Existence and homogenization for semilinear elliptic equations with noncompact nonlinearity

Nonlinear Andysir. Theory. Printed in Great Britain.

Methods

& Applicanons.

Vol. 8. No. 1. pp. 5-15,

0362-546X/84 $3 00 + .OO @ 1984 Pergamon Press Ltd.

1984.

EXISTENCE AND HOMOGENIZATION FOR SEMILINEAR ELLIPTIC EQUATIONS WITH NONCOMPACT NONLINEARITY * DANIELA GIACHETTI Istituto di Matematica, Facolta di Scienze, Universita di Salerno, 84100 Salerno, Italy and MYTHILY RAMASWAMY INRIA, Rocquencourt,

B.P. 105, 781.53 Le Chesnay, France and T.I.F.R. Centre, Bangalore, India (Received

Key words and phrases: Homogenization,

12 November 1982)

G-convergence

0. INTRODUCTION IN THIS paper

we consider

the problem

of the existence Au =f(x,

of a weak solution

for

u)

P>

u E a(Q), where A is a uniformly elliptic A less than the first eigenvalue

linear operator and f is a Caratheodory of A and for some q(x) E L2(Q), f(x,.s)
SZ-0,

f(x, s) 3 As - V(X)

s d 0.

function

such that for

Here we give a very simple existence proof, using a result of [14]. The existence of a classical solution using maximum principle when f is &-Holder continuous in the first variable and locally Lipschitz in the second variable, is proved in [l] and weak solution in [6]. Then we consider the problem of convergence of the solutions for AEuE =f(x,

u,) I

UEE HA(Q), to a solution

(PE)

of the problem Aou = f(x, u) u E HA(Q),

1

(PO)

G when A, are uniformly elliptic linear operators such that A E-+ AC,, in the sense of G-convergence [12], [13], introduced by Spagnolo. The difficulty lies in the fact that f is not a compact perturbation between L2(Q) and L2(Q), since indeed it does not induce any map from L2(Q) to L2(Q). In [ll], homogenization of equations of this type, if f is Lipschitz in the second variable, is treated using maximum principle. Here the convergence of the solutions to a * This work was carried out when the first author was in Paris VI, partially supported by C.N.R., Italia and the second was in INRIA under deputation from T.I.F.R., India. 5

D. GIACHETTI and M.

6

RAMASWAMY

solution of the homogenized problem PO is shown without any regularity assumptions. Since the proof, both in the case of existence and homogenization does not use any characteristic properties of second order operators, it works for higher order also. 1. ASSUMPTIONS

Let Q be a bounded operator in the class

domain

in RN with sufficiently

smooth

&I and let A be an

boundary

laij(X) 1s p, Xaij(X)EiEj 2 cUI&I’, E E RN a.e. in Q Uii=Uji,l~i~j~~,O
w

I and A1 be the first eigenvalue of the operator A. Let f: Q x IR -+ IF! be a Caratheodory function (measurable in x E Q for any t E R and continuous in t for almost every x E S2) satisfying the following condition: There exists n(x) E L2(Q) and A. < Al such that for almost all x E 52 f(x, s) G ?Ls+ q(x)

s 2 0

f(x, s) 2 As - V(X)

SGO

(1.1)

1

and ;ug If@, s) I E L’(Q), We will consider

0 s t < m.

(1.2)

I

(1.3)

the problem Au = f(x, u) 24E a(Q),

and look for a solution

in the following

u~Hb(Q),f(.,u) (Au.o)=l/(.>

sense: EL’(Q),f(.,u)u

u ) u f oreachu

EL’(Q) EH&Q)

and

nL”(Q)

for u = u where

(. , .) denotes

the duality

between

H&Q)

(1.4)

and i

and H-r(R).

2. EXISTENCE

We state the following. THEOREM

u EH~(Q),

1. Under the assumptions in the sense of (1.4).

Proof. The problem

(1.3) is equivalent

(1 .l),

to

(A-A)u=f(x,u)-hi u E H&2).

