The Dirichlet problem for p-harmonic functions on metric spaces

A Bjorn, Jana Björn, N Shanmugalingam

Research output: Contribution to journalArticlepeer-review

Abstract

We study the Dirichlet problem for p-harmonic functions (and p-energy minimizers) in bounded domains in proper, pathconnected metric measure spaces equipped with a doubling measure and supporting a Poincare inequality. The Dirichlet problem has previously been solved for Sobolev type boundary data, and we extend this result and solve the problem for all continuous boundary data. We study the regularity of boundary points and prove the Kellogg property, i.e. that the set of irregular boundary points has zero p-capacity. We also construct p-capacitary, p-singular and p-harmonic measures on the boundary. We show that they are all absolutely continuous with respect to the p-capacity. For p = 2 we show that all the boundary measures are comparable and that the singular and harmonic measures coincide. We give an integral representation for the solution to the Dirichlet problem when p = 2, enabling us to extend the solvability of the problem to L-1 boundary data in this case. Moreover, we give a trace result for Newtonian functions when p = 2. Finally, we give an estimate for the Hausdorff dimension of the boundary of a bounded domain in Ahlfors Q-regular spaces.
Original languageEnglish
Pages (from-to)173-203
JournalJournal für Die Reine und Angewandte Mathematik
Volume556
Publication statusPublished - 2003

Subject classification (UKÄ)

  • Mathematics

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