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Solving Dirichlet problem on unbounded uniform domains by using sphericalization techniques

Riikka Korte, Sari Rogovin, Nageswari Shanmugalingam, Timo Takala

TL;DR

The paper addresses solving the Dirichlet problem for $p$-harmonic functions on unbounded uniform domains within metric measure spaces that are doubling and support a $p$-Poincaré inequality. It develops a sphericalization framework using a metric density $\rho$ to transform unbounded domains into bounded ones, preserving uniformity, doubling, and Poincaré properties, and enabling a Maz'ya-type energy inequality on the transformed space. The main contributions are: (i) existence and uniqueness of $p$-harmonic solutions with boundary traces in the homogeneous Besov space $HB^{1-\theta/p}_{p,p}(\partial\Omega,d,\nu)$, (ii) a precise transfer of Besov trace energies under sphericalization, and (iii) volume-growth criteria linking $p$-parabolicity and $p$-hyperbolicity to boundary behavior at infinity and their effect on uniqueness. The results extend Dirichlet problem theory to unbounded domains in nonsmooth metric spaces and provide concrete density choices and examples for implementation. The work has implications for understanding large-scale boundary behavior via transformed, bounded models and highlights how geometric growth conditions control solvability and uniqueness.

Abstract

Within the setting of metric spaces equipped with a doubling measure and supporting a $p$-Poincaré 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 Poincaré 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.

Solving Dirichlet problem on unbounded uniform domains by using sphericalization techniques

TL;DR

The paper addresses solving the Dirichlet problem for -harmonic functions on unbounded uniform domains within metric measure spaces that are doubling and support a -Poincaré inequality. It develops a sphericalization framework using a metric density to transform unbounded domains into bounded ones, preserving uniformity, doubling, and Poincaré properties, and enabling a Maz'ya-type energy inequality on the transformed space. The main contributions are: (i) existence and uniqueness of -harmonic solutions with boundary traces in the homogeneous Besov space , (ii) a precise transfer of Besov trace energies under sphericalization, and (iii) volume-growth criteria linking -parabolicity and -hyperbolicity to boundary behavior at infinity and their effect on uniqueness. The results extend Dirichlet problem theory to unbounded domains in nonsmooth metric spaces and provide concrete density choices and examples for implementation. The work has implications for understanding large-scale boundary behavior via transformed, bounded models and highlights how geometric growth conditions control solvability and uniqueness.

Abstract

Within the setting of metric spaces equipped with a doubling measure and supporting a -Poincaré 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 Poincaré 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 -harmonic functions in unbounded uniform domains with unbounded boundary when . We also explore the issue of whether such solutions are unique by considering -parabolicity and -hyperbolicity properties of the domain.
Paper Structure (9 sections, 23 theorems, 147 equations)

This paper contains 9 sections, 23 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Definitions and preliminary results
  3. Conformal deformations of the metric and measure
  4. The codimensionality of and
  5. Quasi-preservation of Besov energy
  6. Notions of Dirichlet boundary condition
  7. Solving the Dirichlet problem in unbounded uniform domains: proof of Theorem \ref{['thm:Main']}
  8. Examples
  9. -hyperbolicity and -parabolicity

Key Result

Theorem 1.1

Let $p > 1$ and suppose that $(\Omega, d,\mu)$ satisfies the structural assumptions listed above. For each $f\in HB^{1-\theta/p}_{p,p}(\partial\Omega,d,\nu)$ there is a unique function $u_f\in D^{1,p}(\Omega,d,\mu)$ that is $p$-harmonic in $(\Omega,d,\mu)$ and $T u_f = f$$\nu$-a.e. on $\partial\Omeg

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • ...and 45 more