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

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 $C^\infty(\overline B \,\!^3;S^2)$ if, and only if, the distributional Jacobian of $u$ vanishes identically. This result was originally proved by Bethuel-Coron-Demengel-H\'elein, but we present a different strategy which is motivated by the $W^{2,p}$-case.