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Title: Hamilton-Jacobi theory and heat kernels for Grushin operators
Other Titles: Grushin suan zi de Hamidun-Yakebi li lun he re he
Grushin 算子的哈密頓-雅克比理論和熱核
Authors: Li, Yutian (李玉田)
Department: Department of Mathematics
Degree: Doctor of Philosophy
Issue Date: 2010
Publisher: City University of Hong Kong
Subjects: Elliptic operators.
Differential equations, Elliptic.
Hamilton-Jacobi equations.
Heat equation.
Notes: CityU Call Number: QA329.42 .L49 2010
iv, 88 leaves 30 cm.
Thesis (Ph.D.)--City University of Hong Kong, 2010.
Includes bibliographical references (leaves [77]-84)
Type: thesis
Abstract: In this thesis, we study a class of Grushin operators ∆k=∂2/∂x2 + x2k ∂2/∂y2 ; k 2 N+; which are subelliptic operators. We obtain all the geodesics of the subRimannian geometry induced by ∆k. For any two points in the y-axis, there are infinitely many geodesics connecting them; while for any other two points, there are finitely many geodesic connections. The y-axis is the canonical submanifold whose tangent space recovers the missing direction. We generalize Beals-Gaveau-Greiner's complex Hamilton- Jacobi theory to general step subelliptic operators and apply this to the Grushin operators ∆k. We construct the modified action function f and obtain an integral representation of the heat kernel for step 3 Grushin operator (i.e., ∆2). By computing the critical points of f, the small time aymptotics of the heat kernel is obtained using the method of steepest descent, it has a close relation with the geodesics induced by the operator ∆2. For higher step cases, the modified action functions are also given. In the final part, we give some open problems and future work in this direction.
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Appears in Collections:MA - Doctor of Philosophy

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