Research on the theory of viscosity solutions and its applications

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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.