We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of Laplace-Beltrami operator in some Riemannian manifold. These domains are close to geodesic spheres of small radius centered at a nondegenerate critical point of the scalar curvature.
Nous prouvons l’existence de domaines extrémaux avec volume petit et fixé pour la première valeur propre de l’opérateur de Laplace-Beltrami dans certaines variétés riemanniennes. Ces domaines ressemblent à des sphères géodésiques de rayon petit centrées en un point critique non dégénéré de la courbure scalaire.
Keywords: Extremal domain, Laplace-Beltrami operator, first eigenvalue, scalar curvature, geodesic sphere
@article{AIF_2009__59_2_515_0,
author = {Pacard, Frank and Sicbaldi, Pieralberto},
title = {Extremal domains for the first eigenvalue of the {Laplace-Beltrami} operator},
journal = {Annales de l'Institut Fourier},
pages = {515--542},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {59},
number = {2},
year = {2009},
doi = {10.5802/aif.2438},
mrnumber = {2521426},
zbl = {1166.53029},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2438/}
}
TY - JOUR AU - Pacard, Frank AU - Sicbaldi, Pieralberto TI - Extremal domains for the first eigenvalue of the Laplace-Beltrami operator JO - Annales de l'Institut Fourier PY - 2009 SP - 515 EP - 542 VL - 59 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2438/ DO - 10.5802/aif.2438 LA - en ID - AIF_2009__59_2_515_0 ER -
%0 Journal Article %A Pacard, Frank %A Sicbaldi, Pieralberto %T Extremal domains for the first eigenvalue of the Laplace-Beltrami operator %J Annales de l'Institut Fourier %D 2009 %P 515-542 %V 59 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2438/ %R 10.5802/aif.2438 %G en %F AIF_2009__59_2_515_0
Pacard, Frank; Sicbaldi, Pieralberto. Extremal domains for the first eigenvalue of the Laplace-Beltrami operator. Annales de l'Institut Fourier, Volume 59 (2009) no. 2, pp. 515-542. doi : 10.5802/aif.2438. https://aif.centre-mersenne.org/articles/10.5802/aif.2438/
[1] Sharp local isoperimetric inequalities involving the scalar curvature, Proc. Amer. Math. Soc., Volume 130 (2002) no. 8, pp. 2351-2361 | DOI | MR | Zbl
[2] Domain deformations and eigenvalues of the Dirichlet Laplacian in Riemannian manifold, Illinois Journal of Mathematics, Volume 51 (2007), pp. 645-666 | MR | Zbl
[3] Beweis dass unter allen homogenen Menbranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundtonggibt, Sitzungsber. Bayer. Akad. der Wiss. Math.-Phys. (1923), pp. 169-172 (Munich) | JFM
[4] Variational problems in the theory of elliptic partial differetial equations, Journal of Rational Mechanics and Analysis (1953) no. 2, pp. 137-171 | MR | Zbl
[5] Uber eine von Raleigh formulierte Minimaleigenschaft der Kreise, 94, Math. Ann., 1924 | JFM
[6] Uber Minimaleigenschaften der Kugel in drei und mehr dimensionen, A9, Acta Comm. Univ. Tartu (Dorpat), 1926 | JFM
[7] The Yamabe Problem, Bulletin of the American Mathematical Society, Volume 17 (1987) no. 1, pp. 37-91 | DOI | MR | Zbl
[8] Le profil isopérimétrique d’une variété riemannienne compacte pour les petits volumes, Thèse de l’Université Paris 11, 2006
[9] Constant mean curvature sphere in riemannian manifolds (preprint) | Zbl
[10] Lectures on Differential Geometry, International Press, 1994 | MR | Zbl
[11] Riemannian Geometry, Oxford Science Publications, 1996 | MR | Zbl
[12] Foliation by constant mean curvature spheres, Pacific Journal of Mathematics, Volume 147 (1991) no. 2, pp. 381-396 | MR | Zbl
[13] Eigenvalue variation for the Neumann problem, Applied Mathematics Letters, Volume 14 (2001), pp. 39-43 | DOI | MR | Zbl
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