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

TL;DR

This paper establishes Gevrey class and analytic regularity of solutions to parametric elliptic eigenvalue problems, enabling efficient numerical approximation of eigenpairs across parameter spaces, including cases with infinitely many parameters.

Contribution

Introduces a novel proof technique to demonstrate Gevrey and analytic regularity of solutions, improving numerical scheme convergence analysis for parametric elliptic eigenvalue problems.

Findings

Gevrey class regularity holds for solutions with parameter dependence.

Analytic regularity results improve convergence estimates for numerical methods.

Applicable to problems with infinitely many parameters from random coefficients.

Abstract

We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions where the coefficients (and hence the solution $u$) may depend on a parameter $y$. For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space we propose and analyse 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.