For $N \geq 5$ it is still not known if the $N$-gon which minimizes the first eigenvalue under area constraint (which exists), is the regular one. I have done some numerical computations which suggest that the regular polygons are indeed optimal. You can see the numerical ideas [here][1] (recent version) or [here][2] (old version).


  [1]: http://www.lama.univ-savoie.fr/~bogosel/faber_krahn_polygons.html
  [2]: https://mathproblems123.wordpress.com/2013/12/23/numerical-method-minimizing-eigenvalues-on-polygons/