on Γ−-convergence of functionals with analytic integrand

Annali di Matematica pura ed applicata (IV), Vol. CLXIII (1993), pp. 291-312

On F--Convergence of Functionals with Analytic Integrand (*)(**).

LUIGI GRECO

Sunto. - Si dimostra, sotto opportune ipotesi~ l'analiticit~ dell'integrando del F--limite di una successione di funzionali semplici del C.d.V. ad integrando analitico.

Introduction.

In [9] an example is given of a sequence of functionals

(1) Fh (~, u) = ffh (X, U'(X)) dx (u E W 1' 1 (t~)), t~

with fh (x, z) = ah (x)z 4 + bh (x)z 2, such that (see definition in Section 3) for any open set t~ r R

(2) Fh (t~, u) ~ ]f(u') dx,

but f(.) is not a polynomial (see also [2], [6], [11], [10], [13] for general results of r - - convergence in the Calculus of Variations).

In this paper we study the differentiability properties of the integrand of the/~-- limit of a sequence of simple integral functionals and, in particular, we prove that, under assumptions on a~ (x), bh (x) considered in [9], the function f given in (2) is a real analytic function. Namely, let (~,/~) c R and fh: (~, ~) R ~ R be a sequence of func-

(*) Entrata in Redazione il 9 luglio 1990. Indirizzo dell'A.: Dipartimento di Matematica e Applicazioni ,Renato Caccioppoli,, Via Cin-

tia, complesso Monte S. Angelo, edificio T, 80126 Napoli. (**) This work has been performed as a part of a National Research Project supported by

M.P.I. (40% 1987).

292 LUIGI GRECO: On F--convergence of functionals, etc.

tions such that

(3)

"fh (x, z) measurable in x, convex in z;

fh(x, z) >t ~lz]P;

fh (x, Zo ) R, and assume that (2) holds with Fh given by (1). Then, in Section 4, we prove the following

THEOREM.- Under the above assumptions and, moreover, i f fi~(x, ") is analytic and, for any bounded set H r R, the following conditions hold:

i) 3y>O: - - fh (', z) O: az2fh (x, z) ~ c, for a.e. x e (~, fi), Vz e H, Vh 9 N;

then f(.) is a real analytic function.

Let us now give the plan of the paper. In Sections 1 and 2 we give preliminary results of Convex Analysis, some of which

seem new and interesting in themselves. In Section 3 we prove that Cl-regularity, HSlder continuity of the first derivative

and Ck-regularity of the integrand are preserved by passing to the F--limit. In Section 4 we deal with the analyticity of the F--limit. A first result in this di-

rection is due to L. TARTAR [12]. Finally in Section 5 we give some examples showing that the assumptions of the

Theorem 4.1 cannot be dropped.

1. - Reca l l s and pre l iminary resu l ts on convex ana lys is .

Let us give some definitions and preliminary results of convex analysis (see e.g. [5], [8]).

For f : R --* R, the polar of f is the function f * : R -~ R defined Vy e R by:

f * (y) -- sup {zy - f(z)}. zeR

Let f : R --~R and let z 9 R be such that f(z) is finite. We say that y 9 R is a subgra- dient of f at z ff and only if:

y(t - z) + f(z)

LUIGI GREC0: On F--convergence of functionals, etc. 293

The set (eventually empty) of the subgradients o f f at z is called the subdifferential of f at z and is denoted by af(z). The multifunction af: z --* af(z) is called the subdifferential of f. If f is convex and finite, then Vz e R we have:

3f(z) = [D_ f(z), D + f(z)],

where D_ and D+ are respectively the left and right-hand derivatives; in particular af(z) ~ 0. Moreover af is monotone:

y e af(z), Yl E 3f(zl ) ~ (y - Yl )(z - z~ ) >I O.

Next proposition gives a connection between af and ~f*.

PROPOSITION 1.1. - I f f is convex and finite, then (af) -1 = af*, that is:

y e af(z) r z ~ 3f* (y).

From this it follows

COROLLARY 1.1. - Let f convex and finite, with f * finite. I f f and f * are differentiable, then their derivatives are inverse to each other.

