Liouville Theorem For The Entire Solutions Of 2-Hessian Equation

Fully nonlinear partial differential equation is a kind of very important partial differential equation,which is closely related to convex geometry,differential geome-try,complex geometry,probability theory and other mathematical branches,and has a wide range of applications in optimal transportation,image processing.Therefore,its research has important value both in theory and in practice.Hessian equation is a very important and typical class of fully nonlinear partial differential equations,and it appears in many interesting geometric problems,such as Minkowski problem,Weingarten curvature problem and Special Lagrange equation in Special Lagrange geometry.Therefore,the research on it is of great significance not only to partial differential equation,but also to differential geometry.This article focuses on the uniqueness of the global solution of the 2-Hessian equation in Euclidean space.We prove that under certain conditions,the 2-convex solution of 2-Hessian equation must be a quadratic polynomial if it satisfies the quadratic growth condition,that is,Liouville theorem holds.The main idea of the proof in this paper is to establish Pogorelov type estimate for solutions of 2-Hessian equation.This paper is mainly divided into five parts,and its structure is as follows:In the first part,we mainly introduce the relevant research background,research status and the main content of this paper about k-Hcssian equation.In the second part,we give some preliminary knowledge,such as comparison principle,Evans-Krylov estimate,and so on.In the third part,we prove the main Lemma 3.1.3 in this paper by establishing the Pogorelov type estimate of the solution.In the fourth part,we prove the main Theorem 4.1.1 on the basis of Lemma 3.1.3,that is,Liouville theorem which can obtain the global solution under certain conditions.Finally,in the fifth part,we propose some follow-up questions.