Lower semicontinuity of multiple integrals and convergent integrands

LOWER SEMICONTINUITY OF MULTIPLE INTEGRALS

AND CONVERGENT INTEGRANDS

ZHIPING LI

Abstract. Lower semicontinuity of multiple integrals^{R f}(^{xu P} )^d and^{R f} (^{xu P} )^dare studied. It is proved that the two can derive from each other under certain general hypotheses such as uniform lower compactness property and locally uniform convergence of^f to ^f. The result is applied to obtain some general lower semicontinuity theorems on multiple integrals with quasiconvex integrand ^f, while ^f are not required to be quasiconvex.

Keywords: Lower semicontinuity, convergent integrands, lower precompact- ness, weakly precompact, quasiconvex.

1. Introduction and preliminaries

Let be a measurable space with nite positive nonatomic complete measure , let

ff

:

R

^m

R

^{N !}

R

^f+^1g be extended real-valued functions satisfying certain hypotheses, and let

u

: ^!

R

^m,

P

: ^!

R

^N be measurable functions in two linear topological spaces

U

and

V

respectively.

We consider integral functionals of the form

I

(

uP

) =^Z

f

(

xu

(

x

)

P

(

x

))

d

(1.1) and

I

(

uP

) =

Z

f

(

xu

(

x

)

P

(

x

))

d :

(1.2) The main purpose of this paper is to study, under certain general hy- potheses on

UVf

and

f

, lower semicontinuity theorems of the form

I

(

uP

)lim^!1

I

(

u

P

)

(1.3)

I

(

uP

)lim^!1

I

(

u

P

)

(1.4) and the relationship between them. The study of the relationship between (1.3) and (1.4) was motivated mainly by the needs of convergence analysis for numerical solutions to some problems in calculus of variations. For ex- ample, in many cases, we have lower semicontinuity theorems of the form

Department of Mathematics, Peking University, Beijing 100871, People's Republic of China.

Received by the journal August 7, 1995. Accepted for publication June 3, 1996.

The research was supported by a grant of the National Natural Science Foundation of China.

(1.3), however, for one reason or another, the sequence (

u

P

) obtained by numerical methods is often a minimizing sequence with respect to

I

() rather than with respect to

I

(), and a lower semicontinuity theorem of the form (1.4) is needed for convergence analysis (see Li 1]). In some applica- tions it is equally important to know under what conditions (1.3) can be derived from (1.4) (see^x3).

Some useful results are established in this paper on the relationship be- tween (1.3) and (1.4). These results are then applied to establish some gen- eral lower semicontinuity theorems on multiple integrals with quasiconvex integrands. It is worth noticing that in these theorems

f

are not required to be quasiconvex which allows us to choose

f

more freely in numerical computations.

For better understanding of the background and relevant known results, and for the later use of this paper, we rst introduce some denitions and hypotheses.

Definition 1.1. A function

f

:

R

^m

R

^{N !}

R

^f+^1g is called

LB; measurable, if it is measurable respect to the

-algebra generated by products of measurable subsets of and Borel subsets of

R

^m

R

^N. Definition 1.2. A function

f

:

R

^m

R

^{N !}

R

is a Caratheodory function if

(1):

f

(

uP

) is measurable for every

u

²

R

^m and

P

²

R

^N,

(2):

f

(

x

) is continuous for almost every

x

².

Definition 1.3. A function

f

:

R

^m

R

^{N !}

R

^f+^1g has the lower compactness property on

U V

if any sequence of

f

^;(

xu

(

x

)

P

(

x

)) is weakly precompact in

L

¹() whenever (1):

u

converge in

U

,

P

converge in

V

and (2):

I

(

u

P

)

C <

¹ for all

= 1

2

.

f

has the strong lower compactness property on

U V

if any sequence of

f

^;(

xu

(

x

)

P

(

x

)) is weakly precompact in

L

¹() whenever (1) holds. Here

f

^; = min^f

f

0^g.

