We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings.
Turning now to lower bounds, the minimum of each eigenvalue @math among domains of fixed perimeter or fixed volume is easily seen to be zero, by a pinching'' construction [Section 2.2] GP10b . Geometric lower bounds must therefore involve some other restrictions. An early result is that of Kuttler--Sigillito @cite_2 , who considered planar starlike domains and gave a bound in terms of the radius function and its derivative (see sec:starlike ). One should also mention a recent paper of Jammes @cite_17 , where a lower bound in the spirit of the classical Cheeger inequality is proved for the first nonzero Steklov eigenvalue. See also @cite_13 .
@cite_13 @cite_17 @cite_2
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Let (Mn, g) be a compact Riemannian manifold with boundary and dimensionn⩾2. In this paper we discuss the first non-zero eigenvalue problem (1) Problem (1) is known as the Stekloff problem because it was introduced by him in 1902, for bounded domains of the plane. We discuss estimates of the eigenvalueν1in terms of the geometry of the manifold (Mn, g). In the two-dimensional case we generalize Payne's Theorem [P] for bounded domains in the plane to non-negative curvature manifolds. In this case we show thatν1⩾k0, wherekg⩾k0andkgrepresents the geodesic curvature of the boundary. In higher dimensionsn⩾3 for non-negative Ricci curvature manifolds we show thatν1>k0 2, wherek0is a lower bound for any eigenvalue of the second fundamental form of the boundary. We introduce an isoperimetric constant and prove a Cheeger's type inequality for the Stekloff eigenvalue. We prove a Cheeger inequality for the first positive Steklov eigenvalue. It involves two isoperimetric constants.