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.
Accepted:
DOI: 10.5802/aif.2438
Classification: 53B20
Keywords: Extremal domain, Laplace-Beltrami operator, first eigenvalue, scalar curvature, geodesic sphere
Author's affiliations:
@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 TI - Extremal domains for the first eigenvalue of the Laplace-Beltrami operator JO - Annales de l'Institut Fourier PY - 2009 DA - 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/ UR - https://www.ams.org/mathscinet-getitem?mr=2521426 UR - https://zbmath.org/?q=an%3A1166.53029 UR - https://doi.org/10.5802/aif.2438 DO - 10.5802/aif.2438 LA - en ID - AIF_2009__59_2_515_0 ER -
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/
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