Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 378

Theorem 3. If an internal Neumann problem has a solution, the solution is unique
up to a constant.

Proof. Let u 1 and u 2 be two solutions of an internal Neumann problem
⎧
⎨ Δ u = 0 , Ω ,

∂ n ∂Ω =

(7.87)

⎩

∂ u

f .

= −

The v u 1 u 2 thus must satisfy

⎧
⎨

Δ v = 0 , Ω ,

n ∂Ω =

⎩

∂ v

0 .

∂

The first Green formula yields

∂

2 d

2 ∂ 2 ∂

v v v

∇ v · ∇ v d Ω = + + Ω

∂ x ∂ y ∂ z

Ω Ω

v ∂

v

=

n d S

− v Δ v d Ω = 0 .

∂

∂Ω Ω

Therefore ∂

v

x = ∂

v

y = ∂

v

z =

0, so that

∂ ∂ ∂

u 1 = u 2 + c .

Here c is a constant.

Theorem 4. If an external Neumann problem has a solution, the solution must be
unique.

Proof. Let u 1 and u 2 be two solutions of the external problem
⎧
⎨ Ω , Δ u = 0 ,

∂Ω =

x 2

(7.88)

∂ u

⎩

y 2 z 2

f , lim

r

u = 0 , r = + + .

∂ n

→ ∞

Thus v = u 1 − u 2 must satisfy

⎧
⎨

Ω , Δ v = 0 ,

∂Ω =

∂ v

⎩

0 , lim

r → ∞

v = 0 ,

∂ n

Ω ∂Ω and the n is the external normal of Ω

where the is the region outside of

∂Ω . Once we show that v ∈ Ω , we establish uniqueness. Suppose

on ( M ) ≡ 0, M