J. Mather and A. Fathi defined Mané set and Aubry set, which are the important invariant sets in positive definite Lagrangian system, in different ways. They use variational principle and Weak KAM theory respecti...J. Mather and A. Fathi defined Mané set and Aubry set, which are the important invariant sets in positive definite Lagrangian system, in different ways. They use variational principle and Weak KAM theory respectively. In this paper we provide a proof of the equivalence between the two kinds of definitions, and generalize A. Fathi＇s definition. In the end of the paper, we calculate the Mane set and Aubry set for a single pendulum system.展开更多
Abstract Assume that M is a closed,connected and smooth Riemannian manifold.The authors consider the evolutionary Hamilton-Jacobi equation{∂_(t)u(x,t)+H(x,u(x,t),∂_(x)u(x,t))=0,(x,t∈M×(0,∞),u(x,0)=■(x),where■...Abstract Assume that M is a closed,connected and smooth Riemannian manifold.The authors consider the evolutionary Hamilton-Jacobi equation{∂_(t)u(x,t)+H(x,u(x,t),∂_(x)u(x,t))=0,(x,t∈M×(0,∞),u(x,0)=■(x),where■∈C(M)and the stationary oneH(x,u(x),∂_(x)u(x))=0,where H(x,u,p)is continuous,convex and coercive in p,uniformly Lipschitz in u.By introducing a solution semigroup,the authors provide a representation formula of the viscosity solution of the evolutionary equation.As its applications,they obtain a necessary and sufficient condition for the existence of the viscosity solutions of the stationary equations.Moreover,they prove a new comparison theorem depending on the neighborhood of the projected Aubry set essentially,which is diferent from the one for the Hamilton-Jacobi equation independent of u.展开更多
We consider the construction of the plateau of theα-function in a hyperbolic and positive definite Lagrangian system,and link the boundries of theα-function's plateau with the distribution of c-minimal homoclini...We consider the construction of the plateau of theα-function in a hyperbolic and positive definite Lagrangian system,and link the boundries of theα-function's plateau with the distribution of c-minimal homoclinic orbits to Aubry sets.展开更多
In this paper,we prove that the nearly integrable system of the form H(x,y)=h(y)+εP(x,y),x∈T^(n),y∈T^(n),n≥3 admits orbits that pass through any finitely many prescribed small balls on the same energy level H^(-1)...In this paper,we prove that the nearly integrable system of the form H(x,y)=h(y)+εP(x,y),x∈T^(n),y∈T^(n),n≥3 admits orbits that pass through any finitely many prescribed small balls on the same energy level H^(-1)(E)provided that E>min h,if h is convex,andεP is typical.This settles the Arnold diffusion conjecture for convex systems in the smooth category.We also prove the counterpart of Arnold diffusion for the Riemannian metric perturbation of the flat torus.展开更多
文摘J. Mather and A. Fathi defined Mané set and Aubry set, which are the important invariant sets in positive definite Lagrangian system, in different ways. They use variational principle and Weak KAM theory respectively. In this paper we provide a proof of the equivalence between the two kinds of definitions, and generalize A. Fathi＇s definition. In the end of the paper, we calculate the Mane set and Aubry set for a single pendulum system.
基金supported by the National Natural Science Foundation of China(Nos.12122109,11790272)。
文摘Abstract Assume that M is a closed,connected and smooth Riemannian manifold.The authors consider the evolutionary Hamilton-Jacobi equation{∂_(t)u(x,t)+H(x,u(x,t),∂_(x)u(x,t))=0,(x,t∈M×(0,∞),u(x,0)=■(x),where■∈C(M)and the stationary oneH(x,u(x),∂_(x)u(x))=0,where H(x,u,p)is continuous,convex and coercive in p,uniformly Lipschitz in u.By introducing a solution semigroup,the authors provide a representation formula of the viscosity solution of the evolutionary equation.As its applications,they obtain a necessary and sufficient condition for the existence of the viscosity solutions of the stationary equations.Moreover,they prove a new comparison theorem depending on the neighborhood of the projected Aubry set essentially,which is diferent from the one for the Hamilton-Jacobi equation independent of u.
文摘We consider the construction of the plateau of theα-function in a hyperbolic and positive definite Lagrangian system,and link the boundries of theα-function's plateau with the distribution of c-minimal homoclinic orbits to Aubry sets.
基金supported by National Natural Science Foundation of China (Grant Nos.11790272 and 11631006)supported by National Natural Science Foundation of China (Grant Nos.11790273 and 12271285)。
文摘In this paper,we prove that the nearly integrable system of the form H(x,y)=h(y)+εP(x,y),x∈T^(n),y∈T^(n),n≥3 admits orbits that pass through any finitely many prescribed small balls on the same energy level H^(-1)(E)provided that E>min h,if h is convex,andεP is typical.This settles the Arnold diffusion conjecture for convex systems in the smooth category.We also prove the counterpart of Arnold diffusion for the Riemannian metric perturbation of the flat torus.
基金Supported by National Natural Science Foundation of China(Grant Nos.11271181 and 10801071)the Research Fellowship for Postdoctoral Researchers from the Alexander von Humboldt Foundation+1 种基金the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAAD)the Fundamental Research Funds for Central Universities
文摘We show that the Aubry sets, the Mane sets and the barrier functions are the same for two commuting time-periodic Tonelli Hamiltonians.
