2022 Fiscal Year Final Research Report
Research on the asymptotic problem appearing in dynamical systems and surface evolution equations by the method of viscosity solutions
| Project/Area Number |
19K03580
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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 | The University of Tokyo |
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Principal Investigator |
Mitake Hiroyoshi 東京大学, 大学院数理科学研究科, 准教授 (90631979)
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| Project Period (FY) |
2019-04-01 – 2023-03-31
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| Keywords | ハミルトン・ヤコビ・ベルマン方程式 / 生成伝播モデル方程式 / 平均場ゲーム理論 / 時間分数冪非線形方程式 / 外力付き平均曲率流方程式 |
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| Outline of Final Research Achievements |
During the project period, I focused on problems related to various properties of viscosity solutions of Hamilton-Jacobi-Bellman (HJB) equations appearing in the context of classical mechanics and crystal growth. In particular, I have worked on the following topics: (a) Asymptotic analysis on HJB equations (the large time behavior, homogenization, uniqueness structure of static problems), (b) Analysis on the birth-and-spread model equation appearing in the crystal growth, (c) Analysis on the first-order mean field game (well-posedness for the discount problem, the vanishing discount problem), (d) Study on the time-fractional nonlinear parabolic equation, (e) Study on the forced mean curvature flow equation (the gradient grow-up, large-time behavior for Dirichlet problem, time global Lipschitz estimate on Neumann problem). I got several new and important results and published 12 (peer-reviewed) papers.
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| Free Research Field |
偏微分方程式論
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| Academic Significance and Societal Importance of the Research Achievements |
補助事業期間中に，制御問題，力学系に現れるハミルトン・ヤコビ・ベルマン方程式，平均場ゲーム連立系，界面運動に現れる結晶成長をモデルとした生成伝播モデル方程式，外力付き平均曲率流方程式，土壌中の汚染物質の拡散や，不均質な媒体での拡散現象を記述する時間分数冪非線形方程式に対して計画していた研究を進展することができた．これらの研究において，従来の研究では不十分であった粘性解的手法の開発に成功した．これらは，偏微分方程式論における粘性解理論，弱KAM理論において重要な学術的意義，社会的意義を持つと期待できる．
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