Density of Lipschitz functions in Energy

Sylvester Eriksson-Bique

Published 2020-12-03Version 1

In this paper, we show that density in energy of Lipschitz functions in a Sobolev space $N^{1,p}(X)$ holds for all $p\in [1,\infty)$ whenever the space $X$ is complete and separable and the measure is Radon and finite on balls. Emphatically, $p=1$ is allowed. We also give a few simple corollaries and pose questions for future work. The proof is direct and does not involve the usual flow techniques from prior work. Notable with all of this is that we do not use any form of Poincar\'e inequality or doubling assumption. The techniques are flexible and suggest a unification of a variety of existing literature on the topic.