Work

Boundary Regularity For the Distance Functions, and the Eikonal Equation

Year: 2025

Type: article

Abstract: We study the gain in regularity of the distance to the boundary of a domain in $${\mathbb {R}}^m$$ 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 $${\mathcal {C}}^{1,1}$$ 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.

Source: Journal of Geometric Analysis

DOI: 10.1007/s12220-025-02064-7

Authors: Nikolai Nikolov, Pascal J. Thomas