viscosity solutions of nonlinear partial differential equations with singularities

具有奇点的非线性偏微分方程的粘度解

基本信息

批准号：
10640119
负责人：
ISHII Katsuyuki
金额：
$ 1.86万
依托单位：
Kobe University of Mercantile Marine
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 2001
项目状态：
已结题

项目摘要

In this project, I considered the existence, uniqueness and stability of viscosity solutions of nonlinear partial differential equations (PDE 's in short) with singularities and their applications of some approximate problems. I had some results on the motion of planar polygons with singular curvature and its application to an approximation for the planar motion of a simple closed curve by its curvature. I also showed that a version of an algorithm, which was proposed by Bence, Merryman and Osher in 1992, can be applied to approximate the motion by mean curvature with right-angle boundary condition in a bounded domain.I studied elliptic/parabolic PDE's with nonlinear terms of the spatial gradient. I classifed completely the interaction between the growth properties of nonliner terms and the uniqueness classes for viscosity solutions and proved the existence of viscosity solutions in such classes. I also treated nonlinear second order ellitpic PDE's with subdifferential. Using the definition of the subdifferential, we modified the notion of the usual viscosity solutions and obtained the uniqueness, existence and stability.Maruo mainly studied the radially symmetry of continuous viscosity solutions of Dirichlet problem for nonlinear degenerate elliptic PDE's. He gave the necessary and sufficient condition which assures that the continuous viscosity solutions are radially symmetric. It seems that this condition is optimal. He also obtained the existence and uniqueness of bounded radial viscosity solutions and those of unbounded ones in the whole space.

在这个项目中，我考虑了非线性偏微分方程粘度解决方案的存在，独特性和稳定性（简而言之PDE）具有奇异性及其在某些近似问题上的应用。我对具有奇异曲率的平面多边形的运动有一些结果，并将其应用于简单闭合曲线的平面运动的近似值。我还表明，可以在1992年由Bence，Merryman和Osher提出的一种算法，可以应用于有界域中的平均曲率近似曲率。空间梯度的非线性项。我完全分类了非线器项的生长特性与粘度解决方案的唯一性类别之间的相互作用，并证明了此类类别中的粘度解决方案的存在。我还将非线性二阶Ellitpic PDE用细分处理。使用细分差的定义，我们修改了通常的粘度解的概念，并获得了唯一性，存在和稳定性。Maruo主要研究了非线性变性椭圆形PDE的Dirichlet问题连续粘度解决方案的径向对称性。他给出了必要和充分的条件，以确保连续的粘度溶液在径向对称。看来这种情况是最佳的。他还获得了有限的径向粘度解决方案以及整个空间中无限型粘度解决方案的存在和独特性。