2015/07/01 by Nao Hamamuki, Hamamuki, Nao, Eleftherios Ntovoris +1 · 1 citation
Mathematics · #35A35 #35D40 #35F25 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35A35 #msc:35D40 #msc:35F25
paper · pdf · doi:10.48550/arxiv.1507.00217
arxiv created 2015/07/01 · arxiv updated 2015/07/02
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.