Existence and regularity of solutions to optimal partition problems involving Laplacian eigenvalues
Miguel Ramos, Hugo Tavares, Susanna Terracini · 2012
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, including the following \[ \inf\left\{\Phi(\omega_1,\ldots,\omega_m):=\sum_{i=1}^m \lambda_{k_i}(\omega_i): (\omega_1,\ldots, \omega_m)\in \mathcal{P}_m(\Omega)\right\}, \] 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 \[ \mathcal{P}_m(\Omega)=\left\{(\omega_1,\ldots,\omega_m): \omega_i\subset \Omega \text{ open}, \omega_i\cap \omega_j=\emptyset \forall i eq j\right\}. \] We prove the existence of an open optimal partition $(\omega_1,\ldots, \omega_m)\in \mathcal{P}_m(\Omega)$, proving as well its regularity 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. In order to prove this result, we first treat some general optimal partition problems involving all eigenvalues up to a certain order. The study of this new class of problems is done via a singular perturbation approach with a class of Schr\odinger-type systems which models competition between different groups of possibly sign-changing components. An optimal partition appears in relation with the nodal set of the limiting 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-type monotonicity formulas.