Project/Area Number 
13554003

Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  展開研究 
Research Field 
Basic analysis

Research Institution  Keio University 
Principal Investigator 
KIKUCHI Norio Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (80090041)

CoInvestigator(Kenkyūbuntansha) 
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)
利根川 吉廣 北海道大学, 理学部, 助教授 (80296748)

Project Period (FY) 
2001 – 2002

Project Status 
Completed (Fiscal Year 2002)

Budget Amount *help 
¥10,500,000 (Direct Cost: ¥10,500,000)
Fiscal Year 2002: ¥5,500,000 (Direct Cost: ¥5,500,000)
Fiscal Year 2001: ¥5,000,000 (Direct Cost: ¥5,000,000)

Keywords  discrete Morse flow / Morse(variational) flow / Kashiwagi algorism / Rothe's approximation / local estimate / DeGiorgiNashestimate / Campanato estimate / GehringGiaquintaModica Higher integrability / discrete Morse fow / Morse(variational)flow / 国際研究者交流 / 多国籍 / 多変数変分問題 / 非線形最適化 / 一般臨界点解析 / 最小勾配(モース)流 / 特異点解析 / 超伝導・液晶 
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 timediscrete partial differential equations of ellipticparabolic type, and we have achieved De GiorgiNash 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 LeraySchauder theory with Schauder estimates cannot be directly applied. M.Misawa has treated the regularity and construction of pharmonic 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/
