Analytic and Gevrey Class Regularity for Parametric Elliptic Eigenvalue Problems and Applications

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1874-1900, August 2024.
Abstract. We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions, where the coefficients (and hence the solution) may depend on a parameter. For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space we propose and analyze Gevrey class and analytic regularity of the solution with respect to the parameters. This is made possible by a novel proof technique, which we introduce and demonstrate in this paper. Our regularity result has immediate implications for convergence of various numerical schemes for parametric elliptic eigenvalue problems, in particular, for elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients, e.g., integration by quasi–Monte Carlo methods.