitful research cooperation between the homogenization group in Luleå and other homogenization groups e.g. the homogenization group at Narvik University College and also other applied science groups e.g. the Division of Polymer Engineering at Luleå University of technology. In particular, several doctoral theses, licentiate theses and books have been presented in Luleå and Narvik, see, e.g. [8], [12], [20], [22], [29],[32],[44], [48], [50], [57] and [58]. For more information concerning the homogenization theory and related results and applications we refer to [1], [4], [5],[6], [7], [14], [15], [18], [25], [28], [37], [41], [42], [44], [46], [60] and [62].. 16.

(50) References for chapter 1 and chapter 2..

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(56) [48] L. Simula, Homogenization Theory for Structures of Honeycomb and Chessboard Types, Ph.D. Thesis, Dept. of Math. Luleå University of Technology, Sweden, Luleå 2002. [49] K. R. Sircar and G. Papanicolaou, Stochastic Volatility, Smile & Asymptotics, Applied Mathematical Finance 6(2), 107-145, 1999. [50] N. Svanstedt, G-Convergence and Homogenization of Sequences of Linear and Nonlinear Partial Dierential Operators, Ph.D. Thesis, Dept. of Math. Luleå University of Technology, Luleå, Sweden 1992. [51] N. Svanstedt, A note on bounds for nonlinear multivalued homogenized operators, Apl. Mat., 43(2), 81-92, 1998. [52] P. Suquet, Plasticité et Homogéneisation, Thesis Univ. Paris VI, 1982. [53] D. R. S. Talbot and J. R. Willis, Bounds and self-consistent estimates for the overall properties of nonlinear composites, IMA J. Appl. Math., 39, 215-240, 1987. [54] L. Tartar, Cours peccot au Collège de France, Unpublished, 1977. [55] L. Tartar, Quelques remarques sur l’homogénéisation, In: Functional Analysis and Numerical Analysis, Proc. Japan-France Seminar, 468-482, 1978. [56] L. Tartar, Compensated compactness and applications to partial dierential equations. Nonlinear analysis and mechanics, Heriot-Watt Symposium vol. VI, Research Notes in Mathematics 39, 136-211, Pitman, London 1979. [57] P. Wall, Homogenization of some Partial Dierential Operators and Integral Functionals, Ph.D. Thesis, Dept. of Math. Luleå University of Technology, Luleå, Sweden 1998. [58] N. Wellander, Homogenization of some Linear and Nonlinear Partial Dierential Equations, Ph.D. Thesis, Dept. of Math. Luleå University of Technology, Luleå, Sweden 1998. [59] J. Wiggins, Option Values under Stochastic Volatility, Journal of Financial Economics, 19, 351—372, 1987. 5.

(57) [60] K. Yosida, Functional Analysis, Springer-Verlag, Berlin 1966. [61] E. Zeidler, Nonlinear Functional Analysis and its Applicatios, Vol.2/B, Springer Verlag, New York, 1990. [62] W. P. Ziemer, Weakly Dierentiable Functions, Springer Verlag, Berlin 1989.. 6.

(58) Paper A J. Dasht, J. Byström and P. Wall, A Numerical Study of Stochastic Homogenization, Journal of Analysis and Applications, Vol. 2, No. 3, 159-171, 2004..

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(74) Paper B J. Dasht, J. Engström, A. Kufner and L.-E. Persson, Rothe’s method for parabolic equations on non-cylindrical domains, Submitted, 2005..

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(83) +{, > { 5 3 / dqg vroyh +9, iru n @ 4 rq 3 = Wkh vroxwlrq x4 +{, > { 5 3 > lv wkhq h{whqghg wr 4 e| }hur/ l1h1 x4 +{, @ 3 iru { 5 4 q 3 = Wkhq zh vroyh +9, iru n @ 5 rq 4 1 Frqwlqxlqj wklv surfhvv zh rewdlq d q0wxsoh ri ixqfwlrqv x3 +{, +@

(84) +{,,/ x4 +{, > x5 +{, > ===> xq4 +{, > zlwk xl ghqhg rq l4 dqg hyhq rq l = Zlwk khos ri wkhvh ixqfwlrqv/ zh ghqh d ixqfwlrq xq @ xq +{> w, rq Gk e| xq +{> w, @ xn +{, / { 5 n > w 5 ^nk> +n . 4, k, > n @ 3> = = = > q 4=. +:,. Wklv ixqfwlrq fdq eh frqvlghuhg dv d Urwkh dssur{lpdwlrq ri wkh vroxwlrq ri sureohp +7,1 Rxu sodq lv qrz wkh iroorzlqj= 7.

