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arXiv · 1604.08731

The Dirichlet problem for p-harmonic functions with respect to arbitrary compactifications

Anders Björn·Jana Björn·Tomas Sjödin
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Abstract

We study the Dirichlet problem for p-harmonic functions on metric spaces with respect to arbitrary compactifications. A particular focus is on the Perron method, and as a new approach to the invariance problem we introduce Sobolev-Perron solutions. We obtain various resolutivity and invariance results, and also show that most functions that have earlier been proved to be resolutive are in fact Sobolev-resolutive. We also introduce (Sobolev)-Wiener solutions and harmonizability in this nonlinear context, and study their connections to (Sobolev)-Perron solutions, partly using Q-compactifications.

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Anders Björn, Jana Björn, Tomas Sjödin. 2016-04-29. The Dirichlet problem for p-harmonic functions with respect to arbitrary compactifications. https://doi.org/10.4171/rmi%2F1025

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