Numerical calculation of extremal Steklov eigenvalues in 3D and 4D

TL;DR

This paper introduces a numerical method for optimizing shapes to maximize Steklov eigenvalues in 3D and 4D, extending previous planar studies and employing the Method of Fundamental Solutions for computation.

Contribution

It develops a new numerical approach for shape optimization of Steklov eigenvalues in higher dimensions, expanding beyond planar cases.

Findings

Successfully applied the method to 3D and 4D cases

Demonstrated the effectiveness of the Method of Fundamental Solutions

Extended shape optimization techniques to higher dimensions

Abstract

We develop a numerical method for solving shape optimization of functionals involving Steklov eigenvalues and apply it to the problem of maximization of the $k$-th Steklov eigenvalue, under volume constraint. A similar study in the planar case was addressed in [E. Akhmetgaliyev, C.-Y. Kao and B. Osting, SIAM J. Control Optim. 55(2), 1226-1240, (2017)] using the boundary integral equation method. Here we extend that study to the 3D and 4D cases, using the Method of Fundamental Solutions as forward solver.