Hokkaido Mathematical Journal

Geometric characterization of Monge-Ampère equations

Atsushi YANO

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Abstract

It is well known that a Monge-Ampère equation can be expressed in terms of exterior differential system—Monge-Ampère system, which is the ideal generated algebraically by a contact form and a 2-form and its exterior derivatives on a 5-dimensional contact manifold, and the system is independent of the choice of coordinate system. On the other hand, a single second order partial differential equation of one unknown function with two independent variables corresponds to the differential system on a hypersurface of Lagrange-Grassmann bundle over a 5-dimensional contact manifold obtained by restricting its canonical system to the hypersurface. We observe relations between Monge characteristic systems of Monge-Ampère equation and those of Monge-Ampère system and particularly analyze structure equations of those systems. This observation leads to the result—to characterize Monge-Ampère equation by the property that the certain differential system defined from the Monge characteristic system drops down to the contact manifold.

Article information

Source
Hokkaido Math. J., Volume 41, Number 3 (2012), 409-440.

Dates
First available in Project Euclid: 24 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1351086222

Digital Object Identifier
doi:10.14492/hokmj/1351086222

Mathematical Reviews number (MathSciNet)
MR3012457

Zentralblatt MATH identifier
1256.58001

Subjects
Primary: 58A15: Exterior differential systems (Cartan theory)
Secondary: 53D10: Contact manifolds, general

Keywords
differential system exterior differential system partial differential equation Monge-Ampere equation Goursat equation Monge characteristic system

Citation

YANO, Atsushi. Geometric characterization of Monge-Ampère equations. Hokkaido Math. J. 41 (2012), no. 3, 409--440. doi:10.14492/hokmj/1351086222. https://projecteuclid.org/euclid.hokmj/1351086222


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