Spectral Minimal Partitions for a Family of Tori

ABSTRACT

We study partitions of the two-dimensional flat torus into k domains, with b a real parameter in (0, 1] and k an integer. We look for partitions which minimize the energy, defined as the largest first eigenvalue of the Dirichlet Laplacian on the domains of the partition. We are in particular interested in the way these minimal partitions change when b is varied. We present here an improvement, when k is odd, of the results on transition values of b established by B. Helffer and T. Hoffmann-Ostenhof (2014) and state a conjecture on those transition values. We establish an improved upper bound of the minimal energy by explicitly constructing hexagonal tilings of the torus. These tilings are close to the partitions obtained from a systematic numerical study based on an optimization algorithm adapted from B. Bourdin, D. Bucur, and É. Oudet (2009). These numerical results also support our conjecture concerning the transition values and give better estimates near those transition values.

KEYWORDS:

• Dirichlet-Laplacian eigenvalues
• finite difference method
• minimal partitions
• projected gradient algorithm
• shape optimization

2000 AMS SUBJECT CLASSIFICATION:

• Primary 49Q10
• Secondary 35J05
• 65K10
• 65N06
• 65N25

Acknowledgments

We thank B. Helffer for his suggestions concerning Section 2, and for his help and interest with this project. We thank É. Oudet for his help in implementing the optimization algorithm.

Funding

This work was partially supported by the ANR (Agence Nationale de la Recherche), project OPTIFORM, n°ANR-12-BS01-0007-02, and by the ERC, project COMPAT, ERC-2013-ADG n°339958.

Notes

1 We thank Édouard Oudet for giving us detailed explanations on this point.