Environmental Engineering Reference
In-Depth Information
Example 1.
Demonstrate that the extremum principle does not hold for
u
xx
+
u
yy
+
cu
=
0
(
c
>
0
)
.
Solution
. Note that
0 is not a Laplace equation, hence the extremum
principle is not valid. To demonstrate this, consider a function
Δ
u
+
cu
=
sin
c
sin
c
u
=
2
x
·
2
y
.
u
satisfi
es
Δ
u
+
cu
=
0
and
is eq
ual t
o zero on t
he b
oundary of the square domain
−
2
≤
2
π
,−
2
≤
2
¯
Ω
:
/
c
π
≤
x
/
c
/
c
π
≤
y
/
c
π
.However,
u
(
x
,
y
)=
±
1when
2
2
2
2
1
1in
¯
1
=
/
=
±
/
(
,
)
≤
x
c
π
and
y
c
π
.Also,
u
x
y
Ω
. Therefore the
u
solving
the equation
Δ
u
+
cu
=
0 does not take its minimum and maximum values on the
boundary.
Example 2
.Let
Ω
be a
n
-dimensional bounded region whose boundary is denoted
by
∂Ω
. Show that the necessary condition for the existence of a solution of
Δ
u
=
f
(
x
)
,
x
=(
x
1
,
x
2
, ··· ,
x
n
)
∈
Ω
,
(7.90)
∂Ω
∂
u
u
|
∂Ω
=
c
,
n
d
S
=
A
,
∂
is
f
d
x
=
A
.
Ω
Here
c
and
A
are constants,
f
is a function of
n
variables, the integral is
n
-multiple
and
d
x
=
d
x
1
,
d
x
2
, ··· ,
d
x
n
Proof
.Let
v
=
1 in the second Green formula. An application of the second Green
formula yields
Ω
Δ
∂
u
u
d
x
=
n
d
S
.
∂
∂Ω
f
, we arrive at
Ω
∂Ω
∂
u
Since
Δ
u
=
f
d
x
=
n
d
S
=
A
.
∂
¯
Example 3.
Suppose that
u
∗
∈
C
2
C
1
(
Ω
)
∩
(
Ω
)
is a solution of
⎧
⎨
Δ
u
=
f
(
x
)
,
x
=(
x
1
,
x
2
, ··· ,
x
n
)
∈
Ω
,
(7.91)
∂Ω
∂
u
⎩
u
|
∂Ω
=
c
,
n
d
S
=
A
,
∂
where
c
is a undetermined constant and
A
is a given constant. Show that all solutions
of PDS (7.91) can be written as
u
∗
+
u
=
a
,
a
is a constant
.
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