Geodesic
Estimates of sums of zero multiplicities for eigenfunctions of the Laplace–Beltrami operator
Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 5, pp. 85-90 
V. N. Karpushkin. Estimates of sums of zero multiplicities for eigenfunctions of the Laplace–Beltrami operator. Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 5, pp. 85-90. http://geodesic.mathdoc.fr/item/FPM_2005_11_5_a6/
@article{FPM_2005_11_5_a6,
     author = {V. N. Karpushkin},
     title = {Estimates of sums of zero multiplicities for eigenfunctions of the {Laplace{\textendash}Beltrami} operator},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {85--90},
     year = {2005},
     volume = {11},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2005_11_5_a6/}
}
TY  - JOUR
AU  - V. N. Karpushkin
TI  - Estimates of sums of zero multiplicities for eigenfunctions of the Laplace–Beltrami operator
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2005
SP  - 85
EP  - 90
VL  - 11
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/FPM_2005_11_5_a6/
LA  - ru
ID  - FPM_2005_11_5_a6
ER  - 
%0 Journal Article
%A V. N. Karpushkin
%T Estimates of sums of zero multiplicities for eigenfunctions of the Laplace–Beltrami operator
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2005
%P 85-90
%V 11
%N 5
%U http://geodesic.mathdoc.fr/item/FPM_2005_11_5_a6/
%G ru
%F FPM_2005_11_5_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain an upper estimate $N-\chi(M)$ for the sum $Q_N$ of singular zero multiplicities of the $N$th eigenfunction of the Laplace–Beltrami operator on the two-dimensional, compact, connected Riemann manifold $M$, where $\chi(M)$ is the Euler characteristic of $M$. There are given more strong estimates, but equivalent asymptotically ($N\to\infty$), for the cases of the sphere $S^2$ and the projective plane $\mathbb R^2$. Asymptotically more sharp estimate are shown for the case of a domain on the plane.

[1] Karpushkin V. N., “Kratnosti osobykh tochek sobstvennykh funktsii operatora Laplasa–Beltrami”, Funkts. analiz i ego pril., 29:1 (1995), 80–82 | MR | Zbl

[2] Karpushkin V. N., “O chisle komponent dopolneniya k nekotorym algebraicheskim krivym”, Uspekhi mat. nauk, 57:6 (2002), 185–186 | MR | Zbl

[3] Kurant R., Gilbert D., Metody matematicheskoi fiziki, T. 1, GITTL, M.; L., 1951

[4] Aronszajn N., “A unique continuation theorem for solution of elliptic partial differential equations of second order”, J. Math. Pures Appl., 36 (1957), 235–249 | MR | Zbl

[5] Bers L., “Local behavior of solutions of general linear elliptic equations”, Comm. Pure Appl. Math., 8 (1955), 473–496 | DOI | MR | Zbl

[6] Pleijel Å., “Remarks on Courant's nodal line theorem”, Comm. Pure Appl. Math., 9 (1956), 543–550 | DOI | MR | Zbl