Monge–Ampère equation

From Wikipedia, the free encyclopedia

A Monge-Ampère equation is a second order scalar equation in the plane. Given two independent variables x and y, and one dependent variable z we write the partial derivatives of z with notations

p=z_x,\quad q=z_y,\quad r=z_{xx},\quad s=z_{xy},\quad t=z_{yy}.\,

A Monge-Ampère equation can then be written in local coordinates as

a(rt-s^2)+br+cs+dt+e = 0\,

where a, b, c, d, and e are functions depending on the first order variables x, y, z, p, and q only.

Monge–Ampère equations were first studied by Gaspard Monge in 1784 and later by André-Marie Ampère in 1820.

[edit] See also

[edit] References

  • Gilbarg, D. and Trudinger, N. S. Elliptic Partial Differential Equations of Second Order. Berlin: Springer-Verlag, 1983. ISBN 3540411607 ISBN-13: 978-3540411604
  • A.V. Pogorelov, "Monge–Ampère equation" SpringerLink Encyclopaedia of Mathematics (2001)

[edit] External link

 This mathematics-related article is a stub. You can help Wikipedia by expanding it.
Retrieved from "http://en.wikipedia.org../../../m/o/n/Monge%E2%80%93Amp%C3%A8re_equation_3945.html"
In other languages