Research on viscosity solutions of differential equations and their applications

在我们的项目中得到的结果是：(1)1974年，W. Firey提出了高斯曲率流作为波浪在海滩上滚动的石头磨损过程的数学模型。这个模型假定石头是凸形的。在我们的研究中，我们考虑了石头不凸出的情况。引入了模拟非凸石头在海滩上滚动磨损过程的凸高斯曲率流，建立了基于黏度解的水平集方法。(2)引入了描述凸化高斯曲率流的积分方程的弱解，证明了Cauchy问题弱解的唯一性，并用离散随机逼近证明了它的存在性。(3)研究了一类具有状态约束的一般随机最优控制问题，在较弱的假设条件下，证明了相关值函数的Lipschitz连续性和Holder连续性，证明了值函数在黏性意义上满足相应的Bellman方程，证明了Bellman方程的状态约束问题具有唯一的黏性解。(4)引入了包括Burgers方程在内的一类广泛的一阶偏微分方程的适当粘性解的概念，证明了这类方程可能没有散度形式的适当粘性解的唯一存在性，并在图关于Hausdorff距离的收敛意义上建立了用消失粘性法逼近适当粘性解的收敛性。

The results obtained in our project are : (1) In 1974, W. Firey proposed the Gauss curvature flow as a mathematical model of the wearing process of a stone rolling on the beach by wave. This model assumes that the stone has a convex shape. In our research we considered the case when a stone is not convex. We introduced the convexified Gauss curvature flow which models the wearing process of a nonconvex stone rolling on the beach and established the level set approach based on viscosity solutions method. (2) We introduced weak solutions to the integral equation which describes the convexified Gauss curvature flow, proved that the uniqueness of the weak solution for the Cauchy problem, and proved its existence by a discrete stochastic approximation. (3) We studied a general stochastic optimal control problem with state constraint and proved, under relatively weak assumptions, the Lipschitz continuity and Holder continuity of the associated value function, that the value function satisfy the corresponding Bellman equation in the viscosity sense and that the state constraint problem for the Bellman equation has a unique viscosity solution. (4) We introduced the notion of proper viscosity solutions of a wide class of first-order partial differential equations including the Burgers equation, proved the unique existence of proper viscosity solutions for the class of equations, which may not have divergence form, and established the convergence of the approximation by the vanishing viscosity method to proper viscosity solutions in the sense of convergence of graphs with respect to the Hausdorff distance.