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about minimal partitions and propose numerical computations to check some of their published or unpublished conjectures and exhibit new ones.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 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, Volume 16 (2010) no. 1, pp. 221-246. doi : 10.1051/cocv:2008074. http://www.numdam.org/articles/10.1051/cocv:2008074/
[1] Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains. Comment. Math. Helv. 69 (1994) 142-154. | Zbl
,[2] Inégalités isopérimétriques et applications : domaines nodaux des fonctions propres. Exposé XI, Séminaire Goulaouic-Meyer-Schwartz (1982). | Numdam | Zbl
,[3] Local behavior of solutions of general linear elliptic equations. Commun. Pure Appl. Math. 8 (1955) 473-496. | Zbl
,[4] Computations for nodal domains and spectral minimal partitions. http://w3.bretagne.ens-cachan.fr/math/simulations/MinimalPartitions (2007).
and ,[5] Existence results for some optimal partition problems. Adv. Math. Sci. Appl. 8 (1998) 571-579. | Zbl
, and ,[6] Numerical study of an optimal partitioning problem related to eigenvalues. (In preparation).
, and ,[7] An optimal partition problem for eigenvalues. J. Sci. Comput. 31 (2007) 5-18. | Zbl
and ,[8] An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198 (2003) 160-196. | Zbl
, and ,[9] A variational problem for the spatial segregation of reaction-diffusion systems. Indiana Univ. Math. J. 54 (2005) 779-815. | Zbl
, and ,[10] On a class of optimal partition problems related to the Fucik spectrum and to the monotonicity formula. Calc. Var. 22 (2005) 45-72. | Zbl
, and ,[11] Minimization of the Renyi entropy production in the space-partitioning process. Phys. Rev. E 71 (2005) 46130.
, and ,[12] 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] On nodal domains and minimal spectral partitions. Conference in Montreal (April 2008).
,[14] Converse spectral problems for nodal domains. Mosc. Math. J. 7 (2007) 67-84. | Zbl
and ,[15] On minimal partitions for the disk and the annulus. Provisory notes in February 2007.
and ,[16] Nodal domains and spectral minimal partitions. Ann. Inst. H. Poincaré Anal. Non Linéaire (2007) doi:10.1016/j.anihpc.2007.07.004. | Numdam | Zbl
, and ,[17] Spectral problems with mixed Dirichlet-Neumann boundary conditions: isospectrality and beyond. J. Comput. Appl. Math. 194 (2006) 141-155. | Zbl
, , and ,[18] Problèmes de régularité en optimisation de forme. Ph.D. Thesis, ENS Cachan Bretagne, France (2007).
,[19] Isospectral domains with mixed boundary conditions. J. Phys. A 39 (2006) 2073-2082. | Zbl
, and ,[20] The finite element library Mélina. http://perso.univ-rennes1.fr/daniel.martin/melina (2006).
,[21] On the nodal line of the second eigenfunction of the Laplacian on . J. Differential Geom. 35 (1992) 255-263. | Zbl
,[22] Remarks on Courant's nodal theorem. Comm. Pure. Appl. Math 9 (1956) 543-550. | Zbl
,[23] On the eigenvalues of vibrating membranes. Proc. London Mah. Soc. 3 (1961) 419-433. | Zbl
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