Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach

Pages: 1047 - 1069, Volume 21, Issue 4, August 2008      doi:10.3934/dcds.2008.21.1047

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Mariane Bourgoing - Laboratoire de Mathématiques et Physique Théorique CNRS UMR 6083, Fédération de Recherche Denis Poisson (FR 2964), Université François Rabelais, Tours. Parc de Grandmont, 37200 Tours, France (email)

Abstract: In this article, we continue the study of viscosity solutions for second-order fully nonlinear parabolic equations, having a $L^1$ dependence in time, associated with nonlinear Neumann boundary conditions, which started in a previous paper (cf [2]). First, we obtain the existence of continuous viscosity solutions by adapting Perron's method and using the comparison results obtained in [2]. Then, we apply these existence and comparison results to the study of the level-set approach for front propagations problems when the normal velocity has a $L^1$-dependence in time.

Keywords:  Fully nonlinear second-order parabolic equations, $L^1$ dependence in time, Neumann boundary conditions, existence, level-set approach, viscosity solutions.
Mathematics Subject Classification:  Primary: 35K60, 49L25 ; Secondary: 35B05.

Received: June 2007;      Revised: January 2008;      Available Online: May 2008.