Research on viscosity solutions of differential equations and their applications

(1)1974年，W.Firey提出了高斯曲率流作为波浪作用下石块在海滩上滚动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，我们考虑了石头不是凸的情况。引入了模拟非凸石在海滩上滚动磨损过程的凸化高斯曲率流，建立了基于粘性解方法的水平集方法。(2)引入了描述凸化高斯曲率流的积分方程的弱解，证明了柯西问题弱解的唯一性，并用离散随机逼近的方法证明了弱解的存在性。(3)研究了具有状态约束的一般随机最优控制问题，证明了在相对较弱的假设下，伴随值函数的Lipschitz连续性和Holder连续性，值函数在粘性意义下满足相应的Bellman方程，Bellman方程的状态约束问题有唯一的粘性解。(4)引入了包括Burgers方程在内的一类一阶偏微分方程真粘性解的概念，证明了这类方程(可能没有发散形式)真粘性解的存在唯一性，并证明了在图的关于Hausdorff距离的收敛意义下，消失粘性方法逼近到真粘性解的收敛。

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.