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（因此解$u$）可能依赖于参数$y$。为了在整个参数空间上高效近似评估第一特征对的参数敏感性，我们提出并分析了该解关于参数的Gevrey类正则性和解析正则性。这一成果得益于本文引入并论证的一种新颖证明技巧。我们的正则性结果对参数化椭圆特征值问题数值方法（特别是由随机系数椭圆微分算子产生的无穷维参数椭圆特征值问题）的收敛性具有直接指导意义。