Viscosity solutions of nonlinear degenerate parabolic equations and several applications

For the Cauchy problem of a class of fully nonlinear degenerate parabolic equations, this paper studies the existence, uniqueness and regularity of viscosity solutions; these results apply to Hamilton-Jacobi-Bellman (HJB for short) equation, Leland equation and equations of p-Laplacian type, which find a lot of applications in fluid mechanics, stochastic control theory and optimal portfolio selection and transaction cost problems in finance.; Further studies are done on the properties of viscosity solutions of the above models: (1) Bernstein estimates (especially C 1,α estimates) and convexity of viscosity solutions of the HJB equation; (2) monotonicity in time and in Leland constant of the viscosity solutions to the Leland equation and the relationship between Leland solutions and Black-Scholes solutions; (3) the existence and Lipschitz continuity of the free boundaries of viscosity solutions for fully nonlinear equations u_t + F(Du, D2u) = 0, with p-Laplacian equation as model. Our study extends the application of viscosity solution theory and aids in the qualitative analysis and numerical computation of the above models.; To construct continuous viscosity solutions, we make use of Perron Method and various estimates by virtue of viscosity solution theory; we generalize Bernstein estimates and Kruzhkov's regularization theorem in time from smooth solutions to viscosity solutions; our method applies to initial boundary value problem, though the estimates of uniformly continuous moduli near the boundary need to be obtained and suitable viscosity sub- and super-solutions need to be constructed; to study the Leland equation, we transform it into standard form by Euler transformation and linear translation, then study the property of the viscosity solutions by virtue of comparison principle; to study the properties of the free boundary of equations of p-Laplacian type, we employ comparison principle, reflection principle, moving plane method and the construction of sub and super solutions.