Uniformly Elliptic Equations on Domains with Capacity Density Conditions: Existence of Hölder Continuous Solutions and Homogenization Results

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 Hölder continuous solutions. Additionally, we apply the result to homogenization problems of $\epsilon$-periodic coefficients and present a convergence rate estimate of $L^{\mathrm{\infty }}$ norms.