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

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
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