Density of Lipschitz functions in energy

Abstract

In this paper, we show that the density in energy of Lipschitz functions in a Sobolev space N1,p(X) holds for all p∈[1,∞) whenever the space X is complete and separable and the measure is Radon and positive and finite on balls. Emphatically, p=1 is allowed. We also give a few corollaries and pose questions for future work. The proof is direct and does not involve the usual flow techniques from prior work. It also yields a new approximation technique, which has not appeared in prior work. Notable with all of this work is that we do not use any form of Poincaré inequality or doubling assumption. The techniques are flexible and suggest a unification of a variety of approaches that have appeared in the literature on the topic.