Uniform Regularity Estimates in Homogenization Theory of Elliptic Systems with Lower Order Terms on the Neumann Boundary Problem

In this paper, we mainly employed the idea of the previous paper to study the sharp uniform $W^{1,p}$ estimates with $1<p\le \mathrm{\infty }$ for more general elliptic systems with the Neumann boundary condition on a bounded $C^{1,\eta }$ domain, arising in homogenization theory. Based on the skills developed by Z. Shen and by T. Suslina for different purposes, we also established the $L^2$ convergence rates on a bounded $C^{1,1}$ domain and a Lipschitz domain, respectively. Here we found a “rough” version of the first order correctors (see Theorem 1.3), It allows us to skip the corresponding convergence results on ${\mathbb{R}}^d$ that are the preconditions in T. Suslina’s papers.