We develop a computational method for extremal Steklov eigenvalue problems and apply it to study the problem of maximizing the @math th Steklov eigenvalue as a function of the domain with a volume constraint. In contrast to the optimal domains for several other extremal Dirichlet- and Neumann-Laplacian eigenvalue problems, computational results suggest that the optimal domains for this problem are very structured. We reach the conjecture that the domain maximizing the @math th Steklov eigenvalue is unique (up to dilations and rigid transformations), has @math -fold symmetry, and has at least one axis of symmetry. The @math th Steklov eigenvalue has multiplicity 2 if @math is even and multiplicity 3 if @math is odd.
There are also a few computational studies of extremal Steklov problems. The most relevant is recent work of B. Bogosel @cite_24 . This paper is primary concerned with the development of methods based on fundamental solutions to compute the Steklov, Wentzell, and Laplace-Beltrami eigenvalues. This method was used to demonstrate that the ball is the minimizer for a variety of shape optimization problems. The author also studies the problem of maximizing the first five Wentzell eigenvalues subject to a volume constraint, for which is a special case. Shape optimization problems for Steklov eigenvalues with mixed boundary conditions have also been studied @cite_25 .
@cite_24 @cite_25
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We develop methods based on fundamental solutions to compute the Steklov, Wentzell and Laplace-Beltrami eigenvalues in the context of shape optimization. In the class of smooth simply connected two dimensional domains the numerical method is accurate and fast. A theoretical error bound is given along with comparisons with mesh-based methods. We illustrate the use of this method in the study of a wide class of shape optimization problems in two dimensions. We extend the method to the computation of the Laplace-Beltrami eigenvalues on surfaces and we investigate some spectral optimal partitioning problems. 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. 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. We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results share a common feature: they are proved using methods of complex analysis. In particular, we present modernized proofs of the classical inequalities due to Szego and Weinstock for the first nonzero Neumann and Steklov eigenvalues. We also extend the inequality for the second nonzero Neumann eigenvalue, obtained recently by Nadirashvili and the authors, to nonhomogeneous membranes with log-subharmonic densities. In the homogeneous case, we show that this inequality is strict, which implies that the maximum of the second nonzero Neumann eigenvalue is not attained in the class of simply connected membranes of a given mass. The same is true for the second nonzero Steklov eigenvalue, as follows from our results on the Hersch–Payne–Schiffer inequalities. Copyright © 2009 John Wiley & Sons, Ltd.