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–Pólya–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.