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

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