边界条件含有特征参数的四阶微分算子的自伴性和特征值的依赖性

数学物理学报 ›› 2022, Vol. 42 ›› Issue (3): 671-693.

边界条件含有特征参数的四阶微分算子的自伴性和特征值的依赖性

闫文文,许美珍*()

^{内蒙古工业大学理学院呼和浩特 010051}

收稿日期:2021-08-12 出版日期:2022-06-26 发布日期:2022-05-09
通讯作者: 许美珍 E-mail:xumeizhen1969@163.com
基金资助:
国家自然科学基金(11561051);内蒙古自然科学基金(2021MS01020)

The Self-Adjointness and Dependence of Eigenvalues of Fourth-Order Differential Operator with Eigenparameters in the Boundary Conditions

Wenwen Yan,Meizhen Xu*()

^{College of Sciences, Inner Mongolia University of Technology, Hohhot 010051}

Received:2021-08-12 Online:2022-06-26 Published:2022-05-09
Contact: Meizhen Xu E-mail:xumeizhen1969@163.com
Supported by:
the NSFC(11561051);the NSF of Inner Mongolia(2021MS01020)

摘要/Abstract

摘要：

该文考虑了一类边界条件一端含有特征参数且具有转移条件的四阶微分算子的自共轭性及特征值的依赖性. 通过在适当的Hilbert空间上定义一个与问题相关的线性算子T, 将上述问题的研究转化为对此空间中算子的研究, 并证明算子T是自共轭的. 另外, 在算子T自伴的基础上证明特征值不仅连续依赖而且可微依赖于问题的各个参数, 并给出相应的微分表达式. 特别地, 给出特征值关于特征参数依赖的边界条件系数矩阵的Fréchet导数及关于内部不连续点c左右两侧的一阶导数.

关键词: 四阶微分算子, 转移条件, 自共轭性, 特征值的依赖性, Fréchet导数

Abstract:

In this paper we consider the self-adjointness and the dependence of eigenvalues of a class of discontinuous fourth-order differential operator with eigenparameters in the boundary conditions of one endpoint. By constructing a linear operator T associated with problem in a suitable Hilbert space, the study of the above problem is transformed into the research of the operator in this space, and the self-adjointness of this operator T is proved. In addition, on the basis of the self-adjointness of the operator T, we show that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem, and give the corresponding differential expressions. In particular, giving the Fréchet derivative of the eigenvalue with respect to the eigenparameter-dependent boundary condition coefficient matrix, and the first-order derivatives of the eigenvalue with respect to the left and right sides of the inner discontinuity point c.

Key words: Fourth-order differential operator, Transmission condition, Self-adjointness, Dependence of eigenvalue, Fréchet derivative

中图分类号:

O175.3

闫文文,许美珍. 边界条件含有特征参数的四阶微分算子的自伴性和特征值的依赖性[J]. 数学物理学报, 2022, 42(3): 671-693.

Wenwen Yan,Meizhen Xu. The Self-Adjointness and Dependence of Eigenvalues of Fourth-Order Differential Operator with Eigenparameters in the Boundary Conditions[J]. Acta mathematica scientia,Series A, 2022, 42(3): 671-693.

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