Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 366

Theorem 2. Let Ω be a bounded domain with a piecewise smooth boundary ∂Ω .If

¯

C 1 C 2

u ( M )= u ( x , y , z ) ∈ ( Ω ) ∩ ( Ω ) , at any point M 0 ( x 0 , y 0 , z 0 ) ∈ Ω ,wehave

u 1

r

d S

1
4

) ∂
∂

1

r ∂

u ( M ) 1

4

Δ u ( M )

u ( M 0 )= − ( M − − d Ω , (7.65)

π n ∂ n π r

Ω

∂Ω

where the n stands for the external normal of ∂Ω , M 0 ∈ Ω and

= ( − )

2

+( − )

2

+( − )

2

.

r x x 0 y y 0 z z 0

Equation (7.65) is called the third Green formula and is a fundamental integral for-
mula in studying harmonic functions.
Proof. Consider two functions u

1

= u ( M )= u ( x , y , z ) and v = v ( x , y , z )=

r .Here

2 2 2

r = ( x − x 0 ) +( y − y 0 ) +( z − z 0 ) .

M 0
ε

¯

Both the two functions satisfy the conditions in Theorem 1 for Ω \ Ω .Here

¯

M 0 M 0 M 0

ε

M 0
ε

M 0
ε

ε = Ω ε ∪ ∂Ω

Ω ,the Ω is a sphere of center M 0 and radius ε and the ∂Ω

M 0
ε

¯

M 0
ε

Ω \

is the boundary of Ω . Applying the second Green formula to Ω yields

u

u d

1
r −

1
r Δ

Δ Ω

M 0
ε

¯

Ω \ Ω

u ∂
∂

1
r

d S u ∂

∂

1
r

d S

1

r ∂

u 1

r ∂

u

= − + − .

n ∂ n n ∂ n

∂Ω

M 0
ε

∂Ω

M 0
ε

¯

M 0
ε

Note that the external normal of the inner boundary surface ∂Ω of Ω \ Ω is

M 0
ε

so ∂ n = − ∂ r .Also actually the internal normal of the boundary surface of Ω

1
r =

Δ 0. Thus

u ∂
∂

1
r

d S

Δ u

r

1

r ∂

u u

r 2 d S

1

r ∂

u

d Ω + − + +

r d S

= 0 .

n ∂ n ∂

M 0
ε

M 0
ε

M 0
ε

∂Ω

¯
Ω Ω \ ∂Ω ∂Ω

Note that, by the mean value theorem of integrals,

u
r 2 d S

1
ε

M 0
ε

2

=

2 u

( M

ξ 1 ) ·

4 πε = 4 π u ( M

ξ 1 ) ,

M

ξ 1 ∈ ∂Ω

, (7.66)

M 0
ε

∂Ω

M ξ 2 M ξ 2

1

r ∂

u 1

ε

∂ u

= ∂

u

2 M 0

ε

r d S

= · 4 πε · 4 πε , M

ξ 2 ∈ ∂Ω

. (7.67)

∂ ∂ r ∂ r

M 0
ε

∂Ω