Monge-Ampère Differential Equation

A second-order partial differential equation of the form

(1)

where , , , , and  are functions of , , , , and , and , , , , and  are defined by

			(2)
			(3)
			(4)
			(5)
			(6)

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).

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.