Now we want to give a condition on f in order to have f * e C 1. A function f : R --) R is called strictly convex if and only if, VZl, 22 E R, with z 1 ;~ z2, and W e]0, 1[, we have:

f(~zl + (1 - ~)z2) < )~f(zl) + (1 - ~)f(z2).

The condition we have mentioned is expressed by

PROPOSITION 1.2. - I f f is convex finite, with f * finite, then we have the equivalence

f strictly convex ~f * E C 1 (R).

PROOF. - I f f is strictly convex, then D_ and D+ are strictly increasing (see [7]); hence 0f* = (Of)-1 is univalued, that is f * is differentiable. The derivative is increasing and has a connected graph, hence it is continuous. For the other implication, we have only to invert the previous argument. 9

Now we want to illustrate a ,,dual, condition to HSlder continuity for the first derivative. A continuous function f: R --) R is said to be uniformly convex if and only if there exists ~o: ]0, ~ [~]0, ~ [ such that, Vzl, z2 E R, with zl ~ z2 it results:

(1.1) f ~ 2[ f (Z l ) -~-f(z2)] -- O)([Z1 -- Z2 I)"

It is evident that we have necessarily lira co(t) = 0. More precisely, we are going to t--~ 0 +


prove that o~ verifies the following condition (*):

~(t) (1.2) lira" is finite.

We need a lemma.

LEMMA 1.1. - Let f be a continuous functions verifying (1.1). If Z1, Z 2 E R , with zl ~ z2, and Yl e 8f(zl), Y2 e 8f(z2), then we have

lYl - Y21 >14 ~(Izl - z2 i)

Z -- Z2[

PROOF. - We have:

2[f(zl) + f(z2)]_f( ~ ) _ zl-z24 f(zl) - f ( zl -~- Z2

| zl + z2

f(z2 ) - f ( ~)z l + z2

Z 1 -~-Z 2 Z2 2

We may assume zl > z2. By the convexity of f, if y~ e af(zi) for i = 1, 2, then:

Yl ~> , Y2 ~< Z 1 -~- Z 2 Zl -~- Z 2

zl 2 z2 2

so we get:

9

We prove (1.2). Let zl be a point where D_f is differentiable; let us set Yl = = D_f(zl) and, for z ~ Zl, y = D_f(z). By the previous lemma we have:

lY l -Y l ~> 4~( Iz l - zl ) I z , - z l IZ l -Z l 2 '

from this, as z -o Zl, (1.2) follows. Because of (1.2), ff co(t) = ~tP (t > 0), with ~ > 0, then it must be p I> 2. In the case

oJ(t) = t~t P, we shall speak of p-uniform convexity. 2-uniform convexity is also called strong convexity.

(*) We denote the superior and inferior limits respectively by lim" and lim'.

LUIGI GRECO: On F--convergence of functionals, etc. 295

Let us point out a consequence of Lemma 1.1:

f uniformly convex ~ f * finite.

As a matter of fact, from the lemma, we have:

lira D_f(z)= +~, lira D+f(z )=-~; Z--> + ~ Z-~ -- cc

hence, Vy e R, the function z ~f (z ) - yz is coercive (and continuous). Therefore we have: f * (y) = marx {yz - f(z)}.

REMARK. - It is immediately seen that:

f uniformly convex ~ f strictly convex.

Let us give a sufficient condition for uniform convexity.

PROPOSITION 1.3. - I f f is twice differentiable with f" >~ c > O, then it verifies the condition:

i c (1.3) f ~< ~ [f(zl ) + f(z~ )] - -

and so it results 2-uniformly convex.

PROOF. - Let us set zo = (zl + z2)/2. By the Taylor expansion with remain- der:

f(z) =f(z0) + f'(zo)(Z - Zo) + l f"(~)(z - Zo) 2,

where ~ is an interior point of the interval with endpoints z, zo. By setting z = zl, z = = z2 and by summing, we get:

f ( z l )+f (z2)=2f (zo)+ 1 [f"(~l ) + f"((2)] ( ~L~ )2

and so, since f"~> c, (1.3) holds. []

PROPOSITION 1.4. - Let f be convex and finite; then f is p-uniformly convex (p >i 2) i f and only i f f * is finite and has Hhlder continuous first derivative.

