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Monge-Ampère Differential Equation

A second-order partial differential equation of the form

Hr+2Ks+Lt+M+N(rt-s^2)=0,
(1)

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^2z)/(partialx^2)
(2)
s=(partial^2z)/(partialxpartialy)
(3)
t=(partial^2z)/(partialy^2)
(4)
p=(partialz)/(partialx)
(5)
q=(partialz)/(partialy).
(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

u_(xx)u_(yy)-u_(xy)^2=f(x,y,u,u_x,u_y)
(7)

(Moon and Spencer 1969, p. 171; Zwillinger 1997, p. 134) and

|u_(x_1x_1) u_(x_1x_2) ... x_(x_1x_n); u_(x_2x_1) u_(x_2x_2) ... x_(x_2x_n); | | ... |; u_(x_nx_1) u_(x_nx_2) ... u_(x_nx_n)|=f(x,u,del u)
(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.

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Monge-Ampère Differential Equation

Cite this as:

Weisstein, Eric W. "Monge-Ampère Differential Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Monge-AmpereDifferentialEquation.html

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