Propagation of singularities for weak KAM solutions and barrier functions

TL;DR

This paper investigates how singularities in viscosity solutions to Hamilton-Jacobi equations propagate, providing new insights into their structure and behavior, especially for weak KAM solutions and barrier functions on the n-torus.

Contribution

It introduces a novel characterization of singularity propagation using the level set method and establishes local propagation results for weak KAM solutions in the supercritical case.

Findings

Singularities propagate along generalized characteristics.

Local propagation results for weak KAM solutions are established.

Application to barrier functions reveals detailed singularity behavior.

Abstract

This paper studies the structure of the singular set (points of nondifferentiability) of viscosity solutions to Hamilton-Jacobi equations associated with general mechanical systems on the n-torus. First, using the level set method, we characterize the propagation of singularities along generalized characteristics. Then, we obtain a local propagation result for singularities of weak KAM solutions in the supercritical case. Finally, we apply such a result to study the propagation of singularities for barrier functions.