HÖLDER CONTINUITY OF THE DIRICHLET SOLUTION FOR A GENERAL DOMAIN

Abstract

Let $D$ be a bounded domain in ${\bb R}^n$ . For a function $f$ on the boundary $\partial D$ , the Dirichlet solution of $f$ over $D$ is denoted by $H_D f$ , provided that such a solution exists. Conditions on $D$ for $H_D$ to transform a Hölder continuous function on $\partial D$ to a Hölder continuous function on $D$ with the same Hölder exponent are studied. In particular, it is demonstrated here that there is no bounded domain that preserves the Hölder continuity with exponent 1. It is also also proved that a bounded regular domain $D$ preserves the Hölder continuity with some exponent $\alpha$ , $0 < \alpha < 1$ , if and only if $\partial D$ satisfies the capacity density condition, which is equivalent to the uniform perfectness of $\partial D$ if $n = 2$ .