[Développement asymptotique du profil de Faber–Krahn d'une variété riemannienne compacte]
Nous donnons dans cette Note la preuve d'un résultat bien connu : un développement limité du profil isopérimétrique d'une variété riemannienne donne un développement limité du profil de Faber–Krahn de cette même variété.
The aim of this Note is to give a proof of a well-known fact: an asymptotic expansion of the isoperimetric profile of a Riemannian manifold for small volumes gives an asymptotic expansion of the Faber–Krahn profile for this same Riemannian manifold.
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@article{CRMATH_2008__346_21-22_1163_0,
author = {Druet, Olivier},
title = {Asymptotic expansion of the {Faber{\textendash}Krahn} profile of a compact {Riemannian} manifold},
journal = {Comptes Rendus. Math\'ematique},
pages = {1163--1167},
publisher = {Elsevier},
volume = {346},
number = {21-22},
year = {2008},
doi = {10.1016/j.crma.2008.09.022},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.09.022/}
}
TY - JOUR AU - Druet, Olivier TI - Asymptotic expansion of the Faber–Krahn profile of a compact Riemannian manifold JO - Comptes Rendus. Mathématique PY - 2008 SP - 1163 EP - 1167 VL - 346 IS - 21-22 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.09.022/ DO - 10.1016/j.crma.2008.09.022 LA - en ID - CRMATH_2008__346_21-22_1163_0 ER -
%0 Journal Article %A Druet, Olivier %T Asymptotic expansion of the Faber–Krahn profile of a compact Riemannian manifold %J Comptes Rendus. Mathématique %D 2008 %P 1163-1167 %V 346 %N 21-22 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.09.022/ %R 10.1016/j.crma.2008.09.022 %G en %F CRMATH_2008__346_21-22_1163_0
Druet, Olivier. Asymptotic expansion of the Faber–Krahn profile of a compact Riemannian manifold. Comptes Rendus. Mathématique, Tome 346 (2008) no. 21-22, pp. 1163-1167. doi : 10.1016/j.crma.2008.09.022. http://www.numdam.org/articles/10.1016/j.crma.2008.09.022/
[1] Spectral Geometry: Direct and Inverse Problems, Lecture Notes in Mathematics, vol. 1207, Springer-Verlag, 1986
[2] Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, vol. 115, Academic Press Inc., 1984
[3] Riemannian Geometry – A Modern Introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, 1993
[4] Sharp local isoperimetric inequalities involving the scalar curvature, Proc. Amer. Math. Soc., Volume 130 (2002), pp. 2351-2361
[5] F. Pacard, P. Sicbaldi, Extremal domains for the first eigenvalue of the Laplace–Beltrami operator, Ann. Inst. Fourier, in press
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