Geodesic
Approximations by regular sets and Wiener solutions in metric spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 2, pp. 343-355

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Let $X$ be a complete metric space equipped with a doubling Borel measure supporting a weak Poincaré 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.
Classification : 31C45, 31D05, 35J70, 49J27
Keywords: axiomatic potential theory; capacity; corkscrew; Dirichlet problem; doubling; metric space; nonlinear; $p$-harmonic; Poincaré inequality; quasiharmonic; quasisuperharmonic; quasiminimizer; quasisuperminimizer; regular set; Wiener solution 
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     title = {Approximations by regular sets and {Wiener} solutions in metric spaces},
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Björn, Anders; Björn, Jana. Approximations by regular sets and Wiener solutions in metric spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 2, pp. 343-355. http://geodesic.mathdoc.fr/item/CMUC_2007__48_2_a14/