Closure of Smooth Maps in $W^{1,p}(B^3;S^2)$

TL;DR

This paper establishes that for maps in the Sobolev space $W^{1,p}$ from the 3-ball to the sphere, smooth approximation is possible if and only if the distributional Jacobian vanishes, providing a new proof strategy inspired by the $W^{2,p}$ case.

Contribution

The paper offers a novel proof approach for the approximation of Sobolev maps into spheres, extending the understanding of Jacobian conditions for smooth approximation.

Findings

Smooth approximation holds iff the distributional Jacobian vanishes.

Provides a new proof strategy inspired by the $W^{2,p}$-case.

Clarifies the role of the Jacobian in approximation problems.

Abstract

For every $2 < p < 3$, we show that $u \in W^{1,p}(B^3; S^2)$ can be strongly approximated by maps in $C^\infty(\Bar{B}^3; S^2)$ if, and only if, the distributional Jacobian of $u$ vanishes identically. This result was originally proved by Bethuel-Coron-Demengel-Helein, but we present a different strategy which is motivated by the $W^{2,p}$-case.