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Title: On a class of Grushin operators : a geometric mechanics approach
Other Titles: Guan yu yi lei Grushin xing suan zi de yan jiu : ji he li xue fang fa
關於一類 Grushin 型算子的研究 : 幾何力學方法
Authors: Li, Yutian (李玉田)
Department: Dept. of Mathematics
Degree: Master of Philosophy
Issue Date: 2007
Publisher: City University of Hong Kong
Subjects: Differential equations, Elliptic
Elliptic operators
Notes: CityU Call Number: QA329.42.L5 2007
Includes bibliographical references (leaves [39]-42)
Thesis (M.Phil.)--City University of Hong Kong, 2007
iv, 42 leaves : ill. ; 30 cm.
Type: Thesis
Abstract: We study a class of Grushin operators ¢G = @2 @x2 + x2k @2 @y2 ; k 2 N via the geometric mechanics approach, which is based on analysis of the geometry induced by the operators. By using the shooting method, we reduce the problem of determining the number of geodesics connecting any two points in the induced geometry to a problem of finding how many solutions there are to the Hamilton-Jacobi equation. After a detailed analysis, we obtain the number and length of geodesics connecting any two points. It is shown that there may be more than one geodesics connecting two points arbitrarily near to each other, in some cases, the number of geodesics is even infinite. The most interesting and important case is k = 2, which is related to the quadric harmonic oscillators. In the final part, we discuss some open problems and future work in this direction.
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Appears in Collections:MA - Master of Philosophy

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