Project/Area Number 
13640196

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Ryukoku University 
Principal Investigator 
MATSUMOTO Waichiro Ryukoku University, Faculty of Science and Technology, Professor, 理工学部, 教授 (40093314)

CoInvestigator(Kenkyūbuntansha) 
MORITA Yoshihisa Ryukoku University, Faculty of Science and Technology, Professor, 理工学部, 教授 (10192783)
OKA Hiroe (KOKUBU Hiroe) Ryukoku University, Faculty of Science and Technology, Professor, 理工学部, 教授 (20215221)
YOTSUTANI Shoji Ryukoku University, Faculty of Science and Technology, Professor, 理工学部, 教授 (60128361)
MANDAI Takeshi Osaka ElectroCommunication University, Faculty of technology, Professor, 工学部, 教授 (10181843)
NINOMIYA Kazuhiro Ryukoku University, Fac.Science and Technology, Assistant Professor, 理工学部, 助教授 (90251610)

Project Period (FY) 
2001 – 2003

Project Status 
Completed (Fiscal Year 2003)

Budget Amount *help 
¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2003: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)

Keywords  formal symbol / normal form of systems / Cauchy problem / CaufhyKowalevskaya theorem / strongly hyperbolic systems / system of Fuchs type / Kac problem / minimal curvature energy curve / 偏微分方程式系の標準形 / フックス型偏微分方程式系 / j曲率方程式 
Research Abstract 
We have three themes. The first is to establish the necessary and sufficient condition for the CauchyKowalevskaya theorem for systems of partial differential equations including the generalization to the, Nagumo type. On the original CauchyKowalevskaya theorem, we obtained clearer proof. On the theorem of Nagumo type, we obtained a proof on the necessity and also a proof on the sufficiency in a special case. The second is the characterization of the strong hyperbolicity on systems. On the example which is pointwisely diagonalizable but not strongly hyperbolic by Petrovsky, we have already known that it changes to a strongly hyperbolic system by any hyperbolic perturbation. We showed that this phenomenon occurs generally for the systems with timedependent coefficients. To show this, we apply the solvability of the Cauchy problem for the systems of Fuchs type. We also succeeded the generalization of the structure of the solvability. The third is the solvability of the Cauchy problem for pparabolic systems to the future. Unfortunately, we obtained an idea to solve this problem, but finally we cannot achieve it as the complete form. In these researches, the comparison between the calculation on the noncommutative ring of the meromorphic formal symbols and that on the holomorphic pseudedifferential operators has played an essential role. At first, the Kac problem is not the theme of this project. We obtained a good knowledge on the noncommutative groups through the research on the determinatnt theory on noncommutative ring and it brings a viewpoint on the framework of the existence of the counterexamples on the Kac problem by Sunada. As a result, we obtained concrete example of the domains which change from convex to nonconvex smoothly and for which the Kac problem is affirmative by proving mathematically Watanabe's conjecture by the numerical try. This is the first offer of a nonconvex domain for which the Kac problem is affirmative.
