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Numerical simulations for nodal domainsand spectral minimal partitions

Published online by Cambridge University Press:  19 December 2008

Virginie Bonnaillie-Noël ,
Bernard Helffer  and
Gregory Vial
Virginie Bonnaillie-Noël
Affiliation:
IRMAR, ENS Cachan Bretagne, CNRS, UEB, avenue Robert Schuman, 35170 Bruz, France. Virginie.Bonnaillie@bretagne.ens-cachan.fr; Gregory.Vial@bretagne.ens-cachan.fr
Bernard Helffer
Affiliation:
Laboratoire de Mathématiques, Bâtiment 425, Université Paris-Sud et CNRS, 91405 Orsay Cedex, France. Bernard.Helffer@math.u-psud.fr
Gregory Vial
Affiliation:
IRMAR, ENS Cachan Bretagne, CNRS, UEB, avenue Robert Schuman, 35170 Bruz, France. Virginie.Bonnaillie@bretagne.ens-cachan.fr; Gregory.Vial@bretagne.ens-cachan.fr
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Abstract

We recall here some theoretical results of Helffer et al. [Ann. Inst. H. Poincaré Anal. Non Linéaire (2007) doi:10.1016/j.anihpc.2007.07.004] about minimal partitions and propose numerical computations to check some of their published or unpublished conjectures and exhibit new ones.

Keywords

Eigenmodes of Laplace operatorminimal partitionsnodal domainsfinite element method
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 1 , January 2010 , pp. 221 - 246
Copyright
© EDP Sciences, SMAI, 2008

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References

Alessandrini, G., Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains. Comment. Math. Helv. 69 (1994) 142–154. CrossRef
P. Bérard, Inégalités isopérimétriques et applications : domaines nodaux des fonctions propres. Exposé XI, Séminaire Goulaouic-Meyer-Schwartz (1982).
Bers, L., Local behavior of solutions of general linear elliptic equations. Commun. Pure Appl. Math. 8 (1955) 473–496. CrossRef
V. Bonnaillie-Noël and G. Vial, Computations for nodal domains and spectral minimal partitions. http://w3.bretagne.ens-cachan.fr/math/simulations/MinimalPartitions (2007).
Bucur, D., Buttazzo, G. and Henrot, A., Existence results for some optimal partition problems. Adv. Math. Sci. Appl. 8 (1998) 571–579.
D. Bucur, B. Bourdin and E. Oudet, Numerical study of an optimal partitioning problem related to eigenvalues. (In preparation).
Caffarelli, L.A. and Lin, F.H., An optimal partition problem for eigenvalues. J. Sci. Comput. 31 (2007) 5–18. CrossRef
Conti, M., Terracini, S. and Verzini, G., An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198 (2003) 160–196. CrossRef
Conti, M., Terracini, S. and Verzini, G., A variational problem for the spatial segregation of reaction-diffusion systems. Indiana Univ. Math. J. 54 (2005) 779–815. CrossRef
Conti, M., Terracini, S. and Verzini, G., On a class of optimal partition problems related to the Fucik spectrum and to the monotonicity formula. Calc. Var. 22 (2005) 45–72. CrossRef
Cybulski, O., Babin, V. and Hołyst, R., Minimization of the Renyi entropy production in the space-partitioning process. Phys. Rev. E 71 (2005) 46130. CrossRef
B. Helffer, Domaines nodaux et partitions spectrales minimales (d'après B. Helffer, T. Hoffmann-Ostenhof et S. Terracini). Séminaire EDP de l'École Polytechnique (Déc. 2006).
B. Helffer, On nodal domains and minimal spectral partitions. Conference in Montreal (April 2008).
Helffer, B. and Hoffmann-Ostenhof, T., Converse spectral problems for nodal domains. Mosc. Math. J. 7 (2007) 67–84.
B. Helffer and T. Hoffmann-Ostenhof, On minimal partitions for the disk and the annulus. Provisory notes in February 2007.
B. Helffer, T. Hoffmann-Ostenhof and S. Terracini, Nodal domains and spectral minimal partitions. Ann. Inst. H. Poincaré Anal. Non Linéaire (2007) doi:10.1016/j.anihpc.2007.07.004.
Jakobson, D., Levitin, M., Nadirashvili, N. and Polterovic, I., Spectral problems with mixed Dirichlet-Neumann boundary conditions: isospectrality and beyond. J. Comput. Appl. Math. 194 (2006) 141–155. CrossRef
N. Landais, Problèmes de régularité en optimisation de forme. Ph.D. Thesis, ENS Cachan Bretagne, France (2007).
Levitin, M., Parnovski, L. and Polterovich, I., Isospectral domains with mixed boundary conditions. J. Phys. A 39 (2006) 2073–2082. CrossRef
D. Martin, The finite element library Mélina. http://perso.univ-rennes1.fr/daniel.martin/melina (2006).
Melas, A., On the nodal line of the second eigenfunction of the Laplacian on $\mathbb R^2$ . J. Differential Geom. 35 (1992) 255–263. CrossRef
Pleijel, A., Remarks on Courant's nodal theorem. Comm. Pure. Appl. Math 9 (1956) 543–550. CrossRef
Pólya, G., On the eigenvalues of vibrating membranes. Proc. London Mah. Soc. 3 (1961) 419–433. CrossRef