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arXiv · 2602.15701

Solving Dirichlet problem on unbounded uniform domains by using sphericalization techniques

Riikka Korte·Sari Rogovin·Nageswari Shanmugalingam·Timo Takala
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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.

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Riikka Korte, Sari Rogovin, Nageswari Shanmugalingam, Timo Takala. 2026-02-17. Solving Dirichlet problem on unbounded uniform domains by using sphericalization techniques. https://arxiv.org/abs/2602.15701

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