Spectral Minimal Partitions for a Family of Tori

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 interes...

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Bibliographic Details
Published in:Experimental mathematics Vol. 26; no. 4; pp. 381 - 395
Main Authors: Bonnaillie-Noël, Virginie, Léna, Corentin
Format: Journal Article
Language:English
Published: Taylor & Francis 02.10.2017
Subjects:
Dirichlet-Laplacian eigenvalues
finite difference method
Mathematics
minimal partitions
Numerical Analysis
Primary 49Q10
projected gradient algorithm
Secondary 35J05
shape optimization
Spectral Theory
Projected gradient algorithm
Minimal partitions
Dirichlet-Laplacian eigenvalues
Shape optimization
Finite difference method
ISSN:1058-6458, 1944-950X
Online Access:Get full text
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Summary: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.
ISSN:1058-6458
1944-950X
DOI:10.1080/10586458.2016.1183154