Regularity of all minimizers of a class of spectral partition problems

Hugo Tavares; Alessandro Zilio; Hugo Tavares; Alessandro Zilio

doi:10.3934/mine.2021002

Mathematics in Engineering

2021, Volume 3, Issue 1: 1-31. doi: 10.3934/mine.2021002

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Regularity of all minimizers of a class of spectral partition problems

Hugo Tavares ¹,
Alessandro Zilio ^{2
,
,}

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

Received: 10 February 2020 Accepted: 08 July 2020 Published: 20 July 2020

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\limits_{i = 1}^{m} \left( \sum\limits_{j = 1}^{k_i} \lambda_{j}(\omega_i)^{p_i}\right)^{1/p_i}, \quad \prod\limits_{i = 1}^{m} \left( \prod\limits_{j = 1}^{k_i} \lambda_{j}(\omega_i)\right), \quad \prod\limits_{i = 1}^{m} \left( \sum\limits_{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.
- 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[J]. Mathematics in Engineering, 2021, 3(1): 1-31. doi: 10.3934/mine.2021002

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Abstract

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\limits_{i = 1}^{m} \left( \sum\limits_{j = 1}^{k_i} \lambda_{j}(\omega_i)^{p_i}\right)^{1/p_i}, \quad \prod\limits_{i = 1}^{m} \left( \prod\limits_{j = 1}^{k_i} \lambda_{j}(\omega_i)\right), \quad \prod\limits_{i = 1}^{m} \left( \sum\limits_{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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