Extremality Conditions and Regularity of Solutions to Optimal Partition Problems Involving Laplacian Eigenvalues

Abstract

Let ${\Omega \subset \mathbb{R}^N}$ 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

$${\rm inf}\left\{F({\lambda_{k_{1}}}(\omega_1),\ldots,\lambda_{k_m}(\omega_m)):\ (\omega_1,\ldots, \omega_m) \in \mathcal{P}_m(\Omega)\right\},$$

where ${\lambda_{k_i}(\omega_i)}$ denotes the k _i -th eigenvalue of ${(-\Delta,H^{1}_{0}(\omega_i))}$ counting multiplicities, and ${\mathcal{P}_m(\Omega)}$ is the set of all open partitions of ${\Omega}$, namely

$$\mathcal{P}_m(\Omega)=\left\{(\omega_1, \ldots, \omega_m):\omega_i \subset \Omega \, {\rm open},\ \omega_{i} \cap\omega_j=\emptyset\,\forall i \neq j \right\}.$$

While the existence of a quasi-open optimal partition ${(\omega_1,\ldots, \omega_m)}$ 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 ${\cup_{i=1}^m \partial \omega_{i} \cap \Omega}$ is, up to a residual set, locally a ${C^{1,\alpha}}$ 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.