The assumption of lower compactness property provide us more freedom in applications than the nonnegative assumptions. For example, let

U

be

L

^q(

R

^m), 1

q

+¹, with strong topology and

V

be

L

^p(

R

^N), 1

p

+¹, with weak topology, let

g

:

R

^{m !}

R

^N be such that

j

g

(

xu

)^jp0

c

^j

u

^jq+

b

(

x

)

for some constant

c >

0 and function

b

()²

L

¹(), where

p

⁰= _p^;1p , and let

f

:

R

^m

R

^{N !}

R

satisfy

f

(

xuP

)^h

g

(

xu

)

P

^i;

c

^1j

u

^jq+

b

¹(

x

)

for some constant

c

¹

>

0 and function

b

¹() ²

L

¹(), where ^h

QP

ⁱ =

PNi⁼¹

Q

i

P

i, then it is easy to verify that

f

has the strong compactness prop- erty on

U V

.

Definition 1.4. A sequence of functions

f

:

R

^m

R

^{N !}

R

^f+^1g is said to have the uniform lower compactness property, if

f

^;(

xu

(

x

)

P

(

x

)) are uniformly weakly precompact in

L

¹(), in other words (see 2, 3]),

Esaim: Cocv, June 1996, vol. 1, pp. 169-189

f

(

xu

(

x

)

P

(

x

)) are equi-uniformly integral continuous on ,

i:e:

for any

>

0 there exists

>

0 such that

j Z

0

f

^;(

xu

(

x

)

P

(

x

))

d

^j

<

for all

,

and any measurable subset ⁰ satisfying (⁰)

<

, whenever

u

() converge in

U

,

P

() converge in

V

and

I

(

u

P

)

C <

¹ for all

and

.

Definition 1.5. A sequence of functions

f

:

R

^m

R

^{N !}

R

^f+^1g is said to converge to

f

:

R

^m

R

^{N !}

R

^f+^1g locally uniformly in

R

^m

R

^N, if there exists a sequence of measurable subsets l with (ⁿl) ^!0 as

l

^{! 1} such that for each

l

and any compact subset

G

R

^m

R

^N

f

(

xuP

)^;!

f

(

xuP

)

unifomly on l

G

, as

^!1

:

Definition 1.6. A sequence of functions

f

:

R

^m

R

^{N !}

R

^f+^1g is said to converge to

f

:

R

^m

R

^{N !}

R

^f+^1g locally uniformly in the sense of integration on

U V

, if there exists a sequence of measurable subsets l with (ⁿl)^!0 as

l

^!1 such that

Z

l

nE⁽uPK⁾

f

(

xu

(

x

)

P

(

x

))

d

^!Z

l

nE⁽uPK⁾

f

(

xu

(

x

)

P

(

x

))

d

uniformly in

U V

for each

l

and any xed

K >

0, where

E

(

uPK

) =^f

x

² :^j

u

(

x

)^j

> K

or ^j

P

(

x

)^j

> K

^g

:

Remark 1.7. In numerical computations of singular minimizers, truncation methods turned out to be successful (see 1]). In such applications,

f

can be as simple as

f

(

xuP

) = min^f;

max^f

f

(

xuP

)^gg

which converge to

f

locally uniformly in

R

^m

R

^N if

f

is a Caratheo- dory function, and which have the uniform lower compactness property on

U V

if

f

has the lower compactness property on

U V

.

Remark 1.8. When is a locally compact metric space, lin denition 1.5 and denition 1.6 can be taken to be compact subsets of . In such a case, to say that

f

converge to

f

locally uniformly in

R

^m

R

^N is equivalent to saying that

f

converge to

f

uniformly on every compact subset in

R

^m

R

^N. Especially, if

f

converge to

f

uniformly on every bounded subset in

R

^m

R

^N then

f

converge to

f

locally uniformly in

R

^m

R

^N, if

f

f

are Caratheodory functions and

f

converge to

f

in measure on every bounded subset in

R

^m

R

^N then

f

converge to

f

locally uniformly in the sense of integration on

U V

.

Remark 1.9. To cover more applications, we may consider

f

f

as map- pings from

U V

to^fmeasurale functions

g

: ^!

R

^f+^1gg. This allows us to consider, for example, interpolations of

f

(

xu

(

x

)

P

(

x

)) in nite ele- ment spaces. In fact, denitions and results in this paper can be extended parallely to such mappings without diculty, as we will see that in the proofs of the theorems

f

are related to

f

only as such mappings.

