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arXiv · 1705.02253

Poincar\'e inequalities and Newtonian Sobolev functions on noncomplete metric spaces

Anders Björn·Jana Björn
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Abstract

Let $X$ be a noncomplete metric space satisfying the usual (local) assumptions of a doubling property and a Poincar\'e inequality. We study extensions of Newtonian Sobolev functions to the completion $\widehat{X}$ of $X$ and use them to obtain several results on $X$ itself, in particular concerning minimal weak upper gradients, Lebesgue points, quasicontinuity, regularity properties of the capacity and better Poincar\'e inequalities. We also provide a discussion about possible applications of the completions and extension results to $p$-harmonic functions on noncomplete spaces and show by examples that this is a rather delicate issue opening for various interpretations and new investigations.

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Anders Björn, Jana Björn. 2017-05-05. Poincar\'e inequalities and Newtonian Sobolev functions on noncomplete metric spaces. https://doi.org/10.1016/j.jde.2018.07.029

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