ABSTRACT. In the present paper we consider a second order weakly nonlinear elliptic equation of divergent form with a lower term growing at in_Fruity (with ...
Mathematical Notes, Vol. 68, No. 3, 2000
On the Homogenization in a P e r f o r a t e d C u b e
of Weakly Nonlinear Divergent Operators
H. A. M a t e v o s s i a n and S. V . Pikouline
UDC 517.95
ABSTRACT. I n the present p a p e r we consider a second order weakly nonlinear elliptic equation of divergent form with a lower t e r m growing at in_Fruity (with respect to the unknown function) as a power function. It is proved t h a t a sequence of solutions in the perforated cubes converges to a solution in the nonperforated cube as t h e diameters o f t h e holes tends t o zero, and the rate of convergence depends on the power exponent of the lower term. KEY WORDS: homogenization theory, second order weaklynonlinear elliptic equation, perforated domain, equation with sufficiently strong nonlinear absorbtions.
Problems of homogenization theory for perforated domains have been studied in many monographs and papers (see [1-5]), in which different restrictions (e.g., boundary conditions) are imposed on the behavior of solutions in t h e neighborhood of the holes. However, these restrictions are unnecessary when averaging solutions of equations with sufficiently strong nonlinear absorbtions. A similar phenomenon has been discovered in t h e theory of removable singularities. In this connection we mention the papers [6-11]. We use the following notation: B(x0, R) ----{x E R n : Ix -- Xol _< R} is a closed ball in R '~ of radins R centered at the point xo; S(xo, R ) - - OB(xo, R) = {x e R n : Ix - xol = R} is the sphere in R~; Q(0,1)--{x 9 ~:0_ ao > 0 and the following conditions are fulfilled:
~-~1r ~ ~ ~
%(~)r162 _ ~1r ~,
~ = ~-~t > o,
(3)
i,j~l
for all r : (Cx,--., 50 e R '~, 1r
~b
= ~-~i=1~?; a0 -----c o n s t ;
a > nl(n-
2); ~(z) e -r-/l(Qm(0, 1)) n
L~(Qm(O, 1)). D e f i n i t i o n . The function urn(x) e HI(Q,n(O, 1))N Loo(Qm(O, 1)) is called a generalized solution of problem (1), (2) ff for any function ~b(x) E C~~ 1)) the following integral identity is fi,lfllled
_
~(0,1)
%(x)
Ox~
Translated f r o m Mat~rna~icheskie Zamet$r Original article s u b m i t t e d April 3, 1999.
~x~
~ +
,,,(o,1)
a(x)Ju,,(x)l'-~z~(x)r
=0
Vol. 68, No. 3, pp. 390-398, September, 2000.
0001--4346/2000/6834-0337525.00
(~2000 Kluwer Academic/Plenum Publishers
337
and for any function X(X) e C~176
1)), which v~ni.qhes in a neighborhood of Ukm=1B(yk,rn, Rm), we
have the Zollo~g inclu,ion X(=)(~rn(=) -~(=)) 9 ~ ( Q , ( O , 1)). In general, for problem (1), (2) uniqueness theorems do not hold because no boundary conditions are given at the boundary of the holes of the perforated domain Qm(0, 1). We observe that under certain conditions the limit of solutions of problem (1), (2) exists, and this limit does not depend on the choice of the sequence urn(x); here u~(x) is an arbitrary solution of problem (1), (2). Let u(x) be a solution of the following problem:
z~(=) -- a(=)lu(=)l~-l~(=), u(=) = ~(=),
= e OQ(O, 1).
(4) (5)
as m -+ eo.
(8)
= ~ Q(O, 1),
T h e o r e m 1. Suppose that P ~ = o(rn (1-~)/((~-2)~-~)) Then
i) the solution of problem (1), (2) converges to the solution of problern (4), (5) in the following sense: for any positive constant d we have
sup
=~Q,.,~(O, 1)
I~-(=)- ~(=)1 -* 0
as m --~ oo,
(7)
where
OB(y~,rn, P~)) > d; k = 1,..., m} ;
Orn,,(0,1) = {= e Orn(0,1): ~ ( = , ii) th, total voZu,n~ of the hoZ,, m-+
" B (y~,~, a~)] -lUg=, rn B(y~,rn, P~)I co~ve~es to ~e,'o as V[U~=,
oo.
Corollary. If there exists a set in the perforated cube Qm(O, 1) which is bounded away from all the holes with a fixed constant, then the convergence on this set will be uniform. P r o o f o f T h e o r e m 1. i) In the paper [6] the estimate of the solution of Eqs. (1) is obtained by using the distance to the boundary
I~(=)1 < C(d~t(=,
OQrn(o,
1))) ~/r162
where the positive constant C does not depend on x and rn. By F(x) we denote the fundamental solution of the operator L with singularity at the origin of coordinates 0 that takes positive values in B(0, R) \ {0}. It is known [12] that such a solution exists and satisfies the inequalities C~Ixl =-" O. Hence, our assumption is incorrect, i.e., urn(x) converges to u(x) as m -+ oo in sense (7). For the volume of the ball of radius R in the space R '~ , we have the inequality
V[B(0, n)l _< CR~, where C d e p e n d s only on n . We can estimate the total volume of the holes as follows:
B(yk,m, am) 1~.I~' (zt'rn" (X) -- 'U,(X))[ IVO,,,Cx)II~,.,(~:) ~,(:~)l~O~ -~(x),~ -
341
< pA ~{
OPm-t(x)l~m(x) - ~(x) q (
qOm(x_)
• Iv(,~,(x) - ,.,(:0) I~ + p~,~l,,,,, (:~)2~o,,,(:,:) - "(x)l IVe'~(x)lO ~'
= •2A f~ Iv(,,,,,,(x) __l~',~(=)-~(=)l>t}
z~(x) lq- tO~(x) da: -,,(~,))I'I~,,~(~)
+ c~ .,t/l,,,,,(=)-,,(=)l> ~} IVO,~(x)l" I,.,,(z) - =(z)la+~o~-" (~)dz.
(17)
Now to estimate the second term in the fight-hand side of inequality (17), we set, in Young's inequality, a = lure(z)-u(z)]q+tOPm-2(:r,), b = IVOm(z)l ~ for a = (1 + q)/(o" + q) and choose e = 2Csct/(apO3); using (10), we obtain
< ~oC~/~ -
2
+
l~.(x)_~(z)l~+~e~(z) ~
l'~.,(=)-~(=)l>t}
c~~ I~,,~(=)-,~(=)1>1} IVO,~(x)l~C~+q)/c~-~) dx.
(18)
Using equalities (11), (13) and estimates (14)-(18), we have
fQco,~) L(~,,,(x) - ,.,(:,:))~(x) dx 1 f{ < -2-A
I,,,,,(z)-,.,(~)ll} Iv('''(x)
- u(x))l%,,(x)
- ~,(:~)l'~-tO~(:,:) d:,:
+ 04 _J~l,,,,,(=)-,.,(=)l_~} I ~ ( x ) - ~(x)l~'+~o~(:O
IVO,,,(z)12(~+~)/c ~-~)
l~,.,,(=)-~(=)l>i}
dx
(19)
_>~c~ f( +
lil,.~(z)-,l(=)l~} IV~
_>o.oc~ f{
+ 21A
I',.,,,,.(~)-',.,(~) I_oo.
Indeed,
f{
lu,~(x)-u(=)l_
