2.2. LAPLACE’S EQUATION 33
Proof. Let r :=
1
4
dist(V, ∂U). Choose x, y V , |x y| r. Then
u(x) =
B(x,2r)
u dz
1
α(n)2nrn
B(y,r)
u dz
=
1
2n

B(y,r)
u dz =
1
2n
u(y).
Thus
2nu(y)
u(x)
1
2n
u(y) if x, y V , |x y| r.
Since V is connected and
¯
V is compact, we can cover
¯
V by a chain of
finitely many balls {Bi}i=1,
N
each of which has radius
r
2
and Bi Bi−1 =
for i = 2,... , N. Then
u(x)
1
2n(N+1)
u(y)
for all x, y V .
2.2.4. Green’s function.
Assume now U
Rn
is open, bounded, and ∂U is
C1.
We propose
next to obtain a general representation formula for the solution of Poisson’s
equation
−Δu = f in U,
subject to the prescribed boundary condition
u = g on ∂U.
a. Derivation of Green’s function. Suppose u C2(
¯
U ) is an arbitrary
function. Fix x U, choose ε 0 so small that B(x, ε) U, and apply
Green’s formula from §C.2 on the region := U B(x, ε) to u(y) and
Φ(y x). We thereby compute
(24)

u(y)ΔΦ(y x) Φ(y x)Δu(y) dy
=
∂Vε
u(y)
∂Φ
∂ν
(y x) Φ(y x)
∂u
∂ν
(y) dS(y),
ν
denoting the outer unit normal vector on ∂Vε. Recall next ΔΦ(x y) = 0
for x = y. We observe also
∂B(x,ε)
Φ(y x)
∂u
∂ν
(y) dS(y)
Cεn−1
max
∂B(0,ε)
|Φ| = o(1)
as ε 0. Furthermore the calculations in the proof of Theorem 1 show
∂B(x,ε)
u(y)
∂Φ
∂ν
(y x) dS(y) =
∂B(x,ε)
u(y) dS(y) u(x)
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