A Reilly inequality for the first Steklov eigenvalue

Abstract

Let M be a compact submanifold with boundary of a Euclidean space or a Sphere. In this paper, we derive an upper bound for the first non-zero eigenvalue p1 of Steklov problem on M in terms of the r-th mean curvatures of its boundary ∂ M. The upper bound obtained is sharp. © 2011 Elsevier B.V.