Effective Hamiltonian Of Hamilton-jacobi Equations

Consider the Hamilton-Jacobi equation H(x, P+Du) =λ, there exists one and only one real numberλ∈R such that the equation has a global viscosity solution. It is called the effective Hamiltonian and denoted by H(P).The physical meaning of the effective Hamiltonian is that it represents the eigenvalue of an eigenstate. It is important in the study of asymptotic solutions of Hamilton-Jacobi equations and in the homogenization theory. It is also closely related to the Weak KAM theory and Aubry-Mather theory.Some developments in the field is discussed in Chapter 1. It includes the generalization of theoretical analysis and also the developments in numerical calculations and in applications. Some preparations are also given.Chapter 2 begins with the original proof of existence and uniqueness of effec-tive Hamiltonian given by Lions,Papanicolaou,Varadhan. Then it proceeds with a new geometric proof of the result. Using the geometric proof, some proper-ties of the effective Hamiltonian can be discussed and the close relation between effective Hamiltonian and Aubry-Mather theory can be found.Some equivalent expressions of the effective Hamiltonian will be discussed in Chapter 3. These equivalent expressions comes from the new geometric proof and shows the limit property of the effective Hamiltonian. Two variational ex-pressions are also discussed, which are the theoretical foundation of numerical calculations. The second part of this chapter deals with the basic properties of the effective Hamiltonian which shows the relation between properties of the effective Hamiltonian and properties of the Hamiltonian function.Chapter 4 is devoted to the calculation of the effective Hamiltonian. Some examples are given. Since usually it is impossible to give the explicit expressions of the effective Hamiltonian, numerical calculation methods are discussed. There are basically two kinds of numerical calculation methods:PDE methods and variational methods. Both are discussed briefly.Some applications of the effective Hamiltonian are given in Chapter 5. First the physical meaning is discussed. Then it proceeds with the discussion of the role the effective Hamiltonian played in the study of asymptotic solutions of Hamilton-Jacobi equations and in the homogenization theory. At last, it points out the close relation among the effective Hamiltonian, the Weak KAM theory and Aubry-Mather theory.