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arXiv · 2303.00649

Density of continuous functions in Sobolev spaces with applications to capacity

Sylvester Eriksson-Bique·Pietro Poggi-Corradini
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Abstract

We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if $(X,d,\mu)$ is a locally complete and separable metric measure space, then continuous functions are dense in the Newtonian space $N^{1,p}(X)$. Here the measure $\mu$ is Borel and is finite and positive on all metric balls. In particular, we don't assume properness of $X$, doubling of $\mu$ or any Poincar\'e inequalities. These resolve, partially or fully, questions posed by a number of authors, including J. Heinonen, A. Bj\"orn and J. Bj\"orn. In contrast to much of the past work, our results apply to locally complete spaces $X$ and dispenses with the frequently used regularity assumptions: doubling, properness, Poincar\'e inequality, Loewner property or quasiconvexity.

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Sylvester Eriksson-Bique, Pietro Poggi-Corradini. 2023-03-01. Density of continuous functions in Sobolev spaces with applications to capacity. https://arxiv.org/abs/2303.00649

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