Some homogenization results for strongly nonlinear equations

SOME HOMOGENIZATION NONLINEAR

RESULTS FOR STRONGLY EQUATIONS”

DANIELA GIACHET~I Istituto di Matematica. Facoltti di Scienze. Universita di Salerno, 84100. Salerno. Italy (Received 5 Augw~ 1983: received for publicarion 23 Januar! Key words and phrases: Homogenization.

1981)

strongly nonlinear equations.

INTRODUCTION IN THIS

paper we deal with the problem of the homogenization Au +g(-,u) u E ff;q

of equations of the type

=f

(0.1)

Q),

where fE H-‘(R), g is a noncompact nonlinearity, which satisfies only a sign condition g(x, t)t 5 0 a.e. in 52, without any growth assumption and A is a nonlinear operator in one of the following classes, for which compactness results with respect to H-convergence are known: Au = -div(a(x,

grad u)) where a(x, :) is a Caratheodory vector valued function, Lipschitz and strongly monotone in g

(04

Au = -div(x,

u) grad u) ..,hn.Q AT/” +\ h,,lA..“” 1lS.....‘“1\ ilAlU, I, /J, R “, cl\ \sec; I_-.,. C,r+:,., 2, “I AC W11G‘G 1* \*, ‘, “Gl”Ug;a+r. L” an L,LI>J .Jc;CI,“I,JJ matrices uniformly coercive bounded and equicontinuous in t.

/n

\“.JI

2\

Au = -div(N(x)b(u)

gradu), with b continuous function from Wto Rand N(x) matrix depending only on x.

(0.4)

The analogous problem when A is linear is solved with different methods in [S]. Let us note that no characteristic properties of second order operators are used here and then the proof works for higher order also; moreover no regularity assumption (besides Caratheodory conditions) is made on the nonlinear term g. In the last section we treat the problem of the homogenization from the point of view of the T-convergence of functionals with uncontrolled growth, whose Euler’s equations can look like (0.1). 1. NOTATIONS

AND

PREREQUISITES

Let R be an open bounded set in KY’with a smooth boundary, E a sequence which converges ’ This research was supported by G.N.A.F.A.,

C.N.R. 1201

1102

D. GIACHETTI

to zero, E’, E”.

subsequences

of E. If & and p are two real numbers such that O
we define the class of operators: A(n,

(i) (ii) (iii) (iv)

p, Q) = {A : HA.‘(Q) -+H-‘(R)

: Au = -div(a(x,

gradu)),

a(,~, 6) vector valued CarathCodory function from R x R”+ R”, [0(x, 5) - a(x, q)][;L - ‘713 LYI E - VI’, I+, j) - 4x, r7)1s P/s”- ‘71, u(x, 0) = 0.

We shall denote by of the spaces.

and + respectively the convergence

in the weak and strong topology

Definition

1.1. [18] We shall say that a sequence of operators (AE)EEE of A(cY, p, Q) Hconverges to Ao EA(d, p’, Q)(O < d s p’ < +m), denoting it by A, JAo if for each fE H-‘(R) UC--‘no

in Hd,‘( S2)

-div(u,(x,

grad uE)) = f

in !2

implies a,(~,

grad

u,)

-

QO(X,

Let us note that from the last convergence -div(uO(x,

grad

UO>

in ( L2( 22))“.

we necessarily have

grad uo)) =f

in Q.

When, in the linear case, uE(x, 5’) is the vector whose ith component

is ,$i u’;‘(x) ;” this notion

of the matrices A, = (a:‘) lsi,,s,, to A0 =

is reduced to the well known G- or H-convergence (u$‘)l
The following theorem justifies the notion of H-convergence. 1.1. [la] For each sequence (A E) EEE of operators of A( LY,p, Q), there are an operator A0 E A(cu, p’/cr, S2) and a subsequence E’ such that (AEs)EIEE. H- converges to AU. Moreover we consider a function

THEOREM

g(x,r):

Rx

w+LR

such that for each s > 0, ;uz ]g(x, t) I s h,(x) g is a Carathcodory

E L’( S-2)

function and g(x, r)t 3 0 a.e. in R, for each t E R.

