Existence of Scalar Minimizers for Nonconvex Simple Integrals of Sum Type

Ž .JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 221, 559]573 1998ARTICLE NO. AY985915

Existence of Scalar Minimizers for Nonconvex SimpleIntegrals of Sum Type

Antonio Ornelas*´

´Cima-ue, rua Romao Ramalho 59, P-7000 E¨ora, Portugal˜

Submitted by Arrigo Cellina

Received June 23, 1997

One usually asks, as sufficient condition to guarantee the existence of minimiz-w x b� Ž . Ž .4 Ž .ers x: a, b ª R in AC for the integral H h x9 q w x dt}with h ? havinga

Ž .superlinear growth at ` and w ? being any lower semicontinuous function boundedŽ . Ž . Ž .below}that h ? be convex, that is, h j s h** j at all j in R.

Ž . Ž .The purpose of this paper is to show that the condition h 0 s h** 0 is actuallyenough to guarantee the existence of minimizers. Q 1998 Academic Press

Key Words: calculus of variations; nonconvex integrals.

In order to ensure that the integral

bh x9 t q w x t dt 1� 4Ž . Ž . Ž .Ž . Ž .H

a

has minimizers among the absolutely continuous functions passing throughŽ . Ž .the points a, A , b, B , a condition which turns out to play a distin-

guished role is

h** 0 s h 0 , 2Ž . Ž . Ž .

Ž .where h** ? is, as usual, the largest convex function less than or equal toŽ .h ? . Indeed, for w, h: R ª R lower semicontinuous and bounded below,

with

h jŽ .< <ª q` as j ª `, 3Ž .

< <j

* E-mail: ornelas@dmat.uevora.pt.

559

0022-247Xr98 $25.00Copyright Q 1998 by Academic Press

All rights of reproduction in any form reserved.

ANTONIO ORNELAS´560

w x Ž .it was proved in 10 that 1 always has minimizers if, moreover, theŽ .boundary of each level set of w ? ,

­ s : w s s const , 4� 4Ž . Ž .is a null set; and this is a very weak restriction, since the boundaries oflevel sets have, for usual functions, at most countably many points.

Ž .Whoever is concerned with the minimization of integrals like 1 asŽ .mathematical models for systems of applied science}in which often w ?Ž .is determined only within an experimental error}might argue that 4

always holds true. However}be it for the sake of insight, completeness,Ž .simplicity}one is naturally led to wonder whether hypothesis 4 is really

Ž . Ž .necessary in order to prove the existence of minimizers for 1 under 2 .Ž .The first and crucial step toward the elimination of hypothesis 4 was

w x Ž .performed in Theorem 6 of 10 , showing that 1 still attains its minimumŽ .even when w ? has level sets with positive measure boundary, at least

Ž .provided a price is paid as a compensation for such liberty on w ? , namelyŽ .that the function h ? have an especially simple set

K s j g R: h** j - h j 5� 4Ž . Ž . Ž .Ž .of noncontact points with h** ? }exactly one interval with 0 as one

endpoint.The aim of this paper is to show that no price has to be paid for the

Ž . Ž .desired liberty on w ? : by the theorem proved below, 1 has minimizersŽ . Ž . Ž .provided w ? , h ? satisfy the hypotheses up to 3 . The main tool is the

w xaforementioned Theorem 6 of 10 , used in conjunction with the LiapunovŽ w x.theorem on the range of vector measures see, e.g., 3 . The author thanks

Paolo Marcellini and Nicola Fusco for the development of the techniquesw xof 10 and for stimulating this research.Sufficient conditions for the existence of scalar minimizers for simple

Ž . Ž .integrals like 1 , with general nonconvex h ? , with superlinear growth,Ž .and nonlinear w ? , were first obtained, independently of each other, by

Ž 1 Ž .Aubert and Tahraoui for C strictly monotone w ? with several restric-Ž . w x. Ž Ž . w x.tions on h ? }see 4 , by Marcellini for monotone w ? }see 12 ,

Ž w x.by Raymond under strong smoothness assumptions}see 16 , and byŽ Ž . w x.Cellina and Colombo for concave w ? }see 5 . More recently, Cellina

Ž .and Mariconda analyzed the set GG of ‘‘good’’ functions w ? }that is,Ž . Ž .those for which 1 has minimizers for general h ? }showing that it is a

Ž w x.dense subset of the space of continuous functions bounded below see 6 .w xIn 14 the set GG was characterized as a meager subset of the same space

}in the sense of Baire category}and was shown to contain the class ofŽ .concave]monotone functions. This class consists of all the functions w ?

