Abstract:Let $X$ be a complete metric space equipped with a doubling Borel measure supporting a weak Poincar\'e inequality. We show that open subsets of $X$ can be approximated by regular sets. This has applications in nonlinear potential theory on metric spaces. In particular it makes it possible to define Wiener solutions of the Dirichlet problem for $p$-harmonic functions and to show that they coincide with three other notions of generalized solutions.
Keywords: axiomatic potential theory, capacity, corkscrew, Dirichlet problem, doubling, metric space, nonlinear, $p$-harmonic, Poincar\'e inequality, quasiharmonic, quasisuperharmonic, quasiminimizer, quasisuperminimizer, regular set, Wiener solution
AMS Subject Classification: Primary 31C45; Secondary 31D05, 49J27