2005 Fiscal Year Final Research Report Summary
Research on structures of solutions for geometric variational problems
| Project/Area Number |
15540214
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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 |
Global analysis
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| Research Institution | Tokyo University of Science |
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Principal Investigator |
TACHIKAWA Atsushi Tokyo University of Science (T.U.S.), Faculty of Science and Technology, Professor, 理工学部, 教授 (50188257)
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| Co-Investigator(Kenkyū-buntansha) |
OTSUKI Nobukazu T.U.S., Faculty of Science and Technology, Professor, 理工学部, 教授 (80112895)
KUBAYASHI Tako T.U.S., Faculty of Science and Technology, Professor, 理工学部, 教授 (90178319)
NAGASAWA Takeyuki Saitama University, Faculty of Science, Professor, 理学部, 教授 (70202223)
OGASAWA Masao T.U.S., Faculty of Science and Technology, Assistant, 理工学部, 助手 (50408704)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Variational problems / Partial regularity / VMO / Harmonic maps |
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| Research Abstract |
The aim of this research project is to investigate structures of solutions for geometric variational problems. While we were studying harmonic maps into Finsler manifolds, the necessity of the research on the variational functionals with singularities occurred. So, in this research project, we considered, as important points, regularity of the solutions for variational problems with singularities or weak solutions of partial differential equations with singular coefficients. For this purpose, we investigated partial regularity of minimizers for functionals with VMO (Vanishing Mean Oscillation)-coefficients in cooperation with Prof.Maria Alessandra Ragusa (Universita di Catania (Italy)). As results of cooperation with M.A.Ragusa, we got some results on partial regularity of minimizers. Namely, we proved that if u is a minimizer of certain functional with VMO-coefficients then u satisfies "u is Holder continuous except a subset of the domain whose m-2-ε dimensional Hausdorff measure is 0, where m is the dimension of the domain".
On the other hand, each researchers investigated their own problems : T.Nagasawa studied Helfrich variational problem which is one of mathematical models for shape transformation theory of human red blood cells. The existence of associated gradient flow was proved locally for arbitrary initial data, and globally near spheres. M.Ogawa studied free boundary problems for flows of an incompressible ideal fluid. He showed the unique existence of the solution, locally in time, even if the initial surface and the bottom are uneven.
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