A Numerical Study of Steklov Eigenvalue Problem via Conformal Mapping

TL;DR

This paper introduces a spectral method using conformal mappings to numerically solve Steklov eigenvalue problems and optimize domain shapes to maximize specific eigenvalues in two dimensions.

Contribution

It develops a novel spectral approach based on conformal mappings and Fourier series for solving Steklov eigenvalue problems and shape optimization in 2D.

Findings

Eigenfunctions expanded in Fourier series facilitate discretization.

Gradient ascent effectively finds optimal domains for eigenvalue maximization.

Coefficients of conformal mappings obtained for various eigenvalue indices.

Abstract

In this paper, a spectral method based on conformal mappings is proposed to solve Steklov eigenvalue problems and their related shape optimization problems in two dimensions. To apply spectral methods, we first reformulate the Steklov eigenvalue problem in the complex domain via conformal mappings. The eigenfunctions are expanded in Fourier series so the discretization leads to an eigenvalue problem for coefficients of Fourier series. For shape optimization problem, we use the gradient ascent approach to find the optimal domain which maximizes $k-$th Steklov eigenvalue with a fixed area for a given $k$. The coefficients of Fourier series of mapping functions from a unit circle to optimal domains are obtained for several different $k$