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Prime ends for domains in metric spaces
, 2012
"... Abstract. In this paper we propose a new definition of prime ends for domains in met-ric spaces under rather general assumptions. We compare our prime ends to those of Carathéodory and Näkki. Modulus ends and prime ends, defined by means of the p-modulus of curve families, are also discussed and r ..."
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Abstract. In this paper we propose a new definition of prime ends for domains in met-ric spaces under rather general assumptions. We compare our prime ends to those of Carathéodory and Näkki. 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.
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, 2014
"... In this paper we make a survey of some recent developments of the theory of Sobolev spaces W 1,q(X, d,m), 1 < q <∞, in metric measure spaces (X, d,m). In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on Γ-convergence; this result extends Cheeg ..."
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In this paper we make a survey of some recent developments of the theory of Sobolev spaces W 1,q(X, d,m), 1 < q <∞, in metric measure spaces (X, d,m). In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on Γ-convergence; this result extends Cheeger’s work because no Poincare ́ in-equality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of m. We also discuss the lower semiconti-nuity of the slope of Lipschitz functions and some open problems.
