Uniformly Elliptic Equations on Domains with Capacity Density Conditions: Existence of H\"{o}lder Continuous Solutions and Homogenization Results

TL;DR

This paper investigates uniformly elliptic equations on CDC domains, establishing the existence of Hölder continuous solutions and analyzing homogenization with convergence rate estimates for periodic coefficients.

Contribution

It extends known results by proving Hölder continuity of solutions and providing homogenization convergence rates on CDC domains.

Findings

Existence of globally Hölder continuous solutions on CDC domains

Homogenization convergence rate estimates for periodic coefficients

Summary of properties and inequalities related to CDC domains

Abstract

This is a progress report on study of uniformly elliptic Poisson-type equations on domains with capacity density conditions (CDC domains). We give a brief summary of known facts of CDC domains, including Hardy's inequality, and review a previous work of existence of globally H\"{o}lder continuous solutions. Additionally, we apply the result to homogenization problems of $\epsilon$-periodic coefficients and present a convergence rate estimate of $L^{\infty}$ norms.