no. 1
Numerical simulations for nodal domains and spectral minimal partitions
ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 1, pp. 221-246.

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about minimal partitions and propose numerical computations to check some of their published or unpublished conjectures and exhibit new ones.

DOI : 10.1051/cocv:2008074
Classification : 35P05, 65N25, 65N30, 49Q10
Mots clés : eigenmodes of Laplace operator, minimal partitions, nodal domains, finite element method 
@article{COCV_2010__16_1_221_0,
     author = {Bonnaillie-No\"el, Virginie and Helffer, Bernard and Vial, Gregory},
     title = {Numerical simulations for nodal domains and spectral minimal partitions},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {221--246},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {1},
     year = {2010},
     doi = {10.1051/cocv:2008074},
     mrnumber = {2598097},
     zbl = {1191.35189},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2008074/}
}
TY  - JOUR
AU  - Bonnaillie-Noël, Virginie
AU  - Helffer, Bernard
AU  - Vial, Gregory
TI  - Numerical simulations for nodal domains and spectral minimal partitions
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2010
SP  - 221
EP  - 246
VL  - 16
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2008074/
DO  - 10.1051/cocv:2008074
LA  - en
ID  - COCV_2010__16_1_221_0
ER  - 
%0 Journal Article
%A Bonnaillie-Noël, Virginie
%A Helffer, Bernard
%A Vial, Gregory
%T Numerical simulations for nodal domains and spectral minimal partitions
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2010
%P 221-246
%V 16
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2008074/
%R 10.1051/cocv:2008074
%G en
%F COCV_2010__16_1_221_0
Bonnaillie-Noël, Virginie; Helffer, Bernard; Vial, Gregory. Numerical simulations for nodal domains and spectral minimal partitions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 1, pp. 221-246. doi : 10.1051/cocv:2008074. http://www.numdam.org/articles/10.1051/cocv:2008074/

[1] G. Alessandrini, Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains. Comment. Math. Helv. 69 (1994) 142-154. | Zbl

[2] P. Bérard, Inégalités isopérimétriques et applications : domaines nodaux des fonctions propres. Exposé XI, Séminaire Goulaouic-Meyer-Schwartz (1982). | Numdam | Zbl

[3] L. Bers, Local behavior of solutions of general linear elliptic equations. Commun. Pure Appl. Math. 8 (1955) 473-496. | Zbl

[4] V. Bonnaillie-Noël and G. Vial, Computations for nodal domains and spectral minimal partitions. http://w3.bretagne.ens-cachan.fr/math/simulations/MinimalPartitions (2007).

[5] D. Bucur, G. Buttazzo and A. Henrot, Existence results for some optimal partition problems. Adv. Math. Sci. Appl. 8 (1998) 571-579. | Zbl

[6] D. Bucur, B. Bourdin and E. Oudet, Numerical study of an optimal partitioning problem related to eigenvalues. (In preparation).

[7] L.A. Caffarelli and F.H. Lin, An optimal partition problem for eigenvalues. J. Sci. Comput. 31 (2007) 5-18. | Zbl

[8] M. Conti, S. Terracini and G. Verzini, An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198 (2003) 160-196. | Zbl

[9] M. Conti, S. Terracini and G. Verzini, A variational problem for the spatial segregation of reaction-diffusion systems. Indiana Univ. Math. J. 54 (2005) 779-815. | Zbl

[10] M. Conti, S. Terracini and G. Verzini, On a class of optimal partition problems related to the Fucik spectrum and to the monotonicity formula. Calc. Var. 22 (2005) 45-72. | Zbl

[11] O. Cybulski, V. Babin and R. Hołyst, Minimization of the Renyi entropy production in the space-partitioning process. Phys. Rev. E 71 (2005) 46130.

[12] 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).

[13] B. Helffer, On nodal domains and minimal spectral partitions. Conference in Montreal (April 2008).

[14] B. Helffer and T. Hoffmann-Ostenhof, Converse spectral problems for nodal domains. Mosc. Math. J. 7 (2007) 67-84. | Zbl

[15] B. Helffer and T. Hoffmann-Ostenhof, On minimal partitions for the disk and the annulus. Provisory notes in February 2007.

[16] 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. | Numdam | Zbl

[17] D. Jakobson, M. Levitin, N. Nadirashvili and I. Polterovic, Spectral problems with mixed Dirichlet-Neumann boundary conditions: isospectrality and beyond. J. Comput. Appl. Math. 194 (2006) 141-155. | Zbl

[18] N. Landais, Problèmes de régularité en optimisation de forme. Ph.D. Thesis, ENS Cachan Bretagne, France (2007).

[19] M. Levitin, L. Parnovski and I. Polterovich, Isospectral domains with mixed boundary conditions. J. Phys. A 39 (2006) 2073-2082. | Zbl

[20] D. Martin, The finite element library Mélina. http://perso.univ-rennes1.fr/daniel.martin/melina (2006).

[21] A. Melas, On the nodal line of the second eigenfunction of the Laplacian on ℝ 2 . J. Differential Geom. 35 (1992) 255-263. | Zbl

[22] A. Pleijel, Remarks on Courant's nodal theorem. Comm. Pure. Appl. Math 9 (1956) 543-550. | Zbl

[23] G. Pólya, On the eigenvalues of vibrating membranes. Proc. London Mah. Soc. 3 (1961) 419-433. | Zbl

Cité par Sources :