Changes of configuration in free boundary problems

我们研究了在两个平行平面之间的狭窄缝隙中注入流体所产生的流动，称为Hele-Shaw流，并讨论了初始时间后的流动形态。这是一个用椭圆型方程描述的典型的自由或移动边界问题。我们将潜在的理论方法应用于这一问题，并成功地获得了比以前更准确的流动描述。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，则拐角将以相同的内角保持一段时间，而如果角度大于直角，则拐角在初始时间后立即消失。另一种关键情况是内角等于2π的情况。在这种情况下，我们进行了详细的讨论，得到了角点成为层流流点的良好条件。除了对我们的研究有贡献外，每个研究者都得到了自己的结果。Okada研究了数值调和分析，得到了函数逼近的估计。仓田讨论了非线性椭圆型方程的非负解，得到了某些无界区域的存在定理。Takuwa利用推广的Pohozaev恒等式证明了p-调和映射的唯一性定理。Ishii讨论了Hamilton-Jacobi方程的齐次化，讨论了齐化的收敛速度。Mochizuki讨论了特征函数的大时间渐近性和谱表示。

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. The other critical case is the case that the interior angle equals to 2π. We gave a detailed discussion in this case and obtained good conditions for the corner to be a laminar-flow point.In addition to the contribution to our study, each of the investigators obtained his own results.Okada studied numerical harmonic analysis and obtained estimates on approximations of functions.Kurata discussed nonnegative solutions of nonlinear elliptic equations and obtained the existence theorem for certain unbounded domains.Takakuwa showed the uniqueness theorem for p-harmonic maps by applying the generalized Pohozaev identity.Ishii treated homogenization of Hamilton-Jacobi equations and discussed the rate of convergence of the homogenization.Mochizuki discussed large time asymptotics of eigenfunctions and spectral representation.