Research on the theory of viscosity solutions and its applications

提出并证明了一阶偏微分方程组水平集方法在垂直方向奇异扩散的有效性。建立了包括极小曲面方程在内的完全非线性椭圆型偏微分方程组粘性解的强极大值原理。建立了一个完全非线性一致椭圆型偏微分方程解的例子，在适当的假设下，建立了这类非线性偏微分方程解的极大值原理、Holder正则性和Dirichlet问题的可解性。我们引入了凸化的Gauss曲率流，建立了其推广的水平集方法，并证明了水平集方法中出现的偏微分方程解的存在唯一性。给出了广义凸化高斯流的一个随机逼近格式，并证明了它的收敛性质。我们从数学上证明了当晶体形状为圆柱体时，Berg效应的发生。对于BMO(Bence-Merrima-Osher)格式，我们给出了它收敛于平均曲率流的新证明和收敛速度的最优估计。证明了具有Ornstein-Uhlenbeck算子的抛物型偏微分方程解的渐近解随时间渐近收敛。分析了均匀椭圆型PDE周期均匀化过程中的均匀化和粘度消失的同时效应。证明了某些h路过程的零噪声极限的存在唯一性，建立了具有二次代价的Monge-Kantorovich问题的存在唯一性。在数学金融方面，我们研究了一般因素模型的最优停止时间问题和风险敏感的投资组合优化问题，并构造了它们的最优策略。在相当一般的条件下，我们分析了当p趋于无穷大时p-Laplace方程的解的渐近行为。

We proposed and proved the effectiveness of singular diffusions in the vertical direction in the level set approach to first-order partial differential equations (pde for short). We established the strong maximum principle to viscosity solutions of fully nonlinar elliptic pde including the minimal surface eqaution. We builded an example of fully nonlinear uniformly elliptic pde for which the maximum principle does not holds, and established the maximum principle, Holder regularity, and the solvability of the Dirichlet problem for such nonlinear pde under suitable hypotheses. We introduced the convexified Gauss curvature flow, formulated the level set approach to its generalizations, and established existence and uniqueness of solutions of the pde which appears in the level set approach. We also introduced a stochastic approaximation scheme to the generalized convexified Gauss flow and proved its convergence. We proved on a mathematical basis the occurrence of Berg's effect when the crystal shape is a cylinder. For the BMO (Bence-Merrima-Osher) scheme, we gave a new proof of its convergence to the mean curvature flow and the optimal estimate on the rate of convergence. We proved the convergence the asymptotic solutions as time goes to infinity of solutions of parabolic pde with the Ornstein-Uhlenbeck operator. We analized the simultaneous effects of homogenization and vanishing viscosity in periodic homogenization of uniformly elliptic pde. We proved existence and uniqueness of the limit in the zero-noise of certain h-path processes and established existence and uniqueness of the Monge-Kantorovich problem with a quadratic cost. Regarding mathematical finance, we studied optimal stopping time problems and risk-sensitive portfolio optimization problems for general factor models and constructed their optimal strategies. We analized the asymptotic behavior of solutions of p-Laplace equations as p goes to infinity in a fairly general setting.