Some remarks on singular capillary cones with free boundary
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912229245321216 |
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| author | Pacati, Alberto Tortone, Giorgio Velichkov, Bozhidar |
| author_facet | Pacati, Alberto Tortone, Giorgio Velichkov, Bozhidar |
| contents | We study minimizing singular cones with free boundary associated with the capillarity problem. Precisely, we provide a stability criterion $à$ la Jerison-Savin for capillary hypersurfaces and show that, in dimensions up to $4$, minimizing cones with non-sign-changing mean curvature are flat. We apply this criterion to minimizing capillary drops and, additionally, establish the instability of non-trivial axially symmetric cones in dimensions up to $6$.
The main results are based on a Simons-type inequality for a class of convex, homogeneous, symmetric functions of the principal curvatures, combined with a boundary condition specific to the capillary setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07697 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some remarks on singular capillary cones with free boundary Pacati, Alberto Tortone, Giorgio Velichkov, Bozhidar Differential Geometry Analysis of PDEs We study minimizing singular cones with free boundary associated with the capillarity problem. Precisely, we provide a stability criterion $à$ la Jerison-Savin for capillary hypersurfaces and show that, in dimensions up to $4$, minimizing cones with non-sign-changing mean curvature are flat. We apply this criterion to minimizing capillary drops and, additionally, establish the instability of non-trivial axially symmetric cones in dimensions up to $6$. The main results are based on a Simons-type inequality for a class of convex, homogeneous, symmetric functions of the principal curvatures, combined with a boundary condition specific to the capillary setting. |
| title | Some remarks on singular capillary cones with free boundary |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2502.07697 |