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 numerically study positions of high spots (extrema) of the fundamental sloshing mode of a liquid in an axisymmetric tank. Our approach is based on a linear model and reduces the problem to an appropriate Steklov eigenvalue problem. We propose a numerical scheme for calculating sloshing modes and a novel method for making images of an oscillating fluid.