Ž .such that any s g R belongs to an open interval along which w ? is eitherconcave or monotone}in particular, they cannot have strict local mini-mum points. These results were obtained using tools developed by Amar

EXISTENCE OF SCALAR MINIMIZERS 561

Ž .and Cellina, who have drawn attention to the special case 2 , allowing, inŽ . Ž w x.this case, w ? to have isolated strict local minimum points see 1 .

A general survey of existence results for nonconvex minimization prob-w xlems may be found in 13 .

The main result is the following.

THEOREM 1. Let h, w : R ª R be lower semicontinuous functions suchthat

h jŽ .< <h** 0 s h 0 , ª q` as j ª `Ž . Ž .

< <j

Ž .and w ? is bounded below.Then for any A, B the integral

bh x9 t q w x t dt� 4Ž . Ž .Ž . Ž .H

a

Ž .has a minimizer x ? in the class of the absolutely continuous functionsŽ . Ž .satisfying x a s A, x b s B.

Ž .Proof. a In what follows we shall suppose generally that A F B; ifB were less than A, a similar reasoning would be used.

Ž . Ž .Let h** ? be the bipolar of h ? , namely, the largest convex function lessŽ . Ž .than or equal to h ? , and let ­ h** ? be its subdifferential, in the sense of

Ž w x.convex analysis see, e.g., 17, 9 . It is well known that the set

y1­ h** ? m [ j g R: m g ­ h** j� 4Ž . Ž . Ž .Ž .

is, for each m g R, a compact nonempty interval. Without loss of general-Ž .ity}possibly subtracting an affine function to h ? }we may suppose that

min h** j s h** 0 s 0, 0 s max ­ h** 0 . 6Ž . Ž . Ž . Ž .jgR

We may also suppose, to simplify the notation, that

w xmin w A , B s 0. 7Ž .Ž .

Define the numbers

m [ min ­ h** 0 F m [ 0 s max ­ h** 0 ,Ž . Ž .y0 0

y1a [ min ­ h** ? mŽ . Ž .Ž .y0 y0

y1F b [ 0 \ a F b [ max ­ h** ? 0 . 8Ž . Ž . Ž .Ž .y0 0 0

ANTONIO ORNELAS´562

Ž .Since the set K in 5 is open, there exist at most countably many numbers

m - m F 0 s m - myr y0 0 r

such that

y1a [ min ­ h** ? mŽ . Ž .Ž .r r

y1- b [ max ­ h** ? m , y` F r - r - r F q`,Ž . Ž .Ž .r r y` q`

9Ž .

with

y` - a - b F a F b s 0 s a F b F a - b - q`,yr yr y0 y0 0 0 r r

rq`

b - 0 - a for r s 1, 2, . . . , a , b s K .Ž .Dyr r r rrsry`

Also, there exist unique numbers

q - 0 s q s qr y0 0

such that

w xh** j s q q m j , ;j g a , b , r - r - r . 10Ž . Ž .r r r r y` q`

We may suppose that

h** j - h j , ;j g a , b , ; r g r , r , 11Ž . Ž . Ž . Ž . Ž .r r y` q`

Ž .because if it were not so then we could change h ? }so as to satisfyŽ .11 }without affecting the convexified problem

bmin h** y9 t q w y t dt , y a s A , y b s B. 12� 4Ž . Ž . Ž . Ž . Ž .Ž . Ž .H

a

Ž . Ž . Ž .b Fix any minimizer y ? of the convexified integral 12 . Define

w x w xA9 [ min y a, b , B9 [ max y a, b ,Ž . Ž .w x w xS [ s g A9, B9 : w s s min w A9, B9 ,� 4Ž . Ž .m

ty [ min yy1 S , tq [ max yy1 S ,Ž . Ž .m m m m

13Ž .

sy [ y ty , sq [ y tq .Ž . Ž .m m m m

Ž .Suppose the minimum is finite. We claim first that y ? must be strictlyw yx w q xmonotone along a, t and along t , b . Indeed, if, for example, t - tm m 1 2

EXISTENCE OF SCALAR MINIMIZERS 563

w yx Ž . Ž . Ž Ž .. Ž y.were points in a, t with y t s y t , then necessarily w y t ) w s ,m 1 2 mw x; t g t , t , so that, defining1 2

w x¡y t for t in a, t ,Ž . 1yy t q t y t for t in t , t y t q t ,Ž .2 1 1 m 2 1~y t [Ž .1 y y ys for t in t y t q t , t ,m m 2 1 m

y¢y t for t in t , b ,Ž . m

we would obtain an absolutely continuous function passing throughŽ . Ž . Ž .a, A , b, B and giving to the convexified integral 12 a value strictlylower than its minimum, which is absurd.

