Hokkaido Mathematical Journal

YANO Atsushi,

Geometric characterization of Monge-Ampère equations.

Hokkaido Mathematical Journal, 41 (2012) pp.409-440

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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.

MSC(Primary)58A15
MSC(Secondary)53D10
Uncontrolled Keywordsdifferential system, exterior differential system, partial differential equation, Monge-Ampere equation, Goursat equation, Monge characteristic system