The Sharp $C^{1,1}$ Target Threshold for Minimizing Constraint Maps

Let $K\subset\R^m$ be the closure of a bounded domain whose boundary is a compact embedded hypersurface, and let \( \overline M=\R^m\setminus\operatorname{int}K \) be the allowed target. We prove that if $\partial K$ is of class $C^{1,1}$, then every $\overline M$-valued local minimizer of the Dir