Regularity of minimizers of quasi perimeters with a volume constraint
Qinglan Xia · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2005
We study the regularity of the boundary of sets minimizing a quasi perimeter T(E) = P (E, Ω) + G(E) with a volume constraint.Here Ω is any open subset of R n with n 2, and G is a lower semicontinuous function on sets of finite perimeter satisfying G(E) G(F ) + C|E F | β for all sets E, F of finite perimeter with equal volume.We show that under the condition β > 1 -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 for quasi minimizers of perimeter, we get the classical C 1,α regularity for the reduced boundary of E.