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.