PROOF. - Let f be p-uniformly convex. From Lemma 1.1, if Yi e 9f(zi) (i = 1, 2), then we get

l Y l - - Y2 I ~ 4 /A IZ l - - 2:2 I p -1

So ~f* = (af)-1 is univalued, that is f * is differentiable (actually, being f strictly con-


vex, we already knew that f * e C1). Moreover, since:

Izi - z21

t Yl - Ye 11/@ - 1) ~< (4/x)- ~/(p - ~),

the derivative is HSlder continuous. Now let (d/dy)f* be HSlder continuous with exponent a and constant K. We may

assume zl > z2. By the convexity of f, we have:

zl+3z2)+f(z2)] = 4

f(zl) - f 3zl + z2 f(z2) - f Z1 -t- 3Z 2 (4 ) (4) 3z~ + z2 zl + 3z2

z, 4 z2 4

o 8 4 4 "

By Proposition 1.1 and HSlder continuity we have assumed, we get:

4

and hence:

(z l - z2 )2 + 1D<

8(2K) 1/~

2. - Closure results with respect to weak convergence.

Let us recall that a sequence (fh) of functions in L ~ (~,/7) (~,~ eR, ~


give sufficient conditions for the limit g = g(x, y) to be differentiable with respect to y. So we assume:

(2.1) g~ (., y) *-- g(-, y), Vy 9 R .

PROPOSITION 2.1. - Let gh = gh (x, y) (h 9 N), g = g(x, y) be functions satisfying (2.1), which are measurable and bounded in x and convex in y. Then, Vt~ measurable set, Vy 9 R, we have:

(2.2) Ia I8_ lim'h ~yy gh (x, y) dx >I -~y g(x, y) dx ,

(2.3) lim,,I 8+ f 8+ h 8yy g~ (x, y) dx


subset of R, the following conditions hold:

i) 3K = K(H) > O: _-4-2~. g~ (', Y) 0:

ly- l


Now we are going to prove that, ~r~ 9 R:

(2.7) a 7y gh~(, y-) - - v(', ~).

Actually, for a fixed ~b e L 1 (zt,/3) we have:

f ~gh

I;[ 1 0, 3~ > 0 such that:

JY - Yl < ~ -~ygh~(x, y) - ~ygh~(x, y) @(x)dx


It follows that:

Y

Let E c eI be the set of Lebesgue points, out of I, of the functions:

Y

I V(', t) dr, g(., y) - g(., ~), for y, ~ e Q. Then I ee l = 0 and, Vx e E, Vy, ~ e Q, we have:

Y

(2.9) I ~(x, t) dt = g(x, y) - g(x, ~).

If x e E is such that g(x, .) e C(R), then (2.9) holds Vy, ~ e R. From this it follows that, if x e E and g(x, ") is continuous, it is also differentiable and verifies (2.8). Therefore g(x, .)e CI(R), for a.e. x and (by a compactness argument) we get (2.5). 9

PROPOSITION 2.3. - Let gh = gh (x, y) (h e N) be functions such that gh (x, ") e C k (R), for a.e. x e (~, fl), gh(', Y), (8k/aYk)gh( ", Y) ~ L ~ (~, fi), Vy ~R and assume that, for every H bounded subset of R, the following conditions hold:

i) 3K=K(H)>O: --2-~g~(',Y) 0, 38 = 8(H, ~) > 0:

(y, ~eg, lY -Y l 0:

gh

LUIGI GRECO: On/"--convergence of functionals, etc. 301

apply Proposition 2.2 subsequently to the sequences

a ( (gh), ( ~y gh ) , "" , Oyk_ l gh m Before we state the next proposition, we must introduce the space C ~ (R) of real

analytic functions. A function g = g(y) is said to be analytic if and only if Vy e R it can be developed in a power series with center at y, in a neighbourhood of y.

PROPOSITION 2.4. - Let gh = gh (x, y) (h e N) be functions such that gh (x, ") e C ~ (R), for a.e. x e (~, fl), (ak/ayk)gh(', y) e L ~ (~, ~), Vy e R, Vk e N and assume that, for every H bounded subset of R, 3?, = y(H) > 0:

gh(', y) L ~ ~< ~,kk!, Vy e H .

