Boundary regularity for the distance functions, and the eikonal equation

2024/09/03 by Nikolov, Nikolai, Thomas, Pascal J. · 2 citations
#35B65 #35F20 #35F21 #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2409.01774

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.

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