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.