加权Laplace算子Dirichlet特征值问题的一个万有不等式及其应用

数学物理学报 ›› 2025, Vol. 45 ›› Issue (1): 54-73.

加权Laplace算子Dirichlet特征值问题的一个万有不等式及其应用

杨贵诚¹,温杨哲²,毛井^2,^3,^*()

¹湖北商贸学院武汉 430074
²湖北大学数学与统计学学院, 应用数学湖北省重点实验室武汉 430062
³湖北大学智能感知系统与安全教育部重点实验室武汉 430062

收稿日期:2024-06-03 修回日期:2024-07-26 出版日期:2025-02-26 发布日期:2025-01-08
通讯作者: * 毛井, E-mail:jiner120@163.com
基金资助:
国家自然科学基金(11801496);国家自然科学基金(11926352);霍英东教育基金会高等院校青年教师基金(161004);应用数学湖北省重点实验室和智能感知系统与安全教育部重点实验室(湖北大学)

A Universal Inequality for the Dirichlet Eigenvalue Problem of the Weighted Laplacian and its Application

Yang Guicheng¹,Wen Yangzhe²,Mao Jing^2,^3,^*()

¹Hubei Business College, Wuhan 430074
²Faculty of Mathematics and Statistics, Key Laboratory of Applied Mathematics of Hubei Province, Hubei University, Wuhan 430062
³Key Laboratory of Intelligent Sensing System and Security (Hubei University), Ministry of Education, Wuhan 430062

Received:2024-06-03 Revised:2024-07-26 Online:2025-02-26 Published:2025-01-08
Supported by:
NSFC(11801496);NSFC(11926352);Fok Ying-Tung Education Foundation(161004);Key Laboratory of Intelligent Sensing System and Security(Hubei University),Ministry of Education

摘要/Abstract

摘要：

该文研究了欧氏空间中具有光滑边界的有界区域 $\Omega$ 上加权 Laplace 算子 $\mathbb {L}_\phi$ 的 Dirichlet 特征值问题. 在加权函数 $\phi$ 满足一定约束条件的前提下, 利用变分法, 并在恰当地构造测试函数的基础上, 可以得到该特征值问题的一个万有不等式.

关键词: 加权 Laplace 算子, Dirichlet 特征值问题, Green 公式

Abstract:

In this paper, we study the Dirichlet eigenvalue problem of the weighted Laplace operator $\mathbb {L}_\phi$ on a bounded domain $\Omega$ with a smooth boundary in $n$-dimensional Euclidean space. Under the premise that the weighted function $\phi$ satisfies certain constraints, a universal inequality of the eigenvalue problem can be obtained by using the variational method and constructing the test function appropriately.

Key words: weighted Laplace operator, Dirichlet eigenvalue problem, Green formula

中图分类号:

O186

杨贵诚, 温杨哲, 毛井. 加权Laplace算子Dirichlet特征值问题的一个万有不等式及其应用[J]. 数学物理学报, 2025, 45(1): 54-73.

Yang Guicheng, Wen Yangzhe, Mao Jing. A Universal Inequality for the Dirichlet Eigenvalue Problem of the Weighted Laplacian and its Application[J]. Acta mathematica scientia,Series A, 2025, 45(1): 54-73.

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