Second, we claim that when sy s sq the following may be assumedm mŽ . Ž .about y ? : There exists a unique point s of global minimum of w ? onm

w x Ž . w y qxA9, B9 and y ? is constant, equal to s , along t , t . In fact, them m mfunction

y¡y t for t in a, t ,Ž . my q~s for t in t , t ,y t [Ž . m m m1q¢y t for t in t , bŽ . m

satisfies this property, is absolutely continuous, passes throughŽ . Ž . Ž .a, A , b, B , and gives to the convexified integral 12 the same value asŽ .y ? .

Ž . Ž .c Let us study now the behavior of the relaxed solution y ? along thew y qx y qinterval t , t when s / s . Partition the interior of the set S ,m m m m mŽ .defined in 13 , as

l`y qint S s s , s ,Ž .Dm l l

ls1

with sy - sq , l F q`. Sincel l `

y y1 y q q y1 y qt [ min y s , s - t [ max y s , s ,Ž . Ž .l l l l l l

we may define

y tq y y tyŽ . Ž .l lj [ ,l q yt y tl l

Ž . w y qxand suppose that y ? has constant slope j along the interval t , t }be-l l lŽ .cause otherwise the function y ? , defined by1

y y y qy t q j t y t for t in t , t , 1 F l - l ,Ž . Ž .l l l l l `y t sŽ .1 ½ w xy t for other t in a, b ,Ž .

ANTONIO ORNELAS´564

would verify this property and would also be, by Jensen’s inequality appliedw y qx Ž .on each interval t , t , a minimizer of the convexified integral 12 .l l

Ž .Notice, by the way, that y ? is well defined, in the sense that the intervals1Ž y q.t , t , l s 1, 2, . . . , must necessarily be pairwise disjoint; indeed, other-l l

Ž .wise y ? would have}in going from one connected component of int SmŽ .to another and coming back}to intersect the open set A9, B9 _ S , inm

Ž . Ž .which w ? assumes higher values, and hence y ? would give to theŽ . Ž .convexified integral 12 a strictly higher value than y ? , which is absurd.1

Therefore we may write

l`y1 y qy ? int S s t , t , 14Ž . Ž . Ž .Ž .Dm l l

ls1

Ž . Ž y q.and suppose y9 ? constant along each interval t , t .l lConsider now the boundary ­S . Since ­S ; S , the possibility ofm m m

Ž .similar modifications on y ? as above shows that

y1 y qs g ­S « y ? s is an interval t s , t sŽ . Ž . Ž . Ž .m

having nonempty interior only for at most countably many points sk.Therefore we may write

k`y1 y k q kint y ? ­S s t s , t s , 15Ž . Ž . Ž . Ž . Ž .Ž .Dm

ks1

Ž . Ž .because the image by y ? of an interval along which y ? assumes twodifferent values must necessarily be an interval with nonempty interior.

We now prove the following claim:

Ž . Ž . w x y qIf y t s y t with t - t on a, b and s / s , then1 2 1 2 m m

w x's g S : y t ' s along t , t . 16Ž . Ž .m m m 1 2

Ž .Moreo¨er, A - B and y ? is monotone.

w yx w q xCase 1. t , t g a, t or t , t g t , b .1 2 m 1 2 m

Ž .These situations are impossible because y ? is strictly monotone there.

w y. Ž q xCase 2. t g a, t , t g t , b .1 m 2 m

Ž . w yxCase 2a. y ? decreases strictly along a, t .m

y Ž . Ž . Ž .We have s - y t s y t F A F B. Hence, by 13 and the continuitym 1 2Ž . q Ž . w q xof y ? , A - s ; in particular, since y ? is strictly monotone along t , b ,m m

y Ž . q y Ž . Ž .we have s - A F B F y t - s , and hence s - A s y t s y t sm 2 m m 1 2

