Minimal Orbits And H(?)lder Regularity Of Weak KAM Solutions In Positive Definite Hamilton Systems

For nearly integrable Hamiltonian system with multi degrees of freedom, KAM the-ory verifies the dynamical stability of most orbits, i.e. along these orbits the variation of action variables remains small forever. On the other hand, Nekhoroshev discovered so-called effective stability, in exponential long time scale, along each orbit the varia-tion of action remains small. Since Arnold discovered ([Arn2]) dynamical unstability in a time-periodic nearly integrable system with two degrees of freedom, people have being pursued the object:the typical case in multidimensional problem is topologi-cal instability:there exist a lot of orbits along which the variation of action variable reaches the order 1. This is the famous conjecture of Arnold diffusion.There are two ways in the study of Arnold diffusion, geometric method ([DLS]) and variational method ([Ma4], [CY1], [CY2]). Based on Mather theory, the variational method has been successfully applied to study this problem in a priori unstable system with multi degrees of freedom ([CY1]):it is a generic phenomenon for this kind of systems.In applying Mather theory, the theme of Arnold diffusion is to construct orbits connected two Aubry sets with different cohomology classes. Therefore, the structure of Aubry sets needs to be well understood. In this thesis, we study the existence of the homoclinic orbits of Aubry set. We show that there are infinitely many M-semi-static homoclinic orbits to A(0) provided that the relative homology group of A(0) is nontrivial and there exists a cohomology c at the boundary of the flat such that hc(g)>0 holds for each 0≠g∈H1(M×T,A(0),Z). A closely related topic to Mather theory is weak-KAM theory. It corresponds to the fixed points of so called the Lax-Oleinik operators. Mane set, Aubry set and Mather's barrier function can be expressed in terms of conjugate pairs of weak-KAM solutions. In this paper, we define generalized weak-KAM solutions, with which we define a new barrier function ([CY2]), the regularity of this new barrier functions plays an important role in proving the generic of existence of Arnold-type diffusion orbits.For a priori unstable time-periodic nearly integrable Hamiltonian systems with two degrees of freedom, there is a continuous path in H1(T2,R) such that for each cohomology class c in this path, the c-minimal measure is supported on a normally hyperbolic cylinder([CY1],[CY2]). In this paper, with an order-preserving property obtained for backward and forward minimal configurations for monotone twist map and normally hyperbolic structure of the invariant cylinder, we show that the generalized weak-KAM solutions for these classes can be parameterized by the area bounded by the graph of these solutions and obtain the 1/4-Holder regularity of these solutions in the parameter. So, we can construct the Arnold-type diffusion orbits in a priori unstable generic Hamiltonian systems by using the ideas from [CY1]. The shape of Arnold-type diffusion orbits constructed here is different from the shape of the diffusion orbits constructed ([CY1], [CY2]).