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

Pages: 763 - 800, Volume 21, Issue 3, July 2008      doi:10.3934/dcds.2008.21.763

       Abstract        Full Text (454.4K)       Related Articles

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 are interested in viscosity solutions for second-order fully nonlinear parabolic equations having a $L^1$ dependence in time and associated with nonlinear Neumann boundary conditions. The main contributions of our study are, not only to treat the case of nonlinear Neumann boundary conditions, but also to revisit the theory of viscosity solutions for such equations and to extend it in order to take in account singular geometrical equations. In particular, we provide comparison results, both for the cases of standard and geometrical equations, which extend the known results for Neumann boundary conditions even in the framework of continuous equations.

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

Received: June 2007;      Revised: November 2007;      Available Online: April 2008.