Homogenization in a thin domain with an oscillatory boundary

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by  José M. Arrieta

HOMOGENIZATION IN A THIN DOMAIN WITH AN
OSCILLATORY BOUNDARY
JOSE´ M.ARRIETA∗,† ANDMARCONEC.PEREIRA‡
Abstract. In this paper we analyze the behavior of the Laplace operator with Neumann boundary con-
ditions in a thin domain of the type R(cid:15) ={(x1,x2)∈R2 |x1 ∈(0,1),0<x2 <(cid:15)G(x1,x1/(cid:15))} where the
1 functionG(x,y)isperiodicinyofperiodL. Observethattheupperboundaryofthethindomainpresents
1 ahighlyoscillatorybehaviorand,moreover,theheightofthethindomain,theamplitudeandperiodofthe
0 oscillationsareallofthesameorder,givenbythesmallparameter(cid:15).
2
R´esum´e: Dans cet article, nous analysons le comportement de l’op´erateur de Laplace avec conditions aux
n limitesdeNeumanndansundomainefinedutypeR(cid:15)={(x1,x2)∈R2|x1∈(0,1),0<x2<(cid:15)G(x1,x1/(cid:15))}
a lorsquelafonctionG(x,y)estp´eriodiquedanslavariableydep´eriodeL. Onobservequelalimitesup´erieure
J du domaine fine pr´esente une comportement hautement oscillatoire et, en outre, l’ hauteur du domaine,
8 l’amplitudeetlap´eriodedesoscillationssonttousdumˆemeordre,donn´eparunpetitparam`etre(cid:15).
1
Keywords: Thindomain,oscillatoryboundary,homogenization.

]
P
A
h. 1. Introduction
t
a In this work we study the asymptotic behavior of the solutions of the Neumann problem for the Laplace
m
operator
[ −∆w(cid:15)+w(cid:15) =f(cid:15) in R(cid:15)

1 ∂w(cid:15) (1.1)
v  ∂ν(cid:15) =0 on ∂R(cid:15)
3 with f(cid:15) ∈L2(R(cid:15)), ν(cid:15) =(ν(cid:15),ν(cid:15)) is the unit outward normal to ∂R(cid:15) and ∂ is the outside normal derivative.
0 1 2 ∂ν(cid:15)
The domain R(cid:15) is a thin domain with a highly oscillating boundary which is given by
5
3 R(cid:15) ={(x ,x )∈R2 |x ∈(0,1), 0<x <(cid:15)G (x )}
1 2 1 2 (cid:15) 1
.
1
where G (·) is a function satisfying 0<G ≤G (·)≤G for some fixed positive constants G , G and such
0 (cid:15) 0 (cid:15) 1 0 1
that oscillates as the parameter (cid:15) → 0. We may think, for instance, that the function G is of the form
1 (cid:15)
1 G(cid:15)(x)=a(x)+b(x)g(x/(cid:15)α) where a,b:I (cid:55)→R are piecewise C1-functions defined on I =(0,1), g :R→R is
: an L-periodic smooth function and α≥0, see Figure 1. This includes the case where the function G (·) is a
v (cid:15)
purely periodic function, for instance, G (x) = 2+sin(x/(cid:15)α) but also includes the case where the function
i (cid:15)
X
G is not periodic and the amplitude is modulated by a function. As a matter of fact, we will be able to
(cid:15)
r treat more general cases, see hypothesis (H) below, but to stay the general ideas in the introduction we may
a
consider the proptotype function G (x)=a(x)+b(x)g(x/(cid:15)α).
(cid:15)
The existence and uniqueness of solutions for problem (1.1) for each (cid:15) > 0 is garanteed by Lax-Milgram
Theorem. We are interested here in analyzing the behavior of the solutions as (cid:15)→0, that is, as the domain
gets thinner and thinner although with a high oscillating boundary.
Observe that the domain is thin since R(cid:15) ⊂(0,1)×(0,(cid:15)G ) and its upper boundary oscillates due to the
1
function g(x/(cid:15)α), (if α>0 and g is not a constant function).
