Optimization of spectral functions of Dirichlet-Laplacian eigenvalues

We consider the shape optimization of spectral functions of Dirichlet-Laplacian eigenvalues over the set of star-shaped, symmetric, bounded planar regions with smooth boundary. The regions are represented using Fourier-cosine coefficients and the optimization problem is solved numerically using a quasi-Newton method. The method is applied to maximizing two particular nonsmooth spectral functions: the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue, both of which are generalizations of the Payne-Polya-Weinberger ratio. The optimal values and attaining regions for n=<13 are presented and interpreted as a study of the range of the Dirichlet-Laplacian eigenvalues. For both spectral functions and each n, the optimal attaining region has multiplicity two nth eigenvalue.

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