Journal of the Mathematical Society of Japan

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

Anders BJÖRN and Jana BJÖRN

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We study p -harmonic functions in complete metric spaces equipped with a doubling Borel measure supporting a weak ( 1 , p ) -Poincaré inequality, 1 < p < . 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

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

First available in Project Euclid: 21 May 2007

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Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 31C45: Other generalizations (nonlinear potential theory, etc.) 35B65: Smoothness and regularity of solutions 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 49N60: Regularity of solutions

barrier doubling metric space nonlinear obstacle problem p-harmonic Poincaré inequality regular superharmonic


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.

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