Spectral minimal partitions for a family of tori
Résumé
We study partitions of the rectangular two-dimensional flat torus of length 1 and width b into k domains, with b a 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.
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