Elliptic partial differential equation

Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form

with this naming convention inspired by the equation for a planar ellipse.

In parabolic and hyperbolic equations, characteristics describe lines along which information about the initial data travels. Since elliptic equations have no real characteristic curves, there is no meaningful sense of information propagation for elliptic equations. This makes elliptic equations better suited to describe static, rather than dynamic, processes.[2]

A general second-order partial differential equation in n variables takes the form

This equation is considered elliptic if there are no characteristic surfaces, i.e. surfaces along which it is not possible to eliminate at least one second derivative of u from the conditions of the Cauchy problem.[1]

Unlike the two-dimensional case, this equation cannot in general be reduced to a simple canonical form.[2]