Partial Differential Equations

1.2 The Laplace Equation 13
Lemma 1.2.12. Let U ⊆ R n be an open and bounded subset with smooth boundary. For u ∈ C 2 (U) and
x ∈ U, we have the formula
∫
u(x) = − u(y) ∂G
∫
∂U ∂ν (x, y) dS ∂U(y) − G(x, y)∆u(y) dy,
U
where G is Green’s function for the Laplace equation on U and ∂G
∂ν (x, y) := D yG(x, y) · ν(y) for y ∈ ∂U.
Proof. Green’s formula gives that
∫
∫
− ϕ x (y)∆u(y) dy =
U
∫
=
∂U
∂U
(u(y) ∂ϕx
∂ν (y) − ϕx (y) ∂u )
∂ν (y) dS ∂U (y)
)
(u(y) ∂ϕx (y) − Φ(y − x)∂u
∂ν ∂ν (y) dS ∂U (y).
Adding this to the representing formula of u(x) in Lemma 1.2.11, we then obtain the claim.
Theorem 1.2.13. Let U ⊆ R n be an open and bounded subset with smooth boundary, and let G be Green’s
function for the Laplace equation on U. If u ∈ C 2 (U) is a solution of the boundary value problem (1.9),
then for all x ∈ U, we have
∫
u(x) = − g(y) ∂G
∫
∂ν (x, y) dS ∂U(y) + G(x, y)f(y) dy.
U
Proof. This immediately follows by Lemma 1.2.12.
∂U
Example 1.2.14 (Balls). Let n ≥ 2, and let x ∈ R n \ {0}. Then, ˜x :=
with respect to the inversion through the unit sphere S n−1 := ∂B(0; 1).
x
|x| 2
is called the point dual to x
x
~
x
S n-1
0
Fig. 1.6.
We set
for x ≠ 0 and
ϕ x (y) := Φ ( |x|(y − ˜x) )
{
0 for n = 2,
ϕ 0 (y) :=
1
n(n−2)α(n)
for n ≥ 3,
i.e. ϕ 0 (y) = Φ( y
|y|
) for y ≠ 0. From the explicit formula
{
−
1
ϕ x 2π
log |x| |y − ˜x| for n = 2,
(y) =
1
1
n(n−2)α(n) |x| n−2 |y−˜x|
for n ≥ 3,
n−2