TITLE: certain smoothness of principal eigenvalue of Dirichlet Laplacian on polygons
QUESTION [5 upvotes]: For a given polygon $P_N$, with side lengths $x_1,\cdots,x_N$ and interior angles $\theta_1,\cdots,\theta_N$ let $\lambda(x_1,\cdots,x_N,\theta_1,\cdots,\theta_N)$ denote the least eigenvalue of Dirichlet Laplacian on $P_N$.
Question. Is $\lambda$ as a function of $x_1,\cdots,x_N,\theta_1,\cdots,\theta_N$ smooth in each variable?if not, is it at least twice continuously differentiable?  
Edit 1. The exitence and continuity of the first derivative follows from Theorem 2.5.1 of A. Henrot's book.

REPLY [5 votes]: $P_N$ seems to be real analytic. 
Part L of the main theorem of 

Andreas Kriegl, Peter W. Michor, Armin Rainer: Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators. Integral Equations and Operator Theory 71,3 (2011), 407-416. (pdf)

shows that the eigenvalues can be chosen real analytic after a local blow up of the coordinates. Since eigenvalues can cross each other, the smallest one may change. In each single parameter, the eigenvalues are real analytic.