带奇异位势的加权退化椭圆方程解的唯一延拓性

数学物理学报 ›› 2026, Vol. 46 ›› Issue (1): 174-189.

带奇异位势的加权退化椭圆方程解的唯一延拓性

杜广伟^*(), 韦飞雪()

曲阜师范大学数学科学学院山东曲阜 273165

收稿日期:2024-12-16 修回日期:2025-06-27 出版日期:2026-02-26 发布日期:2026-01-19
通讯作者: 杜广伟 E-mail:guangwei87@mail.nwpu.edu.cn;feixuewei0420@163.com
作者简介:韦飞雪, Email: feixuewei0420@163.com
基金资助:
山东省自然科学基金(ZR2020QA017)

Unique Continuation Properties of Weighted Degenerate Elliptic Equation with Singular Potential

Guangwei Du^*(), Feixue Wei()

School of Mathematical Sciences, Qufu Normal University, Shandong Qufu 273165

Received:2024-12-16 Revised:2025-06-27 Online:2026-02-26 Published:2026-01-19
Contact: Guangwei Du E-mail:guangwei87@mail.nwpu.edu.cn;feixuewei0420@163.com
Supported by:
Natural Science Foundation of Shandong Province(ZR2020QA017)

摘要/Abstract

摘要：

该文研究了如下与广义 Baouendi-Grushin 向量场相关的带奇异位势的二阶变系数加权退化椭圆方程解的唯一延拓性

$$- \sum\limits_{i,j = 1}^N {{X_j}({a_{ij}}(x,y){X_i}u)} + V(x,y)u = 0,$$

其中

$$\lambda {\left| \eta \right|^2}w \leqslant \left\langle {A\eta,\eta } \right\rangle \leqslant {\lambda ^{ - 1}}{\left| \eta \right|^2}w, \ \ A = {({a_{ij}})_{N \times N}},$$

$w$ 是与拟距离相关的权函数. 首先利用加权 Hardy 不等式、Rellich 型恒等式及 Dirichlet 能量估计, 证明了方程弱解频率函数的单调性, 进一步建立了弱解的二重性, 最后得到了方程弱解的唯一延拓性.

关键词: 唯一延拓性, Baouendi-Grushin 向量场, 奇异位势, 频率函数

Abstract:

In this paper, we consider the following second order variable coefficient weighted degenerate elliptic equation with singular potential constituted by Generalized Baouendi-Grushion vector fields

$$ - \sum\limits_{i,j = 1}^N {{X_j}({a_{ij}}(x,y){X_i}u)} + V(x,y)u = 0,$$

where

$$\lambda {\left| \eta \right|^2}w \leqslant \left\langle {A\eta,\eta } \right\rangle \leqslant {\lambda ^{ - 1}}{\left| \eta \right|^2}w,\ \ A = {({a_{ij}})_{N \times N}},$$

$w$ is a weight function related to quasi distance. By using the weighted Hardy inequality, Rellich type identity and the estimates of Dirichlet energy, we first get the monotonicity of the frequency function of the weak solutions. Then the doubling and unique continuation properties of the weak solutions are obtained.

Key words: unique continuation properties, Baouendi-Grushion vector fields, singular potential, frequency function

中图分类号:

O175.2

杜广伟, 韦飞雪. 带奇异位势的加权退化椭圆方程解的唯一延拓性[J]. 数学物理学报, 2026, 46(1): 174-189.

Guangwei Du, Feixue Wei. Unique Continuation Properties of Weighted Degenerate Elliptic Equation with Singular Potential[J]. Acta mathematica scientia,Series A, 2026, 46(1): 174-189.

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