(1.2)

the

problem

(1.3)

has

a solution

Existence

Denoting

and homogenization

for semilinear

elliptic

equations

with noncompact

nonlinearity

by B = A - A, g(x, s) = h - f(x, s),

the problem

is Bu + g(x, u) = 0 I

u E ffh(Q), where

the function

g(x, S) satisfies g(x, s) signs

We note that each function

which satisfies

2 -q(x).

(2.3)

(2.3) can be written

as

g(x, s) = g1(x, s) + gz(x, s) where

gl (x, s) and gz(x, s) are CarathCodory

functions

(2.4)

such that

gl(x, s)s * 0, Jgz(x, s) 1S 217(x) for almost allx E Q] andforsE (note

that v 2 0 because

of -q(x)

6 f(x,

0) s q(x)).

s 20

-g(x,O))-

s ‘co

1

-(g(x.s)

g2(x7s) ={ (g(x,s) by the hypothesis

-g(x,O))_ -g&O))-

(2.5)

In fact we can take

(g(x,.$ -g(x,O))’

gl(x,s) = -(g(x,s) Moreover

1

R

+g(x,O) -tg(x,O)

s 20 s

so.

(1.2)

(2.6) Furthermore

B is coercive

for all A < A,. In fact

the existence of a solution can be obtained using the following version of the theorem of [ 141. Let B : Wa+‘(S2) -+W-m~p’(Q) (1 < p < x, (l/p) + (l/p’) = 1, m 2 1) be uniformly elliptic function satisfying (2.4), linear operator of order 2m and g(x, t) : Q X F! -+ R, a Carathkodory (2.5), (2.6) and fE W-m3p’(S2). Then there exists u lWy.P(i2) such that g(. , u), ug(., u) E L1(S2) and

Therefore

W, u) +

1g( . , u>u= If-u

for all u EW$P(Q) n L”(Q) and for u = U. (See also [6] in the case of an operator of second order and [5] in the case where

q(x) = 0.)

‘% uo a~erugsa

ue uy$qo

11:~

aM

MON ‘(~‘2) us SB lgds pue (1’~) liq pauyap

‘W:H i

3

an

(“n ‘x)X + Wg

0 =

.(On ‘Onov) = (“n ‘WV)

‘%uo.Ils (ig)r 7

u!

Bu0.w (b)r7

e siyxa

uq

(On ‘ . )JOn t

(“n ‘ . )pn

(On ‘.)Jt

(“n ‘.)S

u!

‘(d&w

(Z’E)

3 On

r (On ‘x)J = OnOv 30 uoynlos r2 ‘On 01 sa%awo:, iCIy2aM ygy~ ‘~3~ (“n) icq palouap u@e ‘aDuanbasqns alayi ‘(I ‘5) uralqold ayl30 suogn[os 30 333( %) axranbas yxa .103 ‘0~ t3 v pue 0~

30 anpzAua2!a

1s.y aqi ST ;y a.IayM ‘;y> y 103 (2.1) ‘(1.1) suogdwnssE

ay] lapun

.%u!MoI~~~ ai.p aaold

‘k@H

(T’E)

i (“n

~33,~( s”v) awanbasqns

e s! alay

.awa%aAuoy)

I! aq g’“(“v)

‘E MI~~IO’~HJ,

01 s! alay urge Ino

3n

3

‘x)4 = “n’v

30 uoynlos B ‘% rapysuo3 sn ia7 ‘0~ 01 sa%aAuoD-3 s3v ~eq~ yns ‘MI UI 0~ lolelado ue pue ‘m 30 slowado 30 3sa(sv) aDuanbas yxa 10~ ‘z MSXOXHL

30 uo!iou

ayi say&

([L] ‘[ET] ‘[zT]) waloayi ‘VaM-W~H

a3uanbas

y (s ‘x)2 a.IayM

la7 ‘2 30 acwanbasqns

u! .Qv--s~Jv

%u~Mo~~o~ayL ‘(7.s)~-H

3 s

.(a ‘d ‘a) n = ,y UTslowtado 30 e ,g pue (0) n {N 3 U: (U/I) = 3) = 2 la?

