IJPAM: Volume 12, No. 4 (2004)

ON A CLASS OF MONGE-AMPÉRE BOUNDARY
VALUE PROBLEMS AND ESTIMATES OF BOUNDS

Lakhdar Ragoub
Department of Mathematics
College of Sciences
King Saud University
P.O. Box 2455, Riyadh 11451, SAUDI ARABIA
e-mail: radhkla@hotmail.com


Abstract.The purpose of this paper is to generalize the result of Ma concerning the Monge-Ampère equation written in its general form as $\det u_{ij}=g(\vert{\bf {\nabla\,}}u\vert^{2})h(u)$, where $u_{ij}$ denote the Hessian of $u$, and $g$, $h$ are positive functions. With a Robin and Dirichlet boundary conditions he obtained in $\RR^{2}$ an estimate for the mean curvature of $\partial\Omega$ of $\Omega$, where $\Omega$ is a bounded convex domain. Our goal is to extend the Dirichlet case in $\RR^{N}$ and to investigate the nonlinear boundary condition $\frac{\partial u}{\partial n} + \alpha(u)=0
\,\,\mbox{on}\,\,\partial\Omega$, where ${\alpha}$ is subject to some appropriate conditions. This extension is due to the generalization of the maximum principle in $\RR^{N}$.

Received: February 2, 2004

AMS Subject Classification: 28D10, 35B05, 35B50, 35J25, 35J60, 35J65

Key Words and Phrases: Monge-Ampère equations, maximum principle

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 12
Issue: 4