Verified eigenvalue estimation for elliptic differential operators and its application in non-linear problems

微分算子的特征值问题是工程和数学中的一个基本问题。历史上已经给出了拉普拉斯函数的上界，但下界仍然很难求。本文提出了一种新的求解拉普拉斯特征值下界的算法。该算法以有限元法为基础，利用超圆方程进行求解。这是第一个可以方便地处理一般形状域上的特征值问题的方法。本文还成功地将特征值界应用于任意多边形域上的非线性偏微分方程的解验证。

The eigenvalue problem for differential operators is a basic problem in both engineering and mathematics. The upper bounds for the Laplacian have been given in history, but the lower bounds remain to be very difficult. In this research, a new algorithm is developed to give lower bounds for the eigenvalues of the Laplacian. Such an algorithm is based on the finite element method along with the use of the hypercircle equation. It is the first method that can easily deal with eigenvalue problems on domain of general shapes. The eigenvalue bounds are also successfully applied to solution verification for nonlinear partial differential equations defined on arbitrary polygonal domains.