Environmental Engineering Reference
In-Depth Information
1
R
2
<
R
2
[
u
(
M
0
)(
4
π
−
σ
)+
u
(
M
0
)
σ
]=
u
(
M
0
)
,
4
π
which is absurd. Also note that
R
is arbitrary. Therefore
u
(
M
)
≡
u
(
M
0
)
for all points
in
V
M
R
,ifthe
u
takes its maximum value inside of
Ω
.
Now consider another point
M
∗
inside of
Ω
. We can always find a broken line
that connects
M
0
and
M
∗
. This broken line can be always covered by
a finite number of spheres
K
0
,
inside of
Ω
.Here
K
0
is a sphere with its center located at
M
0
,
K
n
is a sphere with its center in
K
n
−
1
(
K
1
, ···
K
n
,
···
K
N
that are also inside of
Ω
n
=
and
K
N
contains point
M
∗
. In all these spheres,
u
1
,
2
, ··· ,
N
)
(
M
)
≡
u
(
M
0
)
by the
M
∗
)=
above results. In particular,
u
(
u
(
M
0
)
. Therefore,
u
(
M
)
≡
u
(
M
0
)
for all
M
∈
Ω
by the arbitrariness of
M
∗
.
Note also that
u
is continuous in
¯
¯
.This
is a contradiction to the condition in Theorem 4, so
u
can take its maximum value
only on the boundary
Ω
. Therefore
u
(
M
)
≡
u
(
M
0
)
for all
M
∈
Ω
.
In heat conduction, the Laplace equation governs the steady temperature distri-
bution in a domain without an internal heat source or sink. Since the temperature
takes its minimum value on the domain boundary (say, point
M
0
on the boundary)
by the extremum principle, the heat will be conducted inside the domain towards
this point and finally to the outside of the domain. Let
n
be the external normal of
the boundary. The normal derivative of the temperature
∂
∂Ω
u
n
thus must be negative
semi-definite at the point
M
0
. In effect, we can prove that
∂
∂
u
n
is negative definite at
∂
∂
u
point
M
0
. Similarly,
n
is positive definite at the boundary point where
u
takes its
maximum value. All these results are summarized in the following strong extremum
principle.
∂
Theorem 5 (Strong Extremum Principle).
Assume that
u
(
M
)
is a harmonic func-
¯
tion in a closed domain
Ω
, is continuous and non-constant in
Ω
, and takes its min-
imum (maximum) value at point
M
0
on the boundary
∂Ω
. If the normal derivative
∂
u
n
exists at
M
0
, it must be negative (positive) definite.
Proof.
If
u
is a harmonic function,
∂
u
is also a harmonic function. Thereforewithout
loss of generality, we prove the theorem only for the case of minimum value.
Consider a closed sphere
K
R
of radius
R
in
−
Ω
with its spherical surface
∂
K
R
tangential at point
M
0
. By the extremum principle,
u
(
M
0
)
<
u
(
M
)
,
M
∈
K
R
\
M
0
.
Therefore
M
0
≤
∂
u
0
.
∂
n
In order to prove the theorem, we must show that this equality is impossible. Without
loss of generality and for convenience, take the origin as the center of
K
R
. Consider
another sphere
K
R
/
2
of center origin and radius
R
/
2, and an auxiliary function in the
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