NOI.LVZIN39Ot’IOH

‘E

Existence and homogenization

for semilinear elliptic equations with noncompact nonlinearity

By (3.4) we get

By (3.3), (2.9,

we get

and therefore Cl

I/~Ell~h(n) s -

d

= C2, YE E E.

We have also

(3.5) Then,

there

exists a subsequence,

again denoted

by (u,),~~

U&--‘ uo

in

u,+

a.e. in Q,

gl(.

u.

, u3

-+

gl(

by Lebesgue’s

We now prove

theorem,

uO such

that,

H& S2)-weak,

*, ~0)

a.e.

in

Q,

g2(.,u,)+g2(~,uo)a.e.inQ. It is clear,

and

(3.6) I

that

g2(.,G---,g2(.,u0)

in

L2(Q2>.

(3.7)

gd~,uJ-gl(.,u~)

in

L’(Q).

(3.8)

that

In fact, for each 6 > 0,

By (34, (U&E

(3.6) and Vitali’s theorem, we obtain is bounded in HA(Q), we have also

and so, for a subsequence,

Agu,-

x in H-l(Q)-weak.

A&u, - Au, = Aau, is also bounded

(3.8). Since A, E M, V’EE E and the sequence

in the space L’(Q). Apu,+

x

(gl( . >~3

By theorem in

H-1,4(Q)

+ gd . , ~3) 1 of [8], we get Vq < 2.

(3.9)

10

Then

D. GIACHETTI and M. RAMASWAMY

by lemma

14 of [4], x = Aouo,

(3.10)

lim inf (AEuE, u,) 2 (Aouo, u,,). By (3.7), (3.8) and (3.10), G E H:(Q) nL”(Q),

we can now pass to the limit in the following

equation,

for each

and get (3.11)

gz(~,uo,)@=o

(mono> 4) + ogl(~,u(,)@+ I

where B. denotes the operator AtI - J.. Let us call T = - Bou,, - gz(. , UO)= g,(. , u,,). We have T E H-‘(Q) a.e. for LQ)E HA(Q) and so by theorem 1 of [5], TUOE L’(Q) and

n L’(Q)

with Tzq, B 0

which gives, (Bouo, u,,) +

IR

g,( . . ~u)k +

IR

g2(

> uo)u,,

= 0.

(3.12)

By (3.11) and (3.12), (3.2) is proved. We now prove that (AA. u,S(. Obviously

it will be sufficient

kg(. n, = (BFuC, uJ, b, =

. ur) -

to prove @A.

Calling

u,) + (Ac,uo, u,,). uof( . . kl)

(3.13) in

L’(R).

in

L ‘( Q).

that

u,) + U&U,,. u,,). 3u,> -+ kg( . , 4)

g,(. . u,)u,, I 61

we have by (3.7), (3.12).

respectively, lim (a, + b,) = a,, + b,, r-0

lim inf b, 2 b,, E- 0 hence

it follows

that lim a, = a(), lim h, = b(,. E--r0 F-o

(3.10) and Fatou’s

lemma,

D. GIACHETTI

12

and

M.RAMASWAW

Now we want to point out an example in which we do not have almost everywhere convergence in x, for fixed t, of {f’} to f”, but still we can obtain the homogenization result. This example suggests that perhaps we can improve theorem 4, by weakening the hypothesis (iii). 5. Let f&(x, S) = aE(x)f(x,

THEOREM

s). We assume

that

a, are L” functions such that 0 < yl s uE G y2 < or,, in L”(Q)-weak *, at -a f(x, s) is a Caratheodory function satisfying (1.2) and the following condition: of Ao) with ( w h ere A: is the first eigenvalue r](x) E L*(n) and x <($/y~),

Then there exists a subsequence, converges to uo, a solution of

~(x,S)~As++(x)

sao,

f(x, s) 1 XF - i(x)

S G 0.

again denoted

by (uJEEE of solutions

there

exists

of (4. l), which weakly

Aouo = a!(~, uo), (4.2)

z&JE iY& Q).

