Viscosity solutions of nonlinear partial differential equations and a game-theoretic approach
| Project/Area Number |
16K17635
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Mathematical analysis
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| Research Institution | Fukuoka University |
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Principal Investigator |
LIU QING 福岡大学, 理学部, 助教 (70753771)
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
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| Keywords | 粘性解 / 完全非線形偏微分方程式 / 微分ゲーム / 時間分数階微分 / ハイゼンベルグ群 / サブリーマン多様体 / ゲーム理論 / 動的境界条件 / 曲率流方程式 / 凸性の保存 / Heisenberg group / 平均曲率流 / 非線形偏微分方程式 |
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| Outline of Final Research Achievements |
In this research project, we used the game-theoretic approach to study the well-posedness and behavior of solutions of various nonlinear partial differential equations based on viscosity solution theory. In particular, we established a game interpretation for dynamic boundary problems of parabolic equations and constructed approximate solutions of time fractional evolution equations. Moreover, motivated by applications in image processing, we investigated the asymptotic behavior of solutions to power curvature flows. In addition to the study in the Euclidean space, we also used the viscosity solution theory to prove existence and uniqueness of solutions to fully nonlinear parabolic systems on the Heisenberg group.
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| Academic Significance and Societal Importance of the Research Achievements |
本研究では，微分ゲームによる方法を積極に取り入れることにより，非線形偏微分方程式の粘性解理論を発展させることができた．ユークリッド空間のみならず，より一般的な距離空間上でも偏微分方程式の解の一意存在性，正則性と凸性等の問題を解決できた．更に，画像処理や土壌汚染問題などを記述する数理モデルに対し，粘性解理論を用い，解の構成法と漸近挙動を考察し，それらの実際の問題に応用できる基礎的な数学理論を構築し，社会貢献に繋がる研究成果が得られた．
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Report
(4 results)
Research Products
(28 results)
