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.
Regarding other eigenvalue functionals, Dittmar @cite_38 proved that among simply connected planar domains with given conformal radius, the disk minimizes the infinite sum of reciprocals of all squares, @math . Henrot--Philippin--Safoui @cite_32 proved that among convex domains of fixed measure in @math , the product of the first @math nonzero Steklov eigenvalues is maximal for a ball. Their method is based on an isoperimetric inequality for moment of inertia. Edward @cite_9 proved for simply connected domains @math of perimeter @math that the relative sum of squares is minimal for the unit disk: @math .
@cite_38 @cite_9 @cite_32
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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 .