PORTUGALIAE MATHEMATICA Vol. 62, No. 1, pp. 111 (2005) 

On a class of MongeAmpére problems with nonhomogeneous Dirichlet boundary conditionL. Ragoub and F. TchierDepartment of Mathematics, College of Sciences,King Saud University, P.O.\ Box 2455, Riyadh 11451  SAUDI ARABIA Email: 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 MongeAmpére 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 nonhomogeneous 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: MongeAmpére equations; maximum principle. Classification (MSC2000): 28D10, 35B05, 35B50, 35J25, 35J60, 35J65. Full text of the article:
Electronic version published on: 7 Mar 2008.
© 2005 Sociedade Portuguesa de Matemática
