Regularity of the optimal sets for the second Dirichlet eigenvalue

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