Lasry-Lions, Lax-Oleinik and Generalized characteristics

TL;DR

This paper explores the relationships among Lasry-Lions regularization, Lax-Oleinik operators, and generalized characteristics within the framework of Tonelli Hamiltonian dynamics, advancing the understanding of singularity propagation in Hamilton-Jacobi equations.

Contribution

It establishes new connections between regularization techniques and generalized characteristics in the context of variational Hamiltonian dynamics.

Findings

Unified framework for regularization and characteristics

Enhanced understanding of singularity propagation

Connections to Mather and weak KAM theories

Abstract

In the recent works \cite{Cannarsa-Chen-Cheng} and \cite{Cannarsa-Cheng3}, an intrinsic approach of the propagation of singularities along the generalized characteristics was obtained, even in global case, by a procedure of sup-convolution with the kernel the fundamental solutions of the associated Hamilton-Jacobi equations. In the present paper, we exploit the relations among Lasry-Lions regularization, Lax-Oleinik operators (or inf/sup-convolution) and generalized characteristics, which are discussed in the context of the variational setting of Tonelli Hamiltonian dynamics, such as Mather theory and weak KAM theory.