22 2. FOUR IMPORTANT LINEAR PDE
for i = 1,...,n, and so
Δu = v (r) +
n − 1
Hence Δu = 0 if and only if
(5) v +
n − 1
v = 0.
= 0, we deduce
log(|v |) =
1 − n
and hence v (r) =
for some constant a. Consequently if r 0, we have
b log r + c (n = 2)
+ c (n ≥ 3),
where b and c are constants.
These considerations motivate the following
DEFINITION. The function
(6) Φ(x) :=
log |x| (n = 2)
(n ≥ 3),
deﬁned for x ∈
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
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)| ≤
(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 :
→ R and note that the mapping x → Φ(x − y)f(y)
(x = y) is harmonic for each point y ∈
and thus so is the sum of ﬁnitely
many such expressions built for diﬀerent points y.