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

TL;DR

This paper extends the trace operator concept to arbitrary bounded domains and explores the relationships between different solutions of elliptic Dirichlet problems, including existence results for semilinear equations with measure and boundary data.

Contribution

It introduces a generalized trace operator for Sobolev spaces on arbitrary bounded open sets and analyzes solution concepts for elliptic Dirichlet problems in nonsmooth domains.

Findings

Extended trace operator to Sobolev spaces on arbitrary domains

Established relationships between weak, soft, and Perron solutions

Proved existence results for semilinear equations with measure and boundary data

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 $D$.