Numerical studies of the Steklov eigenvalue problem via conformal mappings

摘要

In this paper, spectral methods based on conformal mappings are proposed to solve the Steklov eigenvalue problem and its 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 a gradient ascent approach to find the optimal domain which maximizes kth 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.