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Regularity of all minimizers of a class of spectral partition problems

1 CAMGSD and Departamento de Matemática, Instituto Superior Técnico, Pavilhão de Matemática, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
2 Université de Paris and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions (LJLL), F-75006 Paris, France

This contribution is part of the Special Issue: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday
Guest Editor: Gianmaria Verzini
Link: www.aimspress.com/mine/article/5753/special-articles

Special Issues: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday

We study a rather broad class of optimal partition problems with respect to monotone and coercive functional costs that involve the Dirichlet eigenvalues of the partitions. We show a sharp regularity result for the entire set of minimizers for a natural relaxed version of the original problem, together with the regularity of eigenfunctions and a universal free boundary condition. Among others, our result covers the cases of the following functional costs \[ (\omega_1, \dots, \omega_m) \mapsto \sum_{i=1}^{m} \left( \sum_{j=1}^{k_i} \lambda_{j}(\omega_i)^{p_i}\right)^{1/p_i}, \quad \prod_{i=1}^{m} \left( \prod_{j=1}^{k_i} \lambda_{j}(\omega_i)\right), \quad \prod_{i=1}^{m} \left( \sum_{j=1}^{k_i} \lambda_{j}(\omega_i)\right) \] where $(\omega_1, \dots, \omega_m)$ are the sets of the partition and $\lambda_{j}(\omega_i)$ is the $j$-th Laplace eigenvalue of the set $\omega_i$ with zero Dirichlet boundary conditions.
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Keywords elliptic competitive systems; optimal partition problems; Laplacian eigenvalues; segregation phenomena; extremality conditions; regularity of free boundary problems; blowup techniques

Citation: Hugo Tavares, Alessandro Zilio. Regularity of all minimizers of a class of spectral partition problems. Mathematics in Engineering, 2021, 3(1): 1-31. doi: 10.3934/mine.2021002

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