Homogenization theory of elliptic system with lower order terms for dimension two

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, Hölder 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 [22,23] 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.