First eigenvalue for the \(p\)-Laplace operator. (English) Zbl 0948.35090

Matei, Ana-Maria

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 39, No. 8, 1051-1068 (2000).

This paper deals with the first eigenvalue of the \(p\)-Laplacian operator \(\Delta_p\), \(p\geq 1\), on \(m\)-dimensional Riemannian manifolds. The first result is a generalization of a comparison theorem proved by S. Y. Cheng [Math. Z. 143, 289-297 (1975; Zbl 0329.53035)] for \(p=2\). The next result is a Faber-Krahn-type isoperimetric inequality for the \(p\)-Laplacian in bounded domains of a manifold with positive Ricci curvature, which was obtained in the case of the Laplacian by P. Bérard, D. Meyer [Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 513-541 (1982; Zbl 0527.35020)]. A comparison result of Lichnerowicz-Obata and Cheeger-type estimates are also generalized to the case \(p>1\).

Reviewer: B.Dittmar (Halle)

MSC:

35P15	Estimates of eigenvalues in context of PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

\(p\)-Laplacian; eigenvalue isoperimetric inequality

Citations:

Zbl 0329.53035; Zbl 0527.35020