| Project/Area Number |
17540188
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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 | Saitama University |
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Principal Investigator |
NAGASAWA Takeyuki Saitama University, Graduate School of Science and Engineering, Professor (70202223)
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| Co-Investigator(Kenkyū-buntansha) |
KOIKE Shigeaki Saitama University, Graduate School of Science and Engineering, Professor (90205295)
OHTA Masahito Saitama University, Graduate School of Science and Engineering, Associate professor (00291394)
SAKAMOTO Kunio Saitama University, Graduate School of Science and Engineering, Professor (70089829)
KOHSAKA Yoshihito Muroran Institute of Technology, 工学部, Associate Professor (00360967)
TACHIKAWA Atsushi Tokyo Science University, Department of Mathematics, Professor (50188257)
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| Project Period (FY) |
2005 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,760,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥360,000)
Fiscal Year 2007: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2006: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Keywords | geometric evolution equations / Willmore functional / Helfrich variational problem / center manifold / bifurcation theory / bifurcation equation / gradient flow / 条件付勾配流 / 超曲面の発展方程式 / 安定性 / 正則性 |
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| Research Abstract |
In this research we investigate the gradient flow with constraints for functionals defined for family of curves and surfaces as geometric evolution equations. The gradient flow, which decreases the value of functionals via deformation, is one of the method for finding critical points. Various shapes in the nature should be stable in some sense. Functionals are the measure of stability, and therefore the limit of gradient flow should be stable in this sense.
Nagasawa and Kohsaka consider the Willmore functional for surfaces with prescribed area and enclosed volume (the Helfrich variational problem), and construct the associate gradient flow (the Helfrich flow), and analyze the structure of center manifold near sphere. On the stationary problem for the problem, solutions bifurcating from sphere with more 2, 4, 6 and 8 are constructed by Nagasawa. We reduce the bifurcation equation by use of the group equivariance but not the equivariant branching lemma of the bifurcation theory. Furthermore Nagasawa considers the Helfrich flow for plane curves. There an approximate problem such that the constraints are realized as a singular limit is proposed. It is shown that the uniform estimates for solutions for approximate problem and their convergence.
In many case, equations of gradient flow are parabolic type. Koike investigates the maximum principle and comparison results for fully nonlinear parabolic and elliptic equations. Ohta studies the stability of solutions for evolution equation of hyperbolic type. Sakamoto investigates the CR-structure of manifolds. Kohsaka studies the nonlinear stability of stationary solutions for surface diffusion with boundary conditions. Solutions of geometric variational problem are weak solution of a nonlinear equation. Hence it is important to analyze their regularity. Tachikawa studies the regularity theory of minimal critical points for integral functional with discontinuous coefficients.
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