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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Monge-Ampère equations
relative to a Riemannian metric


Authors: A. Atallah and C. Zuily
Journal: Trans. Amer. Math. Soc. 349 (1997), 3989-4006
MSC (1991): Primary 35J65, 35Q99
MathSciNet review: 1433109
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Abstract: We prove that in a bounded strictly convex open set $\Omega $ in $\mathbb {R}^n$, the problem

\begin{displaymath}\begin {cases} \det \nabla ^2u=f(x),\\ u|_{\partial \Omega }=\varphi , \end {cases}\end{displaymath}

where $f>0,f\in C^\infty (\overline \Omega ), \varphi \in C^\infty (\partial \Omega )$, has a unique strictly convex solution $u\in C^\infty (\overline \Omega )$. This result extends to an arbitrary metric a theorem which has been proved by Caffarelli-Nirenberg-Spruck in the case of the Euclidean metric.


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Additional Information

A. Atallah
Affiliation: Département de Mathématiques, Université de Paris-Sud, Bât. 425, 91405 Orsay Cedex, France

C. Zuily
Affiliation: Département de Mathématiques, Université de Paris-Sud, Bât. 425, 91405 Orsay Cedex, France
Email: claude.zuily@math.u-psud.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01833-3
PII: S 0002-9947(97)01833-3
Received by editor(s): March 6, 1995
Received by editor(s) in revised form: November 28, 1995
Article copyright: © Copyright 1997 American Mathematical Society