The rate of convergence of new Lax-Oleinik type operators for time-periodic positive definite Lagrangian systems

TL;DR

This paper investigates the convergence rate of new Lax-Oleinik operators for time-periodic positive definite Lagrangian systems, providing upper bounds and examples demonstrating the limits of convergence speed.

Contribution

It offers an upper bound estimate for the convergence rate of new Lax-Oleinik operators and constructs examples showing the optimality of this rate.

Findings

Upper bound estimate for convergence rate

Example showing convergence cannot be faster than O(1/t)

Convergence rate depends on the hyperbolic nature of the Aubry set

Abstract

Assume that the Aubry set of the time-periodic positive definite Lagrangian $L$ consists of one hyperbolic 1-periodic orbit. We provide an upper bound estimate of the rate of convergence of the family of new Lax-Oleinik type operators associated with $L$ introduced by the authors in \cite{W-Y}. In addition, we construct an example where the Aubry set of a time-independent positive definite Lagrangian system consists of one hyperbolic periodic orbit and the rate of convergence of the Lax-Oleinik semigroup cannot be better than $O(\frac{1}{t})$.