A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems

Abstract

In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,{\mathbb{R}}^1)$ as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,{\mathbb{R}}^1)$ as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case.

We give a new definition of Lax-Oleinik type operator; add some references

References in corpus (4)

• Lecture notes on Mather's theory for Lagrangian systems
• Weak solutions of the Hamilton-Jacobi equation for time periodic lagrangians
• The link between the shape of the Aubry-Mather sets and their Lyapunov exponents
• Pseudographs and Lax-Oleinik semi-group: a geometric and dynamical interpretation

Cited by in corpus (6)

• The rate of convergence of new Lax-Oleinik type operators for time-periodic positive definite Lagrangian systems
• Weak KAM theory for general Hamilton-Jacobi equations I: the solution semigroup under proper conditions
• Gevrey genericity of Arnold diffusion in a priori unstable Hamiltonian systems
• Weak KAM Theorem for a Class of Infinite-Dimensional Lagrangian systems
• Exponential convergence of 1-graph of the solution semigroup of contact Hamilton-Jacobi equations
• Global Behaviors of weak KAM Solutions for exact symplectic Twist Maps