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

TL;DR

This paper establishes a Poincaré-type inequality for volume-constrained energy-minimizing sets with controlled singularities, leading to a classification of such sets in simple domains and proving their boundary smoothness.

Contribution

It introduces a new inequality for stable sets with volume constraints and classifies all local energy-minimizers in basic geometric domains.

Findings

All local energy-minimizers in a ball, half-space, or wedge are classified.

The relative boundary of energy-minimizing sets is proven to be smooth.

A Poincaré-type inequality is established for sets with controlled singular sets.

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.