2000 Fiscal Year Final Research Report Summary
A versatile study of positive solutions to parabolic and elliptic partial differential equations
| Project/Area Number |
11440039
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B).
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | Tokyo Institute of Technology |
|---|
Principal Investigator |
MURATA Minoru Graduate School of Science and Engineering, Tokyo Institute of Technology Professor, 大学院・理工学研究科, 教授 (50087079)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
KURATA Kazuhiro Tokyo Metropolitan University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (10186489)
AIKAWA Hiroaki Shimane University, Interdisciplinary Faculty of Science and Engineering, Professor, 総合理工学部, 教授 (20137889)
ISHIGE Kazuhiro Nagoya University, Graduate School of Mathematics, Tokyo Institute of Technology Associate Professor, 大学院・多元数理科学研究科, 助教授 (90272020)
SHIGA Hiroshige Graduate School of Science and Engineering, Tokyo Institute of Technology Professor, 大学院・理工学研究科, 教授 (10154189)
UCHIYAMA Kouhei Graduate School of Science and Engineering, Tokyo Institute of Technology Professor, 大学院・理工学研究科, 教授 (00117566)
|
|---|
| Project Period (FY) |
1999 – 2000
|
| Keywords | parabolic PDE / elliptic PDE / positive solution / Cauchy problem / uniqueness / fundamental solution / perturbation theory / Martin boundary |
|---|
| Research Abstract |
M.Murata and K.Ishige studied uniqueness of nonnegative solutions of the Cauchy problem to second order parabolic equations on Riemannian manifolds and domains of R^n, and gave a sharp and general sufficient condition for the uniqueness via asymptotic properties at infinity of the equation and manifolds (to appear in Ann. Scuola Normale Sup. Pisa.). This is a simple and general result which unifies all previous results on the uniqueness.
M.Murata investigated the structure of positive solutions of second order elliptic equations in skew product form, and determined the Martin boundary and Martin kernel by exploiting and developping perturbation theory and estimates of fundamental solutions for parabolic equations. This result was reviewed by such experts as Y.Pinchover, A.Grigor'yan, V.Maz'ya, S.Gardiner, and has been submitted.
K.Ishige studied uniqueness of nonnegative solutions of the Dirichlet and Neumann boundary value problems to second order parabolic equations on domains of R^n, and gave sharp sufficient conditions for the uniqueness via asymptotic properties at infinity of the equation and domains. For the Dirichlet problem, he established an optimal uniqueness theorem (Journal of Differential Equations, 158(1999), 251 -290). As for the Neumann problem, he revealed a large difference from the Dirichlet problem and established a uniqueness theorem for domains belonging to a class wider than that for the Dirichlet probelm.
|
