关于Hamilton-Jacobi方程解的正则性和全局结构的注记——纪念李国平院士吴新谋教授诞辰100周年

数学物理学报 ›› 2010, Vol. 30 ›› Issue (5): 1283-1287.

关于Hamilton-Jacobi方程解的正则性和全局结构的注记——纪念李国平院士吴新谋教授诞辰100周年

王靖华|赵引川|温海瑞

中国科学院数学与系统科学研究院北京 100190|华北电力大学数理系北京 102206|北京理工大学数学系北京 100081

收稿日期:2010-08-02 出版日期:2010-10-25 发布日期:2010-10-25
基金资助:
国家自然科学基金(10871133, 11071246, 10926067)资助

Remarks on the Regularity and Global Structure of Solutions

WANG Jing-Hua, ZHAO Yin-Chuan, WEN Hai-Rui

Academy of Mathematics and System Sciences the Chinese Academy of Sciences, Beijing |100190；Department of Mathematics and Physics, North China Electric Power University, Beijing 102206；School of Sciences, Beijing Institute of Technology, |Beijing 100081

Received:2010-08-02 Online:2010-10-25 Published:2010-10-25
Supported by:
国家自然科学基金(10871133, 11071246, 10926067)资助

摘要/Abstract

摘要：

该文考虑高维Hamilton-Jacobi方程的柯西问题. 作者证明了从任一初始点出发的特征线永不碰到奇异点集合的充分必要条件是初始函数在该点取到最小值.在此基础上, 证明了奇异点集合的道路连通分支和初始函数不取最小值的点集合的道路连通分支之间存在一一对应, 而且解的梯度的间断一旦产生就不会消失. 特别指出, 该文的结果不需要“初始函数的梯度在无穷远趋近于零”这一限制条件, 而文献[12]中重要的命题2.7和主要结果之一的定理3.3是在这一条件下得到的.

关键词: Hamilton-Jacobi 方程, 特征线, 奇异点, 整体结构

Abstract:

The paper is concerned with the Cauchy problem for the Hamilton-Jacobi equations of multi-dimensional space variables. We prove the sufficient and necessary condition for that a characteristic emanating from a given point never touches the set of singularity points is that the initial function attains its minimum at this point. Finally, we prove there exists one-to-one correspondence between the path connected components of the set of singularity points and the path connected components of the set on which the initial function does not attain
its minimum. Each path connected component of the set of the singularity points never terminates at a finite time. In particularly, it is worth pointing out that our results are obtained without assuming that the gradient of the initial function tends to zero at infinity under which the important proposition 2.7 and theorem 3.3, one of the main results in [12] are obtained.

Key words: Hamilton-Jacobi equations, Characteristic, Singularity point, Global structure

中图分类号:

35L60

王靖华, 赵引川, 温海瑞. 关于Hamilton-Jacobi方程解的正则性和全局结构的注记——纪念李国平院士吴新谋教授诞辰100周年[J]. 数学物理学报, 2010, 30(5): 1283-1287.

WANG Jing-Hua, ZHAO Yin-Chuan, WEN Hai-Rui. Remarks on the Regularity and Global Structure of Solutions[J]. Acta mathematica scientia,Series A, 2010, 30(5): 1283-1287.

参考文献

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