On $C^{1,\alpha }$-regularity for critical points of a geometric obstacle-type problem

Abstract

We consider critical points of the geometric obstacle problem on vectorial maps $u:{\mathbb{B}}^2\subset {\mathbb{R}}^2\to {\mathbb{R}}^N$ \[ \int_{\mathbb{B}^2} |\nabla u|^2 \quad \mbox{subject to $u\in {\mathbb{R}}^N\backslash {\mathbb{B}}^N(0)$}. \] Our main result is $C^{1,\alpha }$-regularity for any $\alpha <1$. Technically, we split the map $u=\lambda v$, where $v:{\mathbb{B}}^2\to {\mathbb{S}}^{N-1}$ is the vectorial component and $\lambda =|u|$ the scalar component measuring the distance to the origin. While $v$ satisfies a weighted harmonic map equation with weight ${\lambda }^2$, $\lambda$ solves the obstacle problem for \[ \int_{\mathbb{B}^2} |\nabla λ|^2+λ^2 |\nabla v|^2, \quad \mbox{subject to $\lambda \ge 1$}. \] where $|\nabla v{|}^2\in L^1({\mathbb{B}}^2)$. We then play ping-pong between the increases in the regularity of $\lambda$ and $v$ to obtain finally the $C^{1,\alpha }$-result.