Free boundary problems in potential theory

我们研究了在两个平行平面之间的狭窄间隙内注入流体所产生的流动，称为Hele-Shaw流动，并讨论了初始时间后立即的流动形状。这是一个典型的由椭圆方程描述的自由或移动边界问题。我们将势理论方法应用于该问题，并成功地获得了比以前更精确的流动描述。我们处理了初始域在边界上有一个角的情况。如果内角小于直角，则该角以相同的内角存在一段时间，而如果内角大于直角，则该角在初始时间后立即消失。我们还对直角角和尖角进行了详细的讨论。除了对我们的研究有贡献之外，每个研究者都有自己的研究结果。Mochizuki讨论了广义KPP方程小解的大时间渐近性。Ishii处理了Hamilton-Jacobi方程和高斯曲率流的均匀化问题。Kurata讨论了退化椭圆算子的基本解、特征值渐近性和特征函数。Takakuwa给出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通过分段单调变换讨论了动力系统返回时间的分布。

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.