Moreover let g = g(x, y) be a Carathdodory function for which (2.1) holds. Then we have g(x, ") e C~(R), for a.e. x e (~,~).

PROOF. - By Proposition 2.3, 31 r (a,/?) of measure zero such that, Yx ~t I, g(x, .) e C ~ (R) and VH bounded subset of R, 3~, > 0:

I Ok ~yk g(x, Y) I

302 LUIGI GRECO: On /"--convergence of functionals, etc.

PROPOSITION 3.1 (see [4]) . - If (V, z) is a first countable space, then:

(3.1) F = F- (z) - lim Fh ~ F = F/eq (z) - lim Fh. h

PROPOSITION 3.2 (see [6]). - Let V be a reflexive and separable Banach space, II II its norm and v the weak topology. I f Fh : V---> R U {+ ~} verify, Vu e V, Vh e N

Fh (u) >- ~llulP

(with ~ > 0, p > 1), then (3.1) holds.

In the sequel, we shall consider seguences of functions fh: (~, fi) x R --* R verifying the following conditions (see [6], w 4):

fh(x, z) measurable in x, convex in z;

fh(x, z) >I ~lzlP;

(3.2) fh (x, Zo) ~< M;

Ifh (x, z) dx < A(z)

with ~> 1, p> 1, zoeR, M>0 and A:R-->R. Given fh = fh (x, z), for a fLxed x 9 (~, fi) we shall denote by fh* (x, h) the polar (see

w 1) of the function z ~fh (x, z). If fh verify (3.2)2 and (3.2)3, then we get

(3.3)

(3.4)

(1) yzo - M

LUIGI GREC0: On P--convergence of functionals, etc. 303

THEOREM 3.1. - Let fh (h 9 N), f be functions verifying (3.2) and Fh , F be defined by (3.4) and (3.6). Then (3.5) hold, V~9 c (a, fl) open interva~ if and only if"

(3.7) f~( . ,y )~_~f*( . ,y ) , VyeR.

In this section, we prove some theorems concerning preservation, for the F--limit, of differentiability properties of the integrands of functionals (3.4).

THEOREM 3.2. - Let fh (h 9 N), f be functions verifying (3.2) and Fh, F be defined by (3.4) and (3.6) respectively. Let us assume (3.5), VP c (a, fl) open interval. Moreover let fh (x, .) e C 1 (R) Vh e N, for a.e. x 9 (~, ~) and assume that, for any bounded set HeR, W > 0, 3K = K(H) > 0, 38 = 8(H, ~) > 0 such that:

~zfh(',Z)IIL


by (3.8), Vr o > 0, 3o > 0 such that, set t)~ = ]Xo - ~, Xo + 0[, if z - 5 < 80, then we have(*):

a ~ f (x, y l ) - Oy

and by Proposition 2.1 this yields:

f~ (x, gl ) - ~ f~ (x, Yt) dx < 6~.

Therefore 3h e N, 3x1 e eI:

(3.9) 0+ 0 , f-~ (xl, Yl) - ~y f ~ (xl, Yl ) < 80. 0---y

We are going to prove that, fixed s > 0, 8o can be choosen so that this yields: yl - - Yl < s. By i) and (3.2)~, there exists ~: R -~ R convex such that, Yz e R, Vh e N, for a.e. x e (~, ~), it results fh (x, z)


COROLLARY 3.1. - Let fl = ~ (x, z) be a function verifying (3.2), periodic of period 1 in x, with continuous derivative in y, uniformly with respect to x. Set, for h>l ,

(3.11) fh (x, z) = f~ (hx, z)

and define Fh by (3.4). Then 3f e C 1 (R) such that, VQ r (0.1) open interval:

i f - (W -- H I 'p (~) )

(3.12) Fh(Q, u) ~ f(u')dx (u 9 HI'P(Q)). J

THEOREM 3.3. - Let fh (h 9 N), f be functions verifying (3.2) and Fh, F be defined by (3.4), (3.6). Let us assume (3.5), VQ r (~, fi) open interval. Moreover, let the functions fh(x, ") have equi-HSlder continuous derivatives: fh(X, ")9 CI(R) for a.e. x and 3~9 3K>0:Vz, ~ 9 ( z~) , VheN,

