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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.01774 |
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| _version_ | 1866909650827345920 |
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| author | Nikolov, Nikolai Thomas, Pascal J. |
| author_facet | Nikolov, Nikolai Thomas, Pascal J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01774 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundary regularity for the distance functions, and the eikonal equation Nikolov, Nikolai Thomas, Pascal J. Analysis of PDEs Complex Variables 35F20, 35F21, 35B65 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. |
| title | Boundary regularity for the distance functions, and the eikonal equation |
| topic | Analysis of PDEs Complex Variables 35F20, 35F21, 35B65 |
| url | https://arxiv.org/abs/2409.01774 |