## Journal of the Mathematical Society of Japan

### Boundary regularity for p-harmonic functions and solutions of the obstacle problem on metric spaces

#### Abstract

We study $p$-harmonic functions in complete metric spaces equipped with a doubling Borel measure supporting a weak $(1,p)$-Poincaré inequality, $1. We establish the barrier classification of regular boundary points from which it also follows that regularity is a local property of the boundary. We also prove boundary regularity at the fixed (given) boundary for solutions of the one-sided obstacle problem on bounded open sets. Regularity is further characterized in several other ways.

Our results apply also to Cheeger $p$-harmonic functions and in the Euclidean setting to $A$-harmonic functions, with the usual assumptions on $A$.

#### Article information

Source
J. Math. Soc. Japan, Volume 58, Number 4 (2006), 1211-1232.

Dates
First available in Project Euclid: 21 May 2007

https://projecteuclid.org/euclid.jmsj/1179759546

Digital Object Identifier
doi:10.2969/jmsj/1179759546

Mathematical Reviews number (MathSciNet)
MR2276190

Zentralblatt MATH identifier
1211.35109

#### Citation

BJÖRN, Anders; BJÖRN, Jana. Boundary regularity for p-harmonic functions and solutions of the obstacle problem on metric spaces. J. Math. Soc. Japan 58 (2006), no. 4, 1211--1232. doi:10.2969/jmsj/1179759546. https://projecteuclid.org/euclid.jmsj/1179759546

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