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

TL;DR

This paper introduces a new family of Lax-Oleinik type operators with parameters for positive definite Lagrangian systems, demonstrating improved convergence to weak KAM solutions in both time-periodic and time-independent cases.

Contribution

The paper develops a novel class of Lax-Oleinik operators that converge to weak KAM solutions, surpassing traditional semigroup convergence in certain cases.

Findings

Converges to backward weak KAM solutions in time-periodic systems.

Faster convergence than classical Lax-Oleinik semigroup in time-independent systems.

Provides new tools for analyzing 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.