Y l - Since, by hypothesis and Proposition 1.1, Vh 9 N, for a.e. x 9 (a, fl):

~yfh' (X' Yl)-- -~y f:(2r ~1) ~ K '

then by Proposition 2.1, VD measurable set (1~1 > 0), we get:

-~-yf (x, Yl) -

and, since Q is arbitrary, we have:

l)lJ

a_ . >1 (y l -Y l ) 1/~, 8+ f * ( x, Yl) - -~--- f (x, Yl) (3.13) ~y oy K

for a.e. x 9 (~,/7). Now let z, ~ 9 R. Assume z > ~ and take x e (~, ~), fixed in the sequel, such that

(3.13) is true Vyl, Yl 9 Q and f(x, .) is convex. Let:

(3.14) y e af(x, z), ~ 9 af(x, -5) ;

306 LuIG~ GRECO: On F--convergence of functionals, etc.

then y 1> ~. We want to prove that:

y -~ (3.15) ~ ~< K.

(z - 5)~

This is evident in the case y = ~; therefore let y > ~. For y~, ~ e Q such that Y > Y~ > Yl > Y, we get (3.8) and then, by (3.13), again we get (3.15). By exchanging z and 5, we obtain (ff (3.14) is valid):

ly -~1 ~

LUIGI GREC0: On F--convergence of functionals, etc. 307

for a.e. x 9 (a, fi), Vy 9 Vh 9 By i i i ) f~(x, ") 9 Ck(R): actually, for j = 2 ..... k, we have

8 8J ) 8j Pj-2 7-Z~Z2~(x,'),..., ~zj fh(x, ' )

(3.17) - *(x,.)= [8 8y j fh ~z 2 fh (x, ") 8y fh (x, -)

(with Pj _ 2 = Pj - 2 (ul, ..., uj _ 1 ) polynomial of degree j - 2, independent of fi~ (x, .)). If we set

K ,= max lust ~ O: (ae /&2) fh (x , z) >i c, for a.e. x 9 (a, fl), Vz 9 H. Set (3.11) and define Fh by (3.4), Vh 9 N. Then 3 f e Ck(R) for which (3.12) holds.

308 LUIGI GREC0: On F- -convewence of functionals, etc.

PROOF. - By hypothesis the function

VI,(.,Z) is continuous and so it is bounded on bounded subset of R. So (ak-1/SZk-1)f~ (X, ") is locally Lipsehitz, uniformly with respect to x a.e. in (~,~) and

z) L ~

is continuous. In this way, we can verify the hypothesis of Theorem 3.4 for the sequence (r de-

fined by (3.11). 9

4. - Analyticity of the F-- l imit.

Let us begin with some simple preliminary remarks about the analyticity of the inverse function of an analytic function. Let f and g be inverse functions to each other, f analytic andf ' > 0. Let us put y =f(z); then g'(y) = l / f '(z). Hence, if f '(z) >I c > 0, then we have:

(4.1) (0 t c > 0 and [f(J)(z) I ~ ~,Jj!

(for j = 2, ..., k - 1, k), then:

Pk()~l, . . . , )~k- l '~ ]2"2!, ..., ] ,k - l (k - 1)!, ],k/c!) (4.2) Ig (k) (Y) I ~< c

Let us denote the right-hand side of (4.2) by )~k. As a consequence of (4.1) and (4.2), for the sequence (s we get:

(4.3) I g (k) (y) I ~< )~k, Vk e N .

Let us prove that 2e = Z(c, V) > 0:

(4.4) )~k ~ ~kk!, Vk ~ N.


Let us def'me

(~,z)2 fl (Z) de__f CZ -- S (~Z) i ~- CZ i=2 1 - Tz'

for - 1 < Tz ~< 1 - (c/T + 1) -1/e. The inverse function g is:

y + - - c / r - 2) 2 - 4(c / r + 1)

g~ (Y) = 2r(c/r + 1) '

for y ~< [1 - (c/T + 1) 1/2 ]2. It results fi (0)= 0 and gl(k)(0)= ~k, Yk ~ N. Then there exists ~ > 0 verifying

(4.4), since gl is analytic in a neighbourhood of the origin.

