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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 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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    $\begingroup$ Look in Kato's book on perturbation theory. Analytic deformations --> analytic eigenbranches. $\endgroup$ – Neal Oct 31 '16 at 17:13
  • $\begingroup$ @Neal I actually own Kato's thick book and probed its contents and subject index but couldnt find anything on neither analytic deformation nor eigen branches. Are you sure this topics are discussed in Kato's? $\endgroup$ – BigM Nov 1 '16 at 17:25
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    $\begingroup$ Have you looked in Chapter 7, Analytic perturbation theory? $\endgroup$ – Neal Nov 1 '16 at 19:10
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$P_N$ 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.

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