In this paper we set up a rigorous justification for the reinitializationalgorithm. Using the theory of viscosity solutions, we propose a well-posedHamilton-Jacobi equation with a parameter, which is derived from homogenizationfor a Hamiltonian discontinuous in time which appears in the reinitialization.We prove that, as the parameter tends to infinity, the solution of the initialvalue problem converges to a signed distance function to the evolvinginterfaces. A locally uniform convergence is shown when the distance functionis continuous, whereas a weaker notion of convergence is introduced toestablish a convergence result to a possibly discontinuous distance function.In terms of the geometry of the interfaces, we give a necessary and sufficientcondition for the continuity of the distance function. We also propose anothersimpler equation whose solution has a gradient bound away from zero.
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