Uniqueness Results for Nonlocal Hamilton-Jacobi Equations

TL;DR

This paper introduces a novel approach to establish the uniqueness of solutions for nonlocal Hamilton-Jacobi equations, particularly for expanding fronts, extending previous results and applying to Fitzhugh-Nagumo systems.

Contribution

It provides the first uniqueness proof for a Fitzhugh-Nagumo system and extends existing dislocation dynamics results through new perimeter and interior cone estimates.

Findings

Proved uniqueness of solutions for expanding fronts in nonlocal Eikonal equations.

Extended dislocation dynamics results with a simplified approach.

First uniqueness result for Fitzhugh-Nagumo systems.

Abstract

We are interested in nonlocal Eikonal Equations describing the evolution of interfaces moving with a nonlocal, non monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh-Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts.