Reinforcement problems in the calculus of variations(*)

Ann. Inst. Henri

Vol. 3, n°

Poincaré,

4, 1986, p. 273-284.

Analyse

Reinforcement

linéaire

problems

in the calculus of variations

Emilio ACERBI and

non

Giuseppe

(*)

BUTTAZZO

Scuola Normale Superiore, Piazza dei Cavalieri, 7, 1-56100, Pisa,

Italy

We study the torsion of an elastic bar surrounded by thin layer made of increasingly hard material. In the model increasingly the problem ellipticity constant tends to zero in the outer layer ; the equations considered may be fully nonlinear. Depending on the link between thickness and hardness we obtain three different expressions of the limit problem.

ABSTRACT.

-

an

On etudie la torsion d’une barre elastique enveloppee RESUME. d’une couche tres mince d’un materiau tres dur. Dans le probleme modele la constante d’ellipticite de la couche exterieure est tres petite ; les equations considerees peuvent etre completement non-lineaires. En modifiant la relation entre 1’epaisseur et la rigidite, on obtient trois différents problemes-limites. -

M ots-clés :

Reinforcement, r-convergence. Integral functionals, Non-equicoercive

problems.

I. INTRODUCTION

Several recent papers with the reinforcement setting may be outlined

(see [2]] [3] and the references quoted there) deal problem for an elastic bar, whose mathematical as

(*) Financially supported by

follows.

a

national research

project of

the Italian

Ministry

Education. Annales de l’Institut Henri Poincaré - Analyse non linéaire - 0294-1449 Vol. 3/8b/04~’273,1?, ~ 3,20/@ Gauthier-Villars

111

of

274

E.

Let Q be

thickness a

minimum

a

ACERBI AND G. BUTTAZZO

bounded open set in

goes to zero as G

point

-~

0;

surrounded by set

Qg

a

layer Eg whose

and denote

=

of the functional

in

where G is lower semicontinuous in the LP topology of W 1 ~p(S~), and f (x, z) is non-negative and convex in z. The reinforcement problem consists in studying the behaviour of uE as G tends to zero. Under suitable assumptions we characterize the r-limit of the functionals (1.1) in the LP topology: it is known (see [4]) that this immediately gives informations on the convergence of uE. We prove in particular that

where L

=

only on.f In [3] it

lim was

E1y-p-1,

v

proved

is the outward normal vector to Q an

analogous

and y depends

result valid in the two-dimensional

then related to the torsion of an elastic bar with cross section Q enclosed in an increasingly thin shell made of an increasingly hard material. Using again some techniques of elliptic equations, the results of [3] were generalized in [2] to the many-dimensional case, with

Other similar results may be found in [1 ], sections 1, 3, 5, and in [5], chapter 13. In this paper we use the direct methods of Calculus of Variations and r-convergence, which allow us to give some answers even in the fully nonlinear case.

II. NOTATIONS AND STATEMENT OF THE RESULT 1 In what follows we denote by Q a bounded open subset of fR" with [R be a Lipschitz boundary, and by v its outward normal vector. Let d : function satisfying .

Annales de r Institut Henri Poincaré -

Analyse

non

linéaire

REINFORCEMENT PROBLEMS

For all

E >

0 fix r£

>

0

so

that lim rE F~CI

=

275

0, and set

Our assumptions on lQ ensure that the mapping (a, t ) ~ o- + tv(a) is if G is sufficiently small, hence for every x E ~E there exist invertible a(x) E aSZ and t(x) E ]0, rEd(6(x)) [ such that

If

we will briefly write d(x) in place of Take p > 1, and let f~ satisfy f : f~n x (2.1) for all x E [Rn the function f (x, . ) is convex; (2 . 2) there exists a function cv : [0, + oo [ [0, + oo [ which is continuous, increasing, vanishing at the origin and such that -~

(2 . 3) for all

x E

(2.4) there exists convex

and

z E

fRn

non-negative continuous function y(x, z) which is p-homogeneous as a function of z and satisfies a

where ~ : [0, + ~~o[ vanishes at infinity.

-~

[0,

+

~c

[

is

continuous, decreasing and

Finally consider a functional G : W 1 ~p(S2) ~ [0, + oo] such (2. 5) G is lower semicontinuous in the topology LP(Q) ;

When u E instead of G(u In). For every u E

is such that and G

>

uIn E

we

will

that

simply

write

G(u)

0 set

We want to characterize the r-limit of F~ in the topology LP(Q), depending on the behaviour of r£. Indeed, it is well known that the r-convergence of a sequence of functionals is strictly related to the convergence of their Vol. 3, n° 4-1986.

276

E.

minimum space,

ACERBI AND G. BUTTAZZO

points and minimum values : more precisely, let o mappings from X into ~, and x E X. We set

If these two r-limits agree, their

common

"

X be

value will be denoted

metric

a

by

°

We have:

[II. I ] (see [4 ], Theorem 2 . 6). Let X be a metric space and F, (F~)~> o mappings from X into R such that is equicoercive, I. e. for every A > 0 there exists a compact I ) the family (F~) subset K;_ of X such that THEOREM

-

it) F~(X) lim F~ Then F has

=

F. .

minimum

a

on

X and min f

=

lhn

x

lim

=

and x~ ~ M in X, then M is

lim x

Define for all

a

F~) ; (inf x minimum

moreover

point for

f F.

.

u e

and for every L ~ 0

We

will prove in section III the following result :

THEOREM every

u E

REMARK

[II . 2 ].

-

we

[11.3].

Assume that lim

=

LE

[o,

+

oo ] ; then for

have

-

The result above yields immediately that

does not exist then the functionals

Fg do

not

r-converge in

Annales de l’Institut Henri Poincaré -

~-~o

the

Analyse

topology non

linéaire

277

REINFORCEMENT PROBLEMS

III. PROOF OF THE RESULT

We will later need the LEMMA

[III. 1].

-

Let

following results : a :

(~" --~ (1~

be a continuous

function

and

then

u E

Proof We denote by the letter c any positive constant. Set for every F > 0 Then lim ccy

Vol.

3,

=

n° 4-1986.

0 and

278

E.

LEMMA [III . 2 ]. Let such that for all z E ~n -

Then

if

for

ACERBI AND G. BUTTAZZO

R be

g : (~n -~

C1,

convex,

p-homogeneous

and

every x, y E [R"

Proof. By the homogeneity of g we may assume|x| = | y== 1, hence g(x) x D must p prove rove that for (Dg(y))~~0, we set x 0 we must =

Let ;c be

a

minimum

on {

point of

Multiplying by x and using we obtain 0, that is

= 1 ~ :then

Euler’s theorem

on

p-homogeneous functions

=

tx such that Now take x we from (3.1) that for obtain of x,

=

=

By the convexity of g

we

suitable 11

>

0

have

whence ( y - x, Dg( y) ~

Dg( y) ~ >

so

g( y) ; by eventually taking - x instead a

0 ; analogously 0 and

we

have

finally

But then

Proof of Theorem [I I . 2 ] in the inequality

We begin with the For every £ > 0 set

then 0 Let u E

_

case

L

+

oo .

GL . _

0 outside E£ and moreover ] W 1 °p(SZ) : by the regularity of oSZ we may assume that 1

on

=

Annales de l’InstÜut Henri Poincaré -

u E

Analyse

W 1’p((~n). non

linéaire

279

REINFORCEMENT PROBLEMS

Setting

uE

tion for every

Fix M

Let

3(x)

>

=

we

=

have uE E

and u£

--~

ul~ in

0 and set

dist (x,

Q) :

E

we

~n : ~

have

By (3.2) (3.3) (3.4) and by Lemma [III. 1 ] we obtain

F~(~) ~

lim sup s-o

Letting

M -~ +

oo

prove the

We

now

Vol.

3, n° 4-1986.

in addi-

]0, 1 [

tE

and

t

-->

1

yields

inequality GL

F-.

