Regularity of the Optimal Sets for a Class of Integral Shape Functionals.

We prove the first regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain is obtained as the integral of a cost function (, ) depending on the solution of a certain PDE problem on . The main feature of these functionals is that the minimality of a domain cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions , where is the solution of the PDE with Dirichlet boundary conditions.