2002 Fiscal Year Final Research Report Summary
Research on viscosity solutions of differential equations and their applications
| Project/Area Number |
12440044
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Waseda University (2001-2002)
Tokyo Metropolitan University (2000) |
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Principal Investigator |
ISHII Hitoshi Waseda University, School of Education, Professor, 教育学部, 教授 (70102887)
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| Co-Investigator(Kenkyū-buntansha) |
SAKAI Makoto Tokyo Metropolitan University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70016129)
MOCHIZUKI Kiyoshi Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (80026773)
GIGA Yoshikazu Hokkaido University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70144110)
ISHII Katsuyuki Kobe University of Mercantile Marine, Faculty of Mercantile Marine, Associate Professor, 商船学部, 助教授 (40232227)
KOIKE Shigeaki Saitama University, Faculty of Science, Professor, 理学部, 教授 (90205295)
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| Project Period (FY) |
2000 – 2002
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| Keywords | Viscosity solutions / Optimal control / Curvature flow / Hamilton-Jacobi equations / Gauss curvature flow / Stochastic optimal control / Level set approach / Uniformly elliptic equations |
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| Research Abstract |
The results obtained in our project are : (1) In 1974, W. Firey proposed the Gauss curvature flow as a mathematical model of the wearing process of a stone rolling on the beach by wave. This model assumes that the stone has a convex shape. In our research we considered the case when a stone is not convex. We introduced the convexified Gauss curvature flow which models the wearing process of a nonconvex stone rolling on the beach and established the level set approach based on viscosity solutions method. (2) We introduced weak solutions to the integral equation which describes the convexified Gauss curvature flow, proved that the uniqueness of the weak solution for the Cauchy problem, and proved its existence by a discrete stochastic approximation. (3) We studied a general stochastic optimal control problem with state constraint and proved, under relatively weak assumptions, the Lipschitz continuity and Holder continuity of the associated value function, that the value function satisfy the corresponding Bellman equation in the viscosity sense and that the state constraint problem for the Bellman equation has a unique viscosity solution. (4) We introduced the notion of proper viscosity solutions of a wide class of first-order partial differential equations including the Burgers equation, proved the unique existence of proper viscosity solutions for the class of equations, which may not have divergence form, and established the convergence of the approximation by the vanishing viscosity method to proper viscosity solutions in the sense of convergence of graphs with respect to the Hausdorff distance.
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