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

Let $K\subset {\mathbb{R}}^m$ be the closure of a bounded domain whose boundary is a compact embedded hypersurface, and let $\bar{M}={\mathbb{R}}^m\setminus \mathrm{int}K$ be the allowed target. We prove that if $\mathrm{\partial }K$ is of class $C^{1,1}$, then every $\bar{M}$-valued local minimizer of the Dirichlet energy $E(u;B):={\int }_B\mathrm{\mid }Du{\mathrm{\mid }}^2\text{dd}x$ is locally $W^{2,\mathrm{\infty }}$, and hence locally $C^{1,1}$, on its continuity set. For every $0<\alpha <1$, we also construct a convex body with $C^{1,\alpha }$ boundary and an everywhere continuous global minimizer that is not $C^{1,1}$. Thus $C^{1,1}$ is the sharp target regularity threshold in the Hölder scale.