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 Holder 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, (a) (2-a) . This class is indeed the optimal one for local minimizers of variational functionals with an integrand that depends a-H & ouml;lder continuous on the minimizer itself, and as such can (the boundary of) the isoperimetric set be locally written (with additional constraint).