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 Vε := U − B(x, ε) to u(y) and

Φ(y − x). We thereby compute

(24)

Vε

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)