Optimal regularity of isoperimetric sets with Hölder densities

We establish a regularity result for optimal sets of the isoperimetric problem with double density under mild ($\alpha$-)Hölder regularity assumptions on the density functions. Our main Theorem improves some previous results and allows to reach in any dimension the regularity class $C^{1,\frac{\alpha }{2-\alpha }}$. This class is indeed the optimal one for local minimizers of variational functionals with an integrand that depends $\alpha$-Hölder continuous on the minimizer itself, and as such can (the boundary of) the isoperimetric set be locally written (with additional constraint).