Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866910633337815040 |
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| author | Xia, Chao Zhang, Xuwen |
| author_facet | Xia, Chao Zhang, Xuwen |
| contents | In this paper, we prove a Poincaré-type inequality for any set of finite perimeter which is stable with respect to the free energy among volume-preserving perturbation, provided that the Hausdorff dimension of its singular set is at most $n-3$. With this inequality, we classify all the volume-constraint local energy-minimizing sets in a unit ball, a half-space or a wedge-shaped domain. In particular, we prove that the relative boundary of any energy-minimizing set is smooth. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2106_14780 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball Xia, Chao Zhang, Xuwen Differential Geometry Analysis of PDEs 49Q20, 28A75, 53A10, 53C24 In this paper, we prove a Poincaré-type inequality for any set of finite perimeter which is stable with respect to the free energy among volume-preserving perturbation, provided that the Hausdorff dimension of its singular set is at most $n-3$. With this inequality, we classify all the volume-constraint local energy-minimizing sets in a unit ball, a half-space or a wedge-shaped domain. In particular, we prove that the relative boundary of any energy-minimizing set is smooth. |
| title | Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball |
| topic | Differential Geometry Analysis of PDEs 49Q20, 28A75, 53A10, 53C24 |
| url | https://arxiv.org/abs/2106.14780 |