Definition 1.10. (See 4, 5, 6]) A function

f

:

R

^{mn !}

R

is quasiconvex

if ^Z

0

f

(

P

+

D

(

x

))

dx

f

(

P

) (⁰)

for every

P

²

R

^mn,

²

C

⁰¹(⁰

R

^m), and every open bounded subset ⁰

R

ⁿ, where is the Lebesgue measure on

R

ⁿ. Let

R

ⁿbe open and bounded. A Caratheodory function

f

:

R

^m

R

^mn!

R

is quasiconvex in

P

if there exists a subset

E

with (

E

) = 0 such that

f

(

xu

) is quasiconvex for all

x

²ⁿ

E

and

u

²

R

^m.

Example 1.11. Let

R

²,

n

=

m

= 2. Then the function

f

(

Du

) =^j

Du

^j2;(tr

Du

)²

is quasiconvex (in fact it is polyconvex, see 6, 9, 10]) and is unbounded be- low. In addition,

f

has the strong lower compactness property on

L

^p(

R

²²) with weak topology for all

p >

2. If

V

is taken to be

L

²(

R

²²) with weak topology then

f

has a slightly weaker lower compactness property which can be described by Chacon's biting lemma 5, 7, 8, 3] (see lemma 3.6)

Let

D

(

R

^k) =^f

v

: ^!

R

^{k j}

v

is measurable ^g

:

We assume that

U

D

(

R

^m) and

V

D

(

R

^N) are decomposable, i.e.

if

v

() belongs to one of them, then

T()

v

() belongs to the same space whenever

T

is a measurable subset of , where

T() is the characteristic function of

T

:

T(

x

) =

( 1

if

x

²

T

0

if

x =

²

T:

We assume that

U

and

V

satisfy the following hypotheses on their topolo- gies:

(H1): If

v

()

= 1

2

belong to one of the spaces and converge there to zero and if

T

^!0, then

T ()

v

() also converge to zero.

(H2): The topology in

U

is not weaker than the topology of convergence in measure the topology of

V

is not weaker than the topology induced in

V

by the weak topology of

L

¹(

R

^N).

It is easy to see that if

U

is taken to be

L

^q(

R

^m), 1

q

+¹, with strong topology and

V

is taken to be

L

^p(

R

^m) with weak topology for 1

p <

+¹ or weak topology for

p

= +¹, then (H1) and (H2) are satised. This covers most applications in Sobolev spaces. For applications concerning Orlicz spaces see for example 2].

Esaim: Cocv, June 1996, vol. 1, pp. 169-189

Remark 1.12. Here and throughout this paper, assumptions and state- ments are referred to sets with measure-negligible projections on , i.e.

they hold on a subset ⁰ with ⁰= .

In Reshetnyak's result ( see theorem 1.2 in 11]), is taken to be a local compact metric space,

ff

:

R

^{N !}

R

are nonnegative functions such that for any

>

0 there is a compact set

A

with (ⁿ

A

)

<

and

f

(

xu

)

f

(

xu

) being continuous on

A R

^N,

f

(

x

)

f

(

x

) are convex for almost all

x

², and

f

^!

f

locally uniformly in

R

^N as

^!1, and

V

is taken to be

L

¹(

R

^N) with weak topology.

In the case when

f

(

xu

) is convex, (1.3) was proved (see Ioe 2]) under the hypotheses that

f

satises lower compactness property and

U

,

V

satisfy (H1) and (H2), and (1.4) was proved (see Li 12]), by using (1.3) , under the hypotheses that

f

has the lower compactness property,

f

have the uniform compactness property and

f

^!

f

locally uniformly in

R

^m

R

^N.

The results of the type (1.3) concerning quasiconvexity of

f

(

xu

) and

P

=

Du

can be found in 5, 6]. The following theorem, which will be used in ^x3, was established by Acerbi and Fusco 5, 6].

Theorem 1.13. Let

R

ⁿ be bounded and open. Let

F

(

u

) =

Z

f

(

xuDu

)

dx u

²

W

^1p(

R

^m)

where 1

p <

¹, and where

f

:

R

^m

R

^mn!

