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Furthermore, if (8.21) holds, then h must satisfy
h
0
.0/
0;
(8.23)
and if (8.22) is the case, then it follows that
h
0
.1/
0:
(8.24)
Further information on this important topic can be found in your favorite calculus
textbook. It is important that you get this right. In order to illuminate the properties
(
8.19
), (
8.20
), (8.23), and (8.24), we recommend the reader make some plots of
smooth functions defined on the closed unit interval!
Recall that the unknown function
u
in (
8.16
)-(
8.18
) is a function of two variables.
Are similar properties as those stated above valid in this case? Yes, indeed: The same
results hold with respect to each of the variables!
Consider a smooth function
v
of the spatial position x
2
Œ0; 1 and time t
2
Œ0; T ,
i.e.,
v
D
v
.x; t /. More precisely, we assume that the partial derivatives of all orders
of
v
with respect x and t are continuous, and that
v
W
˝
T
!
IR ;
where
˝
T
D f
.x; t /
j
0<x<1and 0<t<T
g
;
(8.25)
@˝
T
D f
.x; 0/
j
0
x
1
g [ f
.1; t /
j
0
t
T
g [ f
.x; T /
j
0
x
1
g
[ f
.0; t /
j
0
t
T
g
;
(8.26)
and
˝
T
D
˝
T
[
@˝
T
:
(8.27)
That is, ˝
T
denotes the closure of ˝
T
and thus forms a closed set. The reader
should draw a figure illustrating these sets!
Remember that a continuous function defined on a closed set always achieves
both its maximum and minimum value within this set, see, e.g., Ap
ost
ol [4]. Assume
that .x
;t
/
2
˝
T
, an interior point, is a maximum point for
v
in ˝
T
, i.e.,
v
.x; t /
v
.x
;t
/
for all .x; t /
2
˝
T
:
Then, as in the single-variable case,
v
must satisfy
v
x
.x
;t
/
D
0
v
t
.x
;t
/
D
0;
and
(8.28)
v
xx
.x
;t
/
0
v
tt
.x
;t
/
0:
and
(8.29)
This means that if
w
D
w
.x; t / is a smooth function defined on
˝
T
such that
