Deterministic homogenization of elliptic equations with lower order terms

TL;DR

This paper establishes homogenization results for elliptic equations with oscillating coefficients using sigma-convergence, deriving optimal convergence rates without requiring coefficient smoothness, and introduces a Helmholtz-type decomposition in Besicovitch spaces.

Contribution

It provides a novel homogenization approach for elliptic equations with nonsmooth, oscillating coefficients and derives optimal convergence rates in the asymptotic periodic case.

Findings

Homogenization achieved via sigma-convergence method.

Optimal convergence rates for zero order approximation.

Existence of asymptotic periodic correctors for nonsmooth coefficients.

Abstract

For a class of linear elliptic equations of general type with rapidly oscillating coefficients, we use the sigma-convergence method to prove the homogenization result and a corrector-type result. In the case of asymptotic periodic coefficients we derive the optimal convergence rates for the zero order approximation of the solution with no smoothness on the coefficients, in contrast to what has been done up to now in the literature. This follows as a result of the existence of asymptotic periodic correctors for general nonsmooth coefficients. The homogenization process is achieved through a compactness result obtained by proving a Helmholtz-type decomposition theorem in case of Besicovitch spaces.