# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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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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