THEOREM 4.1. - Let fh (h 9 N), f be functions verifying (3.2) and Fh, F be defined respectively by (3.4), (3.6). Let us assume (3.5), Vt~ c (~,fl) open interval. Moreover let fh (X, ") 9 C~~ for a.e. x 9 (a, fl), Vh 9 N and, for any bounded set H oR, let the following conditions be fulfilled:

i) 3 T > O:

ii) 3c = c(H) > O: (c~2 /c~z2) fh (x, z) >I c, for a.e. x 9 (a, fl), Vz 9 H, Vh 9 pC.

Then f(x, .) 9 C~~ for a.e. x e (~,~), since it verifies some estimates like i).

PROOF. - The proof is similar to that of Theorem 3.4: we have only to sharpen some estimates. Since (~/az)fh (x, .), (a/ay)f~ (x, .) are inverse functions to each other, we can use the preliminary remarks. As a matter of fact, like for (3.16), for any bounded set H* oR, there exists a bounded set Hr such that:

a . -~y fh (x, y) E H ,

for a.e. x e (a, ~), Vy 9 H*, Vh 9 N; then by i), 3T1 = T1 (H) > 0:

~A(x , z) ~ ~,~k., Vk 9

and, by ii), for a.e. x 9 (~,~), Vz 9 H, Vh 9 N,

a ~ az2 fh (x, z) >I c > 6.

So 3~>0 such that, for a.e. x 9 gyeH* , Vh, k 9

I I Oy k +1 fh* (x, y) ~< ~k k!.


By applying Proposition 2.4 to the sequence ((a/ay)f~(x,y))h we get (a/ay)f*(x, . )eC~(R), for a.e. xe(~,f l ) and also some estimates like i) and ii); therefore, like in the proof of Theorem 3.4, we can get similar estimates also for f. []

COROLLARY 4.1. - Let f=f (x , z) be a function verifying (3.2) and such that:

i) f(., z) be periodic of period 1;

ii) for any bounded set HeR, 37 > 0:

I I ~ f(x, z) < ~,kk!, for a.e. x, Vz e H, Vk e N;

iii) for any bounded set HeR, 3c = c(H) > O: (a2/aze)fh(x, z) >1 c, for a.e. xe(~,fl), VzeH, VheN.

Let us set (3.11) and define F~ by (3.4) (t9 r (0, 1)). Then 3f e C ~ (R) such that (3.13) holds.

5. - Some example .

EXAMPLE 1. - Let ah, bh : (~,/~) --* [v, tL] (h e N, 0 < v ~< [.r < ~ ) be measurable functions. Let us set, Vh e N:

fh (x, z) = ah (x) z ~ + bh (x) z 2

and define Fh by (3.4). The functions verify (3.2) with p = 4 and so (see Corollary 4.4 [6]) there exists f verifying (3.2) such that, assuming (3.6), (unless passing to a subsequence) we have (3.5). The hypothesis of Theorem 4.1 are fulfilled, hence we get f(x, .)e C~(R), for a.e. x e (~,/~). So we obtain the mentioned result of[12].

EXAMPLE 2. - Let a: R ~ ]0, ~ [ be a measurable function, periodic of period 1, and set, Vh e N, for a.e. x e (0, 1), Vz e R:

1 2 fh(x, Z) = a(hx)z 4 + ~z .

Let us define Fh by (3.4). If a e L 1(0, 1), then condition (3.2) are fulfilled, but if a r L ~ (0, 1), then the hypothesis i) of Theorem 4.1 is not fulfilled and actually the integrand of the homogenized functional may fail to be analytic.

Set, for fLxed y e R:

(5.1) z = z(x, y )~y = a(x)z 8 + z, 1

O(y) = ; z(x, y) dx. 0

LumI GRECO: On F--convergence of functionals, etc. 311

If f is the integrand of the homogenized functional, then we have f '= 0 -~ (see [6]).

