Lipschitz regularity of homogenization with continuous coefficients: Dirichlet problem

TL;DR

This paper establishes uniform Lipschitz regularity estimates for elliptic systems with continuous, Dini mean oscillation coefficients, extending classical homogenization results by weakening regularity assumptions on data.

Contribution

It generalizes previous regularity results by reducing the regularity conditions to integral conditions, broadening applicability in homogenization theory.

Findings

Lipschitz regularity holds under Dini mean oscillation of coefficients

Extension of Avellaneda and Lin's results to weaker data conditions

Applicable to elliptic systems with continuous coefficients in homogenization

Abstract

We study uniform Lipschitz regularity estimates for elliptic systems in divergence form with continuous coefficients, based on rapidly oscillating periodic coefficients derived from homogenization theory. We extend a result by Avellaneda and Lin [Comm. Pure Appl. Math. 40 (1987), pp. 803-847] by minimizing all regularity conditions of the given data to integral conditions. We remark that the coefficients of an elliptic operator have Dini mean oscillation, which corresponds to the results of the latest general regularity theory.