2002 Fiscal Year Final Research Report Summary
Search for an approach to general critical points of variational problems materials science
| Project/Area Number |
13554003
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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 | 展開研究 |
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| Research Field |
Basic analysis
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| Research Institution | Keio University |
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Principal Investigator |
KIKUCHI Norio Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (80090041)
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| Co-Investigator(Kenkyū-buntansha) |
OMATA Seiro Kanazawa University, Faculty of Science, Associate Professor, 理学部, 助教授 (20214223)
SHIMOMURA Syun Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (00154328)
TANI Atsushi Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (90118969)
KASHIWAGI Masahiro Faculty of Chiba Junior College, Associate Professor, 助教授
MISAWA Masashi Kumamoto University, Faculty of Science, Associate Professor, 理学部, 助教授 (40242672)
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| Project Period (FY) |
2001 – 2002
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| Keywords | discrete Morse flow / Morse(variational) flow / Kashiwagi algorism / Rothe's approximation / local estimate / DeGiorgi-Nash-estimate / Campanato estimate / Gehring-Giaquinta-Modica Higher integrability |
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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 the Minimizing Movements method by De Giorgi. This approach can he used in the construction of Morse flows ; starting from initial data we look for minimizers of a series of variational functionals introduced inductively. By using this method, we construct Morse flows to variational problems of harmonic map type. We have noticed this method can still be used under weaker assumptions on the 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, and we have achieved De Giorgi-Nash type Holder estimates and Campanato estimates, and the higher integrability of the gradients of Gehring type. Such estimates are proved to hold independently of the approximation scheme, which enables us to construct Morse flows through Discrete Morse Flows. In conclusion, as a result of this research, we have recognized the characteristics of the Discrete Morse Flows method that can be made use of in the problem of constructing Morse flows for which Leray-Schauder theory with Schauder estimates cannot be directly applied. M.Misawa has treated the regularity and construction of p-harmonic map flows and S.Omata has made mathematical and numerical analysis of Morse flows to several kinds of variational problems. M.Kashiwagi has constructed an algorithm of nonlinear optimizations and composed its software available in http ://srg.pi.cuc.ac.jp/〜kasiwagi/numeric/20040221/
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