EXISTENCE OF SCALAR MINIMIZERS 565

q Ž . w yxB - s . It follows, by the strict monotonicity of y ? along a, t andm mw q x Ž .along t , b , that t s a, t s b. Again, by 13 , S cannot intersectm 1 2 m

Ž y q.s , s , and this leads us to a contradiction: Settingm m

t [ min t g ty , tq : y t s B ,� 4Ž .Ž .0 m m

y¡y t for t in a, t ,Ž . my y y~s for t in t , t q b y t ,y t [Ž . m m m 01

y¢y t y b q t for t in t q b y t , b ,Ž .0 m 0

Ž .we obtain an absolutely continuous function y ? passing through the1Ž . Ž . Ž .points a, A , b, B and giving to the convexified integral 12 a strictly

Ž .lower value than the one given by y ? , which is absurd. Therefore we musthave:

Ž . w yxCase 2b. y ? increases strictly along a, t .m

Ž . Ž . y Ž .We have A F y t s y t - s . By 13 and the strict monotonicity of1 2 mŽ . w q x q Ž . Ž . yy ? along t , b , we have s - A F y t s y t F B - s . But again,m m 1 2 m

Ž .defining y ? exactly as in Case 2a, this leads us to a contradiction, by the1same reasons.

This means Case 2 is impossible.

w y qxCase 3. t , t g t , t .1 2 m m

Žw x. Ž .We must have y t , t ; S , because otherwise the open set t , t _1 2 m 1 2y1Ž . Ž Ž ..y S would contain a nonempty open interval along which w y t )m

Ž Žw x..min w y a, b so that, setting, for example,

w x¡y t for t in a, t ,Ž . 1qy t q t y t for t in t , t y t q t ,Ž .2 1 1 m 2 1~y t [Ž .1 q q qs for t in t y t q t , t ,m m 2 1 m

q¢y t for t in t , b ,Ž . m

again a similar contradiction could be reached.Ž . w xBut then y ? must be constant along t , t , because otherwise we1 2

w x Ž .would arrive at a contradiction: Along t , t the function y ? would1 2Žw x. w y qxassume different values, we would have y t , t ; s , s for some l,1 2 l l

Ž . Ž . Ž .and, by 14 , y9 ? would be a nonzero constant along t , t , and hence1 2Ž . Ž .y t / y t , which is absurd.1 2

Ž .Therefore Case 3 implies 16 .The other cases may be treated similarly, and so the first part of claim

Ž . Ž .16 is proved. If we had A s B then y ? would not be constant alongw x y q Ž .a, b , since s / s ; hence it would be nonmonotone. If y ? werem m

Ž . Ž . Ž .nonmonotone there would exist points t - t with y t s y t but y ?1 2 1 2

ANTONIO ORNELAS´566

w x Ž .nonconstant along t , t , contradicting the first part of claim 16 . The1 2claim is proved.

Ž . Ž Ž .. y qd We claim now that when b ) 0 see 8 or A s B or s s s or0 m mŽ . Ž . Ž Ž ..y ? is nonmonotone or y ? is monotone with k - q` see 15 , there`

Ž . Ž .exists a minimizer z ? of the convexified integral 12 satisfying thefollowing:

Ž Ž .. Ž . w xThere exists s such that w z t G w s , ; t g a, b and there exist0 0w xa9 F b9 in a, b such that

w xz t ' s along a9, b9 ; 17Ž . Ž .0

Ž . Ž .z ? increases strictly with z9 t G b a.e., along at least one of the inter̈ als0w x w x Ž . Ž .a, a9 , b9, b ; z ? decreases strictly, with z9 t F a a.e., along at most oney0

w x w xof the inter̈ als a, a9 , b9, b .

To prove this claim, we will consider several cases. Define the sets:

w xE [ t g a, b : y9 t g a , 0 ,� 4Ž . Ž .y0 y0

w xE [ t g a, b : y9 t g 0, b .� 4Ž . Ž .0 0

Ž .Case 4. b s 0, y ? is monotone with k - q`.0 `

Ž .As a first step, one easily checks that we may construct from y ? , usingŽ .similar methods as above, a new minimizer y ? of the convexified integral1

Ž . Ž .12 which is also monotone but has a number k in 15 reduced by one`

Ž .unit relative to y ? . After k y 1 steps like this, one ends up with a new`

Ž . Ž .minimizer z ? of the convexified integral 12 , which verifies the claim.

