【作者】 李新祥；

【导师】 李大潜； 严军；

【作者基本信息】 复旦大学 ， 应用数学， 2009， 博士

【摘要】 Mather理论是近年来Hamilton系统研究中的一个突破,它在研究著名的Arnold扩散问题中显示了巨大的威力,其理论自身也是十分漂亮和有趣的。用Mather理论来研究Arnold扩散,其基本思路是寻找对应于不同上同调类的Mane集之间的连接轨道。其连接轨道的构造依赖于Mane集的拓扑结构。障碍函数B_c(m),(?)m∈M关于参数c∈H~1(m,R)的连续性在Mane集的拓扑结构的研究中起着关键的作用。另一方面,Hamilton-Jacobi方程的粘性解理论为Mather理论的研究提供了丰富的方法来源,两者相结合必将产生更加强有力的工具。本文主要包含以下几个方面的结果:一,Barrier函数关于参数的连续性和Mather极小测度的遍历性存在着深刻的联系。本文给出了Barrier函数B_c(m),(?)m∈M关于参数c连续的一个充分条件一Maher集唯一遍历。二,本文构造出了一个反例,通过对该反例的具体分析,对Barrier函数关于参数的不连续性做了深入探讨。我们的研究表明:当Mather极小测度有多于1个遍历分支时,Barrier函数关于平均作用量变量通常是不连续的。三,将我们对Barrier函数的研究应用于Hamilton-Jacobi方程粘性解的研究,我们得到了Hamilton-Jacobi方程粘性解在相差一个常数的意义下关于平均作用量变量连续性的充分条件。更多还原

【Abstract】 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.更多还原