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Fall 2002 Nonlinear potential theory on metric spaces
Juha Kinnunen, Olli Martio
Illinois J. Math. 46(3): 857-883 (Fall 2002). DOI: 10.1215/ijm/1258130989


We study nonlinear potential theory on a metric measure space equipped with a doubling measure and supporting a Poincaré inequality. Minimizers, superminimizers and the obstacle problem for the $p$-Dirichlet integral play an important role in the theory. We prove lower semicontinuity of superminimizers and continuity of the solution to the obstacle problem with a continuous obstacle. We also show that the limit of an increasing sequence of superminimizers is a superminimizer provided it is bounded above. Moreover, we consider superharmonic functions and study their relations to superminimizers. Our proofs are based on the direct methods of the calculus of variations and on De Giorgi type estimates. In particular, we do not use the Euler-Lagrange equation and our arguments are based on the variational integral only.


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Juha Kinnunen. Olli Martio. "Nonlinear potential theory on metric spaces." Illinois J. Math. 46 (3) 857 - 883, Fall 2002.


Published: Fall 2002
First available in Project Euclid: 13 November 2009

zbMATH: 1030.46040
MathSciNet: MR1951245
Digital Object Identifier: 10.1215/ijm/1258130989

Primary: 31C45
Secondary: 31C15, 46E35

Rights: Copyright © 2002 University of Illinois at Urbana-Champaign


Vol.46 • No. 3 • Fall 2002
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