22 2. FOUR IMPORTANT LINEAR PDE
for i = 1,...,n, and so
Δu = v (r) +
n − 1
r
v (r).
Hence Δu = 0 if and only if
(5) v +
n − 1
r
v = 0.
If v
= 0, we deduce
log(|v |) =
v
v
=
1 − n
r
,
and hence v (r) =
a
rn−1
for some constant a. Consequently if r 0, we have
v(r) =
b log r + c (n = 2)
b
rn−2
+ c (n ≥ 3),
where b and c are constants.
These considerations motivate the following
DEFINITION. The function
(6) Φ(x) :=
−
1
2π
log |x| (n = 2)
1
n(n−2)α(n)
1
|x|n−2
(n ≥ 3),
defined for x ∈
Rn,
x = 0, is the fundamental solution of Laplace’s equation.
The reason for the particular choices of the constants in (6) will be
apparent in a moment. (Recall from §A.2 that α(n) denotes the volume of
the unit ball in
Rn.)
We will sometimes slightly abuse notation and write Φ(x) = Φ(|x|) to
emphasize that the fundamental solution is radial. Observe also that we
have the estimates
(7) |DΦ(x)| ≤
C
|x|n−1
,
|D2Φ(x)|
≤
C
|x|n
(x = 0)
for some constant C 0.
b. Poisson’s equation. By construction the function x → Φ(x) is har-
monic for x = 0. If we shift the origin to a new point y, the PDE (1) is
unchanged; and so x → Φ(x − y) is also harmonic as a function of x, x = y.
Let us now take f :
Rn
→ R and note that the mapping x → Φ(x − y)f(y)
(x = y) is harmonic for each point y ∈
Rn,
and thus so is the sum of finitely
many such expressions built for different points y.
