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 $(ω_1, \dots, ω_m) \mapsto \sum_{i=1}^{m} \left( \sum_{j=1}^{k_i} λ_{j}(ω_i)^{p_i}\right)^{1/p_i}, \quad \prod_{i=1}^{m} \left( \prod_{j=1}^{k_i} λ_{j}(ω_i)\right), \quad \prod_{i=1}^{m} \left( \sum_{j=1}^{k_i} λ_{j}(ω_i)\right)$ where $(ω_1, \dots, ω_m)$ are the sets of the partition and $λ_{j}(ω_i)$ is the $j$-th Laplace eigenvalue of the set $ω_i$ with zero Dirichlet boundary conditions.

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