Special Session 143: 

A Numerical Study of Steklov Eigenvalue Problem via Conformal Mapping

Chiu-Yen Kao
Claremont McKenna College
USA
Co-Author(s):    Weaam Alhejaili and Chiu-Yen Kao
Abstract:
In this paper, a spectral method based on conformal mappings is proposed to solve Steklov eigenvalue problems and its related shape optimization problem 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 $k$.