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Monge-Ampère Differential Equation

A second-order Partial Differential Equation of the form


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
$\displaystyle r$ $\textstyle =$ $\displaystyle {\partial^2 z\over\partial x^2}$  
$\displaystyle s$ $\textstyle =$ $\displaystyle {\partial^2 z\over\partial x\partial y}$  
$\displaystyle t$ $\textstyle =$ $\displaystyle {\partial^2 z\over\partial y^2}$  
$\displaystyle p$ $\textstyle =$ $\displaystyle {\partial z\over\partial x}$  
$\displaystyle q$ $\textstyle =$ $\displaystyle {\partial z\over\partial y}.$  

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


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

© 1996-9 Eric W. Weisstein