A second-order partial differential equation of the form
|
(1)
|
where ,
,
,
,
and
are functions of , , , , and , and , , , , and are defined by
The solutions are given by a system of differential equations given by Iyanaga and Kawada (1980).
Other equations called the Monge-Ampère equation are
|
(7)
|
(Moon and Spencer 1969, p. 171; Zwillinger 1997, p. 134) and
|
(8)
|
(Gilberg and Trudinger 1983, p. 441; Zwillinger 1997, p. 134).
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References
Caffarelli, L. A. and Milman, M. Monge Ampère Equation: Applications to Geometry and Optimization.. Providence,
RI: Amer. Math. Soc., 1999.Fairlie, D. B. and Leznov, A. N.
"The General Solution of the Complex Monge-Ampère Equation in a Space
of Arbitrary Dimension." 16 Sep 1999. http://arxiv.org/abs/solv-int/9909014.Gilbarg,
D. and Trudinger, N. S. Elliptic
Partial Differential Equations of Second Order. Berlin: Springer-Verlag,
p. 441, 1983.Iyanaga, S. and Kawada, Y. (Eds.). "Monge-Ampère
Equations." §276 in Encyclopedic
Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 879-880, 1980.Moon,
P. and Spencer, D. E. Partial
Differential Equations. Lexington, MA: Heath, p. 171, 1969.Zwillinger,
D. Handbook
of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997.Referenced
on Wolfram|Alpha
Monge-Ampère
Differential Equation
Cite this as:
Weisstein, Eric W. "Monge-Ampère Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Monge-AmpereDifferentialEquation.html
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