Solving a Dirichlet problem for unbounded domains via a conformal transformation

TL;DR

This paper develops a method using conformal transformations to solve the p-Dirichlet problem on unbounded uniform domains with boundary data in Besov spaces, by transforming them into bounded domains where existing results apply.

Contribution

It introduces a class of conformal transformations that render unbounded uniform domains bounded while preserving key analytical properties, enabling the transfer of known results.

Findings

Transformed measures remain doubling

Transformed domains support Poincaré inequalities

Solution to Dirichlet problem for Besov boundary data

Abstract

In this paper, we solve the $p$-Dirichlet problem for Besov boundary data on unbounded uniform domains with bounded boundaries when the domain is equipped with a doubling measure satisfying a Poincar\'{e} inequality. This is accomplished by studying a class of transformations that have been recently shown to render the domain bounded while maintaining uniformity. These transformations conformally deform the metric and measure in a way that depends on the distance to the boundary of the domain and, for the measure, a parameter $p$. We show that the transformed measure is doubling and the transformed domain supports a Poincar\'{e} inequality. This allows us to transfer known results for bounded uniform domains to unbounded ones, including trace results and Adams-type inequalities, culminating in a solution to the Dirichlet problem for boundary data in a Besov class.