# Elliptic partial differential equation

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

$Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0\,$

that satisfies the condition

$B^2 - AC < 0.\$

(Assuming implicitly that $u_{xy}=u_{yx}$. )

Just as one classifies conic sections and quadratic forms based on the discriminant $B^2 - 4AC$, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by $B^2 - AC,$ due to the convention (discussion and explanation here). The above form is analogous to the equation for a planar ellipse:

$Ax^2 + 2Bxy + Cy^2 + \cdots = 0$ , which becomes (for : $u_{xy}=u_{yx}=0$) :
$Au_{xx} + Cu_{yy} + Du_x + Eu_y + F = 0$ , and $Ax^2 + Cy^2 + \cdots = 0$ . This resembles the standard ellipse equation: ${x^2\over a^2}+{y^2\over b^2}-1=0.$

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

$L u =\sum_{i=1}^n\sum_{j=1}^n a_{i,j} \frac{\part^2 u}{\partial x_i \partial x_j} \quad \text{ + (lower-order terms)} =0 \,$, where L is an elliptic operator.

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

$a\frac{\partial^2 u}{\partial x^2} + b\frac{\partial^2 u}{\partial x\partial y} + c\frac{\partial^2 u}{\partial y^2} + d\frac{\partial^2 u}{\partial y\partial z} + e\frac{\partial^2 u}{\partial z^2} \text{ + (lower-order terms)}= 0,$

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

$a\frac{\partial^2 u}{\partial x^2} + c\frac{\partial^2 u}{\partial y^2} + e\frac{\partial^2 u}{\partial z^2} \text{ + (lower-order terms)}= 0.$

This can be compared to the equation for an ellipsoid; ${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}=1.$