Density of continuous functions in Sobolev spaces with applications to capacity

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.