A representation formula for viscosity solutions of nonlocal Hamilton--Jacobi equations and applications

TL;DR

This paper develops a control-based representation formula for viscosity solutions of nonlocal Hamilton--Jacobi equations, enabling analysis of geometric motion, fattening, and long-term behavior of evolving surfaces.

Contribution

It introduces a novel representation formula for nonlocal Hamilton--Jacobi equations using control theory, advancing understanding of geometric motions with nonlocal effects.

Findings

Established a control-based representation formula for solutions.

Analyzed fattening phenomena in geometric motions.

Studied large-time asymptotic behavior of solutions.

Abstract

This paper is concerned with geometric motion of a closed surface whose velocity depends on a nonlocal quantity of the enclosed region. Using the level set formulation, we study a class of nonlocal Hamilton--Jacobi equations and establish a control-based representation formula for solutions. We also apply the formula to discuss the fattening phenomenon and large-time asymptotics of the solutions.