Abstract

We study the Faber - Krahn inequality for the Dirichlet eigenvalue problem of the
Laplacian, first in $R^N$ , then on a compact smooth Riemannian manifold M . For the
latter, we consider two cases. In the first case, the compact manifold has a lower bound
on the Ricci curvature, in the second, the integral of the reciprocal of an isoperimetric
estimator function of the Riemannian manifold is convergent. In all cases, we show
that the first eigenvalue of a domain in $R^N$ , respectively M , is minimal for the ball
of the same volume, respectively, for a geodesic ball of the same relative volume in
an appropriate manifold $M^∗$ . While working with the isoperimetric estimator, the
manifold $M^∗$ need not have constant sectional curvature. In $R^N$ , we also consider the
Neumann eigenvalue problem and present the Szeg¨
o - Weinberger inequality. In this
case, the principal eigenvalue of the ball is maximal among all principal eigenvalues
of domains with same volume.