A Numerical Study of Steklov Eigenvalue Problem via Conformal Mapping
For researchers in spectral geometry and shape optimization, this provides a numerical tool for Steklov eigenvalue problems, though the approach is incremental as it combines existing conformal mapping and spectral techniques.
This paper proposes a spectral method using conformal mappings to solve Steklov eigenvalue problems and shape optimization in 2D, achieving efficient discretization via Fourier series. The method successfully identifies optimal domains that maximize the k-th Steklov eigenvalue under area constraint for various k.
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 $k$