For a convex,coercive continuous Hamiltonian on a closed Riemannian manifold M,we construct a unique forward weak KAM solution of H(x,d_(x)u)=c(H)by a vanishing discount approach,where c(H)is the Ma?écritical val...For a convex,coercive continuous Hamiltonian on a closed Riemannian manifold M,we construct a unique forward weak KAM solution of H(x,d_(x)u)=c(H)by a vanishing discount approach,where c(H)is the Ma?écritical value.We also discuss the dynamical significance of such a special solution.展开更多
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
We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general conv...We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general convergence result for viscosity solutions and adherence of the graph as t →∞.展开更多
We give a rigorous proof of the equivalence of Manes supercritical potential and the minimal action with respect to an associated Jacobi-Finsler metric. As a consequence, we give an explicit representation of the weak...We give a rigorous proof of the equivalence of Manes supercritical potential and the minimal action with respect to an associated Jacobi-Finsler metric. As a consequence, we give an explicit representation of the weak KAM solutions of one-dimensional mechanical systems without the quadratic assumption on the kinetic energy term of the Hamiltonians, and a criterion of the integrability result for such a system of arbitrary degree of freedom by the regularity assumption on Mather＇s a- function is discussed.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.11901560)supported by the National Natural Science Foundation of China(Grant No.11971060)。
文摘For a convex,coercive continuous Hamiltonian on a closed Riemannian manifold M,we construct a unique forward weak KAM solution of H(x,d_(x)u)=c(H)by a vanishing discount approach,where c(H)is the Ma?écritical value.We also discuss the dynamical significance of such a special solution.
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 National Natural Science Foundation of China(Grant Nos.1132510311301106 and 11201288)+1 种基金China Postdoctoral Science Foundation(Grant No.2014M550210)Guangxi Experiment Center of Information Science(Grant No.YB1410)
文摘We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general convergence result for viscosity solutions and adherence of the graph as t →∞.
基金Supported by the National Basic Research Program of China (Grant No. 2007CB814800)Natural Scientific Foundation of China (Grant No. 10971093)
文摘We give a rigorous proof of the equivalence of Manes supercritical potential and the minimal action with respect to an associated Jacobi-Finsler metric. As a consequence, we give an explicit representation of the weak KAM solutions of one-dimensional mechanical systems without the quadratic assumption on the kinetic energy term of the Hamiltonians, and a criterion of the integrability result for such a system of arbitrary degree of freedom by the regularity assumption on Mather＇s a- function is discussed.
