Extremal domains for the first eigenvalue of the Laplace-Beltrami operator

Frank Pacard[1]; Pieralberto Sicbaldi[2]

  • [1] Université Paris Est UFR des Sciences et Technologie Bâtiment P3 - 4e étage 61, avenue du Général de Gaulle 94010 Créteil Cedex (France)
  • [2] Université Paris 12 UFR des Sciences et Technologie Bâtiment P3 - 4e étage 61, avenue du Général de Gaulle 94010 Créteil Cedex (France)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 2, page 515-542
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Pacard, Frank, and Sicbaldi, Pieralberto. "Extremal domains for the first eigenvalue of the Laplace-Beltrami operator." Annales de l’institut Fourier 59.2 (2009): 515-542. <http://eudml.org/doc/10402>.

@article{Pacard2009,
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.},
affiliation = {Université Paris Est UFR des Sciences et Technologie Bâtiment P3 - 4e étage 61, avenue du Général de Gaulle 94010 Créteil Cedex (France); Université Paris 12 UFR des Sciences et Technologie Bâtiment P3 - 4e étage 61, avenue du Général de Gaulle 94010 Créteil Cedex (France)},
author = {Pacard, Frank, Sicbaldi, Pieralberto},
journal = {Annales de l’institut Fourier},
keywords = {Extremal domain; Laplace-Beltrami operator; first eigenvalue; scalar curvature; geodesic sphere; extremal domain},
language = {eng},
number = {2},
pages = {515-542},
publisher = {Association des Annales de l’institut Fourier},
title = {Extremal domains for the first eigenvalue of the Laplace-Beltrami operator},
url = {http://eudml.org/doc/10402},
volume = {59},
year = {2009},
}

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
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 2
SP - 515
EP - 542
AB - 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.
LA - eng
KW - Extremal domain; Laplace-Beltrami operator; first eigenvalue; scalar curvature; geodesic sphere; extremal domain
UR - http://eudml.org/doc/10402
ER -

References

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  1. O. Druet, Sharp local isoperimetric inequalities involving the scalar curvature, Proc. Amer. Math. Soc. 130 (2002), 2351-2361 Zbl1067.53026MR1897460
  2. A. El Soufi, S. Ilias, Domain deformations and eigenvalues of the Dirichlet Laplacian in Riemannian manifold, Illinois Journal of Mathematics 51 (2007), 645-666 Zbl1124.49035MR2342681
  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), 169-172 Zbl49.0342.03
  4. P. R. Garadedian, M. Schiffer, Variational problems in the theory of elliptic partial differetial equations, Journal of Rational Mechanics and Analysis (1953), 137-171 Zbl0050.10002MR54819
  5. E. Krahn, Uber eine von Raleigh formulierte Minimaleigenschaft der Kreise, 94 (1924), Math. Ann. 
  6. E. Krahn, Uber Minimaleigenschaften der Kugel in drei und mehr dimensionen, A9 (1926), Acta Comm. Univ. Tartu (Dorpat) Zbl52.0510.03
  7. J. M. Lee, T. H. Parker, The Yamabe Problem, Bulletin of the American Mathematical Society 17 (1987), 37-91 Zbl0633.53062MR888880
  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 Zbl1165.53038
  10. R. Schoen, S. T. Yau, Lectures on Differential Geometry, (1994), International Press Zbl0830.53001MR1333601
  11. T. J. Willmore, Riemannian Geometry, (1996), Oxford Science Publications Zbl0797.53002MR1261641
  12. R. Ye, Foliation by constant mean curvature spheres, Pacific Journal of Mathematics 147 (1991), 381-396 Zbl0722.53022MR1084717
  13. D. Z. Zanger, Eigenvalue variation for the Neumann problem, Applied Mathematics Letters 14 (2001), 39-43 Zbl0977.58028MR1793700

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