Spectral Minimal Partitions for a Family of Tori

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.

[1]  Corentin L'ena.  Courant-sharp eigenvalues of a two-dimensional torus , 2015, 1501.02558.

[2]  V. Bonnaillie-Noël,et al.  Spectral minimal partitions of a sector , 2013 .

[3]  Beniamin Bogosel,et al.  The method of fundamental solutions applied to boundary eigenvalue problems , 2016, J. Comput. Appl. Math..

[4]  Charles M. Elliott,et al.  A computational approach to an optimal partition problem on surfaces , 2014, 1408.2355.

[5]  Dorin Bucur,et al.  Optimal Partitions for Eigenvalues , 2009, SIAM J. Sci. Comput..

[6]  Susanna Terracini,et al.  Asymptotic estimates for the spatial segregation of competitive systems , 2005 .

[7]  Dorin Bucur,et al.  Uniform Concentration-Compactness for Sobolev Spaces on Variable Domains , 2000 .

[8]  B. Helffer,et al.  the sphere , 2009 .

[9]  B. Helffer,et al.  Aharonov–Bohm Hamiltonians, isospectrality and minimal partitions , 2009 .

[10]  Susanna Terracini,et al.  On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae , 2003 .

[11]  Minimal partitions for anisotropic tori , 2014, 1509.04622.

[12]  Dr. M. G. Worster.  Methods of Mathematical Physics , 1947, Nature.

[13]  L. A. Cafferelli,et al.  An Optimal Partition Problem for Eigenvalues , 2007, J. Sci. Comput..

[14]  B. Helffer,et al.  Nodal Domains and Spectral Minimal Partitions , 2006, math/0610975.

[15]  Dorin Bucur,et al.  Variational Methods in Shape Optimization Problems , 2005, Progress in Nonlinear Differential Equations and Their Applications.

[16]  P. Lockhart,et al.  Introduction to Geometry , 1940, The Mathematical Gazette.

[17]  B. Helffer,et al.  Remarks on two notions of spectral minimal partitions , 2009 .