2002 Fiscal Year Final Research Report Summary
Research on canonical forms to nonlinear elliptic boundary value problems and the global structure of all solutions including singular solutions
| Project/Area Number |
12640225
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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 |
Global analysis
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| Research Institution | Ryukoku University |
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Principal Investigator |
YOTSUTANI Shoji Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (60128361)
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| Co-Investigator(Kenkyū-buntansha) |
MORITA Yoshihisa Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (10192783)
NINOMIYA Hirokazu Ryukoku Univ., Faculty of Sci. and Tech., Associate Professor, 理工学部, 助教授 (90251610)
YANAGIDA Eiji Tohoku Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80174548)
IKEDA Tsutomu Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (50151296)
OKA Hiroe Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (20215221)
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| Project Period (FY) |
2000 – 2002
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| Keywords | elliptic equation / radial solution / canonical form / cross-diffusion / reaction diffusion / nonlocal |
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| Research Abstract |
First, a head investigator S. Yotsutani have obtained canonical forms and structure theorems for radial solutions to semilinear elliptic problems with Y. Kabeya and E. Yanagida.
Radial solutions of semilinear elliptic problems satisfy some boundary value problems for second order differential equations. It is seen that the boundary value problems can be reduced to a canonical form with the Dirichlet, Neumann or Robin boundary condition after suitable change of variables. We get structure theorems to canonical forms to equations with power nonlinearities and various boundary conditions. By using these theorems, it is possible to study the properties of radial solutions of semilinear elliptic equations in a systematic way, and make clear unknown structure of various equations.
Second, it is possible through the canonical forms to convert results for one problem to that of others, and moreover, to find an original methods to investigate singular solutions. As a concrete example, S. Yotsutani clarified the structure of solutions including singular solutions in a unit ball with H. Myogahara and E. Yanagida.
As related problems, we are investigating a limiting equation to a cross-diffusion equation that appears in mathematical biology. We showed that it has different kinds of singular solutions with Y. Lou and W.-M. Ni. This problem is a nonlocal nonlinear elliptic boundary problem, for which no method was known to solve it. We discovered a new method. There are a lot of interesting problems for which the method is applicable. A problem of the Ossen's spiral flow is one of them.
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