(85) Vwhs 4= Ghwhuplqh wkh ixqfwlrqv xn +{, > l1h1 wkh vroxwlrqv ri +9,= Vwhs 5= Ilqg sulrul hvwlpdwhv ri wkh ixqfwlrqv xn +{, = Vwhs 6= Ilqg sulrul hvwlpdwhv iru wkh Urwkh dssur{lpdwlrq xq +{> w, ghqhg e| +:,= Vwhs 7= Wkh ixqfwlrq xq +{> w, lq +:, ghshqgv rq wkh fkrlfh ri k/ l1h1 rq q +wkh glylvlrq ri +3> W , lqwr q sduwv,1 Zh zdqw wr vkrz wkdw wkh qrup ri xq +lq vrph dssursuldwh vsdfh, lv erxqghg lqghshqghqwo| ri q= Frqvhtxhqwo| wklv erxqghg vhtxhqfh +xq , frqwdlqv vrph zhdno| frqyhujlqj vxevhtxhqfh/ dqg qdoo| zh vkrz wkdw wkh olplw lv d vroxwlrq +lq d zhdn vhqvh dv ghqhg odwhu/ vhh ghqlwlrq 514, ri sureohp +7,1 Vwhs 4= Zh zloo orrn iru zhdn vroxwlrqv ri +9,/ l1h1 iru ixqfwlrqv xn 5 Z34>5 + n4 , vxfk wkdw ] xn +{, xn4 +{, . fn +{, xn +{, * +{, g{ k n4 ] dn +{, x3n +{, *3 +{, g{ +;, ] @. n 4. n4. in +{, * +{, g{. iru hyhu| * 5 Z34>5 + n4 , > n @ 4> 5> ===> q 41 E| wkh frqglwlrqv rq dn > fn dqg in / vwdqgdug uhvxowv lq wkhru| iru olqhdu hoolswlf rshudwruv +vhh h1j1 ^6`, jlyhv wkdw wkh vroxwlrq ri +;, h{lvwv dqg lv xqltxh1 Vwhs 5= Vlqfh +;, krogv iru hyhu| * 5 Z34>5 + n4 , > zh fdq fkrrvh lq sduwlfxodu * @ 5kxn dqg rewdlq ] ] 5 x5n x5n4 . +xn xn4 ,5 . 5kdn +x3n , . 5kfn x5n g{ @ 5k in xn g{= n4. n4. Gxh wr +8, zh rewdlq ] ] 3 5 5k dn +xn , g{ 5k n4. n4. 5. 5. +x3n , g{ @ 5k nx3n n =. Pruhryhu/ +8, dovr jlyhv ] ] 5 f x g{ x5n g{> n4 n n n4 zklfk lpsolhv. ] n 4. 5. fn x5n g{ nxn n =. 8. +<,.

(86) Qrwlfh wkdw nn lv wkh O5 0qrup dqg gxh wr wkh idfw wkdw xn +{, @ 3 iru { 5 n q n4 > lw grhv qrw pdwwhu zkhwkhu zh xvh wkh O5 + n4 , ru O5 + n , qrup1 E| Kùoghu lqhtxdolw|/ lw qrz iroorzv iurp +<, wkdw 5. 5 5 5 nxn n nxn4 n . nxn xn4 n . 5k nx3n n 5. +43,. 5k nxn n . 5k nin n nxn n Lq sduwlfxodu 5. 5. 5. nxn n nxn4 n 5k nxn n . 5k nin n nxn n dqg frqvhtxhqwo| iru nxn n A 3> zh kdyh wkdw nxn n5 nxn4 n5 5. 5. nxn n . nxn4 n. 5k nxn n5. . .. 5. 5k nin n nxn n 5. 5. nxn n . nxn4 n nxn n . nxn4 n nin n nin n nxn n5 = @ 5k . 5k 5k . 5k nxn n nxn n5 . nxn4 n5 nxn n. Iru wkh ohiw kdqg vlgh ri wkh deryh lqhtxdolw|/ zh kdyh nxn n5 nxn4 n5 5. 5. nxn n . nxn4 n 5. @4. 5 nxn4 n5 5. 5. nxn n . nxn4 n. 4. nxn4 n nxn n. 5. vlqfh nxn n . nxn4 n 5 nxn n nxn4 n = Frqvhtxhqwo| 4. nin n nxn4 n 5k . 5k > nxn n nxn n. l1h1 nxn n nxn4 n 5k nxn n . 5k nin n = Vxssrvh qrz wkdw 4 5k A 3/ l1h1 k?. 4 ++, q A 5W , 1 5. +44,. Wkhq zh kdyh nxn n . 4 5k nxn4 n . nin n = 4 5k 4 5k. Khqfh lw iroorzv wkdw n nl.4 n [ 4 4 nxn n nx3 n . 5k nil n > 4 5k 4 5k l@4 9. +45,.

