Extremality conditions and regularity of solutions to optimal partition problems involving Laplacian eigenvalues

Let $Ω\subset \mathbb{R}^N$ be an open bounded domain and $m\in \mathbb{N}$. Given $k_1,\ldots,k_m\in \mathbb{N}$, we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, of the following form $\inf\left\{F(λ_{k_1}(ω_1),\ldots, λ_{k_m}(ω_m)):\ (ω_1,\ldots, ω_m)\in \mathcal{P}_m(Ω)\right\},$ where $λ_{k_i}(ω_i)$ denotes the $k_i$--th eigenvalue of $(-Δ,H^1_0(ω_i))$ counting multiplicities, and $\mathcal{P}_m(Ω)$ is the set of all open partitions of $Ω$, namely $\mathcal{P}_m(Ω)=\left\{(ω_1,\ldots,ω_m):\ ω_i\subset Ω\text{ open},\ ω_i\cap ω_j=\emptyset\ \forall i\neq j\right\}.$ While existence of a quasi-open optimal partition $(ω_1,\ldots, ω_m)$ follows from a general result by Bucur, Buttazzo and Henrot [Adv. Math. Sci. Appl. 8, 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 regularity of the partition in the sense that the free boundary $\cup_{i=1}^m \partial ω_i\cap Ω$ is, up to a residual set, locally a $C^{1,α}$ hypersurface. This last result extend the ones in the paper by Caffarelli and Lin [J. Sci. Comput. 31, 2007] to the case of higher eigenvalues.

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