Elliptic PDEs are usually associated with *steady-state problems*.

The simplest example of an elliptic PDE (let it be 2-dimensional) is the *Laplace’s equation*,

which governs incompressible, potential flow (or field).

If we’ll set on the square **0≤x≤1, 0≤y≤1** boundary conditions

then the Laplace’s equation will have the following exact solution:

If in the right side of the equation we have not the zero but some function **f(x, y)**, then such an equation is called *Poisson equation* – it is elliptic as well.

As an example of *Poisson equation* one can give, for instance, the equation for *electric field potential* – in this case the right hand function **f(x, y)**
of the equation is the distribution of electrical charges in the given region, and the unknown function **ɸ(x, y)** is the potential of electric field.

*Poisson equation* for the stream function in two-dimensional rotational flow is an elliptic PDE.

The *steady Navier-Stokes equations* are also elliptic.

A lot of examples of elliptic PDE one can obtain in classical mechanics from equations responsible for *energy preservation*.

Elliptic equations one can find also in quantum mechanics – for instance, steady *Schrodinger equation*.

For second-order elliptic PDE an important *maximum principal* exist.
Namely, both the maximum and minimum values of **ɸ** must occur on the boundary (except the trivial case that **ɸ** is a constant).

The *maximum principle* is useful in testing that computational solutions of elliptic PDEs are behaving properly.

Go to this page for more detailed discussion of elliptic problems.