Density of bounded maps in Sobolev spaces into complete manifolds

TL;DR

This paper studies the density of bounded Sobolev maps into complete manifolds within Sobolev spaces, revealing conditions under which bounded maps are dense, especially highlighting differences when the integrability exponent is an integer.

Contribution

It establishes a quantitative trimming property equivalent to density for integer p and provides conditions ensuring density of smooth maps in Sobolev spaces.

Findings

Density holds for non-integer p.

A trimming property characterizes density for integer p.

Uniform Lipschitz geometry ensures density.

Abstract

Given a complete noncompact Riemannian manifold $N^n$, we investigate whether the set of bounded Sobolev maps $(W^{1, p} \cap L^\infty) (Q^m; N^n)$ on the cube $Q^m$ is strongly dense in the Sobolev space $W^{1, p} (Q^m; N^n)$ for $1 \le p \le m$. The density always holds when $p$ is not an integer. When $p$ is an integer, the density can fail, and we prove that a quantitative trimming property is equivalent with the density. This new condition is ensured for example by a uniform Lipschitz geometry of $N^n$. As a byproduct, we give necessary and sufficient conditions for the strong density of the set of smooth maps $C^\infty (\overline{Q^m}; N^n)$ in $W^{1, p} (Q^m; N^n)$.