Mt;:

then

280

E. ACERBI AND G. BUTTAZZO

To this

aim,

we

must show that

Without loss of generality

so

that

hence

may

(omitting for simplicity

u E

Since M is

we set

then for

assume

the

that for

a

~

u

in

suitable sequence

(Eh)

subscript h)

W 1 °p(S2). By the semicontinuity of G, it will suffice to prove that

Fix M > 0 and set it follows

If

we

and u~

if uE E

M~;

= ~ x E f~" : ~

then

by (2 . 4) (3 . 5)

arbitrary, we may only prove that

for every h~N

a

suitable sequence

vanishing

as

h - +

oo we

Annales de l’Institut Henri Poincaré -

have

Analyse

non

linéaire

REINFORCEMENT PROBLEMS

and

By (3.5)

(3 . 7) iv)

therefore in (3 aEoQ we set

we

281

have

. 6) we may assume also that y(x, . ) is a C 1 function. For every v(a) Dzy(a, v(~)) ; then by Euler’s theorem we have =

By the regularity of ~03A9, the mapping (03C3, t ) - a + tv(a) is invertible if E is sufficiently small, so that there exists rE(~) > 0 such that By using

the

regularity assumptions

on

aQ and

~(r),

one

may

on

03A3~

easily verify

that

uniformly in a E aS2. By simple changes of + briefly DUf;(x) in place of

where q

is

By (3 . 5)

and

a

Vol. 3, n° 4-1986.

variables

suitable bounded function. Then

(3 . 7) it )

we

obtain

we

obtain, writing

282

E. ACERBI AND G. BUTTAZZO

with

cor

vanishing

For all c7

E

Using (3 .10)

and

we

as ~

~

0. Therefore

have

and Lemma

[III . 2]

we

finally get

(3.6) follows from (3. 8) (3.9) and from the

convergence of Ut to

u

in

Proof of Theorem [II . 2 ] in the case L + oo . Since G~ for all B, the inequality F + G~ is trivial. As for the F > G~ we remark that, since E/rp -1 -~ + oo, given any inequality L > 0 we have E > Lrp-1~ for ~ small enough ; therefore for every u E =

By

the first part of the

proof (case

L

+

f)

we

obtain

whence the supremum with respect to L.. apply the result to the study of the asymptotic behaviour (as of the minimum values and minimum points of the functionals

by taking We 1

-~

now

0)

Annales de l’Institut Henri Poincaré -

Analyse

non

linéaire

283

REINFORCEMENT PROBLEMS

Since

fudx is continuous

u H

in the strong

topology of

~n

~ve

have

(see [4 ], Theorem 2 . 3)

Therefore, in order

apply

to

coerciveness of

+

inequalities (2 . 3) (2 . 6),

then

Rn fudx)~>

we

only

(uE) is relatively compact

THEOREM is

Theorem

[III . 3 ].

-

a

For all

E

topology

a E

aSZ and

enough,

tE

Assume 0

L

+ ~ ;

is

follows that for any b

for suitable constants

Vol. 3. n° 4-1986.

bounded,

By

the

and

if (3 .11) holds,

[0, rEd(6) ]

v

equi-

have to show that if uE E

constant c~ such that

is small

Since

must prove the

in

-

if

in the

we

o

relatively compact in By (3.11) it immediately Proof

exists

[II ..1 ],

c we

have

>

then 0 there

E. ACERBI AND G. BUTTAZZO

284

It follows from

so

(3.13), with

t

=

0, that

that

If 03B4 is properly chosen, by (3.12) (3.14) and

whence

J)

|u~ ] Pdx - 0 and ]~u~ W1,p(03A9)

(3.15)

we

obtain

# c..

£~

REFERENCES ATTOUCH, Variational convergence for functions and operators. Applicable Math. Series, Pitman, London, 1984. [2] H. BREZIS, L. CAFFARELLI and A. FRIEDMAN, Reinforcement problems for elliptic equations and variational inequalities. Ann. Mat. Pura Appl., (4), t. 123, 1980,

[1]

[3] [4] [5]

H.

p. 219-246. L. CAFFARELLI and A. FRIEDMAN, Reinforcement problems in elastoplasticity. Rocky Mountain J. Math., t. 10, 1980, p. 155-184. E. DE GIORGI and G. DAL MASO, 0393-convergence and calculus of variations. Mathematical theories of optimization. Proceedings, 1981. Ed. by J. P. Cecconi and T. Zolezzi. Lecture Notes Math., t. 979, Springer, Berlin, 1983. E. SANCHEZ-PALENCIA, Non-homogeneous media and vibration theory. Lecture Notes Phys., t. 127, Springer, Berlin, 1980.

(Manuscrit

reçu le 5 décembre

(revise

le 18 avril

Annales de l’lnstiiut Henri Poincare -

Analyse

1984) 1985)

non

linéaire