Environmental Engineering Reference
In-Depth Information
Proof.
Consider two circles
C
M
0
and
C
M
R
with sufficiently
of center
M
0
and radius
ε
ε
D
is contained in the region of bound-
small
ε
and sufficiently large
R
such that the
∂
ary
C
M
0
and
C
M
R
.Let
D
∗
be the region enclosed by
C
M
0
D
and
C
M
R
.
D
∗
reduces
,
∂
ε
ε
ln
r
, applying the second
to
D
as
ε
→
0and
R
→
∞
. With the harmonic function
v
=
Green formula yields
u
∂
∂
ln
1
r
ln
1
r
∂
d
s
u
u
ln
1
−
=
−
Δ
r
d
σ
.
n
∂
n
D
∗
C
M
0
C
M
0
R
D
∪
∂
ε
∪
Note that
u
∂
∂
ln
1
r
ln
1
r
∂
d
s
u
−
C
M
0
ε
n
∂
n
u
∂
∂
∂
u
=
−
r
(
−
)
−
ln
r
d
s
ln
ε
r
d
s
C
M
0
ε
C
M
0
ε
∂
1
ε
∂
u
=
u
d
s
−
(
ln
ε
)
r
d
s
C
M
0
ε
C
M
0
ε
∂
−
∂
u
=
2
π
u
r
2
πε
ln
ε
,
∂
where
u
and
∂
u
are the mean-values of
u
and
∂
u
on
C
M
0
, respectively. By the
ε
∂
r
∂
r
L'Hôpital's rule, lim
ε
→
0
(
ε
ln
ε
)=
0. Also, lim
ε
→
0
u
=
u
(
M
0
)
. Thus
u
∂
∂
ln
1
r
ln
1
r
∂
d
s
u
lim
ε
→
0
−
=
2
π
u
(
M
0
)
.
C
M
0
ε
n
∂
n
Similar to in Theorem 1, there always exists a constant
k
for a sufficiently large
R
such that, by Eq. (7.132)
d
s
u
∂
∂
ln
1
r
ln
1
r
∂
u
−
C
M
0
R
n
∂
n
r
d
s
1
R
∂
u
=
−
(
)
u
d
s
ln
R
C
M
0
R
C
M
0
R
∂
1
R
k
R
2
k
R
2
2
≤
π
R
+(
ln
R
)
π
R
2
k
R
+
2
k
π
ln
R
R
=
.
ln
R
R
=
By the L'Hôpital's rule, lim
R
0. Therefore
→
∞
u
∂
∂
ln
1
r
ln
1
r
∂
d
s
u
lim
R
−
=
0
.
C
M
0
R
n
∂
n
→
∞
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