Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 377

Theorem 2. If an external Dirichlet problem has a solution, the solution must be
unique and stable.

Proof. Uniqueness: Let u 1 and u 2 be two solutions of the problem, then u = u 1 − u 2

must satisfy

⎧
⎨

Ω ,

Δ u = 0 ,

u | ∂Ω = 0 ,

(7.85)

⎩

x 2

y 2 z 2 lim

r

u = 0 , r = + + .

→ ∞

∈ Ω . Suppose

Uniqueness will be established once we show that u ( M ) ≡ 0, M

u ( M 0 ) = 0at M 0 . Without loss of generality, consider u ( M 0 ) > 0. We can always

find a spherical surface S R with a sufficiently large radius R : x 2 y 2 z 2 R 2 such

+ + =

∂Ω and S R ,and u that M 0 is inside the region Ω R of boundary | S R < u ( M 0 ) by the

condition lim
r

u = 0. The harmonic function u in Ω R thus does not take its maxi-

→ ∞

∂Ω and S R , which contradicts the extremum principle. mum value on its boundary

≡

Therefore, u 0 so the solution is unique.

Stability: Let u 1 and u 2 be solutions of
⎧
⎨

⎧
⎨

Ω , Ω ,

Δ u = 0 , Δ u = 0 ,

u | ∂Ω = f 1 , u | ∂Ω = f 2 ,

and (7.86)

⎩ ⎩

lim
r

u = 0 lim

r

u = 0 ,

→ ∞ → ∞

respectively. Then u = u 1 − u 2 must satisfy

⎧
⎨

Ω ,

Δ u = 0 ,

u | ∂Ω = f 1 − f 2 ,

⎩

lim
r

u = 0 .

→ ∞

Ω , by the condition lim
r → ∞

Let sup
∂Ω

| f 1 − f 2 | < ε . For any point M in u = 0, we can

always find a spherical surface S R with a sufficiently large radius R : x 2 y 2 z 2 R 2 + + =

∂Ω and S R and sup

such that M is inside the region Ω R of boundary

S R |

u | < ε .Bythe

extremum principle, the harmonic function u satisfies

sup

Ω R |

u | = sup

∂Ω R |

u | , ∂Ω R = ∂Ω ∪ S R .

Thus sup

Ω R |

u | < ε . By the arbitrariness of M , we thus obtain

sup
Ω

| u | < ε or sup

Ω

| u 1 − u 2 | < ε .

Therefore, the solution is stable with respect to the boundary values.