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 one time continuously differentiable?