Some Problems In Mather Theory And Weak KAM Theory

Mather theory as a great breakthrough in Hamiltonian systems has shown significant influence in study of Arnold diffusion. It is also an interesting and elegant theory. The basic approach for studying Arnold diffusion is to find out the orbits which connect different Mane sets associating to different cohomology classes. The existence of connecting orbits depend upon the topological structure of the Mane sets. The continuity of the barrier function B_c(m) ((?) m∈M) with respect to c plays the key role in studying of the topological properties of Mane sets. On the other hand, the viscosity solution theory on the Hamilton-Jacobi equation provides us plenty of sources of methods to study the Mather theory. Combining these two methods will surely give us more powerful tools.The main results of this thesis are as follows:1, There are close relationships between the barrier function's continuity with respect to the parameter and the ergodic property of Mather's minimal measures. In this thesis we get a sufficient condition for barrier function's continuity with respect to the parameter------Mather set is uniquely ergodic.2, We construct a counterexample and discuss deeply inside the discontinuity of barrier function with respect to the parameter through analyzing the counterexample carefully. The study indicates that the barrier function is discontinuity with respect to the average action variable when Mather's minimal measure has more than one ergodic components in general cases.3, Applying the results of barrier function to the viscosity solution of Hamilton- Jacobi equation, we get a sufficient condition of the viscosity solution's continuity with respect to the average action variable in the sense of a constant difference.