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

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