# A rigorous setting for the reinitialization of first order level set equations

### Nao Hamamuki

Hokkaido University, Sapporo, Japan### Eleftherios Ntovoris

Cité Descartes - Champs sur Marne, Marne la Vallée, France

## Abstract

In this paper we set up a rigorous justification for the reinitialization algorithm. Using the theory of viscosity solutions, we propose a well-posed Hamilton-Jacobi equation with a parameter, which is derived from homogenization for a Hamiltonian discontinuous in time which appears in the reinitialization. We prove that, as the parameter tends to infinity, the solution of the initial value problem converges to a signed distance function to the evolving interfaces. A locally uniform convergence is shown when the distance function is continuous, whereas a weaker notion of convergence is introduced to establish a convergence result to a possibly discontinuous distance function. In terms of the geometry of the interfaces, we give a necessary and sufficient condition for the continuity of the distance function.We also propose another simpler equation whose solution has a gradient bound away from zero.

## Cite this article

Nao Hamamuki, Eleftherios Ntovoris, A rigorous setting for the reinitialization of first order level set equations. Interfaces Free Bound. 18 (2016), no. 4, pp. 579–621

DOI 10.4171/IFB/374