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.
The literature on the spectral geometry of the Steklov problem is expanding rapidly, and so we had to omit many papers here. We refer to @cite_14 @cite_23 for recent surveys.
@cite_14 @cite_23
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The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometry. While this problem shares some common properties with its more familiar Dirichlet and Neumann cousins, its eigenval- ues and eigenfunctions have a number of distinctive geometric features, which makes the subject especially appealing. In this survey we discuss some recent advances and open questions, particularly in the study of spectral asymptotics, spectral invariants, eigenvalue estimates, and nodal geometry. Abstract Let M be a compact submanifold with boundary of a Euclidean space or a Sphere. In this paper, we derive an upper bound for the first non-zero eigenvalue p 1 of Steklov problem on M in terms of the r -th mean curvatures of its boundary ∂ M . The upper bound obtained is sharp. We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundary components. Our estimates generalize a recent result of Fraser-Schoen, as well as the classical inequalites obtained by Hersch-Payne-Schiffer, whose approach is used in the present paper. We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of boundary mass concentration. We discuss the asymptotic behavior of the Neumann eigenvalues in a ball and we deduce that the Steklov eigenvalues minimize the Neumann eigenvalues. Moreover, we study the dependence of the eigenvalues of the Steklov problem upon perturbation of the mass density and show that the Steklov eigenvalues violates a maximum principle in spectral optimization problems.