Homogenization of generalized second-order elliptic difference operators

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:265

Article

Simas, Alexandre B.¹  Valentim, Fabio J.² 

[1] Univ Fed Paraiba, Dept Matemat, Cidade Univ Campus 1, BR-58051970 Joao Pessoa, Paraiba, Brazil

[2] Univ Fed Espirito Santo, Dept Matemat, Av Fernando Ferrari 514, BR-29075910 Vitoria, ES, Brazil

关键词: Homogenization;    Difference operators;    Differential operators;   

DOI  :  10.1016/j.jde.2018.05.018

来源: Elsevier

【 摘 要 】

Consider a function W(x(1), . . . , x(d)) = Sigma(d)(k=1) W-k(x(k)), where each W-k : R -> R is a strictly increasing right continuous function with left limits. Given a matrix function A = diag{a(1), . . . , a(d)}, let del A del(W) = Sigma k=1(d) partial derivative x(k )(a(k)partial derivative(Wk )) be a generalized second-order differential operator. Our chief goal is to study the homogenization of generalized second-order difference operators, that is, we are interested in the convergence of the sequence of solutions of lambda u(N) - del(N) A(N)del(N)(W)u(N) = f(N) to the solution of lambda u - del A del W-u = f, where the superscript N stands for some sort of discretization. In the continuous case we study the problem in the context of W-Sobolev spaces, whereas in the discrete case we develop the theoretical context in the present paper. The main result is a homogenization result. Under minor assumptions regarding weak convergence and ellipticity of these matrices A(N), we show that every such sequence admits a homogenization. We provide two examples of matrix functions verifying these assumptions: the first one consists of fixing a matrix function A under minor regularity assumptions, and taking a convenient discretization A(N); the second one consists on the case where A(N) represents a random environment associated to an ergodic group, a case in which we then show that the homogenized matrix A does not depend on the realization omega of the environment. Finally, we provide an application geared towards the hydrodynamical limit of certain gradient processes. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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