Local singular characteristics on $\mathbb{R}^2$

TL;DR

This paper proves that for a Tonelli Hamiltonian on , different notions of singular characteristics coincide up to reparameterization, establishing a uniqueness result for singular characteristics in Hamilton-Jacobi equations.

Contribution

It demonstrates the equivalence of various notions of singular characteristics for Tonelli Hamiltonians on , leading to a general uniqueness criterion.

Findings

Different notions of singular characteristics coincide up to bi-Lipschitz reparameterization.

A new uniqueness result for singular characteristics is established.

The results apply to a broad class of Hamilton-Jacobi equations in two dimensions.

Abstract

The singular set of a viscosity solution to a Hamilton-Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for singular characteristics, not restricted to mechanical systems or problems in one space dimension, is missing at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on $\mathbb{R}^2$, two different notions of singular characteristics coincide up to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a uniqueness result for the class of singular characteristics that was introduced by Khanin and Sobolevski in the paper [On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. Arch. Ration. Mech. Anal., 219(2):861-885, 2016].