Ahlfors-regularity for minimizers of a multiphase optimal design problem

We establish an Ahlfors-regularity result for minimizers of a multiphase optimal design problem. It is a variant of the classical variational problem which involves a finite number of chambers E(i) of prescribed volume that partition a given domain Ω ⊂Rn. The cost functional associated with a configuration ({E(i)}i, u) is made up of the perimeter of the partition interfaces and a Dirichlet energy term, which is discontinuous across the interfaces. We prove that the union of the optimal interfaces is (n−1)-Ahlfors-regular via a penalization method and decay estimates of the energy.