Asymptotic behaviour of optimal spectral planar domains with fixed perimeter

We consider the problem of minimizing the kth Dirichlet eigenvalue of planar domains with fixed perimeter and show that, as k goes to infinity, the optimal domain converges to the ball with the same perimeter. We also consider this problem within restricted classes of domains such as n-polygons and tiling domains, for which we show that the optimal asymptotic domain is that which maximises the area for fixed perimeter within the given family, i.e., the regular n-polygon and the regular hexagon, respectively. Physically, the above problems correspond to the determination of the shapes within a given class which will support the largest number of modes below a given frequency. (C) 2013 AIP Publishing LLC.