Regularity of minimizers of quasi perimeters with a volume constraint

Abstract

In this article, we study the regularity of the boundary of sets minimizing a quasi perimeter $T\left( E\right) =P\left( E,\Omega \right) +G\left( E\right)$ with a volume constraint. Here $\Omega$ is any open subset of $\mathbb{R}^{n}$ with $n\geq 2$, $G$ is a lower semicontinuous function on sets of finite perimeter satisfying a condition that $G\left( E\right) \leq G\left( F\right) +C\left| E\div F\right| ^{\beta }$ among all sets of finite perimeter with equal volume. We show that under the condition $\beta >1-\frac{1}{n}$, any volume constrained minimizer $E$ of the quasi perimeter $T$ has both interior points and exterior points, and $E$ is indeed a quasi minimizer of perimeter without the volume constraint. Using a well known regularity result about quasi minimizers of perimeter, we get the classical $C^{1,\alpha }$ regularity for the reduced boundary of $E$.