(87) dqg vlqfh 4@4 5k A 4 zh rewdlq nxn n . 4 4 5k. $ n # n [ nil n = nx3 n . 5k. +46,. l@4. Ohw xv qrz lqwurgxfh wkh vsdfh O4>5 +G, @ O4 3> W > O5 + w , > l1h1 + ] ] 4>5. O. +G, @. W. y @ y +{> w, = nynO4>5 +G, @. 5. 3. w. y +{> w, g{. 4@5. ,. gw ? 4 1. Ghqh wkh ixqfwlrq I +{> w, rq Gk vlploduo| dv x +{> w, lq +:,/ l1h1 I +{> w, @ in +{, lq wkh uhfwdqjoh Un4 > n @ 4> 5> ===> q +uhfdoo wkdw Un4 @ i+{> w, > { 5 n4 > w 5 ++n 4, k> nk`j,1 Wkhq $4@5 #] q ] lk [ 5 nI nO4>5 +Gk , @ I +{> w, g{ gw l@4 +l4,k. @. q ] lk [. l@4 +l4,k. l4. nin n gw @. q [ l@4. k nil n >. dqg lq sduwlfxodu nI nO4>5 +Gk , @. n [. k nil n =. +47,. 4 ++, q A 7W , / 7. +48,. n. l@4. Li zh qrz uhsodfh +44, e| k?. zh kdyh 4@5 ? 4 5k ? 4 dqg . 4 4 5k. n. @ h{s n oq. 4 4 5k. . 5k h{s n 4 5k. h{s +n7k, h{s +q7k, @ h{s +7W , = Frqvhtxhqwo|/ lw iroorzv iurp +46, wkdw nxn n h{s +7W , nx3 n . 5 nI nO4>5 +Gk , = n. :. +49,.

(88) Dgglqj qrz wkh lqhtxdolwlhv +43, iru n @ 4> 5> ===> u> u ? q> zh rewdlq 5. 5. nxu n nx3 n . 5k. u [. u [ n@4. nxn n5 . 5k. n@4. 5. nxn xn4 n . 5k. u [. u [ n@4. 5. nx3n n. nin n nxn n =. n@4. Dyrlglqj wkh wklug whup rq wkh ohiw dqg hvwlpdwlqj wkh uljkw kdqg vlgh e| +49,/ zh kdyh wkdw nxu n5 . 5k. u [. 5. nx3n n. n@4. nx3 n5 . 5k h{s +;W , .5k h{s +7W ,. u [ n@4. u q r [ nx3 n5 . 7 nx3 n nI nO4>5 +Gk , . 7 nI n5O4>5 +Gk , n. n@4. nin n nx3 n . 7k h{s +7W ,. u [ n@4. n. nin n nI nO4>5 +Gk , = n. +4:,. Qrz xvlqj wkh idfw wkdw +iru n u, nI nO4>5 +Gk , nI nO4>5 +Gk , u. n. 5. 5. lq frpelqdwlrq zlwk wkh hvwlpdwh 5 nx3 n nI n nx3 n . nI n / irupxod +47, dqg k. u [ n@4. nx3 n5 @ uk nx3 n5 W nx3 n5 /. zh fdq hvwlpdwh wkh uljkw kdqg vlgh lq +4:, e| nx3 n5 . 5k h{s +;W , .5k h{s +7W ,. u [ n@4. .7 h{s +;W , k .; h{s +;W , k. n. n@4. nin n nx3 n . 7k h{s +7W ,. 5 nx3 n . 5 h{s +;W , k. #. u q r [ nx3 n5 . 7 nx3 n nI nO4>5 +Gk , . 7 nI n5O4>5 +Gk ,. u [. n@4 u [ n@4. u [. u [ n@4. n. nin n nI nO4>5 +Gk , n. nx3 n5. n@4. 5. nx3 n . k. u [ n@4. 5. $ 5. nI nO4>5 +Guk ,. nI nO4>5 +Gk , . 5 h{s +7W , nx3 n k. .7 h{s +7W , nI n5O4>5 +Guk , k. u. u [. nin n. n@4. ;. u [ n@4. nin n.

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