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The Perron method for p-harmonic functions in metric spaces
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-9677-8321
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-1238-6751
University of Cincinnati.
2003 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 195, no 2, 398-429 p.Article in journal (Refereed) Published
Abstract [en]

We use the Perron method to construct and study solutions of the Dirichlet problem for p-harmonic functions in proper metric measure spaces endowed with a doubling Borel measure supporting a weak (1,q)-Poincaré inequality (for some 1q<p). The upper and lower Perron solutions are constructed for functions defined on the boundary of a bounded domain and it is shown that these solutions are p-harmonic in the domain. It is also shown that Newtonian (Sobolev) functions and continuous functions are resolutive, i.e. that their upper and lower Perron solutions coincide, and that their Perron solutions are invariant under perturbations of the function on a set of capacity zero. We further study the problem of resolutivity and invariance under perturbations for semicontinuous functions. We also characterize removable sets for bounded p-(super)harmonic functions.

Place, publisher, year, edition, pages
2003. Vol. 195, no 2, 398-429 p.
Keyword [en]
Dirichlet problem; Metric space; Nonlinear; Perron solution; p-harmonic; Sobolev function
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-18241DOI: 10.1016/S0022-0396(03)00188-8OAI: oai:DiVA.org:liu-18241DiVA: diva2:217146
Note
Original Publication: Anders Björn, Jana Björn and Nageswari Shanmugalingam, The Perron method for p-harmonic functions in metric spaces, 2003, Journal of Differential Equations, (195), 2, 398-429. http://dx.doi.org/10.1016/S0022-0396(03)00188-8 Copyright: Elsevier Science B.V., Amsterdam http://www.elsevier.com/ Available from: 2009-05-13 Created: 2009-05-13 Last updated: 2017-12-13Bibliographically approved

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