Regularity of isoperimetric sets in $\mathbb R^2$ with density

TL;DR

This paper proves improved regularity results for the boundaries of isoperimetric sets in the plane with a certain class of density functions, advancing understanding of geometric measure theory in weighted spaces.

Contribution

It establishes that with a ${ m C}^{0,eta}$ density, isoperimetric boundaries are ${ m C}^{1,rac{eta}{3-2eta}}$, enhancing previous regularity results.

Findings

Boundaries are ${ m C}^{1,rac{eta}{3-2eta}}$ for ${ m C}^{0,eta}$ densities.

Improves regularity from previous known results.

Advances geometric measure theory in weighted Euclidean spaces.

Abstract

We consider the isoperimetric problem in $\mathbb R^n$ with density for the planar case $n=2$. We show that, if the density is ${\rm C}^{0,\alpha}$, then the boundary of any isoperimetric is of class ${\rm C}^{1,\frac \alpha{3-2\alpha}}$. This improves the previously known regularity.