∗ Corresponding author: Jos´e M. Arrieta, Departamento de Matem´atica Aplicada, Facultad de Matem´aticas, Universidad
ComplutensedeMadrid,28040Madrid,Spain. e-mail: [email protected].
†Partiallysupportedby: MTM2009-07540DGES,Spain;PHB2006-003PCandPR2009-0027fromMICINN;andGR58/08,
Grupo920894(BSCH-UCM,Spain).
‡PartiallysupportedbyFAPESP2008/53094-4,CAPESDGU127/07andCNPq305210/2008-4.
1
2 J.M.ARRIETAANDM.C.PEREIRA
Figure 1. The thin domain R(cid:15)
Since the domain R(cid:15) is thin, “approaching” the line segment (0,1) ⊂ R, it is reasonable to expect that
the family of solutions w(cid:15) will converge to a function of just one variable and that this function will satisfy
an equation of the same type of (1.1) but in one dimension, say Lu+u = h in (0,1) with some boundary
conditions, where L is a second order elliptic operator in one dimension. As a matter of fact, if the function
G (x) is independent of (cid:15), (say α = 0 or g ≡ 0), that is, the thin domain does not present any oscillations
(cid:15)
whatsoever:
R(cid:15) ={(x ,x ):0<x <1,0<x <(cid:15)G(x)}
1 2 1 2
the limit equation is given by
 1
−G(x)(G(x)wx)x+w(x)=f(x) x∈(0,1) (1.2)
 w (0)=w (1)=0
x x
seeforinstance[10,11]. Observethatthegeometryofthethindomainentersintothelimitequationthrough
the diffusion coefficient.
Moreover, if we consider 0 ≤ α < 1 and if we assume that a(x)+b(x)g(x/(cid:15)α) → h(x) w-L2(0,1) and
1 → k(x) w-L2(0,1) (observe that h(x)k(x) ≥ 1 a.e. and in general it is not true that
a(x)+b(x)g(x/(cid:15)α)
h(x)k(x)≡1), then the limit problem is
 (cid:18) (cid:19)
1 1
− v +v =f, in (0,1)
h(x) k(x) x
x
 v (0)=v (1)=0
x x
see [2]. Observe that this case contains the previous one. If α=0, then h(x)=a(x)+b(x)g(x)≡G(x) and
k(x)= 1 = 1 , and we recover equation (1.2).
a(x)+b(x)g(x) G(x)
Inthisworkweareinterestedinaddressingthecaseα=1,thatisG (x)=a(x)+b(x)g(x/(cid:15)),wherenone
(cid:15)
of the techniques used to solve the previous ones apply. Observe that this situation is very resonant: the
heightofthedomain,theamplitudeoftheoscillationsattheboundaryandtheperiodoftheoscillationsare
of the same order (cid:15). Moreover we are interested in addressing not only the purely periodic case, that is, the
case where the function G (x) = G(x/(cid:15)) for some L-periodic smooth function G but also the general case
(cid:15)
where the amplitude of the oscillation depend on x in a continuous fashion, that is, in our prototype case,
the situation where a and b are not piecewise constant, but piecewise continuous function.
The purely periodic case can be addressed by somehow standard techniques in homogenization theory, as
developed in [5, 8, 12]. If G (x)=G(x/(cid:15)) where G is an L-periodic C1 function and if we denote by
(cid:15)
Y∗ ={(y ,y )∈R2 : 0<y <L, 0<y <G(y )}
1 2 1 2 1
then the limit equation is shown to be
(cid:40)
−q w +w =f(x), x∈(0,1)
0 xx
(1.3)
w(cid:48)(0)=w(cid:48)(1)=0
HOMOGENIZATION IN THIN DOMAINS 3
where
1 (cid:90) (cid:110) ∂X (cid:111)
q = 1− (y ,y ) dy dy ,
0 |Y∗| ∂y 1 2 1 2
Y∗ 1
and X is the unique solution of
 −∆X =0 in Y∗
 ∂∂∂NXX =0 on B2
=N on B (1.4)
∂N 1 1
 X(cid:90) (0,Xy2d)y=1dXy2(L=,0y2) on B0
Y∗
where B is the lateral boundary, B is the upper boundary and B is the lower boundary of ∂Y∗, that is
0 1 2
B ={(0,y ):0<y <G(0)}∪{(L,y ):0<y <G(L)}
0 2 2 2 2
B ={(y ,G(y )):0<y <L}
1 1 1 1
B ={(y ,0):0<y <L}.
