Optimization of spectral functions of Dirichlet-Laplacian eigenvalues

  • Authors:

  • Braxton Osting

  • Affiliations:

  • Department of Applied Physics and Applied Mathematics, Columbia University, 500 W. 120th St., New York, NY 10027, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2010

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Abstract

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=