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 $\Omega$ is obtained as the integral of a cost function j(u, x) depending on the solution u of a certain PDE problem on $\Omega$ . The main feature of these functionals is that the minimality of a domain $\Omega$ 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 $j(u,x)=-g\left(x\right)u+Q\left(x\right)$ , where u is the solution of the PDE $-\Delta u=f$ with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal u from the inwards/outwards optimality of $\Omega$ and then we use the stability of $\Omega$ with respect to variations with smooth vector fields in order to study the blow-up limits of the state function u. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose $\partial \Omega$ into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of $\partial \Omega$ we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove $C^{\infty }$ regularity of the regular part of the free boundary when the data are smooth.