A short proof of the ${\mathcal{C}}^{1,1}$ regularity for the eikonal equation

Abstract

We give a short and self-contained proof of the interior ${\mathcal{C}}^{1,1}$ regularity of solutions $\phi :\Omega \to \mathbb{R}$ to the eikonal equation $|\nabla \phi |=1$ in an open set $\Omega \subset {\mathbb{R}}^N$ in dimension $N\ge 1$ under the assumption that $\phi$ is pointwise differentiable in $\Omega$.