1991 Fiscal Year Final Research Report Summary
A study on nonlinear degenerate elliptic partial differential equations
| Project/Area Number |
02640150
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | CHUO UNIVERSITY |
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Principal Investigator |
ISHII Hitoshi Chuo Univ. Math. Dept., Professor, 理工学部, 教授 (70102887)
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| Co-Investigator(Kenkyū-buntansha) |
MITSUMATSU Yoshihiko Chuo Univ. Math. Dept., Associate Professor, 理工学部, 助教授 (70190725)
SEKIGUCHI Tsutomu Chuo Univ. Math. Dept., Professor, 理工学部, 教授 (70055234)
MATSUYAMA Yoshio Chuo Univ. Math. Dept., Professor, 理工学部, 教授 (70112753)
KURIBAYASHI Akikazu Chuo Univ. Math. Dept., Professor, 理工学部, 教授 (40055033)
IWANO Masahoro Chuo Univ. Math. Dept., Professor, 理工学部, 教授 (70087013)
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| Project Period (FY) |
1990 – 1991
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| Keywords | Viscosity Solutions / Degenerate Elliptic Partial Differential Equations / Optimal Control / Hamilton-Jacobi Equations / Mean Curvature Flow / 非線形偏微分方程式 |
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| Research Abstract |
We have studied the existence and uniqueness of viscosity solutions of nonlinear elliptic partial differential equations and its applications. The results are the following: We have given a formulation of the maximum principle for semicontinuous functions and proved it. The result asserts the existence of a generalized second order derivatives for semicontinuous functions at their maximum points when they have a special form. The result makes the theory of viscosity solutions quite complete. We have studied generalized flows of surfaces by mean curvature and showed that for the proof of the uniqueness of the generalized flows, it is important to use the functions of 4th power as a test function in the standard argument of uniqueness. We also have shown that if the initial surface is convex then so is the surface in the future. We have studied monotone systems of partial differential equations. We have studied second order elliptic partial differential equations in Hilbert spaces and established the existence and uniqueness theorem of viscosity solutions.
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