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| Let $\Omega\subset \mathbb{R}^N$ be a open bounded domain and $m\in \mathbb{N}$. Given $k_1,\ldots,k_m\in \mathbb{N}$, we consider the following optimal partition problem
\[
\inf\left\{\Phi(\omega_1,\ldots,\omega_m):=\sum_{i=1}^k \lambda_{k_i}(\omega_i):\ (\omega_1,\ldots, \omega_k)\in \mathcal{P}_m(\Omega)\right\},
\]
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 sub-partitions of $\Omega$, namely
\[
\mathcal{P}_m(\Omega)=\{(\omega_1,\ldots,\omega_m):\ \omega_i\subset \Omega \text{ open},\ \omega_i\cap \omega_j=\emptyset\ \forall i\neq j\}.
\]
We prove the existence of an optimal partition $(\omega_1,\ldots, \omega_m)$, proving as well its regularity in the sense that the free boundary $\Omega\cap \cup_{i=1}^m \partial \omega_i$ is, up to a residual set, locally a $C^{1,\alpha}$ hyper-surface.
In order to prove this result, we first show existence and regularity for a general optimal partition problem involving eigenvalues where there is an underlying invariance with respect to orthogonal transformations. This class of problems includes the one with cost function
\[
\Phi_p(\omega_1,\ldots, \omega_m)=\sum_{i=1}^m \left(\sum_{j=1}^{k_i}\left(\lambda_j(\omega_i)\right)^{p}\right)^{1/p},
\]
whose solutions approach the solutions of the original problem as $p\to \infty$. The study of this new class of problems is done via a detailed study of a Schrodinger-type system which models competition between different groups of possibly sign-changing components. An optimal partition appears as the nodal set of the components, as the competition parameter becomes large. The proofs also involve new boundary Harnack principles on NTA and Reifenberg flat domains, as well as an extensive use of Almgren's monotonicity--type formulas. |
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