Volume-constraint local energy-minimizing sets in a ball

Abstract

In this paper, we prove a Poincar\'e inequality for any volume-constraint local energy-minimizing sets, provided its singular set is of Hausdorff dimension at most $n-3$. With this inequality, we prove that the only volume-constraint local energy-minimizing sets in the Euclidean unit ball, whose singular set is closed and of Hausdorff dimension at most $n-3$, are totally geodesic balls or spherical caps intersecting the unit sphere with constant contact angle; for stable sets in a wedge-shaped domain or in a half space, provided the same condition of the singular set, must be spherical. In particular, they are smooth.Comment: 35 pages, 2 figures; added some arguments on stable measure-theoretic capillary hyper surfaces in a half space or in a domain with planar boundaries in section 6; typos correcte