Research On Several Classes Of Fully Nonlinear Hessian Equations

Hessian equation is an important class of fully nonlinear partial differential equa?tions,which have high frequencies in differential geometry;complex geometry;partial differential equation;optimal transport problems and the theory of convex body.Even in recent years,image processing technologies and artificial intelligence theory and so on can be found in its trace.The motivation for researching Hessian equation mainly comes from their wide application in differential geometry.For example,k-Yamabe problem in conformal geometry is equivalent to solving the fully nonlinear elliptic Hessian equation on a compact manifold;Alexandrov problem of predicting curvature and so on.All of these are equivalent to a special case of Hessian equation-Monge-Ampere equation.There are too many to mention.Therefore,research on Hessian equation has important theoretical significance and application value.Hessian equation is a class of fully nonlinear partial differential equations that depend on Hessian matrix eigenvalues of their solutions.This leads to the complexity and difficulty of solving the Hessian equation itself.Since 1976,Shing-Tung Yau developed a powerful method of priori estimates and successfully solved Calabi conjecture,although various forms of Hessian equations researched by Chinese and foreign mathematicians have emerged one after another,a priori estimates method has still been one of the main way to solve their problem.In this paper we study several classes of Hessian equations,by proving C?'s priori estimates of the admissible solution,and using the theorem of EvansKrylov and the theory of Schauder to construct a higher-order estimation of the solution,and apply the continuity method and the topological degree theorem to get the existence of the problem solution.The primary contents of the dissertation are as follows:Firstly,the existence of the C1,1 solutions of a kind of completely nonlinear Hessian equation obstacle problem on compact manifolds was proved.We use the perturbation technique to reconstruct the obstacle problem by introducing a positive penalty term related to ?(>0).Given the structural condition of the equation and assuming that the sub-solution exists,it is proved that ?(>0)has nothing to do with C? a priori estimation,then because of compactness,we got the existence of obstacle problems C1,1 solutions.Secondly,the existence and uniqueness of continuous viscous solutions for a class of second-order partial differential problems are proved by the Perron method on ?(?)Rn bounded domain.The key to this conclusion is that Barles under certain conditions established C? regularity of second-order partial differential equation's viscous sub-solution.Therefore,in the Perron process,the functional classes satisfies the Arzela-Ascoli theorem is appropriately selected,and then the existence of the continuous viscosity solution is obtained.When the principle is established,the uniqueness of the continuous viscosity solution can be further proved.As an application,we considered the k-Hessian equation.In some special cases,we can also get the Ca regularity of the viscosity subsolution.Finally,on MT ? M ×(0,T],establish the permissible solution of the Dirichlet problem of a class of fully nonlinear parabolic Hessian equation C? priori estimation.In order to avoid adding too many geometric assumptions on the manifold boundary,it is assumed that the sub-solution of the equation exists.In the estimation process,we usually consider the viscosity solution at the critical point under the assumption of the operator structure the properties,combined with certain characteristics of the parabolic Hessian equation itself,use the comparison principle and some basic inequalities and techniques to gradually obtain the corresponding estimates.