Computational Methods For Extremal Steklov Problems

TL;DR

This paper introduces a computational approach to extremal Steklov eigenvalue problems, revealing structured optimal domains and conjecturing their symmetry, uniqueness, and eigenvalue multiplicities.

Contribution

It develops a novel computational method for extremal Steklov problems and provides new insights into the structure and symmetry of optimal domains.

Findings

Optimal domains are highly structured and symmetric.

Conjecture of uniqueness and symmetry of maximizing domains.

Eigenvalue multiplicities depend on parity of p.

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 $p\geq3$ is odd.