Extremality conditions and regularity of solutions to optimal partition   problems involving Laplacian eigenvalues
  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 \[ \inf\left\{F(\lambda_{k_1}(\omega_1),\ldots, \lambda_{k_m}(\omega_m)):\ (\omega_1,\ldots, \omega_m)\in \mathcal{P}_m(\Omega)\right\}, \] 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 \[ \mathcal{P}_m(\Omega)=\left\{(\omega_1,\ldots,\omega_m):\ \omega_i\subset \Omega \text{ open},\ \omega_i\cap \omega_j=\emptyset\ \forall i\neq j\right\}. \] While 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, 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 $\cup_{i=1}^m \partial \omega_i\cap \Omega$ is, up to a residual set, locally a $C^{1,\alpha}$ 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. 
Miguel Ramos, Hugo Tavares, Susanna Terracini
math.AP math.OC