2001 Fiscal Year Final Research Report Summary
viscosity solutions of nonlinear partial differential equations with singularities
| Project/Area Number |
10640119
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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 of Mercantile Marine |
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Principal Investigator |
ISHII Katsuyuki Faculty of Mercantile Marine, Department of Nuclear Engineering, Kobe University of Mercantile Marine, Assistant Professor, 商船学部, 助教授 (40232227)
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| Co-Investigator(Kenkyū-buntansha) |
MARUO Kenji Faculty of Mercantile Marine, Department of Nuclear Engineering, Kobe University of Mercantile Marine, Professor, 商船学部, 教授 (90028225)
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| Project Period (FY) |
1998 – 2001
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| Keywords | subdifferential / nonlinear PDE / vviscosity solutions / motion by mean curvature / radially symmetric solutions |
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| Research Abstract |
In this project, I considered the existence, uniqueness and stability of viscosity solutions of nonlinear partial differential equations (PDE 's in short) with singularities and their applications of some approximate problems. I had some results on the motion of planar polygons with singular curvature and its application to an approximation for the planar motion of a simple closed curve by its curvature. I also showed that a version of an algorithm, which was proposed by Bence, Merryman and Osher in 1992, can be applied to approximate the motion by mean curvature with right-angle boundary condition in a bounded domain.
I studied elliptic/parabolic PDE's with nonlinear terms of the spatial gradient. I classifed completely the interaction between the growth properties of nonliner terms and the uniqueness classes for viscosity solutions and proved the existence of viscosity solutions in such classes. I also treated nonlinear second order ellitpic PDE's with subdifferential. Using the definition of the subdifferential, we modified the notion of the usual viscosity solutions and obtained the uniqueness, existence and stability.
Maruo mainly studied the radially symmetry of continuous viscosity solutions of Dirichlet problem for nonlinear degenerate elliptic PDE's. He gave the necessary and sufficient condition which assures that the continuous viscosity solutions are radially symmetric. It seems that this condition is optimal. He also obtained the existence and uniqueness of bounded radial viscosity solutions and those of unbounded ones in the whole space.
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