Boundary regularity for the distance functions, and the eikonal equation

We study the gain in regularity of the distance to the boundary of a domain in $\R^m$. 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 stud...

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Bibliographic Details
Main Authors Nikolov, Nikolai, Thomas, Pascal J
Format Journal Article
LanguageEnglish
Published 03.09.2024
Subjects
Mathematics - Analysis of PDEs
Mathematics - Complex Variables
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Summary:We study the gain in regularity of the distance to the boundary of a domain in $\R^m$. 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.
DOI:10.48550/arxiv.2409.01774