2006 Fiscal Year Final Research Report Summary
Study on Regularity and Singularity of a weak solution to the m-harmonic maps and the evolution
| Project/Area Number |
17540199
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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 | Kumamoto University |
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Principal Investigator |
MISAWA Masashi Kumamoto University, Graduate School of Science and Technology, Professor, 大学院自然科学研究科, 教授 (40242672)
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| Co-Investigator(Kenkyū-buntansha) |
TONEGAWA Yoshihiro Hokkaido University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (80296748)
NAKAJIMA Tohru Shizuoka University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (50362182)
KOBAYASHI Osamu Kumamoto University, Graduate School of Science and Technology, Professor, 大学院自然科学研究科, 教授 (10153595)
FURUSHIMA Mikio Kumamoto University, Graduate School of Science and Technology, Professor, 大学院自然科学研究科, 教授 (00165482)
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| Project Period (FY) |
2005 – 2006
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| Keywords | m-harmonic map / m-harmonic map flow / regularity / singularity / energy concentration / free boundary |
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| Research Abstract |
We obtain the following results and try to publish the papers in some Journal.
(1) Free boundary problem for m-harmonic maps and m-harmonic map flow
We show the existence of the local in time solution of the m-harmonic map flow into smooth compact manifold with free boundary on a closed submanifold of the target manifold, which satisfies the m-harmonic map flow equation in the weak sense and is H"older continuous with its gradient in time-space region up to the boundary of the space domain. The maximal existence time of the solution is estimated below by the m-energy of the initial datum. Also, the singular behavior of the solution at the singular time (maximal existence time) can be characterized by a non-constant m-harmonic maps into the target manifold. The m-harmonic map is defined on m-dimensional sphere, or m-dimensional ball with free boundary, and they are called m-harmonic sphere, or m-harmonic disk, respectively. These solutions are exactly minimal submanifolds in the target manifold.
(2) Finite singularity of the m-harmonic map flow
It is expected that the singular set at the singular time is consist of finitely many points. In the paper, we make device of some formula and try to prove the conjecture. However, we are faced with a serious gap of the proof., which is now studied by us to be overcome. We obtain the formula which says the monotonicity of the scaled energy in the intrinsic way to the m-harmonic Laplace operator and is of its own interest.
(3) A priori estimates for the linearized parabolic system of non-divergence form
We show the a priori estimates in some Sobolev space hold for the linearized parabolic system of the m-harmonic map flow and the existence of a strong solution of the system. The existence result is combined with the Leray-Schauder fixed point theorem aid the reflection method to show the local in time solution of the m-harmonic map flow with free boundary.
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Research Products
(15 results)