1

Furthermore and

(A&u,, u,) + (AOUO,uo)

(4.3) In In order

to prove

aF(M.7

the theorem,

u,)u,+

we need the following

LEMMA. Let {ad and {he} be a sequence (i) 0 s yl s a, s y2 < ~0 and us-(ii) h, are L’ functions converging Then

JR

uo)uo. 1

a(.)&

of a.e. nonnegative

functions

such that

a in L”(B)-weak *. a.e. in Q to h E L’(Q).

we have liminf_/a,h,dxSjahdx.

Proof. Denoting

by xE, the characteristic aE(hE- h) dx = 2=

IR

function

of {x E Q : hE(x) G h(x)}, we have

aXhE - h)xE dx +

a@,-

udhc - h)(l - xc) dx

h>xE dx.

Since aE are bounded in L”(Q) and h F-+ h a.e. and -h(x) s (hF(x) - h(x)xE(x) c 0, by Lebesgue’s theorem, we can pass to the limit in the last relation and get the result. This reasoning is true even if {h,} are not uniformly bounded in L’(R). Then in that case the lim inf can be + m. Proof of theorem 5. The function u$ satisfies the condition of type (1.1) with A as y$t and r~ as yz~. Hence the existence of a solution for (4.1) follows from theorem 1.

12

D. GIACHETTI and M. RAMASIVAMY

Now we want to point out an example in which we do not have almost everywhere convergence in x, for fixed t, of {f’} to f”, but still we can obtain the homogenization result. This example suggests that perhaps we can improve theorem 4, by weakening the hypothesis (iii). THEOREM 5.

Let f&(x, s) = a,(x)~(x,

s). We assume

that

ae are L” functions such that 0 < y1 4 a, G y? < CTJ, in L”(R)-weak *, a, -a ’ d ory function satisfying (1.2) and the following condition: f(x, s) is a C ara th eo ( w h ere A:‘is the first eigenvalue of A,,) with q(x) E L’(Q) and 1 <(Avyz),

Then there exists a subsequence. converges to ug, a solution of

f(X,S)dAsS

i(x)

.f(x,.+&--_(x)

SGO. by (Up) FEN of solutions

again denoted

there

exists

sso,

of (4.1), which weakly

AOUO= @(x, UO). (4.2)

no E &A(Q).

I

Furthermore (AA,

u,) -+ MO+

and

~0)

(4.3) IR

aA.)f(.,

the theorem,

u&,+

In order

to prove

LEMMA.

Let {ad and {A,} be a sequence

we need the following

(i) 0 < yl G aF s y2 < m and up(ii) h, are L’ functions converging Then

Proof.

iR

a (.)A., UO)Q

of a.e. nonnegative

functions

such that

a in L”(R)-weak 1. a.e. in 52 to h E L’(Q).

we have

Denoting

by xF, the characteristic uE(hF - h) dx = 3

function

of {x E 8 : h,(x) G h(x)}, we have

aF(ht-- h)xE dx +

I a& R

aXhp - h)(l - xc) dw

- h>xFdx.

Since aE are bounded in L”(Q) and h, + h a.e. and -h(x) c (hF(x) - h(x)xF(x) c 0, by Lebesgue’s theorem, we can pass to the limit in the last relation and get the result. This reasoning is true even if {hF} are not uniformly bounded in L’(Q). Then in that case the lim inf can be + 0~. Proof of theorem 5. The function a$ satisfies the condition of type (1.1) with A as yzx and n as y2~. Hence the existence of a solution for (4.1) follows from theorem 1.

