Project/Area Number  10640179 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Basic analysis

Research Institution  TOKYO METOROPOLITAN UNIVERSITY 
Principal Investigator 
SAKAI Makoto Tokyo Metropolitan University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70016129)

CoInvestigator(Kenkyūbuntansha) 
HIDANO Kunio Tokyo Metropolitan University, Graduate School of Science, Assistant Professor, 大学院・理学研究科, 助手 (00285090)
KURATA Kazuhori Tokyo Metropolitan University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (10186489)
ISHII Hitoshi Tokyo Metropolitan University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70102887)
MOCHIZUKI Kiyoshi Tokyo Metropolitan University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80026773)
TAKAKUWA Shoichiro Tokyo Metropolitan University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (10183435)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,200,000 (Direct Cost : ¥3,200,000)
Fiscal Year 1999 : ¥1,800,000 (Direct Cost : ¥1,800,000)
Fiscal Year 1998 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  Potential theory / Free boundary problem / Quadrature domain / HeleShaw flow / ポテンシャル論 / 自由境界問題 / 求積領域 / ヘレショウ流れ 
Research Abstract 
We studied a flow produced by injection of fluid into the narrow gap between two parallel planes, which is called a HeleShaw flow, and discussed the shape of the flow for immediately after the initial time. This is a typical free or moving boundary problem described by elliptic equations. We applied potential theoretic methods to the problem and succeeded in obtaining more accurate descriptions of the flow than before. We treated the case that the initial domain has a corner on the boundary. If the interior angle is less than a right angle, then the corner persists for some time with the same interior angle, whereas if the angle is greater than a right angle, then the corner disappears immediately after the initial time. We also gave a detailed discussion about a corner with a right angle and a cusp. In addition to the contribution to our study, each of the investigators obtained his own results. Mochizuki discussed large time asymptotics of small solutions to generalized KPP equation. Ishii treated homogenization of HamiltonJacobi equations and Gaussian curvature flows. Kurata discussed the fundamental solution, eigenvalue asymptotics and eigenfunction of degenerate elliptic operators. Takakuwa showed a compactness theorem for harmonic maps. Hidano discussed nonlinear small data scattering of the wave equation. Hirata discussed the distribution of the return times of the dynamical systems by piecewise monotone transformations.
