Spectral minimal partitions for a family of tori

Corentin Léna (1)

1 Dipartimento di Matematica "Giuseppe Peano" [Torino]

We study partitions of the two-dimensional flat torus (R/Z) × (R/bZ) into k domains, with b a parameter in (0, 1] and k an integer. We look for partitions that minimize an energy defined from the first eigenvalue of the Dirichlet Laplacian on the domains. We are in particular interested in the way these minimal partitions change when b is varied. We recall previous results on transition values by Helffer and Hoffmann-Ostenhof (2014), present a slight improvement when k is odd, and state some conjectures. We support these conjectures by looking for candidates to be minimal partitions using an optimization algorithm adapted from Bourdin, Bucur, and Oudet (2009). Guided by these numerical results, we construct k-partitions that are tilings of the torus by hexagons. We compute their energy and thus obtain an improved upper bound of the minimal energy.

Keywords: Minimal partitions; shape optimization; Laplacian; eigenvalues; eigenfunctions; finite difference method; projected gradient algorithm

Domains: Analysis of PDEs [math.AP]

Corentin Léna. Spectral minimal partitions for a family of tori. 2015. ⟨hal-01131601v1⟩