Homogenization of generalized second-order elliptic difference operators

Abstract

Consider a function W(x1,…,xd)=∑k=1dWk(xk), where each Wk:R→R is a strictly increasing right continuous function with left limits. Given a matrix function A=diag{a1,…,ad}, let ∇A∇W=∑k=1d∂xk(ak∂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 ofλuN−∇NAN∇WNuN=fN to the solution ofλu−∇A∇Wu=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 AN, 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 AN; the second one consists on the case where AN 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 ω of the environment. Finally, we provide an application geared towards the hydrodynamical limit of certain gradient processes.