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](https://books.google.fr/books?id=8GXrQNw2SjYC&printsec=frontcover&dq=Henrot%20Antoine&hl=en#v=onepage&q&f=false). 



certain smoothness of principal eigenvalue of Dirichlet Laplacian on polygons
$P\_N$ 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)](http://www.mat.univie.ac.at/~michor/DC-perturb.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. 