(1.1) (1.2)

Some homogenization results for strongly nonlinear equations

1203

The term g( . , u) does not define a map from HA.?(i-2) to H-’ (52) or to any other space such as L’(Q), for example, since it does not verify any growth condition in U. We have the following existence result: THEOREM 1.2. [19, 5, lo] IffE

A EA(cu,&

H-‘(a),

Q), g satisfies (l.l),

(1.2), there exists

a solution of the problem Au +g(.,u)

=f

(I-3)

U E k&*(n) in the following sense and

u E H$2( &); g( *, u) (Au, u> + og(., I vu E #j*(Q)

ng(*, u) EL’(Q),

u)u = (f, u) u = u.

and

n L”(R)

Henceforth a solution of a problem such as (1.3) will be understood in the sense precised by theorem 1.2. We now state some results which we shall often need in the following THEOREM 1.3. [5] If TE L&(Q)

n H-‘(R)

and u E H,$‘(S2) are such that

T(x) u(x) 2 h(x) for some h E L’(R),

then TM E L’(Q)

a.e. in R

and Tudx = (T,u).

THEOREM 1.4. ([ll, 91) Let A be an operator p > 2 such that if u and f verify

U E zP(R) -div(a(x,

in the class A(cu, p, Q): there is an index fE z-+P(Q)

grad u)) =fin

R,

then u belongs to H’,p(Q) and II4IWP(R) s dllflL4-~.P(n, + II 4W(R)~.

Moreover

the index p and the constant c depend only on LY,/S, R.

We conclude this section by stating a compactness result: THEOREM 1.5. [13] Let (fE)EE~ be a sequence of H- ‘,‘(S2) bounded in the space A(S2) of the

120-l

GLACHETTI

D.

measures

on 9:.

such that fC-fi,

in N-‘,‘(R);

then for each 4 < 2.

fE--+J, in H-‘.q(R) 2. THE

MAIN

RESULT

In this section we get a result of homogenization for nonlinear problems We consider a sequence (A,),,E o f operators of A (a, /?, S2) such that

of the type (1.3).

A, &. Our problem

(2.1)

is to pass to the limit in the equation: A&,

A previous homogenization elliptic operators, [8, theorem

+ g(.> 4

wherefEH_i(R).

=L

(2.2)

result for equations of the type (2.2) with A, linear 31, is generalized by the following theorem.

uniformly

THEOREM 2.1. Under

the hypotheses (l.l), (1.2), (2.1), for each sequence (u,),~~ of solutions of the equations (2.2), there exists a subsequence, again denoted by (I(,),~~, which weakly converges in HA.‘(Q) to ~a, a solution of (2.3)

Aouo + g(. , ug) =f. Furthermore inL’(R).

(2.4)

‘> 1(0)(10 in L’(Q),

(2.5)

g(.,u,)+g(~,r~~) g(.,

+l,--tg(

liio(A,u,, Proof.

From

(2.2) we get (A&E, u,> + og(.,

I

Then

(2.6)

14,) = (AOUO, 110).

u,)u,

=

(f, u,>.

we easily obtain

It follows

that there

is a subsequence, 11E -

11%

again denoted

in Hb,*( Q)

and

by (LL,),~E and I{* EH,$,‘(R), 11E-+

11 *

a.e. in R.

+ This means that for each compact k, k C R, there is a constant cc. such that VO E D(Q) supp 4 C k. I(.f-c.Q)’ s ct!lQ!lr-in,.

VEE E.

such that

Some homogenization

results

for strongly

nonlinear

equations

1105

Thus g(.~. uJx)) Moreover

--, g(~, 1~*(x))

(see [19] for a similar argument), I&,

u,(x)) I s

let us observe that. for any 6 > 0

SUP I&, 4 I + Q!(L UC>c~>>uE(X) ~1!4&1

set E in R, using (1.1)

and then, for any measurable

IE By Vitali’s theorem,

a.e. in R.

]g(.,u,)]

C Eha-l + 6Cl. I

we get, using (1.2) inL’(R).

g(.Yu,)-+g(.?u*)

(2.7)

By Fatou’s lemma: A.9 u*)u* G iiiOinf og(.,

IR

~,)u,Sc~.

I

Since the sequence (AG,),~~ is bounded in H-‘(R), there is ti EHb.‘(Q), such that d_ri A ,u, A ‘.“L*

eventually

(2.8)

passing to a subsequence,

in‘q-1’0) \ iCl.

(2.9)

Now AEuE -f= -g(. , u,) converges in L’(Q) and so is bounded in the space .11(Q) of the measures on Q (see footnote on p. 1204). By theorem 1.5 we get A,u,+A&

in H-1.q(R).

