Spectral minimal partitions for a family of tori

Corentin Léna (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We study the spectral minimal $k$-partitions of the two-dimensional flat torus $\left(\RR/\ZZ\right)\times\left(\RR/b\ZZ\right)\,$, with $b$ a parameter in $b\in (0,1]$. We give a heuristic argument to compute a transition value when $k$ is odd. We support this conjecture by looking for candidates to be minimal partitions using an optimization algorithm adapted from \cite{BouBucOud09}. 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 upper bound of the minimal energy.

Mots clés: Spectral theory; minimal partitions; shape optimization; nodal domains; numerical simulations; finite difference method; projected gradient algorithm; tilings of the torus

Domaines: Equations aux dérivées partielles [math.AP]

Corentin Léna. Spectral minimal partitions for a family of tori. 2014. ⟨hal-00981843⟩