Study On Bernstein's Theorem For Several Differential Equations

In recent years,there are all kinds of rigidity theorem of self-similar solutions of La-grangian mean curvature flow by mathematical researching,self-similar solutions can be di-vided into two situations:Self-shrinking solution and self-expanding solution.Compared with the self-shrinking solution of the rigid theorem,the rigidity theorem of self-expansion solu-tions is more difficult,so the rigidity of self-expanding solution theorem research is still no substantial progress.The main form of this paper equation model is based on Monge-Ampere type equationWe can show that an entire solution is a self-shrinking solution to Lagrangian mean curvature flow in pseudo-Euclidean space.Its corresponding self-similar expansion solutionIn this paper,we study the Bernstein theorem of several differential equations,that is,the Bernstein theorem of the second-order equation,that is,under what conditions can we write the quadratic polynomial.Further improve the self-similar solutions of rigid expanded Theorem content.Article is divided into four chapters,details as follows:The first chapter is the introduction,which mainly introduces the research background and current situation of the Lagrangian mean curvature flow problem,and the outline of the Bernstein theorem of high codimension,and gives the definition and lemma,proposition and main theorem?The second chapter we consider the self-expanding solution of the Lagrangian mean curvature flow in the pseudo-European space,and the dimension of the independent variable is 1Lemma desired results,PDE of Cauchy-Kowalevskya theorem on the nature of the resulting solution Lemma appropriate relaxation,then main results obtained in this section.Then consider that the promotion will be generalized equation,Certain restrictions on the function F,that satisfy the equation(0.3)analytical solution necessarily represented by a quadratic polynomial.the same of partial differential equations Cauchy-Kowalevskya Theorem,the resulting positive conclusions Equations relaxed require-ments,the conclusion that the main theorem of this section.In the third chapter,we consider the self-shrinking solution of the Lagrangian mean curvature flow in the pseudo-European space,and the dimension of the independent variable is 1Bernstein theorem of solution.Our argument for u of t=0 Taylor series expansion of the field,We get the lemma similar to that of Chapter 2.In the same way,we use the Cauchy-Kowalevskya theorem in the general theory of partial differential equations to carry out the appropriate condition relaxation on the lemma,Finally,the main results of this paper are obtained.The generalization of the equations in this chapter is further generalized,Certain restrictions on the function F.Analytic solution derived equation(0.4)is necessarily a quadratic polynomial of the lemma,the general theory of partial differential equations of Cauchy-Kowalevskya theorcem,the positive solution of the lemma is relaxed requirements.So get main conclusions.The fourth chapter considers the Bernstein theeorem of a class of differential equationsThe equations in this section arc discussced in a different way than the previous two chapters.Finally,the main results obtained.As a conclusion of this chapter.And the main work and innovation points are summa-rized description of the problem and hypotheses for further research.