Balanced split sets and Hamilton-Jacobi equations

TL;DR

This paper classifies balanced split sets related to Hamilton-Jacobi equations, clarifying their structure and connection to classical solutions like characteristics and shocks.

Contribution

It identifies and classifies all balanced split sets, revealing when the singular locus is the unique balanced split locus and when other cases occur.

Findings

Classification of all balanced split sets

Identification of cases with unique singular locus

Clarification of viscosity solutions versus classical methods

Abstract

We study the singular locus of solutions to Hamilton-Jacobi equations with a Hamiltonian independent of $u$. In a previous paper, we proved that the singular locus is what we call a balanced split locus. In this paper, we find and classify all balanced split sets, identifying the cases where the only balanced split locus is the singular locus, and the cases where this doesn't hold. This clarifies the relationship between viscosity solutions and the more classical approach of characteristics and shocks.