Density of continuous functions in Sobolev spaces with applications to capacity (2303.00649v3)

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.