Boundary regularity for the distance functions, and the eikonal equation

TL;DR

This paper investigates the regularity properties of distance functions to boundaries in Euclidean space and their relation to the eikonal equation, establishing conditions under which these functions are differentiable or have Lipschitz gradients.

Contribution

It proves that a differentiable signed distance function near a boundary implies the boundary is ,1 regular, and shows solutions to the eikonal equation with full domain differentiability have Lipschitz continuous gradients.

Findings

Differentiability of the signed distance function implies ,1 regularity of the boundary.

Solutions to the eikonal equation that are differentiable everywhere have locally Lipschitz gradients.

Regularity of the boundary influences the smoothness of the distance function.

Abstract

We study the gain in regularity of the distance to the boundary of a domain in $\mathbb R^m$. In particular, we show that if the signed distance function happens to be merely differentiable in a neighborhood of a boundary point, it and the boundary have to be $\mathcal C^{1,1}$ regular. Conversely, we study the regularity of the distance function under regularity hypotheses of the boundary. Along the way, we point out that any solution to the eikonal equation, differentiable everywhere in a domain of the Euclidean space, admits a gradient which is locally Lipschitz.