The Dirichlet problem for p-harmonic functions on the topologist's comb

TL;DR

This paper investigates the Perron method for solving the p-harmonic Dirichlet problem on the topologist's comb, demonstrating invariance of solutions under certain perturbations and jumps at inaccessible points.

Contribution

It introduces new invariance results for Perron solutions on complex domains with perturbations at inaccessible points.

Findings

Perron solutions are invariant under arbitrary perturbations at inaccessible points.

Results allow for jumps and perturbations at countable sets of points.

The study extends understanding of p-harmonic functions on fractal-like domains.

Abstract

In this paper we study the Perron method for solving the p-harmonic Dirichlet problem on the topologist's comb. For functions which are bounded and continuous at the accessible points, we obtain invariance of the Perron solutions under arbitrary perturbations on the set of inaccessible points. We also obtain some results allowing for jumps and perturbations at a countable set of points.