1998 Fiscal Year Final Research Report Summary
Study of singular solutions of nonlinear differential equations
| Project/Area Number |
09640209
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | TOKYO METROPOLITAN UNIVERSITY |
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Principal Investigator |
TAKAKUWA Shoichiro Tokyo Metropolitan University, Graduate School of Science, Assosiate Professor, 大学院・理学研究科, 助教授 (10183435)
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| Co-Investigator(Kenkyū-buntansha) |
TODA Masahito Granduate School of Science TOKYO METROPOLITAN UNIVERSITY,Assosiate Professor, 大学院・理学研究科, 助手 (80291566)
NISHIOKA Kunio Granduate School of Science TOKYO METROPOLITAN UNIVERSITY,Assosiate Professor, 大学院・理学研究科, 助教授 (60101078)
HIDANO Kunio Granduate School of Science TOKYO METROPOLITAN UNIVERSITY,Assosiate Professor, 大学院・理学研究科, 助手 (00285090)
OHNITA Yoshihiro Granduate School of Science TOKYO METROPOLITAN UNIVERSITY,Professor, 大学院・理学研究科, 教授 (90183764)
KURATA Kazuhiro Granduate School of Science TOKYO METROPOLITAN UNIVERSITY,Assosiate Professor, 大学院・理学研究科, 助教授 (10186489)
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| Project Period (FY) |
1997 – 1998
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| Keywords | differntial equation / singular point of solution / harmonic map / gauge theory / nonlinear problem / convergence theorem |
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| Research Abstract |
We first study harmonic maps between two Riemannin manifolds. We consider the case that the dimension n of the domain manifold is greater than 2. We prove that any subset of harmonic maps whose gradients are uniformly bounded in L^n space is compact with respect to C^* topology. As a corollary of this result we obtain the uniform estimates of first derivatives of harmonic maps in higher dimensions. This result is published in "Differential and Integral Equations". We show that the Liouville type for harmonic maps holds in higher dimensions, which is an important ingredient of the proof of the compactness theorem. By applying Liouville type theorem we obtain the estimate of the gradients of singular harmonic maps using the distance from the set of singular points. The paper of this result in in preparation and the reserch of singular harmonic maps is in progress.
Next we study the nonlinear problems in gauge theory. Using Grant-in-Aid for Scientific Reserch we invite Professor Kazuo Akutagawa (Shizuoka Univ.) to give a lecture on Seiberg-Witten theory and its application to geometry. We study the moduli space of Yang-Mills connections on a Riemannian manifold of dimension n <greater than or equal> 5 and prove the compactness of subsets of the moduli space whose curvatures are uniformly bounded in L^n space. The paper of this result is to submitted and the reserch of singular Yang-Mills connections is in progres
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