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

TL;DR

This paper provides a concise, self-contained proof demonstrating that solutions to the eikonal equation are twice differentiable with Lipschitz continuous derivatives inside the domain, assuming pointwise differentiability.

Contribution

It offers a new, simplified proof of the interior $ ext{C}^{1,1}$ regularity for solutions to the eikonal equation under pointwise differentiability assumptions.

Findings

Solutions are $ ext{C}^{1,1}$ regular inside the domain.

The proof is shorter and self-contained.

Regularity holds under pointwise differentiability.

Abstract

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