Homogenization Theory of Elliptic System with Lower Order Terms for Dimension Two

TL;DR

This paper advances the homogenization theory for elliptic systems with lower order terms in two dimensions, establishing key estimates and convergence results, and constructing Green functions for these operators.

Contribution

It extends homogenization results to two-dimensional elliptic systems with lower order terms, including estimates, convergence, and Green function construction, previously unresolved for d=2.

Findings

Established $W^{1,p}$, Hölder, Lipschitz, and $L^p$ estimates for the operator

Proved convergence results and rates for the homogenization process in 2D

Constructed Green functions and analyzed their properties in two dimensions

Abstract

In this paper, we consider the homogenization problem for generalized elliptic systems $$ \mathcal{L}_{\varepsilon}=-\operatorname{div}(A(x/\varepsilon)\nabla+V(x/\varepsilon))+B(x/\varepsilon)\nabla+c(x/\varepsilon)+\lambda I $$ with dimension two. Precisely, we will establish the $ W^{1,p} $ estimates, H\"{o}lder estimates, Lipschitz estimates and $ L^p $ convergence results for $ \mathcal{L}_{\varepsilon} $ with dimension two. The operator $ \mathcal{L}_{\varepsilon} $ has been studied by Qiang Xu with dimension $ d\geq 3 $ in \cite{Xu1,Xu2} and the case $ d=2 $ is remained unsolved. As a byproduct, we will construct the Green functions for $ \mathcal{L}_{\varepsilon} $ with $ d=2 $ and their convergence rates.