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Faber-Krahn isoperimetric inequalities and consequences. (Inégalités isopérimétriques de Faber-Krahn et conséquences.) (French) Zbl 0884.58088

Besse, Arthur L. (ed.), Actes de la table ronde de géométrie différentielle en l’honneur de Marcel Berger, Luminy, France, 12–18 juillet, 1992. Paris: Société Mathématique de France. Sémin. Congr. 1, 205-232 (1996).
The author shows that a complete non-compact Riemannian manifold satisfies a Faber-Krahn inequality if and only if it satisfies a Sobolev one. He also gives some other equivalent properties such as bound on heat kernel or bound on Green functions. He also proves that we have a uniform lower bound for the volume of geodesic balls in terms of the Faber-Krahn inequality.
For the entire collection see [Zbl 0859.00016].

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58C40 Spectral theory; eigenvalue problems on manifolds
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