Uniform regularity estimates in homogenization theory of elliptic systems with lower order terms on the Neumann boundary problem

Abstract

In this paper, we mainly employed the idea of the previous paper [34] to study the sharp uniform W1,p estimates with 1<p≤∞ for more general elliptic systems with the Neumann boundary condition on a bounded C1,η domain, arising in homogenization theory. Based on the skills developed by Z. Shen in [27] and by T. Suslina in [31,32], we also established the L2 convergence rates on a bounded C1,1 domain and a Lipschitz domain, respectively. Here we found a “rough” version of the first order correctors (see (1.12)), which can unify the proof in [27] and [32]. It allows us to skip the corresponding convergence results on Rd that are the preconditions in [31,32]. Our results can be regarded as an extension of [23] developed by C. Kenig, F. Lin, Z. Shen, as well as of [32] investigated by T. Suslina.