R

satises

(1):

f

(

) is a Caratheodory function

(2):

f

(

xu

) is quasiconvex

(3): 0

f

(

xuP

)

a

(

x

) +

C

(^j

u

^jp+^j

P

^jp) for every

x

²

u

²

R

^m and

P

²

R

^mn, where

C >

0 and

a

()²

L

¹().

Then, the functional

u

^!

F

(

u

) is sequentially weakly lower semicontinuous on

W

^1p(

R

^m), i.e. (1.3) holds for

P

=

Du

and

P

=

Du

with

u

* u

in

W

^1p(

R

^m), where and in what follows "

*

" means "converges weakly to".

Ball and Zhang 6] generalized the above result to cover the case when

j

f

(

xuP

)^j

a

(

x

) +

C

(^j

u

^jp+^j

P

^jp)

and proved that (1.3) holds on each ⁿ

E

k, where ^f

E

k^g is a nonincreasing sequence of measurable subsets of with limk^!1

E

k = 0.

In ^x2, It is shown, under certain general hypotheses, such as uniform lower compactness property and locally uniform convergence of

f

to

f

(see theorem 2.1 for details), that (1.3) and (1.4) can somehow be derived from each other, and the results are further developed into a theorem concerning

;-limit (see theorem 2.8, for ;-limit, ;-convergence and their applications in calculus of variations see for example 13, 14]). These results generalized the result of Li 12] which was proved for the case when

f

(

xu

) is convex.

In ^x3, the results established in ^x2 are applied to prove some lower semi- continuity theorems of the form (1.3) and (1.4) for quasiconvex integrands,

which generalize the results of Acerbi and Fusco 5], Ball and Zhang 6] and Li 12].

2. Lower semicontinuity and convergent integrands The following theorem establishes the relationship between the lower semicontinuity theorems of the form (1.3) and those of the form (1.4).

Theorem 2.1. Let be a measurable space with nite positive non- atomic complete measure . Let

U

and

V

satisfy (H1) and (H2). Let ^f

u

^g

u

²

U

and^f

P

^g

P

²

V

be such that

u

^;!

u

in

U

(2.1)

and

P

^;!

P

in

V:

(2.2)

Let

f

^f

f

^g:

R

^m

R

^{N !}

R

^f+^1g satisfy

(i):

f

^f

f

^g are ^LB- measurable

(ii):

f

^;(

xu

(

x

)

P

(

x

))

f

^;(

xu

(

x

)

P

(

x

)), and

f

^;(

xu

(

x

)

P

(

x

)) are weakly precompact in

L

¹()

(iii):

f

^!

f

locally uniformly in

R

^m

R

^N. Then, we have (a):

Z

f

(

xuP

)

d

lim^!1

Z

f

(

xu

P

)

d

(2.3) provided that

Z

0

f

(

xuP

)

d

lim^!1

Z

0

f

(

xu

P

)

d

(2.4) for all measurable subset⁰

and (b):

Z

f

(

xuP

)

d

lim^!1

Z

f

(

xu

P

)

d

(2.5) provided that

f

^f

f

^g satisfy a further hypothesis

(iv):

f

⁺(

xu

(

x

)

P

(

x

))

a

(

x

) +

f

⁺(

xu

(

x

)

P

(

x

)), where

a

() ²

L

¹() is nonnegative

and

Z

f

(

xuP

)

d

lim^!1

Z

f

(

xu

P

)

d :

(2.6) To prove the theorem, we need the following lemmas.

