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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
    Commented Oct 31, 2016 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
    Commented Nov 1, 2016 at 17:25
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    $\begingroup$ Have you looked in Chapter 7, Analytic perturbation theory? $\endgroup$
    – Neal
    Commented Nov 1, 2016 at 19:10

1 Answer 1

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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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