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Signed distance functions and viscosity solutions of discontinuous Hamilton-Jacobi Equations

Jean-François Aujol 1 Gilles Aubert
1 ARIANA - Inverse problems in earth monitoring
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - SIS - Signal, Images et Systèmes
Abstract : In this paper, we first review some properties of the signed distance function. In particular, we examine the skeleton of a curve in ^2 and get a complete description of its closure. We also give a sufficient condition for the closure of the skeleton to be of zero Lebesgue's measure. We then make a complete study of the PDE: du/dt +sign(u_0(x))(|Du|-1)=0 , which is closely related to the signed distance function. The existing literature provides no mathematical results for such PDEs. Indeed, we face the difficulty of considering a discontinuous Hamiltonian operator with respect to the space variable. We state an existence and uniqueness theorem, giving in particular an explicit Hopf-Lax formula for the solution as well as its asymptotic behaviour. This generalizes classical results for continous Hamitonian. We then get interested in a more general class of PDEs: du/dt +sign(u_0(x))H(D- u)=0, with H convex Under some technical but reasonable assumptions, we obtain the same kind of results. As far as we know, they are new for discontinuous Hamiltonians.
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Submitted on : Tuesday, May 23, 2006 - 7:44:47 PM
Last modification on : Friday, February 4, 2022 - 3:12:34 AM
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  • HAL Id : inria-00072081, version 1

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Jean-François Aujol, Gilles Aubert. Signed distance functions and viscosity solutions of discontinuous Hamilton-Jacobi Equations. RR-4507, INRIA. 2002. ⟨inria-00072081⟩

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