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 sourcefree
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 Ω be a compact domain, and
∂
Ω be its boundary. Let
f : ∂Ω → R. Then there is a unique solution to ∇
2
φ = 0 on Ω with φ
∂Ω
= f.
Proof.
Suppose
φ
1
and
φ
2
are both solutions such that
φ
1

∂Ω
=
φ
2

∂Ω
=
f
. Then
let Φ = φ
1
− φ
2
. So Φ = 0 on the boundary. So we have
0 =
Z
Ω
Φ∇
2
Φ d
n
x = −
Z
Ω
(∇Φ) · (∇Φ) dx +
Z
∂Ω
Φ∇Φ · n d
n−1
x.
We note that the second term vanishes since Φ = 0 on the boundary. So we have
0 = −
Z
Ω
(∇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
.
Asking that
φ
∂Ω
=
f
(
x
) is a Dirichlet boundary condition. We can instead
ask
n ·∇φ
∂Ω
=
g
, a Neumann condition. In this case, we have a unique solution
up to a constant factor.
These we’ve all seen in IA Vector Calculus, but this time in arbitrary dimen
sions.