Ž .Case 5. y ? is nonmonotone.

Ž w x.The DuBois]Reymond differential inclusion see 2, Theorem 4.1asserts the existence of a constant q such that

w xh** y9 t g q y w y t q y9 t , ­ h** y9 t a.e. on a, b .Ž . Ž . Ž . Ž .Ž . Ž . Ž .18Ž .

Ž Ž ..It easily follows that the function w y ? must assume its minimum valueŽ y.w s a.e. on E j E . This implies the existence of a null set Z suchm y0 0

Ž .that y E j E _ Z ; S . But since S has zero measure in this case, ity0 0 m mw x Ž .follows from 10, Lemma 2 that y9 t s 0 a.e. on E j E . Hencey0 0

E j E has zero measure.y0 0

Ž .Case 6. b ) 0, y ? is monotone.0

Ž .We may suppose that A - B, because if A s B then y ? is constant.Ž .One easily checks, approximating the function w ? by an appropriate

1 w xsequence of C functions, as in 10, Theorem 6 , that there exists a

EXISTENCE OF SCALAR MINIMIZERS 567

Ž . Ž .minimizer z ? of the convexified integral 4 satisfying

w xz9 t f 0, b a.e. on a, b . 19Ž . Ž . Ž .0

Ž . Ž .If z ? is monotone then it satisfies the above claim. Otherwise z ? isnonmonotone and, as seen in Case 5, again it satisfies the claim.

The claim is proved.

Ž . Ž .e We treat now the only case that was not considered in d , namely:

Case 7.

b s 0, A - B , k s q`,0 `20Ž .

w xy ? increases along a, b .Ž .Ž Ž .. Ž Ž .. w xIn particular, we have w y t G 0, h** y9 t G 0 a.e. on a, b , and

y t s sk , y9 t s 0, w y t s 0 s h** y9 t s h y9 t ,Ž . Ž . Ž . Ž . Ž .Ž . Ž . Ž .; t g ty sk , tq sk , ty sk - tq sk , k s 1, 2, . . . . 21Ž . Ž . Ž . Ž . Ž .Ž .

Ž .The constant q in the DuBois]Reymond differential inclusion 18 musttherefore be s 0. Hence

w xh** y9 t g yw y t q y9 t ­ h** y9 t a.e. on a, b . 22Ž . Ž . Ž . Ž . Ž .Ž . Ž . Ž .In particular, this also yields

h** y9 t g y9 t ­ h** y9 t a.e. on yy1 S ;Ž . Ž . Ž . Ž .Ž . Ž . m

Ž .but since y9 t G 0 a.e. and b s 0, we may write0

y9 t s 0 s h** y9 t s h y9 t a.e. on yy1 S . 23Ž . Ž . Ž . Ž . Ž .Ž . Ž . m

We may partition as follows the nonempty open set`

y1 y qy s , s _ S s c , d , 24Ž .Ž . Ž .Dm m m j jjs1

Ž . Ž .with nonempty intervals c , d along each of which y ? increases strictly,j jŽ .by 16 .

Define the set` ` `

y q y k q k y qP [ t , t _ t s , t s _ t , t _ c , d . 25Ž . Ž . Ž .Ž . D D Dm m l l j jks1 ls1 js1

Ž . Ž . Ž . Ž .The definitions in 13 , 14 , 15 , and 24 imply that

y1 y1P ; ­ y ­S ; y S .Ž . Ž .m m

ANTONIO ORNELAS´568

Ž .In particular, by 23 ,

y9 t s 0, h** y9 t s 0 s h y9 t s w y t a.e. on P .Ž . Ž . Ž . Ž .Ž . Ž . Ž .26Ž .

Therefore, setting

w xD [ t g a, b : h** y9 t - h y9 t ,� 4Ž . Ž .Ž . Ž .we have

w xD ; t g a, b : y9 t ) 0� 4Ž .` `

y q y q; Z j a, t j t , b j t , t j c , d , 27Ž .D Dm m l l j jž / ž /ls1 js1

where Z is a null set.Ž . Ž .Define z ? [ y ? in this case.

