Asymptotic behaviour of optimal spectral planar domains with fixed perimeter

International audienceWe 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 frequen