Boundary regularity for the distance functions, and the eikonal equation

Nikolai Nikolov; Pascal J. Thomas

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arXiv · 2409.01774

Boundary regularity for the distance functions, and the eikonal equation

Nikolai Nikolov·Pascal J. Thomas

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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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Nikolai Nikolov, Pascal J. Thomas. 2024-09-03. Boundary regularity for the distance functions, and the eikonal equation. https://arxiv.org/abs/2409.01774

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