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Transactions of the American Mathematical Society
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
Published electronically: February 28, 2000
MathSciNet review: 1707190
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Abstract:

RÉSUMÉ. 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\vert _{\partial \Omega}=\varphi, \end{split}\right.\tag{1} \end{equation*}

$f\in C^{s_*}(\overline\Omega), \varphi\in C^{s_*+2,\alpha}(\Omega)$, $f$est positive et s'annule sur $\Sigma$ un ensemble fini de points de $\Omega$. On démontre alors sous certaines hypothèses sur $\varphi$ et si $\vert\det \varphi_{ij}-f\vert _{C^{s_*}}$ 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\vert _{\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 $\vert\det \varphi_{ij}-f\vert _{C^{s_*}}$ 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

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02581-2
PII: S 0002-9947(00)02581-2
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