viscosity solutions of nonlinear partial differential equations with singularities

在本项目中，我研究了一类具有奇异性的非线性偏微分方程粘性解的存在性、唯一性和稳定性，以及它们在一些近似问题中的应用。我有一些结果运动的平面多边形奇异曲率和其应用的近似平面运动的一个简单的封闭曲线的曲率。证明了Bence，Merryman和Osher在1992年提出的一种算法的一个版本，可以应用于有界区域中的直角边界条件下的平均曲率近似运动。对非线性项的增长性与粘性解的唯一性类之间的相互作用进行了完整的分类，并证明了粘性解的存在性。我还用次微分处理了非线性二阶椭圆型偏微分方程。利用次微分的定义，我们修正了通常的粘性解的概念，得到了粘性解的唯一性、存在性和稳定性.Maruo主要研究了非线性退化椭圆型方程Dirichlet问题的连续粘性解的径向对称性.他给的必要和充分条件，确保连续粘性解决方案是径向对称的。看来这个条件是最理想的。他还获得了有界径向粘性解的存在性和唯一性，以及在整个空间中无界的。

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.