Weak Solutions Of The Time Periodic Hamilton-Jacobi Equation

In this paper we use knowledge from weak KAM theory and combine with analysis,topology and calculus of variations to Study the relation which is between the Peierls barrier,Ma(?) critical action and viscosity solutions of the time periodic Hamilton-Jacobi equation.In the preface we introduce the origin of the Hamilton system,the relation between KAM theory,weak KAM theory and solutions of Hamilton-Jacobi equations,and the newest results.In chapter 1 we introduce some fundamental knowledge.First we state the calculus of variations and several basic conclusions and then give some very important definitions.Last we introduce the definition of viscosity solution and an important property.In chapter 2 we study problems about the Peierls barrier.We give the definition of the Peierls barrier,and prove its viscosity property.Then we prove that it is differential in Aubry set when the Peierls barrier is a subsolutton of Hamilton-Jacobi equations.In chapter 3 we discuss the problem about the Ma(?) critical action.After give the definition of the Ma(?) critical action and some properties,we find the viscosity property of this function and the relation between the viscosity property, the differentiability and the Aubry set.