Lemma 2.2. Let ^f

u

^g

u

²

U

and ^f

P

^g

P

²

V

satisfy (2.1) and (2.2) re- spectively. Let

E

¹(

K

) =^f

x

² :^j

u

(

x

)^j

> K

^g

E

²(

K

) =^f

x

² :^j

P

(

x

)^j

> K

^g

Esaim: Cocv, June 1996, vol. 1, pp. 169-189

and

E

(

K

) =

E

¹(

K

)

E

²(

K

)

:

(2.7) Then

E

(

K

)^;!0

uniformly for

as

K

^!1

:

Proof. For any

>

0, since

u

²

U

, there exists

K

¹(

)

>

1 such that

f

x

² :^j

u

(

x

)^j

> K

^g

< =

2

⁸

K > K

¹(

)

:

Thus, by (2.1) and (H2), there exists

(

)

>

1 such that

E

¹(

K

)

< =

2

⁸

>

(

) and

K > K

¹(

) + 1

:

(2.8) Since

u

²

U

for each

, we have

Klim^!1

E

¹(

K

) = 0

for each

:

Thus, for

^2f1

2

(

)^g, there exists

K

²(

)

>

1 such that

E

¹(

K

)

< =

2

⁸

^2f1

2

(

)^g and

K > K

²(

)

:

(2.9) Let

K

(

) = max^f

K

¹(

) + 1

K

²(

)^g, then (2.8) and (2.9) give

E

¹(

K

)

< =

2

⁸

1 and

K > K

(

)

:

(2.10) On the other hand, it follows from (2.2) and (H2) that

Z

j

P

(

x

)^j

d

C

for some constant

C >

0. Thus, for any

>

0 there exists

K

(

)

>

1 such that

E

²(

K

)

< =

2

⁸

1 and

K > K

(

)

:

(2.11) Hence the lemma follows from (2.10) and (2.11). ² Lemma 2.3. Let

f

^f

f

^g satisfy the hypotheses in theorem 2.1. Let

f

A :

R

^m

R

^{N !}

R

be dened by

f

A(

xuP

) = min^f

Af

(

xuP

)^g

:

(2.12) Let^f

u

^g

u

²

U

and ^f

P

^g

P

²

V

satisfy (2.1) and (2.2) respectively. Let

fl^g be a sequence of measurable subsets of , the existence of which is guaranteed by the hypothesis (iii) for

f

, such that

(ⁿl)^;!0

as

l

^!1

(2.13)

and

f

^;!

f

uniformly on l

G

(2.14)

for each

l

and any compact set

G

R

^m

R

^N.

Suppose ^Z

f

(

xu

P

)

d

C

for some constant

C >

0.

Then, for any

>

0

A >

1, there exist

l

(

) 1 and

(

Al

) 1 such that

R

l

f

A(

xu

P

)

d

^R

f

(

xu

P

)

d

+

8

l

l

(

) and

(

Al

)

:

^(2.15)

Proof.

R

l

f

A(

xu

P

)

d

= ^R

f

(

xu

P

)

d

+^{R n} l(^;

f

(

xu

P

))

d

+ +^R l(

f

A(

xu

P

)^;

f

(

xu

P

))

d

= ^R

f

(

xu

P

)

d

+

I

¹+

I

²

:

By (2.1), (2.2), (2.13) and (ii), there exists

l

(

)

>

0 such that

I

¹ = ^{R n} l(^;

f

(

xu

P

))

d

R

n

l(^;

f

^;(

xu

P

))

d

< =

2

if

l

l

(

)

:

^(2.16) By (2.12), we have

I

^{2 R} l

nE ⁽K⁾(

f

(

xu

P

)^;

f

(

xu

P

))

d

+ +^R_E ⁽_K⁾(

A

^;

f

(

xu

P

))

d

=

I

²¹+

I

²²

where

E

(

K

) is dened by (2.7).

By lemma 2.2,

E

(

K

) ^! 0 uniformly for

as

K

^{! 1}. Thus it follows from (ii) that there exists

K

(

A

)

>

1 such that

I

^{22 R}_E ⁽_K⁾(

A

^;

f

^;(

xu

P

))

d

< =

4

if

K

K

(

A

)

:

Let

K

=

K

(

A

), then

G

(

K

) =^f

u

²

R

^m :^j

u

^j

K

^{g f}

P

²

R

^N :^j

P

^j

K

^g

is a compact set in

R

^m

R

^N. It follows from (2.14) that there exists

(

Al

)

>

0 such that

j

I

^21j

< =

4

⁸

(

Al

)

:

Thus, we have

j

I

^2j

< =

2

⁸

(

Al

)

:

(2.17)

Thus (2.15) follows from (2.16) and (2.17). ²

Lemma 2.4. Let

f

^f

f

^gsatisfy the hypotheses in theorem 2.1. Let^f

u

^g

u

²

U

and ^f

P

^g

P

²

V

satisfy (2.1) and (2.2) respectively. Let

F

(

A

) =^f

x

² :

f

(

xu

(

x

)

P

(

x

))

> A

^g

:

Suppose

Z

f

(

xu

P

)

d

C

(2.18)

Esaim: Cocv, June 1996, vol. 1, pp. 169-189

for some constant

C >

0.