2 1 1
We refer to [4] for a complete analysis of the purely periodic case of a nonlinear parabolic problem.
If we want to analyze now the case where the function G is given by G (x)=a(x)+b(x)g(x/(cid:15)) and the
(cid:15) (cid:15)
functionsa,baresmoothbutnotnecessarilyconstant,itisintuitivelytruethatthelimitingequationshould
behave like (1.3) with a diffusion coefficient q depending on x somehow. Actually if we look at the thin
0
domain in a small neighborhood of a point ξ ∈ (0,1), we will approximately see a domain with very high
oscillations but locally the domain behaves like the thin domain with the function x → a(ξ)+b(ξ)g(x/(cid:15)).
Therefore, it is expected that if we freeze the coefficients of the limit equation in a fixed point ξ ∈(0,1) we
shouldrecoverequation(1.3). Althoughthisintuitiveargumentwillturnouttobetrue,itdoesnotgiveusa
complete information about the limit equation, specially, the way in which the dependence on x is explicitly
stated in the limit equation. For instance, it is not clear at this stage whether the limit equation should be
−(q(x)w ) +w = f or −q(x)w +w = f with q(x) = 1 (cid:82) {1− ∂X(x)(y ,y )}dy dy , or maybe
x x xx |Y∗(x)| Y∗(x) ∂y1 1 2 1 2
− 1 (r(x)w ) +w =f where r(x)=(cid:82) {1− ∂X(x)(y ,y )}dy dy , or maybe other. Observe that all
|Y∗(x)| x x Y∗(x) ∂y1 1 2 1 2
these equations coincide if we consider the purely periodic case.
In order to accomplish our goal and obtain the limit equation in the general case, we propose a technique
that consists in solving first the piecewise periodic case, that is, the case where the functions a(x) and b(x)
are piecewise constant and then do an approximation argument to obtain the limit equation in the general
case. This is a subtle argument since we are approximating first the functions a and b by piecewise constant
functions, say aδ(x) , bδ(x) so that |a(x)−aδ(x)|+|b(x)−bδ(x)|≤δ and obtain the limit equation for δ >0
fixed,passingtothelimitas(cid:15)→0. Then,inthislimitequation,whichwilldependonδ,wepasstothelimit
asδ →0. Butthelimitweareinterestedinistakingfirstδ →0for(cid:15)>0fixed,soweobtainthedomaingiven
by the function a(x)+b(x)g(x/(cid:15)), and then we pass to the limit as (cid:15)→0. But, a priori there is no garantee
that these two limits commute. We will actually show that these two limits commute by proving that the
solutions of problem (1.1) in two domains Ω(cid:15) ={(x,y):0<x<1,0<y <G (x)=a(x)+b(x)g(x/(cid:15))} and
(cid:15)
Ω˜(cid:15) ={(x,y):0<x<1,0<y <G˜ (x)=a˜(x)+˜b(x)g(x/(cid:15))}, are close in the H1 norm uniformly in (cid:15) when
(cid:15)
a,b and a˜,˜b are close. This result, which can be regarded as a domain perturbation result but uniformly
in (cid:15), guarantee that the two limits commute and will show that the limit problem is given as above. We
remark that this domain perturbation result is not deduced from standard and known results on domain
perturbation techniques since we are able to compare the solutions of an elliptic problem in two families of
domains which also depend on a parameter and the way this domains depend on (cid:15) is not smooth at all.
We strongly believe that this technique of solving first the piecewise periodic case and then use an
approximationargument,uniformintheparameter(cid:15),canbeusedinotherproblemstoobtaintheappropriate
homogenized limit for the non periodic case.