Existence

and homogenization

Now using the assumptions

for semilinear

elliptic

equations

with noncompact

nontmeariry

on f, we can split g(x,

sj = AS- f(x, S)

as in (2.4), g(x, s) = where

gl and g2 satisfy

(2.5).

We introduce

g& Then

(4.1) can be written

Denoting

g1(x,

by B, =A, - xy2, the above

some other

equation

+ %YZ - a,)~,

II& along

s G?

with

gz(x,

the functions

(1.2)

jR

+ g&4

g; and g’f as g2(x,

sb,.

obvious

splitting

+ g&4

= 0.

for 1~0)

(4.4)

2 0 and gh and g’, satisfy conditions

similar

gxu,> u, s C% *Rl(lldUF i C6. j

and the a.e.

convergence

gr(u,) 3 gl(u0)

in

-

44

of ~(uJ

to of

gives,

L’(Q).

Since (3.9) and (3.10) are true in this case also, using weak, *, we can pass to the limit in (B&LI,. @) + 2 j 4~2

s>,

is

The operator B, is again coercive and A( yz -a,) to (2.5). So from (4.4) we get as before,

The last estimate theorem 3,

+

s) = g,(x, S)UE,.6(x, s) =

as (for A 20;

By,

s>

as in

(4.5)

(4.5) and the fact that uF-

+ j&)4

= -

a in L”

jgi@J$

and get @GU, @) + 2 j 4~2 Now we can prove

as in theorem ,”

vh~o,

Moreover

UC,) +

by the previous

J

J

lemma

-

~14

+ j ag,(uo)@

3, using theorem r

uo(

y2 -

a>41

+

J

= -

j ag2(4@.

1 of [5], r ag,(uo)k

=

-

J

(4.6)

~g2(@4,.

and by (3.10)

ug,(uo)uo, ug,(u,)u, 3 I I lim inf (A,u,, u,) L (A+q,, u,,)

lim inf

respectively.

Combining

these with (4.6) (again

by the same arguments)

(4.3) follows.

5. REMARKS

1. We want to point out that the difficulty arises because f is not a compact since it does not even induce a map from L2 to L2 or from HA to H-‘.

perturbation,

14

D. GIACHETTI and M. RAMASWAMY

2. In [ll], the homogenization of the same type of equation was proved when fis Lipschitz in the second variable, using the maximum principle. 3. The present proof goes through for higher order operators also as we have not used any property characteristic of the second order operators. 4. A, can be taken to be nonsymmetric also. In that case we take Af, as the best coercivity constant, characterized by G = tlf$-$ Then

{A?}converges

to A: as in [3], defined

-,

by,

A’I= t”f$$ since the characterisation 5. If A,:A”,

is the same as that of the first eigenvalue

(s for “strong”

A,u+Aou

and it means

for all u EH&Q)),

strongly in H&Q), since A,5 A0 and by the theorem, (A&, u,) 6. In fact we have proved in theorem 3, also the homogenization AEuE + g(uJ

in the symmetric

(AwI,

case.

then

u,+

~0

~0).

result

for the equation,

= f.

u, E &(Q), where fE H-‘(Q), A, z A0 in M and g(x, t) satisfies Caratheodory function satisfying g(x,t)

=$3(x,4

the hypotheses

([9])

- Di(a~(X)DjUJ u,-

uo

xE+xO

I then there

exists a sequence

= xE, E E E U {O),

in

H&Q)-weak,

in

w -i,q, q < 2,

a$DiU,Dju,+

of matrices

I

aj:DiuoDjuo>

P, such that {Ps

grad U, = P,grad where r,+ 0 strongly in (L’(Q))%. The matrices P, are obtained

a.e. in Q, for all t E R and

for 0 < t < cQ.