4 < 2.

(2.10)

Passing now to the limit in the equation (2.2): r

(Aoti, u) +

Jg( *, u*)u = (f, u>

VU

R

EH$~(Q) nL”(Q).

(2.11)

We can state (2.11) for u = U* too; indeed, passing to the limit in (2.2), we get, in the sense of distribution f-Aoci=g(.,u*).

Since T=f-Aou=g(.,u*) Tu” =g(.,u*)u*

by theorem

Elf-‘(R) 20

nL*(n),

a.e.inR.

1.3, we get (A&-f,

u*) +

JR

g(.,u*)u*

=o.

(2.12)

We have now to identify u * and Li;to this end, following the outline of [4], let w be an arbitrary function in Hb.‘(S2) and for a fixed q > 0 and p > 2, w? EHb+‘(Q), such that IIw - WallHpcn, c II.

1206

D. GIACHETTI

If we denote by u, the solution of

by theorem

1.4 we get also UC--‘ w?j in Hbvp(S2) for somep > 2.

Since A, is monotone,

(2.13)

we have

0 s (Aeu, - AcuE, u, - u,> = (A&B, u,) - (A&c, = (f, UC>- j--d*,

u,> - (Aowq, u, - u,)

u,)u, - (A CUE,u,) - (A ow9, u, - 0,).

By Fatou’s lemma, (2.1) and (2.13) we have passing to the limit for E--, 0: O~Cf,u*)--lnp(.,u*)u*-GZari,w,)-(Aow,,u*-w,).

(2.14)

We can now pass the limit in (2.14) for q+ 0: Os(f,u*)--l,g(.,u*)u*-(A,ri,w)-(&w,u*-w).

Using (2.12) we get 0 c (A&

u* - w) - (Aow, I(* - w).

We take now w = u* + t@, t E W, $ EH$*(SZ) (Aoi-

and pass to the limit for t--, 0:

AOu*, @) = 0.

Since @ is arbitrary A,,u = Aou*

which means also c zz u*

(2.15)

Since (2.7), (2.8) implyg(. , u*), u*g(. , u*) E L’(Q), by (2. 111,(2. 12) (2.15), u* = uo, where uo is a solution of (2.3). We prove now (2.9, (2.6). As before, let us call, for q > 0 and p > 2 fixed, u,, a function in H$P(S2), such that 11%- ~0llH’jV2,s r7 and zE the solution of the problem AEzE = Aou,, z, E H,+‘(Q).

By theorem

1.4,

z,-

uq in H$P(S2);

Some homogenization

using again the monotonicity and then for n--, 0, we get

1207

results for strongly nonlinear equations

of the operators A, and passing to the limit, at first for E+ 0 (2.16)

u,) 3 (A,-,uo, ug).

li_i inf(A,u,, By Fatou’s lemma

(2.17) Moreover

and

lim (f, E-+0

u,>= (f, UO>= (AOUO, UO>+

I g(

(2.18)

a, UO>UO.

cl

Since for any sequences ((IE)EE~, (bt)aeEof positive real numbers such that lim inf uE 3 a, E-0 1:-:-c‘

_ I.

III11 1111 u 6 9

u,

E--r0

lim (a, + b,) = a + b, E--r0

we have hi

aE =

a,

limb,=

b,

E'O

from (2.16), (2.17), (2.18), we obtain (2.6) and ~_~Ing(..u,)u,=Ing(.,uo)uo. Since

uEg(.,uE)30,

uEg(-,uE)

-+

uog(.,uo)

a.e.

in R and lim.Jo

g

(*,u,)u,=

E'O

g( . , uO)uo, we get (2.5) by a classical lemma from integration theory (see for example [14 theorem 5, chapter VI]).

Jn

Remark

2.1. Let us apply the result of theorem (2n)/n - 2); we get, using the same notations

u,Moreover,

uo

2.1. to the function

g(t) = Itl’-‘t,

s3

in H$‘( Q).

by (2.5),

II4lLJ(Q) + IIuoll L’(R). Since L’(R) is an uniformly convex space, we obtain in this case also (even if the embedding H$‘(Q) CLS(S2) is not compact): u,+

~0

in L’(Q).

1108

D.

