The Dirichlet problem at the Martin boundary of a fine domain

TL;DR

This paper extends the Perron-Wiener-Brelot method to solve the Dirichlet problem at the Martin boundary specifically for fine domains in Euclidean space, broadening potential theory applications.

Contribution

It introduces a new approach for solving the Dirichlet problem at the Martin boundary within fine domains, enhancing classical potential theory techniques.

Findings

Successfully generalized the Perron-Wiener-Brelot method

Provided a framework for boundary value problems in fine domains

Expanded potential theory to more irregular domains

Abstract

We develop the Perron-Wiener-Brelot method of solving the Dirichlet problem at the Martin boundary of a fine domain in $\RR^n$ ($n\ge2$).