Illinois Journal of Mathematics

Nonlinear potential theory on metric spaces

Juha Kinnunen and Olli Martio

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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.

Article information

Illinois J. Math., Volume 46, Number 3 (2002), 857-883.

First available in Project Euclid: 13 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 31C45: Other generalizations (nonlinear potential theory, etc.)
Secondary: 31C15: Potentials and capacities 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems


Kinnunen, Juha; Martio, Olli. Nonlinear potential theory on metric spaces. Illinois J. Math. 46 (2002), no. 3, 857--883. doi:10.1215/ijm/1258130989.

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