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.
Let 0 = λ1 < λ2 ≤ λ3 ≤ … be the Stekloff eigenvalues of a plane domain. The paper is concerned with formulas for ∑∞2λ(–2)j in simply and doubly connected domains. In the simply connected case it is proven that the disk minimized this sum. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) Study of the zeta function associated to the Neumann operator on planar domains yields an inequality for Steklov eigenvalues for planar domains. In this paper we establish isoperimetric inequalities for the product of some moments of inertia. As an application, we obtain an isoperimetric inequality for the product of the @math first nonzero eigenvalues of the Stekloff problem in @math .