2.2. LAPLACE’S EQUATION 37

DEFINITION. Green’s function for the half-space R+

n

is

G(x, y) := Φ(y − x) − Φ(y − ˜) x (x, y ∈ R+,

n

x = y).

Then

Gyn (x, y) = Φyn (y − x) − Φyn (y − ˜) x

=

−1

nα(n)

yn − xn

|y − x|n

−

yn + xn

|y − ˜|n x

.

Consequently if y ∈

∂R+,n

∂G

∂ν

(x, y) = −Gyn (x, y) = −

2xn

nα(n)

1

|x −

y|n

.

Suppose now u solves the boundary-value problem

(32)

Δu = 0 in

R+n

u = g on

∂R+.n

Then from (30) we expect

(33) u(x) =

2xn

nα(n)

∂R+n

g(y)

|x − y|n

dy (x ∈

R+)n

to be a representation formula for our solution. The function

K(x, y) :=

2xn

nα(n)

1

|x −

y|n

(x ∈ R+,y

n

∈

∂R+)n

is Poisson’s kernel for R+,

n

and (33) is Poisson’s formula.

We must now check directly that formula (33) does indeed provide us

with a solution of the boundary-value problem (32).

THEOREM 14 (Poisson’s formula for half-space). Assume g ∈

C(Rn−1)∩

L∞(Rn−1),

and deﬁne u by (33). Then

(i) u ∈

C∞(R+) n

∩

L∞(R+),n

(ii) Δu = 0 in

R+,n

and

(iii) lim

x→x0

x∈Rn

+

u(x) =

g(x0)

for each point

x0

∈

∂R+.n