GIACHETTI

Remark 2.2. As in [8] when ‘4 is linear? we can quite easily prove . the same rec111r 2~ .._ in thmrpm ..- _-.... ----_ -I ..I__._... if we replace g by a sequence of functions (gOEE~ which satisfies the following condition

1.1,

x-

gE(,r, r) is measurable

in x for all I E W. VE E E.

For almost all x E 52, all I, r0 E R, V(EE Elg E(~r,t) - g&

to)

depending

i s Q,,,,(If neither

- toI), OJbeing a function

not

on &nor on x such that

w(lr-tol)~Owhenir-foj-,O. For almost gE(X, f) -+

(2.19)

all x E 52, and for tfixed in !R g0(x,

r>

as

E+

0,

gE(x, r)t 2 0 a.e. in R, Vr E R. sup jg,(x, t) ) s h,(x) E L’(Q). jllSS Moreover in some examples we can still obtain we do not have almost everywhere convergence Indeed let (gE)eEE be a sequence such that

g&r. f) = a&)g&, a,(x) E L”(Q), 0 < a&)

-

the previous homogenization result, in x for fixed t of (gE)EEE to g,].

even

if

4, YI

s a,(x) s y2 < + m, VIEE E.

(2.20)

a0 in L”( S2) weak*,

gE(x. t) satisfies (2.19). Then the proof 31 for equations

of the homogenization result is a mild variation of that given of the type (1.3) with a linear operator A as principal part.

Remark 2.3. We want to point out that the results we obtained problems: for example no truncation argument is used. In the following section. results when the principal 3. HOMOGENIZATIOS

are not related

in [S, theorem

to second

order

we shall prove with slightly different methods some homogenization part A does not belong to the class A (CY,/I. Q). PROBLEMS

WITH

QUASI-LINEAR

PRINCIPAL

PART

This section will be concerned with the problem of the homogenization for strongly nonlinear elliptic equations whose principal part A belongs to some particular classes lvhich are not included in A (N. p, Q) operators A. since it The proof is different from that used in theorem 2.1 for monotone consists essentially on reducing the problem to another one with linear principal part and using the results of [8]. Let us introduce the class of matrices M(LY, p) = {N(x) E (L”(X))“‘, i:V(x)~] =Gp1/1,

N(x)U VJ. E W”.

2 cu///‘. a.e. in R”}.

Some homogenization

and the class of matrices depending on a parameter

t:

Vt fixed,

:M(CY,p, e> = {N(X, f) E 1M(a; p>. iiV(,r,t)-iV(x,s)/~e(It-s1),

esr4r,

1209

results for strongly nonlinear equations

v,r E an,

vs,

vt E 3

increasing, continuous in 0).

We shall consider at first the class of the operators Au = -div(N(

. , u) grad u)

where N(x, t) belongs to M(cY, /3, 0). Our problem is to pass the limit in the equation -div(N,(.,u,)

gradu,)

K, E H&2)

+g(.,

u,) =f

(3.1)

EE E.

Let us note that the existence of a solution u, of (3.1) is proved in [19,5]. In order to define the H-limit of the operators -div(N,( *, u,) grad u,), we shall recall the following compactness result with respect to H-convergence. THEOREM

3.1. [2] If for each E E E, N,(x, t) belongs to M(cY, /?, 6) and I$~is a solution of -div(N,(.

, b)

grad &)

=f

GEE fG’P>

there exists a subsequence E’ and a family of matrices No(x, t), defined for each t E W, belonging to AZ(cu,p’/(v), for each r E R, equicontinuous with respect to t, uniformly in X, such that @E,- $0 in H$*( S2)

Now we are able to prove the following homogenization (3.1).

result for equation

of the type

3.2. If for each EE E, NE(x, t) belongs to the class M(cu, ,l3,0) and g(x. t) satisfies (1.2), each sequence (u,)~~E of solutions of (3.1) has a subsequence, still denoted by weakly convergent to no, which verifies

THEOREM

(l.l), (UE)EEET

-div(No(.

, uo) grad uo) + g( ., ua) =f

(3.2)

L40E H;,?(n) where No(x, t) is the matrix defined by theorem 3.1. Moreover g(*, UA-+g(*,

u0>

inL’(Q)

(3.3)

N,( . , u,) grad u, grad u E= lim E-+0I R

g(*,uE)uE+g(.,uO)uO

inL’(Q).

No(.,uo)

graduograduo

(3.4) (3.5)

1210

D.

