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We investigate a class of parametric elliptic eigenvalue problems withhomogeneous essential boundary conditions where the coefficients (and hence thesolution $u$) may depend on a parameter $y$. For the efficient approximateevaluation of parameter sensitivities of the first eigenpairs on the entireparameter space we propose and analyse Gevrey class and analytic regularity ofthe solution with respect to the parameters. This is made possible by a novelproof technique which we introduce and demonstrate in this paper. Ourregularity result has immediate implications for convergence of variousnumerical schemes for parametric elliptic eigenvalue problems, in particular,for elliptic eigenvalue problems with infinitely many parameters arising fromelliptic differential operators with random coefficients.

Analytic and Gevrey class regularity for parametric elliptic eigenvalue problems and applications