4 J.M.ARRIETAANDM.C.PEREIRA
Following this agenda, we solve first the piecewise periodic case, that is, the case where the function
G (x)=aδ(x)+bδ(x)g(x/(cid:15)) and aδ, bδ are piecewise constant. We consider the points 0=ξ <ξ <...<
(cid:15) 0 1
ξ < ξ = 1 and assume that the functions aδ and bδ are constant in each of the interval (ξ ,ξ ), say
N−1 N i−1 i
aδ(x) = a , b(x) = b . We show that the limit equation is of the same form (1.3) in each of the intervals
i i
(ξ ,ξ ), that is,
i−1 i
−q w +w =f(x), x∈(ξ ,ξ ), ,i=1,...,N (1.5)
i xx i−1 i
where
1 (cid:90) (cid:110) ∂X (cid:111)
q = 1− i(y ,y ) dy dy ,
i |Y∗| ∂y 1 2 1 2
i Y∗ 1
i
and X is the unique solution of (1.4) in the cell Y∗ where
i i
Y∗ ={(y ,y )∈R2 : 0<y <L, 0<y <a +b g(y )}.
i 1 2 1 2 i i 1
Moreover, equation (1.5) is suplemented with Neumann boundary conditions at x=0, x=1 and with some
“matching” condition at the points ξ , i=1,...,N −1, which are continuity of the function and some kind
i
of Kirchoff type conditions, see Section 3 for details. Actually, if we look at the variational formulation of
the limit equation, we obtain
(cid:90) (cid:110) (cid:111) (cid:90)
qδ(x)pδ(x)w (x)ϕ (x)+pδ(x)w(x)ϕ(x) dx= pδ(x)f(x)ϕdx (1.6)
x x
I I
where qδ(x)=qi and pδ(x)= |YLi∗| in (ξi−1,ξi).
Now we will be able to pass to the limit in this equation as δ →0 and obtain the limit problem:
 1
−p(x)(r(x)wx)x+w =f(x), x∈(0,1) (1.7)
 w(cid:48)(0)=w(cid:48)(1)=0
where
1 (cid:90) (cid:110) ∂X(x) (cid:111)
r(x)=p(x)q(x)= 1− (y ,y ) dy dy ,
L ∂y 1 2 1 2
Y∗(x) 1
|Y∗(x)|
p(x)=
L
and X(x) is the unique solution of (1.4) in the basic cell Y∗(x) which depends on the variable x and it is
given by
Y∗(x)={(y ,y )∈R2 : 0<y <L, 0<y <a(x)+b(x)g(y )}.
1 2 1 2 1
Equation (1.7) is the limit equation we were looking for.
Finally, we would like to remark that although we will treat the Neumann boundary condition problem,
we may also impose different conditions in the lateral boundaries of the thin domain R(cid:15), while preserving
the Neumann type boundary condition in the upper and lower boundary. That is for problem (1.1) we may
consider conditions of the Dirichlet type, w(cid:15) = 0, or even Robin, ∂w(cid:15) +βw(cid:15) = 0 in the lateral boundaries
∂N
{(0,y):0<y <(cid:15)G (0)},{(1,y):0<y <(cid:15)G (1)}. Thelimitproblemwillpreservethisboundarycondition.
(cid:15) (cid:15)
That is, in problem (1.7) we will obtain the conditions w =0 or ∂w +βw =0 at x=0 and x=1.
∂N
We describe the contents of the paper. In Section 2 we set up the notation, obtain some technical results
thatwillbeneededintheproofsandstatethemainresult. InSection3weobtaintheresultforthepiecewise
periodic case. In Section 4 we show the continuous dependence result on the domain, uniform in (cid:15), that
will be the key argument to obtain the limit problem. In Section 5 we provide a proof of the main result.
Finally, we include an Appendix where we analyze the behavior of the basic function X solution of (1.4) as
we perturb G.
Acknowledgments. Part of this work was done while the second author was visiting the Applied Math
Department at Complutense University in Madrid. He express his kind gratitude to the Department.
HOMOGENIZATION IN THIN DOMAINS 5
Wewould also liketo thankE. Zuazua, C.Castro, M. EugeniaP´erez, M. Lobo andD. G´omez forthe allthe
discussion and good suggestions on this problem.
