Approximations by regular sets and Wiener solutions in metric spaces

Anders Björn; Jana Björn

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 2, page 343-355
  • ISSN: 0010-2628

Abstract

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Let X be a complete metric space equipped with a doubling Borel measure supporting a weak Poincaré inequality. We show that open subsets of X can be approximated by regular sets. This has applications in nonlinear potential theory on metric spaces. In particular it makes it possible to define Wiener solutions of the Dirichlet problem for p -harmonic functions and to show that they coincide with three other notions of generalized solutions.

How to cite

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Björn, Anders, and Björn, Jana. "Approximations by regular sets and Wiener solutions in metric spaces." Commentationes Mathematicae Universitatis Carolinae 48.2 (2007): 343-355. <http://eudml.org/doc/250233>.

@article{Björn2007,
abstract = {Let $X$ be a complete metric space equipped with a doubling Borel measure supporting a weak Poincaré inequality. We show that open subsets of $X$ can be approximated by regular sets. This has applications in nonlinear potential theory on metric spaces. In particular it makes it possible to define Wiener solutions of the Dirichlet problem for $p$-harmonic functions and to show that they coincide with three other notions of generalized solutions.},
author = {Björn, Anders, Björn, Jana},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {axiomatic potential theory; capacity; corkscrew; Dirichlet problem; doubling; metric space; nonlinear; $p$-harmonic; Poincaré inequality; quasiharmonic; quasisuperharmonic; quasiminimizer; quasisuperminimizer; regular set; Wiener solution; axiomatic potential theory; capacity; corkscrew; Dirichlet problem; doubling; metric space; nonlinear; -harmonic},
language = {eng},
number = {2},
pages = {343-355},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Approximations by regular sets and Wiener solutions in metric spaces},
url = {http://eudml.org/doc/250233},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Björn, Anders
AU - Björn, Jana
TI - Approximations by regular sets and Wiener solutions in metric spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 2
SP - 343
EP - 355
AB - Let $X$ be a complete metric space equipped with a doubling Borel measure supporting a weak Poincaré inequality. We show that open subsets of $X$ can be approximated by regular sets. This has applications in nonlinear potential theory on metric spaces. In particular it makes it possible to define Wiener solutions of the Dirichlet problem for $p$-harmonic functions and to show that they coincide with three other notions of generalized solutions.
LA - eng
KW - axiomatic potential theory; capacity; corkscrew; Dirichlet problem; doubling; metric space; nonlinear; $p$-harmonic; Poincaré inequality; quasiharmonic; quasisuperharmonic; quasiminimizer; quasisuperminimizer; regular set; Wiener solution; axiomatic potential theory; capacity; corkscrew; Dirichlet problem; doubling; metric space; nonlinear; -harmonic
UR - http://eudml.org/doc/250233
ER -

References

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