| Project/Area Number |
11440059
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Keio University |
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Principal Investigator |
KIKUCHI Norio Keio Univ. Fac. of Sci. and Tech., Professor, 理工学部, 教授 (80090041)
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| Co-Investigator(Kenkyū-buntansha) |
MISAWA Masashi Kumamoto Univ., Fac. of Sci., Associate Professor, 理学部, 助教授 (40242672)
SHIMOMURA Shun Keio Univ., Fac. of Sci. and Tech., Professor, 理工学部, 教授 (00154328)
TANI Atushi Keio Univ., Fac. of Sci. and Tech., Professor, 理工学部, 教授 (90118969)
星野 慶介 千葉工業大学, 工学部, 講師
OMATA Seiro Kanazawa Univ., Fac. of Sci., Associate Professor, 理学部, 助教授 (20214223)
ISHIKAWA Shirou Keio Univ., Fac. of Sci. and Tech., Associate Professor (10051913)
前島 信 慶應義塾大学, 理工学部, 教授 (90051846)
田村 要造 慶應義塾大学, 理工学部, 助教授 (50171905)
利根川 吉広 (利根川 吉廣) 北海道大学, 理学部, 助教授 (80296748)
金井 雅彦 名古屋大学, 理学部, 教授 (70183035)
戸瀬 信之 慶應義塾大学, 経済学部, 教授 (00183492)
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| Project Period (FY) |
1999 – 2001
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| Project Status |
Completed (Fiscal Year 2001)
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| Budget Amount *help |
¥4,200,000 (Direct Cost: ¥4,200,000)
Fiscal Year 2001: ¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2000: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Keywords | discrete Morse flow / Morse(variational)flow / difference partial differential equation of elliptic-parabolic type / Rothe's approximation / local estimate / De Giorgi-Nash estimate / Campanato estimate / Gehring-Giaquinta-Modica higher integrability / 多変数変分問題 / 非線形最適化 / 最小勾配(モース)流 / Navier-Stokes方程式 / 調和写像とブラウン運動 / 数理ファイナンス / 調和解析と測定理論 / 特異点・漸近解析 / 変分問題 / モース流 / 調和写像 / 非線形波動 / 差分-偏微分方程式 / 離散モース流 / Rothe近似 / 重調和写像 / p-調和写像 / ナヴィア・ストウクス方程式 / 液晶・超伝導 |
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| Research Abstract |
For the construction of Morse (variational) flows, we introduced the Discrete Morse Flows method, which has been adopted and named Minimizing Movements method by De Giorgi. This approach has some merits in construction of Morse flows, since we can make the most of minimizers to a series of variational functionals inductively introduced starting with initial data. By using this method, we construct Morse flows to variational problems of harmonic map type, in the treatment of which we have noticed the method is available under the weaker assumptions on initial and boundary data. We are trying to approach the construction problems of Morse flows to harmonic map variational problems between metric manifolds, which have no smoothness of the coefficients of the second order differential operators. In the analysis of Discrete Morse Flows, we carry out local estimates of time-discrete partial differential equations of elliptic-parabolic type, so that we have achieved De Giorgi-Nash type Holder estimates and Campanato estimates, and the higher integrability to the gradients of Gehring type. Such kinds of estimates are proved to hold independently of the approximation schemes, which enables to construct Morse flows through Discrete Morse Flows. In conclusion of a series of these research work, we have recognized the characteristics of Discrete Morse Flows method that it can be made use of to the construction problems of Morse flows, to which Leray-Schauder with Schauder estimates cannot be directly applied.
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