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.