*Published Paper*

**Inserted:** 1 oct 2020

**Last Updated:** 25 apr 2023

**Journal:** Ann. Inst. H. PoincarĂ© Anal. Non LinĂ©aire

**Year:** 2022

**Abstract:**

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 \[ \mathcal F_\Lambda(\Omega)=\lambda_2(\Omega)+\Lambda \mathcal L^d(\Omega), \] 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 $D\cap\partial\Omega_-\setminus\partial\Omega_+$.

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