4Partial differential equations

IB Methods

4.1 Laplace’s equation
Definition
(Laplace’s equation)
.
Laplace’s equation on
R
n
says that a (twice-
differentiable) equation φ : R
n
C obeys
2
φ = 0,
where
2
=
n
X
i=1
2
x
2
i
.
This equation arises in many situations. For example, if we have a conservative
force
F
=
−∇φ
, and we are in a region where the force obeys the source-free
Gauss’ law
·F
= 0 (e.g. when there is no electric charge in an electromagnetic
field), then we get Laplace’s equation.
It also arises as a special case of the heat equation
φ
t
=
κ
2
φ
, and wave
equation
2
φ
t
2
= c
2
2
φ, when there is no time dependence.
Definition
(Harmonic functions)
.
Functions that obey Laplace’s equation are
often called harmonic functions.
Proposition.
Let be a compact domain, and
be its boundary. Let
f : R. Then there is a unique solution to
2
φ = 0 on with φ|
= f.
Proof.
Suppose
φ
1
and
φ
2
are both solutions such that
φ
1
|
=
φ
2
|
=
f
. Then
let Φ = φ
1
φ
2
. So Φ = 0 on the boundary. So we have
0 =
Z
Φ
2
Φ d
n
x =
Z
(Φ) · (Φ) dx +
Z
ΦΦ · n d
n1
x.
We note that the second term vanishes since Φ = 0 on the boundary. So we have
0 =
Z
(Φ) · (Φ) dx.
However, we know that (
Φ)
·
(
Φ)
0 with equality iff
Φ = 0. Hence
Φ is constant throughout Ω. Since at the boundary, Φ = 0, we have Φ = 0
throughout, i.e. φ
1
= φ
2
.
φ|
=
f
(
x
) is a Dirichlet boundary condition. We can instead