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
Mot clés : Domaines extrémaux, opérateur de Laplace-Beltrami, première valeur propre, courbure scalaire, sphère géodésique
@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},
zbl = {1166.53029},
mrnumber = {2521426},
language = {en},
url = {http://archive.numdam.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 - http://archive.numdam.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 http://archive.numdam.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. http://archive.numdam.org/articles/10.5802/aif.2438/
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