Strong density for higher order Sobolev spaces into compact manifolds

TL;DR

This paper proves the density of smooth Sobolev maps into compact manifolds under certain homotopy conditions, extending approximation results in higher order Sobolev spaces.

Contribution

It establishes the density of smooth maps in Sobolev spaces into compact manifolds for higher order derivatives, with new results on maps with singularities.

Findings

Smooth maps are dense in Sobolev spaces when certain homotopy groups are trivial.

Maps with controlled singularities are dense without homotopy restrictions.

Extension of density results to higher order Sobolev spaces and singular maps.

Abstract

Given a compact manifold $N^n$, an integer $k \in \mathbb{N}_*$ and an exponent $1 \le p < \infty$, we prove that the class $C^\infty(\overline{Q}^m; N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{k, p}(Q^m; N^n)$ when the homotopy group $\pi_{\lfloor kp \rfloor}(N^n)$ of order $\lfloor kp \rfloor$ is trivial. We also prove the density of maps that are smooth except for a set of dimension $m - \lfloor kp \rfloor - 1$, without any restriction on the homotopy group of $N^n$