Existence and regularity of minimizers for some spectral functionals with perimeter constraint

In this paper we prove that the shape optimization problem $$\min \{{\lambda }_k(\mathrm{\Omega }):\mathrm{\Omega }\subset {\mathbb{R}}^d,\mathrm{\Omega }\text{open},P(\mathrm{\Omega })=1,\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }<+\mathrm{\infty }\},$$ has a solution for any $k\in \mathbb{N}$ and dimension $d$. Moreover, every solution is a bounded connected open set with boundary which is $C^{1,\alpha }$ outside a closed set of Hausdorff dimension $d-8$. Our results are more general and apply to spectral functionals of the form $f({\lambda }_{k_1}(\mathrm{\Omega }),\dots ,{\lambda }_{k_p}(\mathrm{\Omega }))$, for increasing functions $f$ satisfying some suitable bi-Lipschitz type condition.