Elliptic pde

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




A

u

x
x


+
2
B

u

x
y


+
C

u

y
y


+
D

u

x


+
E

u

y


+
F
u
+
G
=
0
,



{\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




u

x


=



∂
u


∂
x





{\displaystyle u_{x}={\frac {\partial u}{\partial x}}}
,




u

x
y


=




∂

2


u


∂
x
∂
y





{\displaystyle u_{xy}={\frac {\partial ^{2}u}{\partial x\partial y}}}
and similarly for




u

x
x


,

u

y


,

u

y
y




{\displaystyle u_{xx},u_{y},u_{yy}}
. A PDE written in this form is elliptic if





B

2


−
A
C
<
0
,


{\displaystyle B^{2}-AC<0,}
with this naming convention inspired by the equation for a planar ellipse.
The simplest nontrivial examples of elliptic PDE's are the Laplace equation,



Δ
u
=

u

x
x


+

u

y
y


=
0


{\displaystyle \Delta u=u_{xx}+u_{yy}=0}
, and the Poisson equation,



Δ
u
=

u

x
x


+

u

y
y


=
f
(
x
,
y
)
.


{\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





u

x
x


+

u

y
y


+

(lower-order terms)

=
0


{\displaystyle u_{xx}+u_{yy}+{\text{ (lower-order terms)}}=0}
through a change of variables.

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