Equi-integrable approximation of Sobolev mappings between manifolds

TL;DR

This paper proves that sequences of smooth maps between compact Riemannian manifolds with equi-integrable Sobolev energy can be strongly approximated by smooth maps, extending density results to higher-order and fractional Sobolev spaces.

Contribution

It extends the density of smooth maps in Sobolev spaces to higher-order and fractional cases, providing new approximation techniques and proofs based on Jacobian continuity.

Findings

Strong approximation of Sobolev maps by smooth maps in higher-order spaces

Extension of density results to fractional Sobolev spaces

Use of weak Jacobian continuity in approximation proofs

Abstract

We show that limits of sequences of smooth maps between compact Riemannian manifolds with equi-integrable $W^{1, p}$-Sobolev energy can always be strongly approximated by smooth maps, giving a counterpart of Hang's density result in $W^{1, 1}$ for the Sobolev space $W^{1, p}$ with integer $p \ge 2$. Our result extends to higher-order Sobolev spaces and is straightforward in fractional Sobolev spaces. We also provide a proof based on the weak continuity of Jacobians in the cases where the cohomological criterion of Bethuel, Demengel, Colon and H\'elein applies.