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

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.

16 pages, 1 figure

References in corpus (1)

• Prime ends for domains in metric spaces

Cited by in corpus (5)

• The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary
• Poincaré inequalities and Newtonian Sobolev functions on noncomplete metric spaces
• The Perron method for $p$-harmonic functions in unbounded sets in ${\mathbf{R}}^n$ and metric spaces
• Boundary regularity for $p$-harmonic functions and solutions of obstacle problems on unbounded sets in metric spaces
• The Perron method associated with finely $p$-harmonic functions on finely open sets