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

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