Inequalities for the Steklov eigenvalues

This paper studies eigenvalues of some Steklov problems. Among other things, we show the following sharp estimates. Let Ω be a bounded smooth domain in an n(⩾2)-dimensional Hadamard manifold an let 0 = λ0 < λ1 ⩽ λ2 ⩽ …  denote the eigenvalues of the Steklov problem: Δu = 0 in Ω and (∂u)/(∂ν) = λu on ∂Ω  . Then ∑i=1nλi-1⩾(n2|Ω|)/(|∂Ω|) with equality holding if and only if Ω is isometric to an n-dimensional Euclidean ball. Let M be an n(⩾ 2)-dimensional compact connected Riemannian manifold with boundary and non-negative Ricci curvature. Assume that the mean curvature of ∂M is bounded below by a positive constant c and let q1 be the first eigenvalue of the Steklov problem: Δ2u = 0 in M and u =  (∂2u)/(∂ν2) − q(∂ u)/(∂ν) = 0 on ∂M. Then q1 ⩾ c with equality holding if and only if M is isometric to a ball of radius 1/c in Rn.