GIACHETTI

Proof. It is easy to prove, using the coercivity of the matrices N,(x. f) and the sign condition on the function g(~, t) that any sequence of solutions (uJEcE of (3.1) satisfies

II41H;~2 (Q)6 ;llfllH-~w From this relation we obtain that there exists a subsequence and U* E H$‘(S2) such that (d&E u,-

of (u,),~E,

still denoted

as

Ll* in Hb,‘( Q).

Then, using the same arguments if the proof of theorem 4.1 of [2], we get N,( . ) u,) x ‘V,(

) u*).

Now the homogenization

result given in [S, theorem 31 for semilinear problems implies that u * = UO,where ug is a solution of (3.2). The relations (3.3), (3.4), (3.5) follow from [8, theorem 31: they are proved as in theorem 2.1 of the present paper. Now let us consider the class of the operators Au = -div(N(

.)b(u) grad u)

where N(x) is a matrix depending only on x and b is a continuous We can prove the following THEOREM

function from ‘R to R.

3.3. Let (N E) EEL be a sequence in M(cr, p) such that NE2 No

and let b,, b0 be functions from W to W, continuous, bo

b,Then if g verifies (l.l),

in L”(Q) weak*.

(1.2) each sequence (I(,)~~~ of solutions of the equations -div(N,(

.)bE(uE) grad u,) + g(. . u,) = f

u, E fl;qn>,

has a subsequence,

satisfying 0 <(/I s b,(t) c ;/2< + z and

E E E u {O},

still denoted by (U,)EEE weakly convergent

to ug, solution of (3.6)0.

Proof. The existence of solutions for (3.6),, E E E U {0}, is guaranteed Following the outline of [3, theorem 2.41, let us put

d,(r) = [b,(s)

(3.6)~

ds,

by [lY. 51.

VtEs?.

The functions 6,, strictly increasing, are bijections from R to R, which are uniformly Lipschitz. C’ functions as their inverse 6;‘. The functions .zEdefined by z, = UUE),

VE E E u (0)

(3.7)

.T..

Some homogenization

are in HA,‘(Q) since u, are in H$‘(Q)

l‘ll

results for strongly nonlinear equations

and, (see [17]),

grad z, = bE(ue) grad u,

a.e. in Q.

By (3.7) we can reduce (3.6), to the problems with linear principal part -div(N(

.) grad z,) + g( *, d,‘(zE))

=f

(3.8)

z, E Hp(n).

The sequence defined by E E E u (0)

g,(x, t) = g(xt WN,

satisfies clearly the assumptions (2.19) of the remark 2.2, because then the conclusion follows directly from the homogenization (b)cEE;

of the properties of result of [8, theorem

Obviously we can replace in both of theorem 3.2 and 3.3 the function g(x, t) by a sequence satisfying (2.19) or (2.20) of the remark 2.2. 4. CONVERGENCE

OF THE

MINIMA

OF FUNCTIONALS GROWTH

WITH

UNCONTROLLED

Let us now consider the problem of the homogenization from the point of view of the Iconvergence of functionals, whose Euler’s equation can look like (1.3), because of a lower order term with uncontrolled growth. More precisely let G(x, s) : R x R + R, g(x, s) : S2 x W + R be such that G(x, s), g(x, S) are Caratheodory

functions,

G(x,O) = 0, ac ,,(x,s)

=g(x,s)

g(x, s)s 2 0 ;;(Ig(x.s)I Moreover

(x,s)

E R x w,

(4.1)

a.e. in R, Vs E W, =H,(x)

=‘(Q).

let us assume that 3u,, E

H,)*(Q) :

I, G( .,uo) < +=.

(4.2)

If h E H-*(R) and (f)E EELis a sequence of functions from R x W x following assumptions are satisfied:

R”

into R, for which the

fE(x, S, 5) is measurable in x and continuous in c = (s, Q,

(4.3)

fc(x, S, 5) is convex in f,

(4.4)

3kik2 3 0, a(x), b(x) E L’(Q), a(x) 20

a.e. in R, such that

kil f12 + a(x) SfE(X,S, 5) c k*I Cl2 + b(x),

(4.5)

V&E Et,

t We confine ourselves to this case (growth of order 2) as we did in Section 1 for the operators (Ar)~.l results given in this section for (fJce~ hold also for growth of order 1 s p < + m.

but the

1217

D. GIACHETTI

for each EE E fixed, let us consider

the functional

From the assumption (1.2) 5;E(~) ?PC(u) is bounded from below and moreover any minimizing sequence a point u, E HA.‘(n) which reaches Let us call now

defined

on HA,‘(Q)

by

is not identically -r. from (1.5) and the positivitv of G. from (1.3) it is I.s.c. in the weak topology of H?).:(Q); is bounded in HA.‘(Q) (see 1.5). Then the existence of the minimum of SE follows.

and recall the following: Definirion

4.1.

