Solving Dirichlet problem on unbounded uniform domains by using sphericalization techniques

TL;DR

This paper extends the existence theory for Dirichlet problems of p-harmonic functions to unbounded uniform domains in metric spaces by employing sphericalization techniques, addressing challenges posed by unbounded domains.

Contribution

It introduces the use of sphericalization to establish existence of solutions in unbounded domains, a method not previously applicable in this setting.

Findings

Existence of solutions in unbounded uniform domains with unbounded boundary.

Analysis of solution uniqueness related to p-parabolicity and p-hyperbolicity.

Application of sphericalization techniques to unbounded domain problems.

Abstract

Within the setting of metric spaces equipped with a doubling measure and supporting a $p$-Poincar\'e inequality, establishing existence of solutions to Dirichlet problem in a bounded domain in such a metric space is accomplished via direct methods of calculus of variation and the use of a Maz'ya type inequality, which is a consequence of the Poincar\'e inequality. However, when the domain and its boundary are unbounded, such a method is unavailable. In this paper, using the technique of sphericalization developed in the prior paper~[32], we establish the existence of solutions to the Dirichlet boundary value problem for $p$-harmonic functions in unbounded uniform domains with unbounded boundary when $1<p<\infty$. We also explore the issue of whether such solutions are unique by considering $p$-parabolicity and $p$-hyperbolicity properties of the domain.