Portugaliae Mathematica   EMIS ELibM Electronic Journals PORTUGALIAE
Vol. 62, No. 1, pp. 1-11 (2005)

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On a class of Monge--Ampére problems with non-homogeneous Dirichlet boundary condition

L. Ragoub and F. Tchier

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

Abstract: We assume in the plane that $\Omega$ is a strictly convex domain, with its boundary $\partial\Omega$ sufficiently regular. We consider the Monge--Ampére equations in its general form $\det u_{ij}=g(|{\bf{\nabla}}u|^{2})h(u)$, where $u_{ij}$ denotes the Hessian of $u$, and $g$, $h$ are some given functions. This equation is subject to the non-homogeneous Dirichlet boundary condition $u=f$. A sharp necessary condition of solvability for this equation is given using the maximum principle in $\R^{2}$. We note that this maximum principle is extended to the $N$-dimensional case and two different applications have been given to illustrate this principle.

Keywords: Monge--Ampére equations; maximum principle.

Classification (MSC2000): 28D10, 35B05, 35B50, 35J25, 35J60, 35J65.

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Electronic version published on: 7 Mar 2008.

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