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

Carron, Gilles

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].

Reviewer: M.Puta (Timişoara)

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

Keywords:

isoperimetric inequalities; Sobolev inequality; Faber-Krahn inequality