First of all, we recall the basic notions and results concerning shape optimization problems for the eigenvalues of the Dirichlet Laplacian.

Then we focus on the study of the regularity of the optimal shapes and on the link with the regularity of related free boundary problems.

The main topic of the talk is the regularity of the optimal sets for a "degenerate" functional, namely the second Dirichlet eigenvalue in a box. Given $D\subset \mathbb{R}^d$ an open and bounded set of class $C^{1,1}$, we consider the following shape optimization problem, for $\Lambda>0$,\begin{equation}\label{eq:main}\min{\Big\{\lambda_2(A)+\Lambda |A| : A\subset D,\text{ open}\Big\}},\end{equation}where $\lambda_2(A)$ denotes the second eigenvalue of the Dirichlet Laplacian on $A$.

In this talk we show that any optimal set $\Omega$ for $\eqref{eq:main}$ is equivalent to the union of two disjoint open sets, $\Omega^\pm$, which are $C^{1,\alpha}$ regular up to a (possibly empty) closed singular set of Hausdorff dimension at most $d-5$, which is contained in the one-phase free boundaries.

In particular, we are able to prove that the set of two-phase points, that is, $\partial \Omega^+\cap \partial \Omega^-\cap D$, is contained in the regular set.

This is a joint work with Baptiste Trey and Bozhidar Velichkov.