Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 386

2. In Ω

1

0 < G ( M , M 0 ) <

r .

(7.107)

4 π

Proof . Applying the extremum principle to PDS (7.102) yields

1

g ( M , M 0 ) > 0 r G ( M , M 0 ) <

r ,

M ∈ Ω .

4 π

By the definition of G ( M , M 0 ) (Eq. (7.103)), we have

⎧
⎨

V M 0

( , )= , ∈ Ω \ ,

Δ G M M 0 0 M

ε

G ( M , M 0 ) | ∂Ω = 0 ,

⎩

1
4π r −

G ( M , M 0 ) | S M 0

ε

= g ( M , M 0 ) > 0 ,

where V M 0
ε

is a sphere of center M 0 and radius ε , ε is a sufficiently small constant

and S M 0
ε

is the boundary of the V M 0
ε

V M 0
ε

. By the extremum principle, for M ∈ Ω \

we have

0 < G ( M , M 0 ) .

Since ε can be arbitrarily small, for M ∈ Ω we obtain

1

0 < G ( M , M 0 ) <

r . 4 π

3.

∂ G

n d S

= − 1 .

∂

∂Ω

and S M 0 Proof. Consider the domain enclosed by ∂Ω in Ω ,where G ( M , M 0 ) is

ε

a harmonic function. By Theorem 1 in Section 7.4.3, we have

∂ G

n d S

= 0 ,

∂

S M 0
ε

∂Ω ∪

where n is the external normal on the boundary surface. Thus

1
4

d S

∂ G ∂ G ∂

∂

∂ g

n d S

= −

n d S

= − +

n d S

∂ ∂ n π r ∂

∂Ω

S M 0
ε

S M 0
ε

S M 0
ε

1

r 2 d S

1

∂ (

r )

1
4

1
4

= = −

d S

π ∂ r π

S M 0
R

S M 0
R