JournalsifbVol. 7 , No. 3DOI 10.4171/ifb/128

Regularity of minimizers of quasi perimeters with a volume constraint

  • Qinglan Xia

    University of Texas at Austin, USA
Regularity of minimizers of quasi perimeters with a volume constraint cover

Abstract

In this article, we study the regularity of the boundary of sets minimizing a quasi perimeter T(E)=P(E,Ω)+G(E)T\left( E\right) =P\left( E,\Omega \right) +G\left( E\right) with a volume constraint. Here Ω\Omega is any open subset of Rn\mathbb{R}^{n} with n2n\geq 2, GG is a lower semicontinuous function on sets of finite perimeter satisfying a condition that G(E)G(F)+CE÷FβG\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 β>11n\beta >1-\frac{1}{n}, any volume constrained minimizer EE of the quasi perimeter TT has both interior points and exterior points, and EE 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 C1,αC^{1,\alpha } regularity for the reduced boundary of EE.