A study on nonlinear degenerate elliptic partial differential equations

点击翻译按钮获取中文摘要

We have studied the existence and uniqueness of viscosity solutions of nonlinear elliptic partial differential equations and its applications. The results are the following: We have given a formulation of the maximum principle for semicontinuous functions and proved it. The result asserts the existence of a generalized second order derivatives for semicontinuous functions at their maximum points when they have a special form. The result makes the theory of viscosity solutions quite complete. We have studied generalized flows of surfaces by mean curvature and showed that for the proof of the uniqueness of the generalized flows, it is important to use the functions of 4th power as a test function in the standard argument of uniqueness. We also have shown that if the initial surface is convex then so is the surface in the future. We have studied monotone systems of partial differential equations. We have studied second order elliptic partial differential equations in Hilbert spaces and established the existence and uniqueness theorem of viscosity solutions.