# Discontinuous viscosity solutions of first order Hamilton-Jacobi equations

## TL;DR

This paper establishes the uniqueness and existence of discontinuous viscosity solutions for a class of first order Hamilton-Jacobi equations with initial data having finite jump discontinuities, and analyzes their evolution.

## Contribution

It proves the uniqueness and existence of discontinuous viscosity solutions for Hamilton-Jacobi equations with jump discontinuities in initial data, using comparison theorems and barrier effects.

## Key findings

- Uniqueness of solutions with finite jump discontinuities.
- Existence of viscosity solutions under given conditions.
- Characterization of the evolution of discontinuities over time.

## Abstract

We consider the simplest example of a time-dependent first order Hamilton-Jacobi equation, in one space dimension and with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. We show that if the initial function has a finite number of jump discontinuities, the corresponding discontinuous viscosity solution of the corresponding Cauchy problem on the real line is unique. Uniqueness follows from a comparison theorem for semicontinuous viscosity sub- and supersolutions, using the barrier effect of spatial discontinuities of a solution. We also prove an existence theorem, as well as a comparison theorem for viscosity solutions with different initial data. In addition, we describe several properties of the evolution of the jump discontinuities.