The Dirichlet problem for pharmonic functions on metric spaces
(2003) In Journal für Die Reine und Angewandte Mathematik 556. p.173203 Abstract
 We study the Dirichlet problem for pharmonic functions (and penergy 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 pcapacity. We also construct pcapacitary, psingular and pharmonic measures on the boundary. We show that they are all absolutely continuous with respect to the pcapacity. For p = 2 we show that all the boundary measures are... (More)
 We study the Dirichlet problem for pharmonic functions (and penergy 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 pcapacity. We also construct pcapacitary, psingular and pharmonic measures on the boundary. We show that they are all absolutely continuous with respect to the pcapacity. 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 Qregular spaces. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/314922
 author
 Bjorn, A; Björn, Jana ^{LU} and Shanmugalingam, N
 organization
 publishing date
 2003
 type
 Contribution to journal
 publication status
 published
 subject
 in
 Journal für Die Reine und Angewandte Mathematik
 volume
 556
 pages
 173  203
 publisher
 De Gruyter
 external identifiers

 wos:000182105900009
 scopus:0041668015
 ISSN
 00754102
 language
 English
 LU publication?
 yes
 id
 bedfae940a7943d3868a485dd127a80a (old id 314922)
 alternative location
 http://www.degruyter.com/journals/crelle/2003/556_173.html
 date added to LUP
 20070824 11:37:48
 date last changed
 20180107 08:39:49
@article{bedfae940a7943d3868a485dd127a80a, abstract = {We study the Dirichlet problem for pharmonic functions (and penergy 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 pcapacity. We also construct pcapacitary, psingular and pharmonic measures on the boundary. We show that they are all absolutely continuous with respect to the pcapacity. 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 Qregular spaces.}, author = {Bjorn, A and Björn, Jana and Shanmugalingam, N}, issn = {00754102}, language = {eng}, pages = {173203}, publisher = {De Gruyter}, series = {Journal für Die Reine und Angewandte Mathematik}, title = {The Dirichlet problem for pharmonic functions on metric spaces}, volume = {556}, year = {2003}, }