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

Mariane Bourgoing

doi:10.3934/dcds.2008.21.1047

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2008, Volume 21, Issue 4: 1047-1069. Doi: 10.3934/dcds.2008.21.1047

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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

Mariane Bourgoing^1,

1.
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

Received: June 2007

Revised: January 2008

Published: October 2008

Abstract / Introduction Related Papers Cited by

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.

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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

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