The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary

Abstract

In this paper we develop the Perron method for solving the Dirichlet problem for the analog of the p-Laplacian, i.e. for p-harmonic functions, with Mazurkiewicz boundary values. The setting considered here is that of metric spaces, where the boundary of the domain in question is replaced with the Mazurkiewicz boundary. Resolutivity for Sobolev and continuous functions, as well as invariance results for perturbations on small sets, are obtained. We use these results to improve the known resolutivity and invariance results for functions on the standard (metric) boundary. We also illustrate the results of this paper by discussing several examples.

References in corpus (3)

• Prime ends for domains in metric spaces
• The Dirichlet problem for p-harmonic functions on the topologist's comb
• The Mazurkiewicz distance and sets that are finitely connected at the boundary

Cited by in corpus (13)

• Prime ends for domains in metric spaces
• The annular decay property and capacity estimates for thin annuli
• The Mazurkiewicz distance and sets that are finitely connected at the boundary
• Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular metric spaces
• The Perron method for $p$-harmonic functions in unbounded sets in ${\mathbf{R}}^n$ and metric spaces
• The Dirichlet problem for p-harmonic functions with respect to arbitrary compactifications
• Prime ends in the Heisenberg group ${\mathbb{H}}_1$ and the boundary behavior of quasiconformal mappings
• Resolutivity and invariance for the Perron method for degenerate equations of divergence type
• Boundary regularity for $p$-harmonic functions and solutions of obstacle problems on unbounded sets in metric spaces
• Prime ends in metric spaces and boundary extensions of mappings
• The variational 1-capacity and BV functions with zero boundary values on metric spaces
• The prime end capacity of inaccessible prime ends, resolutivity, and the Kellogg property
• The Perron method associated with finely $p$-harmonic functions on finely open sets