Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Existence of local sufficiently smooth solutions to the complex Monge-Ampère equation


Author: Saoussen Kallel-Jallouli
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 3227-3242
MSC (2000): Primary 32W20
DOI: https://doi.org/10.1090/S0002-9947-03-03399-3
Published electronically: December 12, 2003
MathSciNet review: 2052948
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the $C^{\infty }$ local solvability of the $n$-dimensional complex Monge-Ampère equation $\det \left( u_{i\overline{j}}\right) =f\left( z,u,\nabla u\right) $, $f\geq 0$, in a neighborhood of any point $z_{0}$where $f\left( z_{0}\right) =0$.


References [Enhancements On Off] (What's this?)

  • 1. S. Alinhac, P. Gérard: Opérateurs pseudo-différentiels et théorème de Nash-Moser, Inter Editions et Editions du CNRS,1991.
  • 2. E. Bedford and B.A. Taylor: The Dirichlet problem for a complex Monge-Ampère equation, Inventiones Mathematicae, 37, 1-44 1979. MR 56:3351
  • 3. L. Bers, F. John, M. Schechter: Partial differential equations, Lectures in Applied Mathematics, vol 3, Amer. Math. Soc., 1957. MR 29:346
  • 4. D. Gilbarg, N.S. Trudinger: Elliptic partial differential equations of second order, second edition, Springer Verlag Berlin, Heidelberg, New York 1983. MR 86c:35035
  • 5. J. Hong and C.Zuily: Existence of $C^{\infty }$ local solutions for the Monge-Ampère equation, Invent. Math. 89, 645-661, 1987. MR 88j:35056
  • 6. L. Hörmander: On the Nash-Moser implicit function theorem, Annales Academia Scientiarum.Fennicae, Serie A I, Math. 10, 67-97, 1984.
  • 7. S. Kallel-Jallouli: Existence of $C^{\infty }$ local solutions of the complex Monge-Ampère equation, Proc. Amer. Math. Soc., 131, 1103-1108, 2003.
  • 8. C.S. Lin: The local isometric embedding in $\mathbb{R} ^{3}$ of 2-dimensional Riemannian manifolds with nonnegative curvature, J. Differential Geometry, 21, 213-230, 1985. MR 87m:53073
  • 9. J. Moser: A new technique for the construction of solutions of nonlinear partial differential equations, Proc. Natl. Acad. Sc. USA 47, 1842-1831, 1961. MR 24:A2695
  • 10. G. Nakamura, Y. Maeda: Local smooth isometric embedding of low dimensional Riemannian manifolds into euclidean spaces, Transactions of the A.M.S., 313, 1-51, 1989. MR 90f:58171
  • 11. S.L. Sobolev: On a theorem of functional analysis, Mat. Sb. (N.S.) 4, 471-497, 1938. (Russian); English transl., Amer. Math. Soc. Transl (2) 34, 39-68, 1963.
  • 12. C.J. Xu: Régularité des solutions pour les équations aux dérivées partielles quasi linéaires non elliptiques du second ordre, C. R. Acad. Sci. Paris Sér. 1 Math. 300, 267-270, 1985.
  • 13. C.J. Xu, C. Zuily: Smoothness up to the boundary for solutions of the nonlinear and nonelliptic Dirichlet problem, Trans. of the Amer. Math. Soc. 308, 243-257, 1988. MR 90c:35034

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 32W20

Retrieve articles in all journals with MSC (2000): 32W20


Additional Information

Saoussen Kallel-Jallouli
Affiliation: Faculté des Sciences, Campus Universitaire, 1060 Tunis, Tunisie
Email: Saoussen.Kallel@fst.rnu.tn

DOI: https://doi.org/10.1090/S0002-9947-03-03399-3
Received by editor(s): January 15, 2002
Received by editor(s) in revised form: February 5, 2003, and March 19, 2003
Published electronically: December 12, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society