2007 Fiscal Year Final Research Report Summary
Analytical study for mean curvature flow by using an numerical algorithm
| Project/Area Number |
17540117
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kobe University |
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Principal Investigator |
ISHI Katsuyuki Kobe University, Graduate school of Matirime Sciences, Associate Professor (40232227)
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| Co-Investigator(Kenkyū-buntansha) |
MARUO Kenji Kobe University, Graduate school of Matiritne Scienozs, Professor (90028225)
KAGEYAMA Yasuo Kobe University, Graduate school of Malirime Sciences, Lecturer (70304136)
KUWAMURA Masataka Kobe University, Graduate scool of Human development and Enviromnent, Associate Professor (30270333)
NAITO Yuki Kobe University, Graduate school of Technology, Associate Professor (10231458)
ADACHI Tadayoshi Kobe University, Grudate school of Science, Professor (30281158)
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| Project Period (FY) |
2005 – 2007
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| Keywords | Mean curvature flow / Reaction-diffustion equations / Semilinear heat equations / Semilinear elliptic equations / Stark Hamiltonian / Resolvnet estimates |
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| Research Abstract |
Ishii studied the numerical algorithm of the mean curvature flow propsed by Benced , Merriman and Other in 1992. He prove the convergence of this algorithm such that the mean curvature flow equation is somehow directly derived form the algorithm. He also derived the optimal rate of convergence in the case of the smooth and compact motion by mean curvature. Ishii also studied the approximation of the noun curvature flow via the Allen -Cahn equation. He obtinaed the optimal rate of convergence to the smooth and compact motion by mean curvature.
Maruo considered the structure of unbouned and radial viscosity solutions far semilinear elliptic equation He completely classified the structure of solutions in term of the asymptotic behavior at infinity of solutions
Naito investigated the sel-similar solutions for semiliear heat equations with power nonlinearity and showed that the solutions behaves asymptotically like self-similar solutions. In the case of Soholev critical nonlinearity, he gave a sufficient condition for the existence of solutions accruing Type II blow up. He considered some semilinear elliptic equations. He obtained multiple existence of solutions.
Kuwamura studied some reaction-diffusion equations with gradient/skew-gradient structure. Noting that such equations has Hamiltonian structure, he derived a necessary condition for the existence of nontrivial pattern wioth space periodicity in terms of Turing stability. By using some ODE with time-delay, he proposed a mathematical model fo an environmen problem.
Adachi refined the resolvent estimates for N-body Stark Hamiltonian, which is known as the limit absorbing principle, by using some localization technique.
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