Project/Area Number  10640119 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Kobe University of Mercantile Marine 
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)

CoInvestigator(Kenkyūbuntansha) 
MARUO Kenji Faculty of Mercantile Marine, Department of Nuclear Engineering, Kobe University of Mercantile Marine, Professor, 商船学部, 教授 (90028225)
富田 義人 神戸商船大学, 商船学部, 教授 (50031456)

Project Fiscal Year 
1998 – 2001

Project Status 
Completed(Fiscal Year 2001)

Budget Amount *help 
¥2,900,000 (Direct Cost : ¥2,900,000)
Fiscal Year 2001 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 2000 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1999 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1998 : ¥800,000 (Direct Cost : ¥800,000)

Keywords  subdifferential / nonlinear PDE / vviscosity solutions / motion by mean curvature / radially symmetric solutions / 退化放物型偏微分方程式 / 退化楕円型偏微分方程式 
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 rightangle 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.
