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 E. Oudet (2009). These numerical results also support our conjecture concerning the transition values and give better estimates near those transition values.

Spectral Minimal Partitions for a Family of Tori

Léna, Corentin
2017

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 E. Oudet (2009). These numerical results also support our conjecture concerning the transition values and give better estimates near those transition values.
File in questo prodotto:
Non ci sono file associati a questo prodotto.
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/3460395
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 5
  • ???jsp.display-item.citation.isi??? 4
social impact