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

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

where A, B, C, D, E, F, and G are functions of x and y and where ${\displaystyle u_{x}={\frac {\partial u}{\partial x}}}$, ${\displaystyle u_{xy}={\frac {\partial ^{2}u}{\partial x\partial y}}}$ and similarly for ${\displaystyle u_{xx},u_{y},u_{yy}}$. A PDE written in this form is elliptic if

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

with this naming convention inspired by the equation for a planar ellipse. Equations with ${\displaystyle B^{2}-AC=0}$ are termed parabolic while those with ${\displaystyle B^{2}-AC>0}$ are hyperbolic.

The simplest examples of elliptic PDE's are the Laplace equation, ${\displaystyle \Delta u=u_{xx}+u_{yy}=0}$, and the Poisson equation, ${\displaystyle \Delta u=u_{xx}+u_{yy}=f(x,y).}$ In a sense, any other elliptic PDE in two variables can be considered to be a generalization of one of these equations, as it can always be put into the canonical form

${\displaystyle u_{xx}+u_{yy}+{\text{ (lower-order terms)}}=0}$

through a change of variables.^[1][2]