Density of Lipschitz functions in Energy

TL;DR

This paper proves that Lipschitz functions are dense in energy within Sobolev spaces on certain metric measure spaces, using a novel, direct approximation method that does not rely on common assumptions like Poincaré inequalities or doubling measures.

Contribution

It introduces a new approximation technique for Lipschitz functions in Sobolev spaces that works without Poincaré or doubling conditions, unifying various existing results.

Findings

Lipschitz functions are dense in energy in $N^{1,p}(X)$ for all $p\in [1,\infty)$.

The proof is direct and does not use flow techniques or Poincaré inequalities.

A new flexible approximation method is developed that broadens the scope of existing theories.

Abstract

In this paper, we show that the 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 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 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.