## Illinois Journal of Mathematics

### Nonlinear potential theory on metric spaces

#### Abstract

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

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

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258130989

Digital Object Identifier
doi:10.1215/ijm/1258130989

Mathematical Reviews number (MathSciNet)
MR1951245

Zentralblatt MATH identifier
1030.46040

#### Citation

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