Then, for any

>

0 and

K

1, there exist

A

(

)

>

1 and

(

K

)

>

1 such that

F

(

A

)

E

(

K

) +

if

A

A

(

) and

(

K

)

^(2.19) where

E

(

K

) is dened by (2.7).

Proof. By (iii), there is a sequence of measurable subsets^fl^g in such that

(ⁿl)^;!0

as

l

^!1

(2.20) and

f

^;!

f

uniformly on l

G

(2.21)

for each

l

and any compact set

G

R

^m

R

^N.

For any

>

0, by (2.20), there is

l

¹(

)1 such that

(ⁿl)

< =

2

if

l

l

¹(

)

:

(2.22) By (2.18),

R

l

nE ⁽K⁾

f

(

xu

P

)

d

= ^R l

nE ⁽K⁾(

f

(

xu

P

)^;

f

(

xu

P

))

d

+^{R n(} l

nE ⁽K⁾⁾(^;

f

(

xu

P

))

d

+

C

=

I

¹+

I

²+

C

It follows from (ii) that

I

^{2 Z}

n(

l

nE ⁽K⁾⁾(^;

f

^;(

xu

P

))

d

Z

(^;

f

^;(

xu

P

))

d

C

¹

(2.23)

for some constant

C

¹

>

0.

It follows from (2.21) that there exists

(

lK

)

>

1 such that

j

I

^{1j R} l

nE ⁽K^{) j}

f

(

xu

P

)^;

f

(

xu

P

)^j

d

1

⁸

(

lK

)

:

(2.24)

Thus we have

Z

l

nE ⁽K⁾

f

(

xu

P

)

d

C

²

⁸

(

lK

)

(2.25) where

C

²=

C

+

C

¹+ 1 is a constant.

Denote ^; = ^f

x

² :

f

(

xu

(

x

)

P

(

x

))

<

0^g and

f

⁺ = max^f

f

0^g then, by (2.25)

R

l

nE ⁽K⁾

f

⁺(

xu

P

)

d

R

l

nE ⁽K⁾(^;

f

^;(

xu

P

))

d

+

C

²

= ^R( l

nE ⁽K^{))\ ;}(^;

f

(

xu

P

))

d

+

C

²

= ^R( l

nE ⁽K^{))\ ;}(

f

(

xu

P

)^;

f

(

xu

P

))

d

+^R( l

nE ⁽K^{))\ ;}(^;

f

(

xu

P

))

d

+

C

²

R

l

nE ⁽K^)j

f

(

xu

P

)^;

f

(

xu

P

)^j

d

+^R (^;

f

^;(

xu

P

))

d

+

C

²

:

It follows from this and (2.23), (2.24) that

Z

l

nE ⁽K⁾

f

⁺(

xu

P

)

d

C

³

⁸

(

lK

)

(2.26) where

C

³=

C

¹+

C

²+ 1 is a constant.

Now (2.26) implies that there exists

A

(

)

>

0 such that

f

x

²lⁿ

E

(

K

) :

f

(

xu

(

x

)

P

(

x

))

> A

^g

< =

2

if

A

A

(

) and

(

lK

)

:

(2.27) Since

F

(

A

)

E

(

K

)(ⁿl)

F

(

lKA

)

where

F

(

lKA

) =^f

x

²lⁿ

E

(

K

) :

f

(

xu

(

x

)

P

(

x

))

> A

^g

we have

F

(

A

)

E

(

K

) + (ⁿl) +

F

(

lKA

)

:

Taking

l

=

l

¹(

) and

(

K

) =

(

l

¹(

)

K

), by (2.22) and (2.27), we

conclude that (2.19) is true. ²

Proof of Theorem 2.1.

