I am interested in the regularity of the solutions of free boundary and free discontinuity problems, which typically arise in the theory of Shape Optimization involving spectral functionals depending on the Dirichlet or Neumann Laplacian with perimeter or measure term. The analysis of the qualitative properties of the optimal sets is often an essential part of the existence theory, as one may see in the following paper.

We prove the existence of an optimal set and the C^{1,α} regularity of its boundary, for spectral functionals under a perimeter constraint, a model problem being

\min\Big\{\lambda_k(\Omega):\ Per(\Omega)=1,\ \Omega\subset\mathbb{R}^d,\ |\Omega|<\infty\Big\},

where λ_{k} is the k-th eigenvalue of the Dirichlet Laplacian on Ω. The key of our regularity result stands in the careful analysis of the sets of positive curvature in viscosity sense and an adaptation of the Bucur's Reduction Lemma. The combination of these techniques leads to the quasi-minimality, with respect to the perimeter, of the optimal sets, which then gives the regularity of the optimal domains.

In the paper

we prove the long-standing open question concerning the Lipschitz continuity of the eigenfunctions on the domains, which are solutions of the problem

\min\Big\{\lambda_k(\Omega):\ \Omega\subset\mathbb{R}^d,\ |\Omega|=1\Big\},

where λ

_{k} is the k-th eigenvalue eigenvalue of the Dirichlet Laplacian. We refine the techniques of Alt-Caffarelli and Briancon-Hayouni-Pierre and we prove that the eigenfunctions, which are quasi minimizers of the Dirichlet integral, are Lipschitz continuous. We then reason by approximation, using sequence of optimal sets, solutions of a specific obstacle problem, for which we can deduce the quasi-minimality of the corresponding eigenfunctions.

Our analysis proves, in particular, that there is a solution of the problem

\min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega):\ \Omega\ open,\ \Omega\subset\mathbb{R}^d,\ |\Omega|=1\Big\},

and all the first k eigenfunctions are Lipschitz. We are currently working on the regularity of the free boundary of the optimal sets.

In the paper

#### (with Giuseppe Buttazzo and Edouard Oudet, to appear in Calc.Var.PDE)

we consider a free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered; it is of the form

\displaystyle\sup_{\int_D\theta\,dx=m}\ \inf_{u\in H^1_0(D)}\int_D\Big(\frac{1+\theta}{2}\vert|\nabla u\vert|^2-fu\Big)\,dx.

We prove the existence of an optimal reinforcement and that it has some higher integrability properties. This problem is related to the classical elastic-plastic torsion problem. Some numerical simulations can be found

here .