Ž . Ž .f Let us now modify the minimizer z ? of the convexified integralŽ . Ž . Ž .12 on each of the intervals in d or on the right-hand side of 27 , in

Ž . Ž .order to construct still another minimizer x ? of 12 satisfying further:

w xh** x9 t s h x9 t a.e. on a, b , 28Ž . Ž . Ž .Ž . Ž .Ž . Ž .so that, in particular, x ? also solves the original problem 1 . Set

D [ t g D : z9 t ) 0 , D [ t g D : z9 t - 0 ,� 4 � 4Ž . Ž .q y

and notice that we always havey1

­ h** ? h**9 z9 D ; 0, q` ,Ž . Ž . Ž .Ž . Ž .Ž .q29Ž .

y1 X­ h** ? h** z9 D ; y`, 0 ,Ž . Ž . Ž .Ž . Ž .Ž .y

Ž . Ž .namely in each one of the cases considered in d and in e , be it A F Bor A ) B.

Let us concentrate our attention for the moment on one of the intervalsŽ .along which the relaxed minimizer z ? is strictly monotone. To fix ideas

and to simplify notation, we suppose in what follows that this interval isw x Ž . w xsimply a, b , z ? is strictly increasing along a, b , and the boundary

Ž . Ž .conditions are z a s A - z b s B.Ž . Ž . Ž . Ž .Consider the intervals a , b as in 9 , 10 , and 11 , so that 0 - a -r r r

b , r s 1, 2, . . . , and define the measurable setsr

w xE [ t g a, b : z9 t g a , b ,� 4Ž . Ž .r r r

1w xS [ z E s s g A , B : g a , b ,Ž . Ž .r r r ry1½ 5z 9 sŽ . 30Ž .

`

w xE s E , S [ z E , E [ a, b _ E, S [ z E ,Ž . Ž .D r 0 0 0rs1

EXISTENCE OF SCALAR MINIMIZERS 569

y1 Ž . y1 w xwhere z 9 s denotes the derivative of the inverse function z : A, Bw x w x w xª a, b of the function z: a, b ª A, B at the point s, so that

y1y1 y1 w xs s z t m t s z s , z 9 z t s z9 t a.e. on a, b .Ž . Ž . Ž . Ž .Ž . Ž .

Define the function

w x y1¨ : A , B ª 0, q` , ¨ s [ z 9 s a.e.,Ž . Ž . Ž .0 0

Ž wand notice that, by the change of variables formula see, e.g., 11, Theoremx.277, p. 197 ,

1B1 q w s q h** ¨ s dsŽ . Ž .H 0½ 5ž /¨ sŽ .A 0

bs 1 q w z t q h** z9 t dt. 31� 4Ž . Ž . Ž .Ž . Ž .H

a

According to the Liapunov theorem on the range of vector measuresŽ w x.see, e.g., 3, 15, 7 , for each r g N there exists a measurable function ¨ :rw x Ž . Ž .A, B ª 0, q` such that, denoting by x ? the characteristic functionSr

of the set S , equal to 1 on S and 0 outside,r r

1 1y1 Xx s g ­ ­ h** ? h** x s , 32Ž . Ž . Ž . Ž .Ž .S Sr rž /ž /¨ s ¨ sŽ . Ž .r 0

B Bx s ¨ s ds s x s ¨ s ds, 33Ž . Ž . Ž . Ž . Ž .H HS r S 0r r

A A

B Bx s ¨ s w s ds s x s ¨ s w s ds. 34Ž . Ž . Ž . Ž . Ž . Ž . Ž .H HS r S 0r r

A A

In particular,

1 1w xh** s h for a.e. s g A , B . 35Ž .ž / ž /¨ s ¨ sŽ . Ž .r r

Define a new increasing function

q`sw x w xt : A , B ª a, q` , t s [ a q x s ¨ s ds .Ž . Ž . Ž .ÝH S rr

A rs0

ANTONIO ORNELAS´570

Ž .Notice that t A s a and, by the properties of measures,

q`B

t B s a q x s ¨ s dsŽ . Ž . Ž .ÝH S rrA rs0

q`B

s a q x s ¨ s dsŽ . Ž .Ý H S rrArs0

q`B

s a q x s ¨ s dsŽ . Ž .Ý H S 0rArs0

q`B

s a q x s ¨ s dsŽ . Ž .ÝH S 0rž /A rs0

B y1s a q z 9 s ds s b.Ž .HA

w x w xThis shows that t : A, B ª a, b is absolutely continuous and hasderivative

q`

w xt 9 s s x s ¨ s ) 0 a.e. on A , B ;Ž . Ž . Ž .Ý S rrrs0

Ž .in particular, t ? is strictly increasing and hence has a well-defined inversew x w xfunction x: a, b ª A, B strictly increasing and absolutely continuous

Ž . Ž .with x a s A, x b s B and derivative

y1 w xx9 t s t 9 x t a.e. on a, b . 36Ž . Ž . Ž .Ž .Ž .

