Boundary regularity for the distance functions, and the eikonal equation

by Pascal Thomas (IMT)

Monday Apr 28, 2025, 2:00 PM → 3:00 PM Europe/Paris

Salle Pellos (1R2-207)

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.

https://arxiv.org/abs/2409.01774