Computational Methods For Extremal Steklov Problems

Abstract

We develop a computational method for extremal Steklov eigenvalue problems and apply it to study the problem of maximizing the $p$-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 $p$-th Steklov eigenvalue is unique (up to dilations and rigid transformations), has p-fold symmetry, and an axis of symmetry. The $p$-th Steklov eigenvalue has multiplicity 2 if $p$ is even and multiplicity 3 if $p\ge 3$ is odd.

14 pages, 8 figures

References in corpus (3)

• Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces
• Steklov eigenvalues and quasiconformal maps of simply connected planar domains
• Spectral geometry of the Steklov problem

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• A numerical study of the Dirichlet-to-Neumann operator in planar domains
• Spectral properties of the Dirichlet-to-Neumann operator for spheroids
• Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems
• The Steklov problem for exterior domains: asymptotic behavior and applications
• Steklov eigenvalues of nearly hyperspherical domains
• A Numerical Study of Steklov Eigenvalue Problem via Conformal Mapping