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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