2.2. LAPLACE’S EQUATION 39

Combining this calculation with estimate (36), we deduce

|u(x)−g(x0)|

≤ 2ε,

provided |x −

x0|

is suﬃciently small.

c. Green’s function for a ball. To construct Green’s function for the

unit ball B(0, 1), we will again employ a kind of reflection, this time through

the sphere ∂B(0, 1).

DEFINITION. If x ∈

Rn

− {0}, the point

˜ x =

x

|x|2

is called the point dual to x with respect to ∂B(0, 1). The mapping x → ˜ x

is inversion through the unit sphere ∂B(0, 1).

We now employ inversion through the sphere to compute Green’s func-

tion for the unit ball U =

B0(0,

1). Fix x ∈

B0(0,

1). Remember that we

must ﬁnd a corrector function φx = φx(y) solving

(37)

Δφx

= 0 in

B0(0,

1)

φx

= Φ(y − x) on ∂B(0, 1);

then Green’s function will be

(38) G(x, y) = Φ(y − x) −

φx(y).

The idea now is to “invert the singularity” from x ∈

B0(0,

1) to ˜ x / ∈

B(0, 1). Assume for the moment n ≥ 3. Now the mapping y → Φ(y − ˜) x is

harmonic for y = ˜. x Thus y →

|x|2−nΦ(y

− ˜) x is harmonic for y = ˜, x and so

(39)

φx(y)

:= Φ(|x|(y − ˜)) x

is harmonic in U. Furthermore, if y ∈ ∂B(0, 1) and x = 0,

|x|2|y

−

˜|2

x =

|x|2 |y|2

−

2y · x

|x|2

+

1

|x|2

=

|x|2

− 2y · x + 1 = |x −

y|2.

Thus (|x||y −

˜|)−(n−2)

x = |x −

y|−(n−2).

Consequently

(40)

φx(y)

= Φ(y − x) (y ∈ ∂B(0, 1)),

as required.