The First Initial-boundary Value Problem For Hessian Equations Of A Class Of Parabolic Type On Riemannian Manifolds

The Hessian equation is a class of fully nonlinear partial differential equation.And this kind of equation depends only on the eigenvalues of the Hessian matrix of the solution formally.When dealing with some problems in differential geometry,it can be transformed into solving a fully nonlinear Hessian equation by some calculations,such as k?Yamabe problem in conformal geometry,Minkowski problem,the prescribed curvature problem and so on.Therefore,the study of the existence and regularity of solutions of this class of equation has very important theoretical significance and application value in solving these geometric problems.Based on the uniqueness and importance of the Hessian equation,a priori estimate of admissible solutions for a class of Hessian equations on Riemannian manifolds is generalized according to the research ideas of related references.The research includes gradient estimation,|ut| estimation and the interior estimation of second derivatives.Our main works are as follows:Firstly,we establish the gradient estimates for the parabolic Hessian equations in MT=M×(O,T].As the generalization of this class of equations,the tensor A[u]and the non-homogeneous terms ?include u and ?u in the equation we study in this paper.With some assumptions,we add growth conditions to A and ?and then we get the gradient estimates.Secondly,we estimate the t in the equation we study.In this part of the proof,a constant term will appear when we establish the L?estimation.In order to eliminate this term,we propose a technical hypothesis,which is Auii=0 and ?u=0 According to the existing methods,similarly,we finish the estimate of |ut| in the equation by constructing a function Wunder the assumptions about the solution exists.Finally,we prove the interior estimates of the second derivatives of admissible solutions in detail.In order to obtain a more perfect interior estimation,we construct a barrier function by using strict sub solution,considering the situation that the function Wreaches its maximum value in MT,and draw corresponding conclusions through the maximum principle.