  This paper is dedicated to the regularity of the optimal sets for the second
eigenvalue of the Dirichlet Laplacian. Precisely, we prove that if the set
$\Omega$ minimizes the functional \[ \mathcal
F_\Lambda(\Omega)=\lambda_2(\Omega)+\Lambda |\Omega|, \] among all subsets of a
smooth bounded open set $D\subset \mathbb{R}^d$, where $\lambda_2(\Omega)$ is
the second eigenvalue of the Dirichlet Laplacian on $\Omega$ and $\Lambda>0$ is
a fixed constant, then $\Omega$ is equivalent to the union of two disjoint open
sets $\Omega_+$ and $\Omega_-$, which are $C^{1,\alpha}$-regular up to a
(possibly empty) closed set of Hausdorff dimension at most $d-5$, contained in
the one-phase free boundaries $D\cap \partial\Omega_+\setminus\partial\Omega_-$
and $D\cap\partial\Omega_-\setminus\partial\Omega_+$.