Tup lg(x, s)i E L’(R) s result:

it is a

+gz(x,r)

such that gi(x, t)t B 0, jg2(x, t)I =Zh(x), h(x) E L*(Q),

7. Let us recall the following if

of [14], namely,

in the

following

are bounded

in L2(Q)“,

and

uo + r,

way:

for

a vector

h E [w”, we define

Existence

PJ

and homogenization

= grad wff, where

for semilinear

w$ is the solution

elliptic equations

with noncompact

nonlinearity

15

of

-Di(a&(x)Diw$J

= 1$1 DXa~{x) .Aj)

in

Q,

wj-A*xEH$Q). This result is in fact a definition of a “corrector” for grad uE, because we have replaced grad U, by the sum of two terms, one of which is P, grad ug, simple to calculate and the other is rs, going to 0 strongly. In the case where the a0 are given by a;(x) =aij(x/~) Ulj(y) Y-periodic for i, j = 1 . . . N. we can construct P, as follows:

where the functions xj The above result is (3.13), the hypotheses hence the existence of Acknowledgements-The

VAE [WN,VXE

51,

P,(x)

A;(grad,X,) (x/E)

= A -

$

are defined over Y as in [2]. a variant of the case q = 2 which is proved in [13], [9]. By (3.9) and of this theorem are satisfied in the case of theorem 3, 4 and 5 and correctors follows.

authors

want to thank

L. Boccardo

and F. Murat

for very useful discussions

REFERENCES 1. AMANN H., Nonlinear elliptic equations with nonlinear boundary conditions, in New Deuelopmenrs in Differentipl Equations (Edited by ECKHAUS) North-Holland, Amsterdam (1976). 2. BENSOUSSAN A.. LIONS J. L. & PAPANICOLAOU G., Asymptotic Analysis for Periodic Sfruc~ures. North-Holland. Amsterdam (1978). 3. BOCCARDO L. & MARCELLINI P.. Sulla convergenza delle soluzioni di disequazioni variazionali. Annali Mat. purer uppl. 110. 137-159 (1976). 4. BOCCARDO L. & MURAT F., Nouveaux rCsultats de convergence dans des problkmes unilatCraux, in Nonlineur Partial Differential Equations and Their Applications. Coil&e de France Seminar, Vol. II (Edited by H. BK~.ZIS & .I. L. LIONS) Research Notes in Mathematics. No 60. Pitman. London, pp. 64-85 (1982). 5. BREZIS H. & BROWDER F. E.. Some properties of higher order Sobolev spaces. J. math. pures appl.. to appear. 6. HESS P., Variational inequalities for strongly nonlinear elliptic operators, J. marh. pures uppl. 52, 285-298 (1973). 7. MURAT F.. H-convergence, SCminaire d’Analyse fonctionnelle et numCrique de I’Universite! d’AlgCrie (1977. 1978). 8. MURAT F., L’injection du cone positif de H-’ dans W-’ 4 est compacte pour tout 4 < 2. J. marh. pures uppl. 60. 309-322 (1981). 9. MURAT F., HomogCnCisation d’inCquations variationnelles avec contraintes unilatCrales. Seminaires 2 la Scuola Normale Superiore di Pisa, unpublished (November 1978). 10. NATANSON I. P., Theory of Functions of a Real Variable. F. Ungar. New York (1961). 11. RAMASWAMY M.. On a nonlinear elliptic equation with periodic coefficients, SIAM J. uppl. Mmh. 40. 409-418 (1981). 12. SPAGNOLO S., Sulla convergenza di equazioni paraboliche ed ellittiche. Annali Scu. norm sup. Piscl 22. 577-597 (1968). 13. TARTAR L.. Cours Peccot au ColEge de France (March, 1977). 14. WEBB J. R. L., Boundary value problems for strongly nonlinear elliptic equations, J. London math. Sot. 21. 123-132 (1980).