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 (recent version) or here (old version).
The local minimality for $n \in \{5,6\}$ is established using validated computing in the following preprint.