We shall prove that, for a suitable ar L2(0, 1), there exists a sequence (y~) such that Yn '-> 0 and

(5.2) lira ~(5) (y~) = + ~. n

By a direct calculation we get:

aSz _ 360a 2 198a2z 4 - 57az 2 + 1 ay 5 (3az 2 + 1) 2 '

which, for y ;~ 0, is bounded (in modulus) by 2145 lYl-4, and then, in R \{0}:

1

0(5) = ~ 35 z(x, .) dx. J

0

Let tl < t2 be the (positive) solutions of 198t 2 - 57t + 1 = 0. If az 2 et]tl, t2[, then we have 05z/ayS~ O. Clearly by (5.1) it follows that:

] ti312 + t~l~ t~/2 + t~/2 [ az 2 ~t It1, t2 [ ~ y ~ a 1/2 ' a 1/2 9

Assume that the range of a is countable and may be ordered in an increasing diver- gent sequence al < a2 < ... < ai < .... The intervals ](t~/2 + t~/2)/a 1/2, (t~/2 + + t1/2)/ai1/2[ are pairwise disjoint if, Yi e N, we have:

ai+l > t2 ( t2+l12 a--7-- ~[ t -~ l ] "

Then we can find a sequence (y~) such that y~ $ 0 and, for a.e. x E (0,1), Yn ~ N, a(x)z2(x, y~) r t2[, so that (aS /~yS)z(x, y~) >10. Since, as n--) ~ , (aS /ayS) 9 9 Z(X, Yn)'-> 360a2(x) a.e. in (0,1), if ar L2(0, 1), then we get (5.2). As 0'(0)= 1, this implies that f is not 6-times differentiable at 0.

Clearly we can find a function a having the necessary qualities.

EXAMPLE 3 . - Let:

1 an(l:

~1z6

Z6-~

ff x e [0, 1 /2 [+Z,

1 4 ~-z , i f xe [1 /2 ,1 [+Z,

f~ (x, z) = f~ (hx, z) .


Let us set ~(z )=z 5+z 3, ~=~-1 and

1 O(y) = -~ (~(y) + y 1/6), "r, = O- ~.

If, as usual, f is the integrand of the homogenized functional, then r~ = f ' . I t is easily seen that

~(0) = ~'(0) = rj (2) (0) = .~(3)(0) = r, (4) (0) = O, ~(5) (0) = 5! 2 5 ,

V(z) - 2 5 z 5 lira - +

z-->0~ Z 6

but

Then f is not 7-times continuously differentiable at 0. In this example the hypothesis ii) of the Theorem 4.1 is not fulfilled, since

~yi fh (x, O) - O.

REFERENCES

[1] L. V. AHLFORS, Compex Analysis, McGraw-Hill, New York (1953). [2] H. ATTOUCH, Variational Convergence for Functions and Operators, Pitman Advanced

Publishing Program (1984). [3] A. AUSLENDER, Optimisasion: Mgthodes Numdriques, Masson, Paris (1976). [4] E. DE GIORGI - T. FRANZONI, SU un tipo di convergenza variazionale, Atti Acad. Naz. Lin-

cei, Rend. C1. Sc. Mat. (8), 58 (1975), pp. 842-850. [5] I. EKELAND - R. TEMAM, Convex Analysis and Variational Problems, North-Holland

(1978). [6] P. MARCELLINI - C. SBORDONE, Dualit4 e perturbazioni di funzionali integrali, Ricerche

Mat., 26 (1977), pp. 383-421. [7] A. W. ROBERTS - D. E. VARBERG, Convex Functions, Academic Press (1970). [8] R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press (1970). [9] C. SBORDONE, Sur un limite d'intdgrales polynomiales du caleul des variations, J. Math.

Pures Appl. (9), 56 (1977), pp. 67-77. [1O] C. SBORDONE, SU alcune applicazioni di un tipo di convergenza variazionale, Ann. Sc.

Norm. Sup. Pisa C1. Sci. (4), 2 (1975), pp. 617-638. [11] L. TARTAR, Cours Peccot au college de France, Paris (1977). [12] L. TARTAR, Personal communication. [13] V. V. ZglKOV, Questions of convergence, duality, and averaging for functionals of the cal-

culus of variations, Math. USSR Izvestiya, 23, no. 2 (1984).

on Γ−-convergence of functionals with analytic integrand

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