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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 dirichletDirichlet 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 timetwice continuously differentiable?

Edit 1. The exitence and continuity of the first derivative follows from Theorem 2.5.1 of A. Henrot's book.

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?

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.

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certain smoothness of principal eigenvalue of Dirichlet Laplacian on polygons

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?