Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball
arXiv:2106.14780 · doi:10.1515/acv-2022-0106
Abstract
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 . 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.
27 pages, 2 figures; major revision: title has been changed and the proof has been rewritten