Monge-Ampère Differential Equation

A second-order Partial Differential Equation of the form

$$Hr+2Ks+Lt+M+N(rt-s^2)=0,$$

where $H$, $K$, $L$, $M$, and $N$ are functions of $x$, $y$, $z$, $p$, and $q$, and $r$, $s$, $t$, $p$, and $q$ are defined by

$$r = {\partial^2 z\over\partial x^2}$$

$$s = {\partial^2 z\over\partial x\partial y}$$

$$t = {\partial^2 z\over\partial y^2}$$

$$p = {\partial z\over\partial x}$$

$$q = {\partial z\over\partial y}.$$

The solutions are given by a system of differential equations given by Iyanaga and Kawada (1980).

References

Iyanaga, S. and Kawada, Y. (Eds.). ``Monge-Ampère Equations.'' §276 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 879-880, 1980.