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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Problème de Dirichlet pour une équation de Monge-Ampère réelle elliptique dégénérée en dimension $n$


Author: Amel Atallah
Journal: Trans. Amer. Math. Soc. 352 (2000), 2701-2721
MSC (1991): Primary 35J25, 35J70, 35Q99
DOI: https://doi.org/10.1090/S0002-9947-00-02581-2
Published electronically: February 28, 2000
MathSciNet review: 1707190
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Abstract: On considère dans un ouvert borné $\Omega$ de $\mathbb {R}^n$, à bord régulier, le problème de Dirichlet \begin{equation*} \left \{ \begin {split} & \det u_{ij}=f(x)\text { dans }\Omega , & u|_{\partial \Omega }=\varphi , \end{split}\right .\tag {1} \end{equation*} où $f\in C^{s_*}(\overline \Omega ), \varphi \in C^{s_*+2,\alpha }(\Omega )$, $f$ est positive et s’annule sur $\Sigma$ un ensemble fini de points de $\Omega$. On démontre alors sous certaines hypothèses sur $\varphi$ et si $|\det \varphi _{ij}-f|_{C^{s_*}}$ est assez petit, que le problème (1) possède une solution convexe unique $u\in C^{[s_*-3-n/2]}(\overline \Omega )$. Abstract. We consider in a bounded open set $\Omega$ of $\mathbb {R}^n$, with regular boundary, the Dirichlet problem \begin{equation*} \left \{ \begin {split} & \det u_{ij}=f(x)\text { in }\Omega , & u|_{\partial \Omega }=\varphi , \end{split}\right .\tag {1} \end{equation*} where $f\in C^{s_*}(\overline \Omega ), \varphi \in C^{s_*+2,\alpha }(\Omega )$, $f$ is positive and vanishes on $\Sigma$, a finite set of points in $\Omega$. We prove, under some hypothesis on $\varphi$ and if $|\det \varphi _{ij}-f|_{C^{s_*}}$ is sufficiently small, that the problem (1) has a unique convex solution $u\in C^{[s_*-3-n/2]}(\overline \Omega )$.


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Additional Information

Amel Atallah
Affiliation: Université de Paris-Sud, Département de Mathématiques, Bât. 425, 91405 Orsay, Cedex, France
Address at time of publication: Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire le Belvedere, 1060 Tunis, Tunisie
Email: sami.baraket@fst.rnu.tn

Keywords: Equation de Monge-Ampere, probleme de Dirichlet, equation elliptique degeneree
Received by editor(s): April 17, 1995
Published electronically: February 28, 2000
Article copyright: © Copyright 2000 American Mathematical Society