Local strict singular characteristics II: existence for stationary equation on $\mathbb{R}^2$

TL;DR

This paper establishes the existence and properties of strict singular characteristics for Hamilton-Jacobi equations in two dimensions, providing rigorous proofs and insights into their differentiability and selection criteria.

Contribution

It offers a rigorous proof of the existence of strict singular characteristics in 2D Hamilton-Jacobi equations and analyzes their differentiability and selection properties.

Findings

Existence of strict singular characteristics is proven.

Strict singular characteristics are right-differentiable.

A selection rule for the momentum p(t) is established.

Abstract

The notion of strict singular characteristics is important in the wellposedness issue of singular dynamics on the cut locus of the viscosity solutions. We provide an intuitive and rigorous proof of the existence of the strict singular characteristics of Hamilton-Jacobi equation $H(x,Du(x),u(x))=0$ in two dimensional case. We also proved if $\mathbf{x}$ is a strict singular characteristic, then we really have the right-differentiability of $\mathbf{x}$ and the right-continuity of $\dot{\mathbf{x}}^+(t)$ for every $t$. Such a strict singular characteristic must give a selection $p(t)\in D^+u(\mathbf{x}(t))$ such that $p(t)=\arg\min_{p\in D^+u(\mathbf{x}(t))}H(\mathbf{x}(t),p,u(\mathbf{x}(t)))$.