Advanced

The Dirichlet problem for p-harmonic functions on metric spaces

Bjorn, A; Björn, Jana LU and Shanmugalingam, N (2003) In Journal für Die Reine und Angewandte Mathematik 556. p.173-203
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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
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
0075-4102
language
English
LU publication?
yes
id
bedfae94-0a79-43d3-868a-485dd127a80a (old id 314922)
alternative location
http://www.degruyter.com/journals/crelle/2003/556_173.html
date added to LUP
2007-08-24 11:37:48
date last changed
2018-01-07 08:39:49
@article{bedfae94-0a79-43d3-868a-485dd127a80a,
  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.},
  author       = {Bjorn, A and Björn, Jana and Shanmugalingam, N},
  issn         = {0075-4102},
  language     = {eng},
  pages        = {173--203},
  publisher    = {De Gruyter},
  series       = {Journal für Die Reine und Angewandte Mathematik},
  title        = {The Dirichlet problem for p-harmonic functions on metric spaces},
  volume       = {556},
  year         = {2003},
}