[7] We say that

converges

to a functional

(i) for each sequence

the sequence

FO on Hi.*(n),

(FJECE,

denoting

defined

it simply

(u,)~~E in Hb*‘(S2) such that u,lim inf FE( 0,) 2 Fo( u)

by (4.7),

r-(Hb~‘(Q),

I$,)

by FEE FO, if: u in HA.‘(Q),

;

E+O

(ii) for each u EHd*‘(R),

there

exists a sequence u,--

u

(u,)~~E in H~2(Q)

such that

in H:,‘(Q),

lim Fe(uE) = F,,(u). E-O In [15. 6] a compactness result (which justifies the previous definition) is proved to T--convergence of functionals like (4.7): for each sequence (fE)EE~ satisfying there exists a subsequence (fE.)E.E~. and f. in the same class, such that

Fe(u)= j,foL U, DLL) for

with respect (4.3)-(1.5).

each u E H,!?(Q),

FE,L, F,,. Remark 4.1. Let us point out that in HA.‘(Q) I--limit of the sequence ( FE)EEE. the r-(Lx(Q). (j)

for each sequence

nL=(Q) s)-limit

we can define, in addition to the as a functional @a, satisfying

(u~)~~E in HA**(Q) n L”(S2) such that uE+

u in L”(R).

lim inf FE(uE) 2 @O(U).

E-0 (jj)

for each u EZfb.‘(R) such that

n L”(Q)

there u,-+

exists 0

a sequence

in L”(R).

iii0 Fc(uE)= @o(u). Then

F. = @o, as proved

in [6, 151.

(uE)EEE in H&.‘(R)

nf.-(Q)

Some homogenizatton

results for strongly nonlinear equations

1213

Remark 3.2. We note that the functional G(o) =-j-o G( , u,) IS a continuous perturbation with r’espect to L” topology. Indeed. for each sequence 0,. which converges to cl in L”(Q). we

have

and by Lebeseue’s theorem lim G( *, u,) = G( ., u). iR 6 IR Let us now suppose that FE K Fo; we want to prove that each sequence (u,),~E of minimizing points for S,(u) converges to a minimizing point for

So(u)=

I fo(., u, Do) + I G( ., 0) - (h, 0). R

Q

By remarks 4.1. and 4.2., we get Fo+c=r-(L=(R),s)-Frr$F,+C) = I--(H:.*(Q),

w) hi (F, + G).

Since by the equicoercivity of the functionals (4.7) and the positivity of J‘o G( *. u). we easily get that each sequence (u,),~~ of minimizing points for a,(u) is bounded in ff6.‘(Q), by a well known theorem (see [7]), the sequence u, converges weakly in HA-*(Q) to a minimizing point ua of 90(u). REFERENCES 1. BENSOUSSAN A., LIONSJ. L. & PAPA~KOLAOUG., Asymptotic Analysis for Periodic Structures. Sorth-Holland. Amsterdam (1978). 2. BOCCARDOL. & MURAT F., Homogeneisation de probltmes quasi-liniaires, Atti del convegno Srudio dei nmh/mmi./imitrnl~//'nnn/;ri fr,n~innn/~ IR.m.nmnn~

1

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D. GLACHETTI

16. SP~\GNOLO S., Convergence in Energy for Elliptic Operator. Numerical Solutions of Partial Differential Equations. III Synspade (1975), (Edited bv Hubbard) Academic Press (1976). 17. STAMPACCHM E., Equations eiliptiques du second ordre k coefficients discontinus, Seminaire de Mathkmatique SuoCrieures No. 16. Les Presses de I’Universitd de MontrCal. %fontrCal (1966). _ 18. T~RT’AR L., Cours Peccot au College de France (1977). 19. WEBB J. R. L. Boundary value problems for strongly nonlinear elliptic equations. /. Land. Math. Sot. 21 123-132 (1980).