Without loss of generality, we assume that

Z

f

(

xu

P

)

d

C

for some constant

C >

0. It follows from (iii) that there exists a sequence of measurable subsets^fl^g of such that

l l⁺¹

⁸

l

and lim_l

!1

(ⁿl) = 0

(2.28) and

f

^;!

f

uniformly on l

G

(2.29)

for each

l

and any compact set

G

R

^m

R

^N.

Esaim: Cocv, June 1996, vol. 1, pp. 169-189

Let

E

(

K

) be dened by (2.7). It follows from lemma 2.2 that there exists an increasing sequence ^f

K

i^gsuch that

1

X

i⁼¹ sup

1<^1f

E

(

K

i)^g

<

¹

:

(2.30) Let

i

>

0

i

= 1

2

be a decreasing sequence of numbers satisfying limi^!1

i = 0. Let

A

i =

A

(

i

=

2ⁱ)

l

i =

l

(

i) i = max^f

(

l

i

K

i)

(

i

A

i

l

i)^g and

F

i =

F

(

i

A

i) = ^f

x

² :

f

(

xu

ⁱ(

x

)

P

ⁱ(

x

))

> A

i^g

where

A

() (

) are dened by lemma 2.4 and

l

() (

_

) are dened by lemma 2.3. Then, by lemma 2.3, we have

Z

l

i

f

Aⁱ(

xu

ⁱ

P

ⁱ)

d

^Z

f

ⁱ(

xu

ⁱ

P

ⁱ)

d

+

i ⁸

i

(2.31) and by lemma 2.4, we have

1

X

i⁼¹

F

i

1

X

i⁼¹(

E

ⁱ(

K

i) +

i

=

2ⁱ)

<

¹

:

(2.32) Let

H

j = ((ⁿl^j)(i j

F

i)) and

G

j = ⁿ

H

j. It follows from (2.28) and (2.32) that

G

j

G

j⁺¹

⁸

j

and lim_j

!1

(ⁿ

G

j) = 0

:

(2.33) Thus, by the denition of

f

Aⁱ and

F

i, we have

R

G^j

f

(

xu

ⁱ

P

ⁱ)

d

= ^R l i

nFⁱ

f

(

xu

ⁱ

P

ⁱ)

d

+^R( l i

nFⁱ⁾ⁿG^j(^;

f

(

xu

ⁱ

P

ⁱ))

d

R

l

i

nFⁱ

f

(

xu

ⁱ

P

ⁱ)

d

+^{R n}_Gj(^;

f

^;(

xu

ⁱ

P

ⁱ))

d

R

l

i

nFⁱ

f

Aⁱ(

xu

ⁱ

P

ⁱ)

d

+^R_Hj(^;

f

^;(

xu

ⁱ

P

ⁱ))

d

⁸

i

j:

It follows from this and (2.31) that

R

G^j

f

(

xu

ⁱ

P

ⁱ)

d

R

f

ⁱ(

xu

ⁱ

P

ⁱ)

d

+^R_H^j(^;

f

^;(

xu

ⁱ

P

ⁱ))

d

+

i

⁸

i

j:

(2.34) Let

i

^!1 in (2.34). By (2.4), we have

R

G^j

f

(

xuP

)

d

lim_i^!1R

f

ⁱ(

xu

ⁱ

P

ⁱ)

d

+limi^!1RH^j(^;

f

^;(

xu

ⁱ

P

ⁱ))

d :

(2.35) By (ii) and (2.33), we have

jlim^!1(sup_i

1 Z

H^j(^;

f

^;(

xu

ⁱ

P

ⁱ))

d

) = 0

:

It follows from this and (2.33), (2.35) that

R

f

(

xuP

)

d

= limj^!1RG^j

f

(

xuP

)

d

lim_i^!1R

f

ⁱ(

xu

ⁱ

P

ⁱ)

d :

This completes the proof of (a).

