Sobolev mappings: Lipschitz density is not an isometric invariant of the target

TL;DR

This paper investigates the density of Lipschitz mappings in Sobolev spaces of mappings from a manifold to a metric space, showing that this density property depends on the specific isometric embedding of the target space.

Contribution

It demonstrates that Lipschitz density in Sobolev spaces is not an isometric invariant of the target space, revealing a dependence on the embedding.

Findings

Lipschitz density can vary with different isometric embeddings.

The property of Lipschitz density is not preserved under isometric isomorphisms.

The result impacts the understanding of Sobolev mappings into metric spaces.

Abstract

If $M$ is a compact smooth manifold and $X$ is a compact metric space, the Sobolev space $W^{1,p}(M,X)$ is defined through an isometric embedding of $X$ into a Banach space. We prove that the answer to the question whether Lipschitz mappings ${\rm Lip}\,(M,X)$ are dense in $W^{1,p}(M,X)$ may depend on the isometric embedding of the target.