Pleijel’s nodal domain theorem for Neumann and Robin eigenfunctions  [ Théorème nodal de Pleijel pour les fonctions propres de Neumann et de Robin ]
Annales de l'Institut Fourier, à paraître, p. 1-19
Nous montrons que le cas d’égalité dans le théorème de Courant n’est réalisé que pour un nombre fini de valeurs propres du laplacien de Neumann, dans un ouvert borné connexe de n à bord C 1,1 , lorsque n2. Ce résultat est analogue au théorème démontré par Pleijel en 1956 pour le laplacien de Dirichlet. Nous montrons de plus que la méthode de démonstration et le résultat peuvent être étendus à une classe de conditions au bord de Robin.
We show that equality in Courant’s nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in an open, bounded, and connected subset of n with a C 1,1 boundary, when n2. This result is analogous to the theorem proved by Pleijel in 1956 for the Dirichlet Laplacian. We also show that the argument and the result extend to a class of Robin boundary conditions.
Reçu le : 2016-12-14
Révisé le : 2017-11-09
Accepté le : 2018-02-02
Classification:  35P05,  35P15,  35P20,  58J50
Mots clés: valeurs propres de Neumann, valeurs propres de Robin, domaines nodaux, théorème de Courant, théorème de Pleijel
@unpublished{AIF_0__0_0_A7_0,
     author = {L\'ena, Corentin},
     title = {Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Léna, Corentin. Pleijel’s nodal domain theorem for Neumann and Robin eigenfunctions. Annales de l'Institut Fourier, à paraître, pp. 1-19.

[1] Bérard, Pierre; Helffer, Bernard Inégalités isopérimétriques et applications, Ann. Sci. Éc. Norm. Supér., Tome 15 (1982) no. 3, pp. 513-541 | MR 690651

[2] Bérard, Pierre; Helffer, Bernard The weak Pleijel theorem with geometric control, J. Spectr. Theory, Tome 6 (2016) no. 4, pp. 717-733 | Article | Zbl 1372.35194

[3] Van Den Berg, Michiel; Gittins, Katie On the number of Courant-sharp Dirichlet eigenvalues, J. Spectr. Theory, Tome 6 (2016) no. 4, pp. 735-745 | Article

[4] Bonnaillie-Noël, Virginie; Helffer, Bernard Nodal and spectral minimal partitions – the state of the art in 2016, Shape optimization and spectral theory, De Gruyter (2017), pp. 353-397 | MR 3681154 | Zbl 1373.49050

[5] Bourgain, Jean On Pleijel’s nodal domain theorem, Int. Math. Res. Not. (2015) no. 6, pp. 1601-1612 | MR 3340367

[6] Charron, Philippe A Pleijel-type theorem for the quantum harmonic oscillator, J. Spectr. Theory, Tome 8 (2018) no. 2, pp. 715-732 | Zbl 06898063

[7] Charron, Philippe; Helffer, Bernard; Hoffmann-Ostenhof, Thomas Pleijel’s theorem for Schrödinger operators with radial potentials, Ann. Math. Qué., Tome 42 (2018) no. 1, pp. 7-29 | Article | Zbl 1384.35048

[8] Courant, Richard Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke, Gött. Nachr., Tome 1923 (1923), pp. 81-84 | Zbl 49.0342.01

[9] Courant, Richard; Hilbert, David Methods of mathematical physics. Vol. I, Interscience Publishers (1953), xv+561 pages | MR 0065391 | Zbl 0051.28802

[10] Donnelly, Harold Counting nodal domains in Riemannian manifolds, Ann. Global Anal. Geom., Tome 46 (2014) no. 1, pp. 57-61 | Article

[11] Grisvard, Pierre Elliptic problems in nonsmooth domains, Pitman Publishing Inc., Monographs and Studies in Mathematics, Tome 24 (1985), xiv+410 pages | MR 775683 | Zbl 0695.35060

[12] Hardt, Robert; Hoffmann-Ostenhof, Maria; Hoffmann-Ostenhof, Thomas; Nadirashvili, Nikolai Critical sets of solutions to elliptic equations, J. Differ. Geom., Tome 51 (1999) no. 2, pp. 359-373 http://projecteuclid.org/euclid.jdg/1214425070 | MR 1728303 | Zbl 1144.35370

[13] Helffer, Bernard; Hoffmann-Ostenhof, Thomas A review on large k minimal spectral k-partitions and Pleijel’s theorem, Spectral theory and partial differential equations, American Mathematical Society (Contemporary Mathematics) Tome 640 (2015), pp. 39-57 | Article | MR 3381015 | Zbl 1346.35132

[14] Helffer, Bernard; Hoffmann-Ostenhof, Thomas; Terracini, Susanna Nodal domains and spectral minimal partitions, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Tome 26 (2009) no. 1, pp. 101-138 | Article | MR 2483815

[15] Helffer, Bernard; Persson Sundqvist, Mikael On nodal domains in Euclidean balls, Proc. Am. Math. Soc., Tome 144 (2016) no. 11, pp. 4777-4791 | Article | MR 3544529

[16] Henrot, Antoine Extremum Problems for Eigenvalues of Elliptic Operators, Birkhäuser, Frontiers in Mathematics (2006), x+202 pages | MR 2251558 | Zbl 1109.35081

[17] Peetre, Jaak A generalization of Courant’s nodal domain theorem, Math. Scand., Tome 5 (1957), pp. 15-20 | MR 0092917

[18] Pleijel, Åke Remarks on Courant’s nodal line theorem, Commun. Pure Appl. Math., Tome 9 (1956), pp. 543-550 | MR 0080861 | Zbl 0070.32604

[19] Polterovich, Iosif Pleijel’s nodal domain theorem for free membranes, Proc. Am. Math. Soc., Tome 137 (2009) no. 3, pp. 1021-1024 | Article | MR 2457442

[20] Reed, Michael; Simon, Barry Methods of Modern Mathematical physics. II. Fourier Analysis, Self-Adjointness, Academic Press (1975), xv+361 pages | MR 0493420 | Zbl 0308.47002

[21] Reed, Michael; Simon, Barry Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press (1978), xv+396 pages | MR 0493421 | Zbl 0401.47001

[22] Rozenblum, G. V.; Shubin, Mikhail A.; Solomyak, Mikhaĭl Z. Spectral Theory of Differential Operators, Partial differential equations. VII, Springer (Encyclopaedia of Mathematical Sciences) Tome 64 (1994) | Article | Zbl 0805.35081

[23] Steinerberger, Stefan A geometric uncertainty principle with an application to Pleijel’s estimate, Ann. Henri Poincaré, Tome 15 (2014) no. 12, pp. 2299-2319 | Article | MR 3272823

[24] Toth, John A.; Zelditch, Steve Counting nodal lines which touch the boundary of an analytic domain, J. Differ. Geom., Tome 81 (2009) no. 3, pp. 649-686 http://projecteuclid.org/euclid.jdg/1236604347 | MR 2487604

[25] Weinstock, Robert Calculus of variations. With applications to physics and engineering, Dover Publications, Dover Books on Advanced Mathematics (1974), x+326 pages (reprint of the 1952 edition) | MR 0443487 | Zbl 0296.49001