Front tracking and iterated minmax for Hamilton-Jacobi equation in one space variable

TL;DR

This paper provides an alternative proof for the viscosity solution of one-dimensional Hamilton-Jacobi equations using front tracking and iterated minmax, enhancing understanding of solution singularities.

Contribution

It introduces a new proof method combining front tracking with iterated minmax for Hamilton-Jacobi equations in one dimension, clarifying solution singularities.

Findings

Alternative proof of viscosity solutions in 1D

Improved understanding of solution singularities

Application of front tracking to Hamilton-Jacobi equations

Abstract

The viscosity solution of the Hamilton-Jacobi equation was constructed by an "iterated minimax" procedure. Using Dafermos' front tracking method, we give another proof of this construction in the case of Hamilton-Jacobi equations in one space dimension. This allows us to get a better understanding in this case of the singularities of the viscosity solution.