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Frank Pacard; Pieralberto Sicbaldi
Extremal domains for the first eigenvalue of the Laplace-Beltrami operator
(Domains extrémaux pour la première valeur propre de l’opérateur de Laplace-Beltrami)
Annales de l'institut Fourier, 59 no. 2 (2009), p. 515-542, doi: 10.5802/aif.2438
Article: subscription required (your ip address: 184.73.7.143) | Reviews MR 2521426 | Zbl 1166.53029
Class. Math.: 53B20
Keywords: Extremal domain, Laplace-Beltrami operator, first eigenvalue, scalar curvature, geodesic sphere
[1] O. Druet, “Sharp local isoperimetric inequalities involving the scalar curvature”, Proc. Amer. Math. Soc. 130 (2002) no. 8, p. 2351-2361
Article | MR 1897460 | Zbl 1067.53026
[2] A. El Soufi & S. Ilias, “Domain deformations and eigenvalues of the Dirichlet Laplacian in Riemannian manifold”, Illinois Journal of Mathematics 51 (2007), p. 645-666
Article | MR 2342681 | Zbl 1124.49035
[3] G. Faber, “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), p. 169-172, Munich JFM 49.0342.03
[4] P. R. Garadedian & M. Schiffer, “Variational problems in the theory of elliptic partial differetial equations”, Journal of Rational Mechanics and Analysis (1953) no. 2, p. 137-171 MR 54819 | Zbl 0050.10002
[5] E. Krahn, Uber eine von Raleigh formulierte Minimaleigenschaft der Kreise 94, Math. Ann., 1924 JFM 51.0356.05
[6] E. Krahn, Uber Minimaleigenschaften der Kugel in drei und mehr dimensionen A9, Acta Comm. Univ. Tartu (Dorpat), 1926 JFM 52.0510.03
[7] J. M. Lee & T. H. Parker, “The Yamabe Problem”, Bulletin of the American Mathematical Society 17 (1987) no. 1, p. 37-91
Article | MR 888880 | Zbl 0633.53062
[8] S. Nardulli, “Le profil isopérimétrique d’une variété riemannienne compacte pour les petits volumes”, Thèse de l’Université Paris 11, 2006
[9] F. Pacard & X. Xu, “Constant mean curvature sphere in riemannian manifolds”, preprint
[10] R. Schoen & S. T. Yau, Lectures on Differential Geometry, International Press, 1994 MR 1333601 | Zbl 0830.53001
[11] T. J. Willmore, Riemannian Geometry, Oxford Science Publications, 1996 MR 1261641 | Zbl 0797.53002
[12] R. Ye, “Foliation by constant mean curvature spheres”, Pacific Journal of Mathematics 147 (1991) no. 2, p. 381-396
Article | MR 1084717 | Zbl 0722.53022
[13] D. Z. Zanger, “Eigenvalue variation for the Neumann problem”, Applied Mathematics Letters 14 (2001), p. 39-43
Article | MR 1793700 | Zbl 0977.58028
Frank Pacard; Pieralberto Sicbaldi
Extremal domains for the first eigenvalue of the Laplace-Beltrami operator
(Domains extrémaux pour la première valeur propre de l’opérateur de Laplace-Beltrami)
Annales de l'institut Fourier, 59 no. 2 (2009), p. 515-542, doi: 10.5802/aif.2438
Article: subscription required (your ip address: 184.73.7.143) | Reviews MR 2521426 | Zbl 1166.53029
Class. Math.: 53B20
Keywords: Extremal domain, Laplace-Beltrami operator, first eigenvalue, scalar curvature, geodesic sphere
Résumé - Abstract
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.
Bibliography
Article | MR 1897460 | Zbl 1067.53026
[2] A. El Soufi & S. Ilias, “Domain deformations and eigenvalues of the Dirichlet Laplacian in Riemannian manifold”, Illinois Journal of Mathematics 51 (2007), p. 645-666
Article | MR 2342681 | Zbl 1124.49035
[3] G. Faber, “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), p. 169-172, Munich JFM 49.0342.03
[4] P. R. Garadedian & M. Schiffer, “Variational problems in the theory of elliptic partial differetial equations”, Journal of Rational Mechanics and Analysis (1953) no. 2, p. 137-171 MR 54819 | Zbl 0050.10002
[5] E. Krahn, Uber eine von Raleigh formulierte Minimaleigenschaft der Kreise 94, Math. Ann., 1924 JFM 51.0356.05
[6] E. Krahn, Uber Minimaleigenschaften der Kugel in drei und mehr dimensionen A9, Acta Comm. Univ. Tartu (Dorpat), 1926 JFM 52.0510.03
[7] J. M. Lee & T. H. Parker, “The Yamabe Problem”, Bulletin of the American Mathematical Society 17 (1987) no. 1, p. 37-91
Article | MR 888880 | Zbl 0633.53062
[8] S. Nardulli, “Le profil isopérimétrique d’une variété riemannienne compacte pour les petits volumes”, Thèse de l’Université Paris 11, 2006
[9] F. Pacard & X. Xu, “Constant mean curvature sphere in riemannian manifolds”, preprint
[10] R. Schoen & S. T. Yau, Lectures on Differential Geometry, International Press, 1994 MR 1333601 | Zbl 0830.53001
[11] T. J. Willmore, Riemannian Geometry, Oxford Science Publications, 1996 MR 1261641 | Zbl 0797.53002
[12] R. Ye, “Foliation by constant mean curvature spheres”, Pacific Journal of Mathematics 147 (1991) no. 2, p. 381-396
Article | MR 1084717 | Zbl 0722.53022
[13] D. Z. Zanger, “Eigenvalue variation for the Neumann problem”, Applied Mathematics Letters 14 (2001), p. 39-43
Article | MR 1793700 | Zbl 0977.58028
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310