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 $k$. |
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