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
