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 Holder continuous function on $\partial D$ to a Holder continuous function on $D$ with the same Holder exponent are studied. In particular, it is demonstrated here that there is no bounded domain that preserves the Holder continuity with exponent 1. It is also also proved that a bounded regular domain $D$ preserves the Holder 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 $n = 2$ .