# Elliptic PDEs

Laplace's equation is a common example of the elliptic equation:

$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0;$$
since the discriminant is negative (B² - 4AC = 0² - (4)(1)(1) = -4).

Here are some more examples of the elliptic equations:

(1) Potential flow: flow around an airfoil (without separation);
(2) Deformation of a membrane damped over its edges.

Similar to the parabolic equations, this type of equations has also smooth solutions.