Illinois Journal of Mathematics
- Illinois J. Math.
- Volume 46, Number 3 (2002), 857-883.
Nonlinear potential theory on metric spaces
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.
Illinois J. Math., Volume 46, Number 3 (2002), 857-883.
First available in Project Euclid: 13 November 2009
Permanent link to this document
Digital Object Identifier
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. https://projecteuclid.org/euclid.ijm/1258130989