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

Abstract

We consider the Dirichlet problem and the weak Dirichlet problem on a general, possibly nonregular bounded domain, for elliptic linear equation with uniformly elliptic divergence form operator. We investigate carefully the relationship between weak, integral, probabilistic, soft and the Perron-Wiener-Brelot solutions of the problem. We also study the relationship between solutions of the Dirichlet problem and weak Dirichlet problem. To this end, we extend the usual notion of the trace operator and consider the Dirichlet problem with Martin boundary.