Author: |
Tran The Dung Tran The Dung |
---|---|

Thesis Title: |
Geometric flows for elastic functionals of curves and the applications Geometric flows for elastic functionals of curves and the applications |

Advisor: |
林俊吉
Lin, Chun-Chi |

Committee: |
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 |

Approval Date: | 2022/06/24 |

Degree: |
博士 Doctor |

Department: |
數學系 Department of Mathematics |

Thesis Publication Year: | 2022 |

Academic Year: | 110 |

Language: | 英文 |

Number of pages: | 167 |

Keywords (in English): | 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 |

Research Methods: | Theorical research |

DOI URL: | http://doi.org/10.6345/NTNU202200953 |

Thesis Type: | Academic thesis/ dissertation |

Reference times: | Clicks: 119 Downloads: 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
1 An elastic flow for nonlinear spline interpolations in R^n 12
1.1 The introduced problem and the main result 12
1.2 The analytical problem and the short-time existence of Theorem 1.1.1 17
1.2.1 The linear problem 21
1.2.2 The proof of Theorem 1.2.1 27
1.3 Converting solutions and diffeomorphisms 32
1.4 The long-time existence of Theorem 1.1.1 42
1.4.1 Uniform bounds 42
1.4.2 The proof of the long-time existence of Theorem 1.1.1 51
2 A second-order elastic flow for path planning in R^2 55
2.1 The introduced problems and the main results 55
2.2 The short-time existence of Theorem 2.1.1 59
2.2.1 The linear problem 61
2.2.2 The proof of the short-time existence of Theorem 2.2.1 64
2.3 The long-time existence of Theorem 2.1.1 69
2.4 The proof of Theorem 2.1.2 79
3 A fourth-order elastic flow for area-constrained planar elasticae with free boundaries on a straight line and fixed non-zero contact angles 82
3.1 The introduced problem and the main result 82
3.2 The analytical problem and the short-time existence of Theorem 3.1.1 88
3.2.1 The linear problem 90
3.2.2 The proof of Theorem 3.2.1 94
3.3 Converting local solutions and diffeomorphisms 99
3.4 The long-time existence of Theorem 3.1.1 104
3.4.1 Equations of evolution 104
3.4.2 The inequalities 114
3.4.3 The proof of the long-time existence of Theorem 3.1.1 120
4 A second-order elastic flow for planar elasticae with a free boundary on a straight line and fixed contact angles 131
4.1 The introduced problem and the main result 131
4.2 The short-time existence of Theorem 4.1.1 134
4.2.1 The linear problem 135
4.2.2 The proof of the short-time existence of Theorem 4.2.1 136
4.3 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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