Let Ω⊂RN\documentclass[12pt]{minimal}
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\begin{document}$${\Omega \subset \mathbb{R}^N}$$\end{document} be an open bounded domain and m∈N\documentclass[12pt]{minimal}
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\begin{document}$${m \in \mathbb{N}}$$\end{document}. Given k1,…,km∈N\documentclass[12pt]{minimal}
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\begin{document}$${k_1,\ldots,k_m \in \mathbb{N}}$$\end{document}, we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, of the following form
infF(λk1(ω1),…,λkm(ωm)):(ω1,…,ωm)∈Pm(Ω),\documentclass[12pt]{minimal}
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\begin{document}$${\rm inf}\left\{F({\lambda_{k_{1}}}(\omega_1),\ldots,\lambda_{k_m}(\omega_m)):\ (\omega_1,\ldots, \omega_m) \in \mathcal{P}_m(\Omega)\right\},$$\end{document}where λki(ωi)\documentclass[12pt]{minimal}
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\begin{document}$${\lambda_{k_i}(\omega_i)}$$\end{document} denotes the ki-th eigenvalue of (-Δ,H01(ωi))\documentclass[12pt]{minimal}
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\begin{document}$${(-\Delta,H^{1}_{0}(\omega_i))}$$\end{document} counting multiplicities, and Pm(Ω)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{P}_m(\Omega)}$$\end{document} is the set of all open partitions of Ω\documentclass[12pt]{minimal}
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\begin{document}$${\Omega}$$\end{document}, namely
Pm(Ω)=(ω1,…,ωm):ωi⊂Ωopen,ωi∩ωj=∅∀i≠j.\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{P}_m(\Omega)=\left\{(\omega_1, \ldots, \omega_m):\omega_i \subset \Omega \, {\rm open},\ \omega_{i} \cap\omega_j=\emptyset\,\forall i \neq j \right\}.$$\end{document}While the existence of a quasi-open optimal partition (ω1,…,ωm)\documentclass[12pt]{minimal}
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\begin{document}$${(\omega_1,\ldots, \omega_m)}$$\end{document} follows from a general result by Bucur, Buttazzo and Henrot [Adv Math Sci Appl 8(2):571–579, 1998], the aim of this paper is to associate with such minimal partitions and their eigenfunctions some suitable extremality conditions and to exploit them, proving as well the Lipschitz continuity of some eigenfunctions, and the regularity of the partition in the sense that the free boundary ∪i=1m∂ωi∩Ω\documentclass[12pt]{minimal}
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\begin{document}$${\cup_{i=1}^m \partial \omega_{i} \cap \Omega}$$\end{document} is, up to a residual set, locally a C1,α\documentclass[12pt]{minimal}
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\begin{document}$${C^{1,\alpha}}$$\end{document} hypersurface. This last result extends the ones in the paper by Caffarelli and Lin [J Sci Comput 31(1–2):5–18, 2007] to the case of higher eigenvalues.
