Trace operator and the Dirichlet problem for elliptic equations on arbitrary bounded open sets

Abstract

We consider the Dirichlet problem on general, possibly nonsmooth bounded domain, for elliptic linear equation with uniformly elliptic divergence form operator. We investigate carefully the relationship between weak, soft and the Perron-Wiener-Brelot solutions of the problem. To this end, we extend the usual notion of the trace operator to Sobolev space $H^1(D)$ with $D$ being an arbitrary bounded open subset of $\mathbb{R}^d$. In the second part of the paper, we prove some existence results for the Dirichlet problem for semilinear equations with measure data on the right-hand side and $L^1$-data on the Martin boundary of $D$.