Regularity of minimal surfaces with capillary boundary conditions
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914817795686400 |
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| author | De Masi, Luigi Edelen, Nick Gasparetto, Carlo Li, Chao |
| author_facet | De Masi, Luigi Edelen, Nick Gasparetto, Carlo Li, Chao |
| contents | We prove $\varepsilon$-regularity theorems for varifolds with capillary boundary condition in a Riemannian manifold. These varifolds were first introduced by Kagaya-Tonegawa \cite{KaTo}. We establish a uniform first variation control for all such varifolds (and free-boundary varifolds generally) satisfying a sharp density bound and prove that if a capillary varifold has bounded mean curvature and is close to a capillary half-plane with angle not equal to $\tfracπ{2}$, then it coincides with a $C^{1,α}$ properly embedded hypersurface. We apply our theorem to deduce regularity at a generic point along the boundary in the region where the density is strictly less than $1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_20796 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularity of minimal surfaces with capillary boundary conditions De Masi, Luigi Edelen, Nick Gasparetto, Carlo Li, Chao Differential Geometry Analysis of PDEs We prove $\varepsilon$-regularity theorems for varifolds with capillary boundary condition in a Riemannian manifold. These varifolds were first introduced by Kagaya-Tonegawa \cite{KaTo}. We establish a uniform first variation control for all such varifolds (and free-boundary varifolds generally) satisfying a sharp density bound and prove that if a capillary varifold has bounded mean curvature and is close to a capillary half-plane with angle not equal to $\tfracπ{2}$, then it coincides with a $C^{1,α}$ properly embedded hypersurface. We apply our theorem to deduce regularity at a generic point along the boundary in the region where the density is strictly less than $1$. |
| title | Regularity of minimal surfaces with capillary boundary conditions |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2405.20796 |