Elliptic Partial Differential Equations

An elliptic partial differential equation is a general partial differential equation of second order

that satisfies the condition

(Assuming implicitly that . )

Just as one classifies conic sections and quadratic forms based on the discriminant, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by due to the convention (discussion and explanation here). The above form is analogous to the equation for a planar ellipse:

, which becomes (for : ) :

, and . This resembles the standard ellipse equation:

In general, if there are n independent variables x₁, x₂, ..., x_n, a general linear partial differential equation of second order has the form

, where L is an elliptic operator.

For example, in three dimensions (x,y,z) :

which, for completely separable u (i.e. u(x,y,z)=u(x)u(y)u(z) ) gives

This can be compared to the equation for an ellipsoid;