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研究生：	Tran The Dung Tran The Dung
論文名稱：	Geometric flows for elastic functionals of curves and the applications Geometric flows for elastic functionals of curves and the applications
指導教授：	林俊吉 Lin, Chun-Chi
口試委員：	林俊吉 Lin, Chun-Chi 司靈得 Spector, Daniel 孟悟理 Ulrich Menne 崔茂培 Tsui, Mao-Pei 蔡東和 Tsai, Dong-Ho 謝明修 Hsieh, Min-Hsiu
口試日期：	2022/06/24
學位類別：	博士 Doctor
系所名稱：	數學系 Department of Mathematics
論文出版年：	2022
畢業學年度：	110
語文別：	英文
論文頁數：	167
英文關鍵詞：	Geometric flow, elastic flow, fourth-order parabolic equation, second-order parabolic equation, elastic spline, spline interpolation, curve fitting, path planning, free boundary problem, contact angles, Holder spaces
研究方法:	Theorical research
DOI URL：	http://doi.org/10.6345/NTNU202200953
論文種類：	學術論文
相關次數：	點閱：104 下載：0
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In this thesis, the method of geometric flow is applied to prove the existence of global solutions to the problem of nonlinear spline interpolations for closed/non-closed curves and the problem of area-constrained planar elasticae with free boundaries on a straight line. Among them, this method applies the theory of either fourth-order parabolic PDEs/PDE or second-order parabolic PDEs/PDE with certain imposed boundary conditions. The results of this study demonstrate the existence of global solutions and sub-convergence of the elastic flow. Furthermore, the geometric flow method provides a new approach to the problem of nonlinear spline interpolations.

Contents
Acknowledgements                                                                                                      i
Abstract 					                                                                                   ii	
List of Figures                                                                                                                  vi

Introduction                                                                                                                     1
Math issues/difficulties, solutions, and the contribution                                      5
Organizations                                                                                                                  10

 An elastic flow for nonlinear spline interpolations in R^n                              12
1 The introduced problem and the main result                                                  12
2 The analytical problem and the short-time existence of Theorem 1.1.1    17
2.1 The linear problem                                                                                                21
2.2 The proof of Theorem 1.2.1                                                                                 27
3 Converting solutions and diffeomorphisms                                                      32
4 The long-time existence of Theorem 1.1.1                                                          42
4.1 Uniform bounds                                                                                                     42
4.2 The proof of the long-time existence of Theorem 1.1.1                                51

A second-order elastic flow for path planning in R^2                                        55
1 The introduced problems and the main results                                                  55 
2 The short-time existence of Theorem 2.1.1                                                         59
2.1 The linear problem                                                                                                  61
2.2 The proof of the short-time existence of Theorem 2.2.1                               64
3 The long-time existence of Theorem 2.1.1                                                           69
4 The proof of Theorem 2.1.2                                                                                      79

 A fourth-order elastic flow for area-constrained planar elasticae with free boundaries on a straight line and fixed non-zero contact angles                         82                                      
1 The introduced problem and the main result                                                     82
2 The analytical problem and the short-time existence of Theorem 3.1.1     88
2.1 The linear problem                                                                                                 90
2.2 The proof of Theorem 3.2.1                                                                                  94
3 Converting local solutions and diffeomorphisms                                              99
4 The long-time existence of Theorem 3.1.1                                                        104
4.1 Equations of evolution                                                                                        104
4.2 The inequalities                                                                                                    114
4.3 The proof of the long-time existence of Theorem 3.1.1                             120

A second-order elastic flow for planar elasticae with a free boundary on a straight line and fixed contact angles                                                                                      131
1 The introduced problem and the main result                                                  131
2 The short-time existence of Theorem 4.1.1                                                      134
2.1 The linear problem                                                                                              135
2.2 The proof of the short-time existence of Theorem 4.2.1                           136
3 The long-time existence of Theorem 4.1.1                                                      140
Conclusion and future works                                                                                    150 

Bibliography                                                                                                                  155
Appendix                                                                                                                        158
A. Parabolic Holder space                                                                                          159
B. Technical lemmas from the literature                                                                 160
C. Technical lemmas                                                                                                    165
                                

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