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.
1
Hokkaido University, Sapporo, Japan
2
Cité Descartes - Champs sur Marne, Marne la Vallée, France
Nao Hamamuki; Eleftherios Ntovoris. A rigorous setting for the reinitialization of first order level set equations. Interfaces and free boundaries, Tome 18 (2016) no. 4, pp. 579-621. doi: 10.4171/ifb/374
@article{10_4171_ifb_374,
author = {Nao Hamamuki and Eleftherios Ntovoris},
title = {A rigorous setting for the reinitialization of first order level set equations},
journal = {Interfaces and free boundaries},
pages = {579--621},
year = {2016},
volume = {18},
number = {4},
doi = {10.4171/ifb/374},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/374/}
}
TY - JOUR
AU - Nao Hamamuki
AU - Eleftherios Ntovoris
TI - A rigorous setting for the reinitialization of first order level set equations
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EP - 621
VL - 18
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/374/
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