8.1 Minimizing (Maximizing) Sequence
455
J 0 ≥ J 1 , the rest may be inferred, there is J i ≥ J i+1 , thus there is
J 0 ≥ J 1 ≥ J 2 ≥ · · · ≥ J n ≥ · · · ≥ J
(8.1.2)
where, J is the true minimum of the functional.
The function sequence {u n }(n = 1, 2, . . .) that can make the various order approximate extrema like the inequality (8.1.2) tend to the true minimum is called the minimizing sequence of the functional. The minimizing sequence provides the upper
limit of minimum of the functional.
Similarly, for a maximum problem, if the function sequence {u n }(n = 0, 1, 2, . . .)
can make the various order approximate extrema be suitable to the following
inequality
J 0 ≤ J 1 ≤ J 2 ≤ · · · ≤ J n ≤ · · · ≤ J
(8.1.3)
where, J is the true maximum of the functional, then the function sequence is called
the maximizing sequence of the functional. The maximizing sequence provides the
lower limit of the functional.
Now put forward such the problem, if the limit of the various order approximate
extremum of a functional is the true extremum of the functional, namely
lim
n→∞
J n = J
(8.1.4)
ask whether the limit of minimizing (or maximizing) sequence is the true extremal
function? namely whether must the limit
lim
n→∞
u n = y(x)
(8.1.5)
hold? where, y(x) is the extremum function that makes the functional reach the
minimum (or maximum).
The answer is conditional. In 1870, Weierstrass pointed out first with a special case
that the limit of the minimizing (or maximizing) sequence is not unconditional true
extremal function. In other words, only under the certain conditions, the minimizing
(or maximizing) sequence can converges to the extremal function that makes the
functional reach minimum (or maximum). When J [y] is a functional equivalent to
a positive definite operator, under the certain conditions, its every minimizing (or
maximizing) sequence converges to the extreme function that makes the functional
reaches minimum (or maximum). The following example gives a counterexample in
which the correct minimization sequence of extremum function can not be obtained.
Example 8.1.1 The Dirichlet problem. Find the minimum of the functional
J [u] =
¨
D
(u
2
x + u
2
y )dxdy
(1)
455
J 0 ≥ J 1 , the rest may be inferred, there is J i ≥ J i+1 , thus there is
J 0 ≥ J 1 ≥ J 2 ≥ · · · ≥ J n ≥ · · · ≥ J
(8.1.2)
where, J is the true minimum of the functional.
The function sequence {u n }(n = 1, 2, . . .) that can make the various order approximate extrema like the inequality (8.1.2) tend to the true minimum is called the minimizing sequence of the functional. The minimizing sequence provides the upper
limit of minimum of the functional.
Similarly, for a maximum problem, if the function sequence {u n }(n = 0, 1, 2, . . .)
can make the various order approximate extrema be suitable to the following
inequality
J 0 ≤ J 1 ≤ J 2 ≤ · · · ≤ J n ≤ · · · ≤ J
(8.1.3)
where, J is the true maximum of the functional, then the function sequence is called
the maximizing sequence of the functional. The maximizing sequence provides the
lower limit of the functional.
Now put forward such the problem, if the limit of the various order approximate
extremum of a functional is the true extremum of the functional, namely
lim
n→∞
J n = J
(8.1.4)
ask whether the limit of minimizing (or maximizing) sequence is the true extremal
function? namely whether must the limit
lim
n→∞
u n = y(x)
(8.1.5)
hold? where, y(x) is the extremum function that makes the functional reach the
minimum (or maximum).
The answer is conditional. In 1870, Weierstrass pointed out first with a special case
that the limit of the minimizing (or maximizing) sequence is not unconditional true
extremal function. In other words, only under the certain conditions, the minimizing
(or maximizing) sequence can converges to the extremal function that makes the
functional reach minimum (or maximum). When J [y] is a functional equivalent to
a positive definite operator, under the certain conditions, its every minimizing (or
maximizing) sequence converges to the extreme function that makes the functional
reaches minimum (or maximum). The following example gives a counterexample in
which the correct minimization sequence of extremum function can not be obtained.
Example 8.1.1 The Dirichlet problem. Find the minimum of the functional
J [u] =
¨
D
(u
2
x + u
2
y )dxdy
(1)