# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## Problème de Dirichlet pour une équation de Monge-Ampère réelle elliptique dégénérée en dimension $n$HTML articles powered by AMS MathViewer

by Amel Atallah
Trans. Amer. Math. Soc. 352 (2000), 2701-2721 Request permission

## 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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• 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
• Received by editor(s): April 17, 1995
• Published electronically: February 28, 2000
• 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
• MathSciNet review: 1707190