1999 Fiscal Year Final Research Report Summary
Research in viscosity solutions using the method of Functional Analysis.
| Project/Area Number |
10640169
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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 |
Basic analysis
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| Research Institution | Kobe University of Mercantile Marine |
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Principal Investigator |
MARUO Kenji Kobe Univ. Mercan. Marine, Faculty of Mercan. Marine, Professor, 商船学部, 教授 (90028225)
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| Co-Investigator(Kenkyū-buntansha) |
INOUE Tetuo Kobe Univ. Mercan. Marine, Faculty of Mercan. Marine, Professor, 商船学部, 教授 (50031448)
ISHII Katsuyuki Kobe Univ. Mercan. Marine, Faculty of Mercan. Marine, Assistant Professor, 商船学部, 助教授 (40232227)
TOMITA Yoshihito Kobe Univ. Mercan. Marine, Faculty of Mercan. Marine, Professor, 商船学部, 教授 (50031456)
MIYAKODA Tuyako Osaka Univ., Faculty of Technology, Assistant, 工学部, 助手 (80174150)
KAGEYAMA Yasuo Kobe Univ. Mercan. Marine, Faculty of Mercan. Marine, Assistant, 商船学部, 助手 (70304136)
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| Project Period (FY) |
1998 – 1999
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| Keywords | Viscosity Solution / Degenerate Elliptic Equation / Existence Theorem / Uniqueness Theorem / Semilinear / Quasilinear / Radial Solution |
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| Research Abstract |
We consider the Dirichelet problem for a semilinear degenerate elliptic equation (DP) :
-g(|x|)Δu+f(|x|, u(x)) = 0, and Boundary Condition
where N【greater than or equal】2 and g(|x|), f(|x|, u) are continuous. We discuss the problem (DP) under the following assumption : 1)g is nonnegative. 2)f is strictly monotone for u. We first define a standard viscosity solution by the viscosity solution such that if g(|x|) = 0 then f(|x|, u(x)) = 0. Then we can prove that the any continuous standard viscosity solution is the radial solution and it is unique. We add an assumption : 3)∫ィイD1a-0ィエD1gィイD1-1ィエD1(s)ds = ∞ or ∫ィイD2a+0ィエD2gィイD1-1ィエD1(s)ds = ∞ for any a : g(a) = 0. Then We obtain that any continuous viscosity solution is the radial solution and it is unique. If the assumption 3) is not satisfied there exist examples such that the continuous viscosity solutions are not unique. Here, the domain is a bounded boall in n-dimension space.
We next state the existence and uniqueness of the continuous unbounded viscosity solution in RィイD12ィエD1. We use the order of the infinite neighborhood of the solution as the boundary condition. We know that the existence or nonexistence of the solution are dependent on a kind of the order of the solution. Moreover, we get the results which the uniqueness or non-uniqueness are also dependent on a kind of the order of the solution. In case, we assume that g, f is sufficiently smooth.
We now show the existence of a continuous viscosity solution to quasi-semilinear degenerate elliptic problem. Here, g(|x|, u), f(|x|, u) are continuous and f is strictly monotone for u. Moreover, we assume there exists an implicite function of f = 0 and the implicite function holds some smoothness. Then we can prove the existence of the continuous viscosity solution. But it is difficult to prove the uniqueness of the solution.
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Research Products
(4 results)
