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 L ε = − div( A ( x/ε ) ∇ + V ( x/ε )) + B ( x/ε ) ∇ + c ( x/ε ) + λI with dimension two. Precisely, we will establish the W 1 ,p estimates, H¨older estimates, Lipschitz estimates and L p convergence results for L ε with dimension two. The operator L ε has been studied by Qiang Xu with dimension d ≥ 3 in [22, 23] and the case d = 2 is remained unsolved. As a byproduct, we will construct the Green functions for L ε with d = 2 and their convergence rates.