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On the intermediate integral
for Monge-Ampère equations


Author: Jeanne Nielsen Clelland
Journal: Proc. Amer. Math. Soc. 128 (2000), 527-531
MSC (1991): Primary 35A30; Secondary 58A15
DOI: https://doi.org/10.1090/S0002-9939-99-05136-9
Published electronically: July 8, 1999
MathSciNet review: 1641669
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Abstract | References | Similar Articles | Additional Information

Abstract: Goursat showed that in the presence of an intermediate integral, the problem of solving a second-order Monge-Ampère equation can be reduced to solving a first-order equation, in the sense that the generic solution of the first-order equation will also be a solution of the original equation. An attempt by Hermann to give a rigorous proof of this fact contains an error; we show that there exists an essentially unique counterexample to Hermann's assertion and state and prove a correct theorem.


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

  • 1. R. Bryant, S. Chern, R. Gardner, H. Goldschmidt, and P. Griffiths, Exterior Differential Systems, Math. Sci. Res. Inst. Publ. 18, Springer-Verlag, New York, 1991.MR 92h:58007
  • 2. R. Bryant and P. Griffiths, Characteristic cohomology of differential systems II: Conservation laws for a class of parabolic equations, Duke Math J. 78 (1995) 531-676.MR 96d:58158
  • 3. E. Goursat, Leçons sur i'intégration des équations aux dérivées partielles du second ordre, vol. I, Gauthier-Villars, Paris, 1890.
  • 4. R. Hermann, Geometry, Physics, and Systems, Marcel Dekker, Inc., New York, 1973.MR 58:13104

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Additional Information

Jeanne Nielsen Clelland
Affiliation: Department of Mathematics, University of Colorado, Boulder, Colorado 80309
Email: Jeanne.Clelland@Colorado.edu

DOI: https://doi.org/10.1090/S0002-9939-99-05136-9
Keywords: Method of the intermediate integral, Monge-Amp\`ere equations, exterior differential systems
Received by editor(s): April 6, 1998
Published electronically: July 8, 1999
Additional Notes: This research was supported in part by NSF grant DMS-9627403.
Communicated by: Lesley M. Sibner
Article copyright: © Copyright 1999 American Mathematical Society

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