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 \(j(u,x)=-g(x)u+Q(x)\), 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.