Next, we prove (b). Without loss of generality, we assume that

Z

f

(

xu

P

)

d

C <

¹

:

(2.36) It follows from (ii), (iv) and (2.36) that

Z

f

(

xu

P

)

d

C

¹

<

¹

:

By (2.6), for any

>

0, there exists ^

(

)

>

0 such that

Z

f

(

xuP

)

d

^Z

f

(

xu

P

)

d

+

⁸

^(

)

:

(2.37) By (ii), for any

>

0, there exists ¹(

)

>

0 such that

j Z

0

f

^;(

xu

P

)

d

^j

<

^{8 0 1}(

)

(2.38)

j Z

0

f

^;(

xu

P

)

d

^j

<

^{8 0 1}(

)

:

(2.39) By (iv) and (ii), for any

>

0, there exists ²(

)

>

0 such that

Z

0

f

⁺(

xu

P

)

d

Z

0

f

⁺(

xu

P

)

d

+

Z

0

a

(

x

)

d

Z

0

f

(

xu

P

)

d

+

⁸⁰ and ^{0 2}(

)

:

(2.40) Let

E

(

K

) =^f

x

² :^j

u

(

x

)^j

> K

or ^j

P

(

x

)^j

> K

^g

:

By lemma 2.2, for any

>

0 there exists

K

( )

>

0 such that

E

(

K

)

<

⁸

K

K

( )

:

(2.41) By (iii), there exists a sequence of measurable subsets l such that (ⁿl) ^!0 as

l

^{! 1} and

f

^!

f

uniformly on l

G

for each xed

l

and compact set

G

R

^m

R

^N. Thus, for any

>

0 there exists

l

( )1 such that

(ⁿl)

<

⁸

l

l

( )

(2.42) and for any

>

0,

l

1 and compact set

G

R

^m

R

^N there exists

(

lG

)

>

0 such that

j

f

(

xvQ

)^;

f

(

xvQ

)^j

<

(2.43) for all

(

lG

) and (

xvQ

)²l

G

.

Esaim: Cocv, June 1996, vol. 1, pp. 169-189

Now, by taking

= min^{f 1}(

) ²(

)^g

K

=

K

( )

l

=

l

( )

G

=^f(

vQ

)²

R

^m

R

^N :^j

v

^j

K

and ^j

P

^j

K

^g and

(

) = max^f

^(

)

(

lG

)^g

we get from (2.37) { (2.43) that

Z

f

(

xuP

)

d

Z

f

(

xu

P

)

d

+

Z

l

nE ⁽K⁾

f

(

xu

P

)

d

+

Z

n(

l

nE ⁽K⁾⁾

f

⁺(

xu

P

)

d

+

Z

l

nE ⁽K⁾

f

(

xu

P

)

d

+

Z

n(

l

nE ⁽K⁾⁾

f

⁺(

xu

P

)

d

+ 3

Z

f

(

xu

P

)

d

+ 4

⁸

>

(

)

:

This and the arbitrariness of

imply (2.5). ²

Corollary 2.5. If the hypothesis (iii) is replaced by the hypothesis

(iii)^0:

f

^!

f

locally uniformly in the sense of integration on

U V

, in theorem 2.1, the conclusions of the theorem still hold.

Proof. Since in the proof of theorem 2.1 the hypothesis (iii) was only used to show that there exists

(

lK

)

>

0 such that

j Z

l

nE ⁽K⁾

f

(

xu

P

)^;

f

(

xu

P

)

d

^j

< =

4

for

(

lK

), the result follows. ²

Corollary 2.6. If the hypothesis (ii) is replaced by the hypothesis

(ii)^0:

f

has the strong lower compactness property and ^f

f

^g have the uniform lower compactness property,

in theorem 2.1, the conclusions of the theorem still hold.

Proof. The proof of the part (a) of theorem 2.1 remains valid, since (ii)⁰ and the assumption^R

f

(

xu

(

x

)

P

(

x

))

d

C <

¹ give (ii).

The proof of the part (b) of theorem 2.1 still holds, because (ii)⁰, (iv) and the assumption^R

f

(

xu

(

x

)

P

(

x

))

d

C <

¹ imply

Z

f

(

xu

(

x

)

P

(

x

))

d

C

¹

<

¹

and hence (ii). ²