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

Chao Xia, Xuwen Zhang

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 $n-3$. 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.

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

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.

Paper Structure

This paper contains 9 sections, 18 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Sets of finite perimeter
  5. Regularity results for local minimizers
  6. Cut-off functions
  7. Poincaré-type inequality for stable sets
  8. Volume-constraint local minimizers in a wedge-shaped domain
  9. Volume-constraint local minimizers in a ball

Key Result

Theorem 1.2

Let $\Omega$ be $\mathbf{R}^{n}_+$ or $\mathbf{B}^{n}$. Let $E\subset\Omega$ be a local minimizer for the free energy functional Free-energyFunctional under volume constraint among sets of finite perimeter. Then $M=\overline{\partial E\cap\Omega}$ is (up to a modification of sets of measure zero for

Figures (2)

  • Figure 1: Notations
  • Figure 2: Notations for wedge

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2: DePM17
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5: Euclidean volume growth
  • Theorem 2.6: DePM15
  • Lemma 3.1: cut-off functions
  • proof
  • ...and 26 more