二维 Grushin 算子的 Dirichlet 特征值问题——献给陈化教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (2): 473-492.

二维 Grushin 算子的 Dirichlet 特征值问题——献给陈化教授 70 寿辰

陈洪葛¹^,^*(), 李金宁²()

¹ 华中师范大学数学与统计学学院, 非线性分析教育部重点实验室, 数学物理湖北省重点实验室武汉 430079
² 重庆大学数学与统计学院重庆 401331

收稿日期:2025-11-29 修回日期:2025-12-12 出版日期:2026-04-26 发布日期:2026-04-27
通讯作者: 陈洪葛 E-mail:hongge_chen@whu.edu.cn;lijinning@whu.edu.cn
作者简介:李金宁，Email：lijinning@whu.edu.cn
基金资助:
国家自然科学基金(12201607);国家自然科学基金(12571249)

Dirichlet Eigenvalue Problem for 2D Grushin Operators

Hongge Chen¹^,^*(), Jinning Li²()

¹ School of Mathematics and Statistics, Key Laboratory of Nonlinear Analysis & Applications (Ministry of Education), Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079
² College of Mathematics and Statistics, Chongqing University, Chongqing 401331

Received:2025-11-29 Revised:2025-12-12 Online:2026-04-26 Published:2026-04-27
Contact: Hongge Chen E-mail:hongge_chen@whu.edu.cn;lijinning@whu.edu.cn
Supported by:
NSFC(12201607);NSFC(12571249)

摘要/Abstract

摘要：

该文研究了二维 Grushin 算子 $\triangle_X=\partial_{x_{1}}^2+x_{1}^2\partial_{x_{2}}^2$ 在有界开集 $\Omega\subset \mathbb{R}^2$ 上的 Dirichlet 特征值问题. Grushin 算子是非等度正则情形下的一类重要的 Hörmander 算子, 其奇异退化点集 $H$ 的二维 Lebesgue 测度为零, 这使得 Métivier 的渐近公式不再适用. 该文通过利用全局热核的显式表达及对 Dirichlet 热核误差项的精细估计, 建立了带对数项的 Weyl 渐近法则 $\lambda_k \sim \frac{4\pi}{s_{\Omega}(0)} \frac{k}{\ln k}$, 并揭示了奇异退化点集 $H$在 $x_2$ 轴投影的一维 Lebesgue 测度 $s_{\Omega}(0)$ 是刻画该算子 Dirichlet 特征值渐近性的几何谱不变量.

关键词: Grushin 算子, Dirichlet 特征值, Weyl 法则

Abstract:

This paper investigates the Dirichlet eigenvalue problem for the 2D Grushin operator $\triangle_X=\partial_{x_{1}}^2+x_{1}^2\partial_{x_{2}}^2$ on a bounded open set $\Omega$ in $\mathbb{R}^2$. The Grushin operator is an important class of Hörmander operators in the non-equiregular case, where the 2D Lebesgue measure of its singular degenerate set $H$ is zero ($|H|=0$), making Métivier's asymptotic formula no longer applicable. By utilizing the explicit expression of the global heat kernel and refined estimates for the error term of the Dirichlet heat kernel, we establish a Weyl asymptotic law with a logarithmic term: $\lambda_k \sim \frac{4\pi}{s_{\Omega}(0)} \frac{k}{\ln k}$. Furthermore, we show that $s_{\Omega}(0)$, the 1D Lebesgue measure of the projection of the singular degenerate set $H$ onto the $x_2$-axis, is a geometric spectral invariant characterizing the asymptotics of the Dirichlet eigenvalues for this operator.

Key words: Grushin operator, Dirichlet eigenvalue, Weyl's Law

中图分类号:

O175.9

陈洪葛, 李金宁. 二维 Grushin 算子的 Dirichlet 特征值问题——献给陈化教授 70 寿辰[J]. 数学物理学报, 2026, 46(2): 473-492.

Hongge Chen, Jinning Li. Dirichlet Eigenvalue Problem for 2D Grushin Operators[J]. Acta mathematica scientia,Series A, 2026, 46(2): 473-492.

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