KAM Tori With Prescribed Frequency For Reversible Systems And Weak KAM Theory

We are devoted to studying the persistence of invariant tori for reversible systems and weak integrability for Hamiltonian systems by means of KAM theory and weak KAM theory. Firstly, we use KAM theory to study the persistence of invariant tori for reversible systems; secondly, we use weak KAM theory to obtain weak KAM solutions of Hamilton-Jacobi equations with homogeneous Neumann boundary conditions. The main contents are arranged as follows:In Chapter 1, we first introduce basic knowledge of both KAM theory and weak KAM theory; and then, we recall the relevant development and remarkable breakthrough obtained through KAM theory and weak KAM theory; finally we summarize the main work in brief.In Chapter 2, we consider the persistence of hyperbolic lower dimensional invariant tori with prescribed frequency for reversible system. By introducing an artificial external parameter and KAM iteration, we prove that if the frequency mapping ω has nonzero Brouwer degree at Diophantine frequency wo, then the perturbed reversible system still has a hyperbolic lower dimensional invariant torus with ω0 as its frequency.Chapter 3 concerns the persistence of elliptic lower dimensional invariant tori for reversible systems. Based on a new technique of KAM iteration to separate non-degeneracy condition from KAM iteration, we obtain a formal KAM theorem for reversible systems, without assuming any non-degeneracy condition or small divisor condition. Moreover, we apply the formal KAM theorem to obtain some interesting results about the persistence of KAM tori with prescribed frequency for reversible systems.In Chapter 4, we study the initial-value problem with homogeneous Neumann boundary conditions for Hamilton-Jacobi equation u(t,x)+H(x,Du(t,x))＝0, on non-compact manifold Ω. Using weak KAM theory, we give a proof of the existence of weak KAM solutions, or viscosity solutions, for convex Hamilton-Jacobi equation with homogeneous Neumann boundary conditions.