Ž . Ž . Ž . Ž .Since t s t s m s s x t , we have, by 35 and 36 ,

bw x t q h x9 t dt� 4Ž . Ž .Ž . Ž .H

a

1Bs w s q h t 9 s dsŽ . Ž .H ½ 5ž /t 9 sŽ .A

q` q` 1B Bs x s ¨ s w s ds q x s ¨ s h dsŽ . Ž . Ž . Ž . Ž .Ý ÝH HS r S rr r ž /¨ sŽ .A A rrs0 rs0

q` q` 1B Bs x s ¨ s w s ds q x s ¨ s h** dsŽ . Ž . Ž . Ž . Ž .Ý ÝH HS r S rr r ž /¨ sŽ .A A rrs0 rs0

EXISTENCE OF SCALAR MINIMIZERS 571

q` q` 1B Bs x s ¨ s w s ds q x s ¨ s h** dsŽ . Ž . Ž . Ž . Ž .Ý ÝH HS 0 S 0r r ž /¨ sŽ .A A 0rs0 rs0

q`B

s x s ¨ s w s dsŽ . Ž . Ž .ÝH S 0rž /A rs0

q` 1Bq x s ¨ s h** dsŽ . Ž .ÝH S 0rž / ž /¨ sŽ .A 0rs0

1Bs w s q h** ¨ s dsŽ . Ž .H 0½ 5ž /¨ sŽ .A 0

bs w z t q h** z9 t dt.� 4Ž . Ž .Ž . Ž .H

a

Ž . �w x 4g Consider the family a , b , i s 1, 2, . . . of intervals on the right-i iŽ . Ž .hand side of 27 , along each of which the minimizer z ? of the convexified

Ž .integral 12 is strictly monotone.For each i, consider the problem of minimization of the integral

biw x t q h x9 t dt 37� 4Ž . Ž . Ž .Ž . Ž .H

ai

Ž .among the absolutely continuous function x ? passing through the pointsŽ Ž .. Ž Ž .. Ž .a , z a , b , z b . Since z ? is a minimizer of the convexified integrali i i iŽ . Ž . w x12 , its restriction z ? to a , b has to be a minimizer of the convexifiedi i i

Ž . Ž .integral associated to 37 . But the strict monotonicity of z ? allows us toiŽ . Ž .construct from z ? , as in f , another strictly monotone absolutely contin-iŽ . w xuous function x ? on a , b satisfyingi i i

b bi iXw x t q h x t dt s w z t q h** z9 t dt ,� 4� 4Ž . Ž . Ž . Ž .Ž . Ž .Ž . Ž .H Hi ia ai i 38Ž .

x a s z a , x b s z b .Ž . Ž . Ž . Ž .i i i i i i

w xFinally, defining the new function ¨ : a, b ª R,

xX t for t in a , b , i s 1, 2, . . . ,Ž . Ž .i i i¨ t [Ž . ½ z9 t for other tŽ .

and

tx t [ a q ¨ t dt ,Ž . Ž .H

a

ANTONIO ORNELAS´572

Ž . Ž .one easily checks that x ? is a minimizer of the original integral 1 ,Ž . Ž . Ž . Ž .because h** ? F h ? and x9 t s z9 t s 0 a.e. on P.

Remark. One easily checks that the same method may be used to proveŽ .the existence of minimizers when w ? is unbounded below, provided it

satisfies, for example,

< < pw s G yg y g s , p G 1.Ž . 1 2

When p ) 1 one should}as usual in the application of the direct methodŽ . Ž .}ask h ? to satisfy a stronger growth condition than 3 , such as

< < qh j G yg q g j ,Ž . 3 4

with either q ) p and g s 1, or q s p and g ) 0 large enough relative to4 4Ž w x.b y a and g see, e.g., 9, 7, 8 .2

ACKNOWLEDGMENTS

Thanks are due to SISSA}where this research was performed}for its stimulatingresearch environment, in particular, to Arrigo Cellina; and to Gianni Dal Maso, coordinator

Ž .at SISSA of the European Union HCM Human Capital and Mobility program}whosepostdoc fellowship financially supported this work.

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