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

A Numerical Study of Steklov Eigenvalue Problem via Conformal Mapping