  We prove a sharp upper bound and a lower bound for the first nonzero
eigenvalue of the Wentzell-Laplace operator on compact manifolds with boundary
and an isoperimetric inequality for the same eigenvalue in the case where the
manifold is a bounded domain in a Euclidean space. We study some fourth order
Stekolv problems and obtain isoperimetric upper bound for the first eigenvalue
of them. We also find all the eigenvalues and eigenfunctions for two kind of
fourth order Stekolv problems on a Euclidean ball.
