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 ${\rm C}^{0,α}$, then the boundary of any isoperimetric is of class ${\rm C}^{1,\frac α{3-2α}}$. This improves the previously known regularity.

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