Long time behaviour for the reinitialization of the distance function

TL;DR

This paper investigates the long-term behavior of solutions to a class of Hamilton-Jacobi equations, demonstrating convergence to the signed distance function from the initial zero level set as time approaches infinity.

Contribution

It provides a rigorous proof of uniform convergence for the viscosity solutions of reinitialization equations, including the distance function case, over long time periods.

Findings

Viscosity solutions converge uniformly as time tends to infinity.

The limit is the signed distance function from the initial zero level set.

The results apply to a class of non-coercive Hamilton-Jacobi equations.

Abstract

In this article we study the long-time behaviour of a class of non-coercive Hamilton-Jacobi equations, that includes, as a notable example, the so called reinitialization of the distance function. In particular we prove that its viscosity solution converges uniformly as $t\rightarrow +\infty$ to the signed distance function from the zero level set of the initial data.