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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| Vydáno v: | Experimental mathematics Ročník 26; číslo 4; s. 381 - 395 |
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| Jazyk: | angličtina |
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Taylor & Francis
02.10.2017
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| ISSN: | 1058-6458, 1944-950X |
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| 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 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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| AbstractList | 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, definedas the largest first eigenvalue of the Dirichlet Laplacian on the domains of the partition. We are inparticular interested in the way these minimal partitions change when b is varied. We present herean improvement, when k is odd, of the results on transition values of b established by B. Helffer andT. Hoffmann-Ostenhof (2014) and state a conjecture on those transition values. We establishan improved upper bound of the minimal energy by explicitly constructing hexagonal tilings of thetorus. 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 betterestimates near those transition values. 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. |
| Author | Bonnaillie-Noël, Virginie Léna, Corentin |
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| Keywords | Projected gradient algorithm Minimal partitions Dirichlet-Laplacian eigenvalues Shape optimization Finite difference method |
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| References | cit0001 Bucur [Bucur et al. 98] D. (cit0007) 1998; 8 cit0010 Coxeter [Coxeter 89] H. S. M. (cit0012) 1989 cit0008 cit0009 cit0017 cit0004 cit0015 cit0005 Courant [Courant and Hilbert 53] R. (cit0011) 1953 cit0016 cit0002 cit0013 cit0003 cit0014 |
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into k domains, with b a real parameter in (0, 1] and k an integer. We look for partitions which minimize... 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... |
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| SubjectTerms | Dirichlet-Laplacian eigenvalues finite difference method Mathematics minimal partitions Numerical Analysis Primary 49Q10 projected gradient algorithm Secondary 35J05 shape optimization Spectral Theory |
| Title | Spectral Minimal Partitions for a Family of Tori |
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