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.

翻译：本文研究了一类具有齐次本质边界条件的参数化椭圆特征值问题，其中系数（进而解$u$）可能依赖于参数$y$。为有效评估整个参数空间上第一特征对的参数灵敏度，我们提出并分析了解关于参数的Gevrey类正则性与解析正则性。这一成果得益于本文引入并展示的一种新型证明技术。我们的正则性结果对数种参数化椭圆特征值问题数值格式的收敛性具有直接意义，特别适用于由随机系数椭圆微分算子导出的无穷维参数化椭圆特征值问题。