Monge-Ampère Equation

Posted on January 2, 2025 • 1 min read • 106 words

Test the LaTeX feature with complex mathematical formulas.

The Monge–Ampère equation is a fully nonlinear, degenerate elliptic equation that draws its name from its initial formulation in two dimensions, by the French mathematicians Monge and Ampère, about two hundred years ago.

The equation form:

$$\begin{gather*} rt - s^2 = ar + 2bs + ct + \phi, \\ r = \frac{\partial^2 z}{\partial x^2}, \; s = \frac{\partial^2 z}{\partial x\partial y}, \; t = \frac{\partial^2 z}{\partial y^2}, \\ p = \frac{\partial z}{\partial x}, \; q = \frac{\partial z}{\partial y} \end{gather*}$$

The type of a Monge–Ampère equation depends on the sign of the expression: $\Delta = \phi + ac + b^2$ .