2006 Fiscal Year Final Research Report Summary
Study of the structure of solutions of partial differential equations in the complex domain
| Project/Area Number |
16540170
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | Sophia University |
|---|
Principal Investigator |
OUCHI Sunao Sophia University, Faculty of Science and Technology, Professor, 理工学部, 教授 (00087082)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
UCHIYAMA Koichi Sophia University, Faculty of Science and Technology, Professor, 理工学部, 教授 (20053689)
TAHARA Hidetoshi Sophia University, Faculty of Science and Technology, Professor, 理工学部, 教授 (60101028)
YOSHINO Kunio Sophia University, Faculty of Science and Technology, Associate Professor, 理工学部, 助教授 (60138378)
HIRATA Hiroshi Sophia University, Faculty of Science and Technology, Lecturer, 理工学部, 講師 (20266076)
GOTO Satoshi Sophia University, Faculty of Science and Technology, Assistant, 理工学部, 手 (00286759)
|
|---|
| Project Period (FY) |
2004 – 2006
|
| Keywords | complex partial differential equations / singularity of solutions / asymptotic behavior / formal solutions / generalized functions / p-Laplace equation / multisummability / Gevrey estimate |
|---|
| Research Abstract |
Main study is asymptotic behaviors of solutions with singularity of partial differential equations (PDE) in the complex domain. The followings were shown.
(1)We studied the relation between formal power series solutions of PDE and genuine solutions, by applying the theory of multisummability of formal power series. We found 2-classes of PDE and showed that formal power series solutions are multisummable for these classes.
(2)It was shown for some important class of PDE that asymptotic behaviors of singular solutions are represented by functions defined by Mellin type integral and the remainder estimate is given by so-called Gevrey-type.
(3)We tried to transform PDE to those of simple form. In order to so we introduce an equivalence class of PDE and an equation with infinite variables. The solvability of this equation means equivalence. The obtained result is applicable to existence and non existence of singular solutions.
(4)Nonlinear total characteristic type of PDE was introduced and Results about existence and nonexistence of singular solutions were obtained.
(5)Non trivial radical solutions of p-Laplacian was constructed.
We also studied the properties of generalized functions, by using fundamental solutions of heat equations, and obtained results about Paley Wiener Theorem, positive definite functions. Function spaces of Gelfand-Silov.
|
