Generalized characteristics and Lax-Oleinik operators: global theory

TL;DR

This paper develops an intrinsic proof for the existence of generalized characteristics in autonomous Tonelli systems, using sup-convolutions, and demonstrates global propagation of singularities along these characteristics.

Contribution

It introduces a new intrinsic proof method for generalized characteristics and establishes global singularity propagation results in Tonelli systems.

Findings

Existence of generalized characteristics proved using sup-convolutions

Global propagation of singularities along generalized characteristics established

Convexity estimates for the fundamental solution derived

Abstract

For autonomous Tonelli systems on $\R^n$, we develop an intrinsic proof of the existence of generalized characteristics using sup-convolutions. This approach, together with convexity estimates for the fundamental solution, leads to new results such as the global propagation of singularities along generalized characteristics.