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,\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/a527ddbd45f7fd074228a8fc8ac1ed7a4f78a73d)
where A, B, C, D, E, F, and G are functions of x and y and where
,
and similarly for
. A PDE written in this form is elliptic if
![{\displaystyle B^{2}-AC<0,}](//wikimedia.org/api/rest_v1/media/math/render/svg/0afb7b67ee67220df802ed8f6eac775118ab409c)
with this naming convention inspired by the equation for a planar ellipse. Equations with
are termed parabolic while those with
are hyperbolic.
The simplest examples of elliptic PDEs are the Laplace equation,
, and the Poisson equation,
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}](//wikimedia.org/api/rest_v1/media/math/render/svg/01dc01b03756954e6587f7eaa0b29ef83e236cbf)
through a change of variables.[1][2]
Elliptic equations have no real characteristic curves, curves along which it is not possible to eliminate at least one second derivative of
from the conditions of the Cauchy problem.[1] Since characteristic curves are the only curves along which solutions to partial differential equations with smooth parameters can have discontinuous derivatives, solutions to elliptic equations cannot have discontinuous derivatives anywhere. This means elliptic equations are well suited to describe equilibrium states, where any discontinuities have already been smoothed out. For instance, we can obtain Laplace's equation from the heat equation
by setting
. This means that Laplace's equation describes a steady state of the heat equation.[2]
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]
We derive the canonical form for elliptic equations in two variables,
.
and
.
If
, applying the chain rule once gives
and
,
a second application gives
![{\displaystyle u_{xx}=u_{\xi \xi }{\xi ^{2}}_{x}+u_{\eta \eta }{\eta ^{2}}_{x}+2u_{\xi \eta }\xi _{x}\eta _{x}+u_{\xi }\xi _{xx}+u_{\eta }\eta _{xx},}](//wikimedia.org/api/rest_v1/media/math/render/svg/2ab01da664bdc0420ded886ee126e085f27a657f)
and
![{\displaystyle u_{xy}=u_{\xi \xi }\xi _{x}\xi _{y}+u_{\eta \eta }\eta _{x}\eta _{y}+u_{\xi \eta }(\xi _{x}\eta _{y}+\xi _{y}\eta _{x})+u_{\xi }\xi _{xy}+u_{\eta }\eta _{xy}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/82506293b60b2fe513f09607a5a1884e0b443352)
We can replace our PDE in x and y with an equivalent equation in
and ![{\displaystyle \eta }](//wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64)
![{\displaystyle au_{\xi \xi }+2bu_{\xi \eta }+cu_{\eta \eta }{\text{ + (lower-order terms)}}=0,\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/04ebe6f61e9264c864f594531f6d2c42510c6287)
where
![{\displaystyle a=A{\xi _{x}}^{2}+2B\xi _{x}\xi _{y}+C{\xi _{y}}^{2},}](//wikimedia.org/api/rest_v1/media/math/render/svg/8ba450ff3908c536732fcd2d9b53e10d4d3b170e)
and
![{\displaystyle c=A{\eta _{x}}^{2}+2B\eta _{x}\eta _{y}+C{\eta _{y}}^{2}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/68e7b1ba86d0da2ea121662341724dbf1c085498)
To transform our PDE into the desired canonical form, we seek
and
such that
and
. This gives us the system of equations
![{\displaystyle a-c=A({\xi _{x}}^{2}-{\eta _{x}}^{2})+2B(\xi _{x}\xi _{y}-\eta _{x}\eta _{y})+C({\xi _{y}}^{2}-{\eta _{y}}^{2})=0}](//wikimedia.org/api/rest_v1/media/math/render/svg/00897e60f24e5ff8de14da784b0ca1223b91193c)
![{\displaystyle b=0=2A\xi _{x}\eta _{x}+2B(\xi _{x}\eta _{y}+\xi _{y}\eta _{x})+2C\xi _{y}\eta _{y},}](//wikimedia.org/api/rest_v1/media/math/render/svg/b2cf01f610e274eacd071e2d9b2b9cfd0255a3f4)
Adding
times the second equation to the first and setting
gives the quadratic equation
![{\displaystyle A{\phi _{x}}^{2}+2B\phi _{x}\phi _{y}+C{\phi _{y}}^{2}=0.}](//wikimedia.org/api/rest_v1/media/math/render/svg/b5ea41166920366a1c1ede56c3ab61eedcc58431)
Since the discriminant
, this equation has two distinct solutions,
![{\displaystyle {\phi _{x}},{\phi _{y}}={\frac {B\pm i{\sqrt {AC-B^{2}}}}{A}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/8de27079db28d1741060b9e20b2767928ebbc9da)
which are complex conjugates. Choosing either solution, we can solve for
, and recover
and
with the transformations
and
. Since
and
will satisfy
and
, so with a change of variables from x and y to
and
will transform the PDE
![{\displaystyle Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+Fu+G=0,\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/a527ddbd45f7fd074228a8fc8ac1ed7a4f78a73d)
into the canonical form
![{\displaystyle u_{\xi \xi }+u_{\eta \eta }+{\text{ (lower-order terms)}}=0,}](//wikimedia.org/api/rest_v1/media/math/render/svg/88ffc50db76ced2eb8729ec9fe731e52bf34979c)
as desired.
A general second-order partial differential equation in n variables takes the form
![{\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}\;{\text{ + (lower-order terms)}}=0.}](//wikimedia.org/api/rest_v1/media/math/render/svg/35911d74e4e9cc21952ecb8e059d0ef8defb4d5a)
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]
Zauderer, Erich (1989). Partial Differential Equations of Applied Mathematics. New York: John Wiley&Sons. ISBN 0-471-61298-7.