Uniform regularity for degenerate elliptic equations in perforated domains

TL;DR

This paper studies degenerate elliptic equations with oscillating coefficients in perforated domains, establishing convergence rates and uniform regularity estimates, which are crucial for understanding spectral problems in such complex geometries.

Contribution

It introduces new quantitative convergence rates and uniform regularity estimates for degenerate elliptic equations in perforated domains, advancing the analysis of spectral problems.

Findings

Established a quantitative convergence rate.

Obtained uniform weighted Lipschitz estimates.

Derived $W^{1,p}$ estimates for solutions.

Abstract

This paper is concerned with a class of degenerate elliptic equations with rapidly oscillating coefficients in periodically perforated domains, which arises in the study of spectrum problems for uniformly elliptic equations in perforated domains. We establish a quantitative convergence rate and obtain the uniform weighted Lipschitz and $W^{1,p}$ estimates.