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In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Carath\'eodory and N\"akki. Modulus ends and prime ends, defined by means of the \p-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.Comment: 46 pages, 4 figure

We use the Perron method to construct and study solutions of the Dirichlet problem for p-harmonic functions in proper metric measure spaces endowed with a doubling Borel measure supporting a weak (1,q)-Poincaré inequality (for some 1q<p). The upper and lower Perron solutions are constructed for functions defined on the boundary of a bounded domain and it is shown that these solutions are p-harmonic in the domain. It is also shown that Newtonian (Sobolev) functions and continuous functions are resolutive, i.e. that their upper and lower Perron solutions coincide, and that their Perron solutions are invariant under perturbations of the function on a set of capacity zero. We further study the problem of resolutivity and invariance under perturbations for semicontinuous functions. We also characterize removable sets for bounded p-(super)harmonic functions.Original Publication:Anders Björn, Jana Björn and Nageswari Shanmugalingam, The Perron method for p-harmonic functions in metric spaces, 2003, Journal of Differential Equations, (195), 2, 398-429.http://dx.doi.org/10.1016/S0022-0396(03)00188-8Copyright: Elsevier Science B.V., Amsterdamhttp://www.elsevier.com

We study p-harmonic functions in complete metric spaces equipped with a doubling Borel measure supporting a weak (1, p)-Poincaré inequality, 1 < p < ∞. We establish the barrier classification of regular boundary points from which it also follows that regularity is a local property of the boundary. We also prove boundary regularity at the fixed (given) boundary for solutions of the onesided obstacle problem on bounded open sets. Regularity is further characterized in several other ways.Our results apply also to Cheeger p-harmonic functions and in the Euclidean setting to A -harmonic functions, with the usual assumptions on A .

In this paper we develop the Perron method for solving the Dirichlet problem for the analog of the p-Laplacian, i.e. for p-harmonic functions, with Mazurkiewicz boundary values. The setting considered here is that of metric spaces, where the boundary of the domain in question is replaced with the Mazurkiewicz boundary. Resolutivity for Sobolev and continuous functions, as well as invariance results for perturbations on small sets, are obtained. We use these results to improve the known resolutivity and invariance results for functions on the standard (metric) boundary. We also illustrate the results of this paper by discussing several examples.