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On the intermediate integral for Monge-Ampère equations
Author(s):
Jeanne
Nielsen
Clelland
Journal:
Proc. Amer. Math. Soc.
128
(2000),
527-531.
MSC (1991):
Primary 35A30;
Secondary 58A15
Posted:
July 8, 1999
MathSciNet review:
1641669
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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:
- 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:
10.1090/S0002-9939-99-05136-9
PII:
S 0002-9939(99)05136-9
Keywords:
Method of the intermediate integral,
Monge-Amp\`ere equations,
exterior differential systems
Received by editor(s):
April 6, 1998
Posted:
July 8, 1999
Additional Notes:
This research was supported in part by NSF grant DMS-9627403.
Communicated by:
Lesley M. Sibner
Copyright of article:
Copyright
1999,
American Mathematical Society
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