2021 Fiscal Year Final Research Report
Study on Hamilton-Jacobi flows with initial data of pathological functions
| Project/Area Number |
18K03360
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12020:Mathematical analysis-related
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| Research Institution | University of Toyama |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
山口 範和 富山大学, 学術研究部教育学系, 准教授 (50409679)
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| Project Period (FY) |
2018-04-01 – 2022-03-31
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| Keywords | Hamilton--Jacobi flow / 至る所微分不可能性 / 逆問題 / 正則効果 |
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| Outline of Final Research Achievements |
The most important aim of this work is to derive nowhere differentiable property of the initial data from some behavior of the Hamilton-Jacobi flows issued from these data. This is a sort of inverse problem. I have achieved this aim by solving this problem affirmatively. The results could be now read in an electric paper of Mathematische Annalen via Open Access. Furthermore, as the partition of the interval [0,1] which is first chosen, we can admit not only the one of equal division but also almost all ones. In the previous papers, we only considered the partition of equal division.
In these senses, I believe that I have succeeded in this work.
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| Free Research Field |
数理解析学
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| Academic Significance and Societal Importance of the Research Achievements |
今回の研究結果から得られる学術的な意義は、Hamilton--Jacobi 方程式の初期値問題の解として定義される Hamilton--Jacobi flow における正則効果がかなり「弱い」ことへの再認識である。なぜなら、この正則効果が強ければ、初期値の至る所微分不可能性などの悪い性質も時間とともに一気に解消されて、Hamilton--Jacobi flow から初期値の至る所微分不可能性を見出すことはできなくなると考えられるからである。
これは、今後 Hamilton--Jacobi 方程式の初期値問題の正則効果へのひとつの見方を与えると考え重要な意義であると確信する。
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