Free boundary problems in potential theory

势论中的自由边界问题

• 批准号: 10640179
• 负责人: SAKAI Makoto
• 金额: $ 2.05万
• 依托单位: TOKYO METOROPOLITAN UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640179/
• 关键词: Potential theory; Free boundary problem; Quadrature domain; Hele-Shaw flow

We studied a flow produced by injection of fluid into the narrow gap between two parallel planes, which is called a Hele-Shaw 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 Hamilton-Jacobi 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.

本文研究了将流体注入两个平行平面之间的狭窄间隙中所产生的流动，称之为Hele-Shaw流动，并讨论了初始时刻之后立即出现的流动形状。这是一个典型的椭圆型方程描述的自由或移动边界问题。我们应用潜在的理论方法的问题，并成功地获得比以前更准确的描述的流动。我们处理的情况下，初始域的边界上有一个角落。如果内角小于直角，则角以相同的内角持续一段时间，而如果角大于直角，则角在初始时间之后立即消失。我们还详细讨论了直角拐角和尖点问题，除了对我们的研究有所贡献外，每个研究者都得到了自己的结果：Mochizuki讨论了广义KPP方程小解的大时间渐近性，石井讨论了Hamilton-Jacobi方程的均匀化和高斯曲率流，Kurata讨论了基本解，Takakuwa证明了调和映射的紧性定理，Hidano讨论了波动方程的非线性小数据散射，Hirata用分段单调变换讨论了动力系统返回时间的分布。

项目成果

H. Ishii (with D. Chopp and L. C. Evans): "Waiting time effects for Gauss curvature flows"Indina Univ. Math. J.. 48. 311-334 (1999)

H. Ishii（与 D. Chopp 和 L. C. Evans）：“高斯曲率流的等待时间效应”Indina Univ。

K. Hidano: "Nonlinear small data scattering for the wave equation in RィイD14+1ィエD1"J. Math. Soc. Japan. 50. 253-292 (1998)

K. Hidano：“D14+1D1 中的波动方程的非线性小数据散射”J. Soc。 50. 253-292 (1998)

S.Takakuwa: "A compactness theorem for harmonic maps," Differential and Integral Equation,. 11. 169-178 (1998)

S.Takakuwa：“调和映射的紧性定理”，微分和积分方程。

K. Mochizuki (with K. Mukai and Q. Huang): "Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values"Nonlinear Analysis. 39. 33-45 (2000)

K. Mochizuki（与 K. Mukai 和 Q. Huang）：“具有缓慢衰减初始值的拟线性抛物线方程的大时间行为和寿命”非线性分析。

K. Kurata (with M. Imai and I. Ohnishi): "A minimizing Problem for the First Dirichlet Eigenvalue and Numerical Results"in Potential theory and related problems, Koukyuroku. 1116. 121-138 (1999)

K. Kurata（与 M. Imai 和 I. Ohnishi）：“第一个狄利克雷特征值和数值结果的最小化问题”，潜在理论和相关问题，Koukyuroku。