Regularity of all minimizers of a class of spectral partition problems

We study a rather broad class of optimal partition problems with respect to monotone and coercive functional costs that involve the Dirichlet eigenvalues of the partitions. We show a sharp regularity result for the entire set of minimizers for a natural relaxed version of the original problem, together with the regularity of eigenfunctions and a universal free boundary condition. Among others, our result covers the cases of the following functional costs $$({\omega }_1,\dots ,{\omega }_m)\mapsto \sum _{i=1}^m{(\sum _{j=1}^{k_i}{\lambda }_j({\omega }_i{)}^{p_i})}^{1\mathrm{/}{p}_i},\prod _{i=1}^m(\prod _{j=1}^{k_i}{\lambda }_j({\omega }_i)),\prod _{i=1}^m(\sum _{j=1}^{k_i}{\lambda }_j({\omega }_i))$$ where $(\omega_1, \dots, \omega_m)$arethesetsofthepartitionand$\lambda_{j}(\omega_i)$isthe$j$-thLaplaceeigenvalueoftheset$\omega_i$ with zero Dirichlet boundary conditions.