The First Initial-boundary Value Problem For A Class Of Parabolic K-Hessian Equations

Hessian equations are a class of fully nonlinear partial differential equations, which are depended on the eigenvalues of Hessian matrix. In this thesis, we investigate the first initial boundary value problem of a class of parabolic Hessian equations-utS k(λ(D~2u))= ψ(x, t, u), which is widely applied to the problem of curve surface deformation, such as intermediate curvature flow and volume preserving.In a smooth domain Ω, we proved the existence of the admissible solution under some basic hypotheses. The proof of the main result includes the following two parts. In the first part, we established the estimation of u and (?)t/(?)u by comparison principle and maximum principle. Next, we established the internal estimation of Du through the test function and the properties of symmetric function f(λ). In the second part, we established a prior estimation of D~2u on (?)pQ_T by estimating the pure tangential, mixed normal-tangential and pure normal second derivatives. Next, we estimated D~2u on (?)Q_T. The existence of the admissible solutions is proved.