# Elliptic partial differential equation

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

${\displaystyle Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+F=0\,}$

that satisfies the condition

${\displaystyle B^{2}-AC<0.\ }$

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

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

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

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

${\displaystyle Lu=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}\quad {\text{ + (lower-order terms)}}=0\,}$, where L is an elliptic operator.

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

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

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

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

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