Signed distance functions and viscosity solutions of discontinuous Hamilton-Jacobi Equations - Laboratoire Jean-Alexandre Dieudonné Accéder directement au contenu
Rapport Année : 2002

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

Jean-François Aujol
Gilles Aubert
  • Fonction : Auteur

Résumé

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.
Fichier principal
Vignette du fichier
RR-4507.pdf (444.44 Ko) Télécharger le fichier
Loading...

Dates et versions

inria-00072081 , version 1 (23-05-2006)

Identifiants

  • HAL Id : inria-00072081 , version 1

Citer

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

Partager

Gmail Facebook X LinkedIn More