A study on nonlinear degenerate elliptic partial differential equations

研究了非线性椭圆型偏微分方程粘性解的存在唯一性及其应用。结果如下：我们给出了半连续函数的最大值原理的一个公式，并证明了它。结果证明了当函数具有特殊形式时，半连续函数在其极大值点处存在广义二阶导数。这一结果使得粘性解的理论变得相当完整。我们用平均曲率研究了曲面的广义流，证明了在唯一性的标准论证中，用四次函数作为检验函数是证明广义流唯一性的重要方法。我们还证明了，如果初始曲面是凸的，那么未来的曲面也是凸的。我们研究了单调偏微分方程组。研究了Hilbert空间中的二阶椭圆型偏微分方程，建立了粘性解的存在唯一性定理。

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.