Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 379

0at M 0 ∈ Ω . Without loss of generality, consider v

v ( M 0 ) = ( M 0 ) > 0. By the con-

dition lim
r

v = 0, there always exists a spherical surface S R with a sufficiently large

→ ∞

radius R : x 2 y 2 z 2 R 2 such that M 0 is inside the region ∂Ω

+ + =

Ω R of boundary

and S R and sup

S R |

v | < v ( M 0 ) . By the extremum principle, the harmonic function in

∂Ω . This requires a strictly pos-

Ω R must take its maximum value at the boundary

∂ v

itive

n somewhere on the boundary by the strong extremum principle. However, ∂

∂Ω =

∂ v

0andsup

S R |

v | < v ( M 0 ) so v does not take its maximum on the boundary

∂ n

of region Ω R , which is contrary to the extremum principle. Thus we obtain v ≡ 0so

the solution is unique.

Theorem 5. If a Robin problem
⎧
⎨

, ∂Ω ,

=

Δ u 0

∂

u ∂Ω =

(7.89)

u

⎩

n + σ

f , σ > 0 .

∂

has a solution, the solution must be unique.

= −

Proof. Let u 1 and u 2 be two solutions of PDS (7.89). v u 1 u 2 must thus satisfy

⎧
⎨ Δ v = 0 ,

∂

v ∂Ω =

v

⎩

n + σ

0 , σ > 0 .

∂

First prove v . By the extremum

principle, v can take its minimum and maximum values only on the boundary

≡ c (constant). Suppose that v ≡ c (constant) in Ω

,
say at points M 0 and M 1 , respectively. By the strong extremum principle, we have
∂

∂Ω

v ∂Ω = 0, so that, by the boundary condition ∂

v ( M 0 ) v

<

n + σ

0, v ( M 0 )=

∂ n ∂

1
σ

∂ v M 0 )

∂

(

− > 0. Thus the minimum value of the harmonic function v is larger than

n

zero in ¯

( ) <

0, so that the maximum value of v is
smaller than zero. We thus arrive at a contradiction. Therefore, v

Ω . Similarly, we can show v M 1

c in ¯ ≡ Ω . Also, by

the boundary condition,
∂

v ∂Ω =

∂

c ∂ Ω = σ

v c

n + σ n + σ

v | ∂Ω = 0 ,

∂ ∂

so that c = 0and v ≡ 0. Therefore the solution is unique.

¯

u in the first Green formula. We can readily use C 2 C 1

Remark .Let v = ( Ω ) ∩ ( Ω ) to

=

show the uniqueness of boundary-value problems of 0ofthefirstandthethird

kinds and the uniqueness up to a constant of the second boundary-value problems
of

Δ u

=

Δ u 0.