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The Dirichlet problem for p-harmonic functions on metric spaces
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.ORCID iD: 0000-0002-9677-8321
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.ORCID iD: 0000-0002-1238-6751
2003 (English)In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, no 556, 173-203 p.Article in journal (Refereed) Published
Abstract [en]

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 PoincarΘ 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 L1 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.

Place, publisher, year, edition, pages
2003. no 556, 173-203 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-23017Local ID: 2394OAI: oai:DiVA.org:liu-23017DiVA: diva2:243330
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2017-12-13

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http://www.degruyter.de/journals/crelle/2003/pdf/556_173.pdf

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Björn, AndersBjörn, Jana

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