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arXiv · 2106.14780

Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball

Chao Xia·Xuwen Zhang
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Abstract

In this paper, we prove a Poincar\'e-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.

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Chao Xia, Xuwen Zhang. 2021-06-28. Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball. https://doi.org/10.1515/acv-2022-0106

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