2. Basic facts, notation and main result
We consider the one parameter family of functions G :I →(0,∞), where I =(0,1), (cid:15)∈(0,(cid:15) ) for some
(cid:15) 0
(cid:15) >0. We will assume the following hipothesis
0
(H) There exist two positive constants G , G such that
0 1
0<G ≤G (x)≤G , ∀x∈I, ∀(cid:15)∈(0,(cid:15) ). (2.1)
0 (cid:15) 1 0
Moreover, the functions G (·) are of the type G (x)=G(x,x/(cid:15)), where the function
(cid:15) (cid:15)
G:I×R −→(0,+∞)
(2.2)
(x,y) −→G(x,y)
is L-periodic in the second variable, that is, G(x,y+L)=G(x,y) and piecewise C1 with respect to
the first variable, that is, there exists a finite number of points 0=ξ <ξ <···<ξ <ξ =1
0 1 N−1 N
suchthatthefunctionG:(ξ ,ξ )×R→(0,+∞)isC1 andsuchthatG, G andG areuniformly
i i+1 x y
bounded in (ξ ,ξ )×R and have limits when we approach ξ and ξ .
i i+1 i i+1
One important example of a function satisfying the above conditions is
N
(cid:88)
G(x,y)=a(x)+ b (x)g (y)
r r
r=1
where the functions a, b ,..,b are piecewise C1 in I = (0,1) and the functions g ,..,g are all C1 and
1 N 1 N
L-periodic.
We define the thin domain as
R(cid:15) ={(x ,x )∈R2 |x ∈I, 0<x <(cid:15)G (x )}.
1 2 1 2 (cid:15) 1
In this work we study the asymptotic behavior of the solutions of the Neumann problem for the Laplace
operator
−∆w(cid:15)+w(cid:15) =f(cid:15) in R(cid:15)

∂w(cid:15) (2.3)
 =0 on ∂R(cid:15)
∂ν(cid:15)
with f(cid:15) ∈L2(R(cid:15)) and where ν(cid:15) =(ν(cid:15),ν(cid:15)) is the unit outward normal to ∂R(cid:15) and ∂ is the outside normal
1 2 ∂ν(cid:15)
derivative.
From Lax-Milgran Theorem, we have that problem (2.3) has a unique solution for each (cid:15) > 0. We are
interested here in analyzing the behavior of the solutions as (cid:15)→0, that is, as the domain gets thinner and
thinner although with a high oscillating boundary.
To study the convergence of (2.3) in the thin oscillating domain R(cid:15), we consider the equivalent linear
elliptic problem
 ∂2u(cid:15) 1 ∂2u(cid:15)
−∂x 2 − (cid:15)2∂x 2 +u(cid:15) =f(cid:15) in Ω(cid:15)
1 2 (2.4)
 ∂∂xu(cid:15)N1(cid:15)+ (cid:15)12∂∂xu(cid:15)N2(cid:15) =0 on ∂Ω(cid:15)
1 2
where f(cid:15) ∈L2(Ω(cid:15)) satisfies
(cid:107)f(cid:15)(cid:107) ≤C (2.5)
L2(R(cid:15))
for some C >0 independent of (cid:15) and now N(cid:15) =(N(cid:15),N(cid:15)) is the outward unit normal to ∂Ω(cid:15) and Ω(cid:15) ⊂R2 is
1 2
given by
Ω(cid:15) ={(x ,x )∈R2 |x ∈I, 0<x <G (x )}. (2.6)
1 2 1 2 (cid:15) 1
Observe that the equivalence between the problems (2.3) and (2.4) is obtained by changing the scale of
thedomainR(cid:15) throughthetransformation(x,y)→(x,(cid:15)y),(see[10]formoredetails). Moreover,thedomain
Ω(cid:15) is not “thin”anymorebut presentsverywild oscillationsat thetop boundary, althoughthe presenceof a
6 J.M.ARRIETAANDM.C.PEREIRA
high diffusion coefficient in front of the derivative with respect the second variable balance the effect of the
wild oscillations.
ThedomainΩ(cid:15) isrelatedtotheonesanalyzedinthepapers[6,1,9]butthefactthatinourcasewehave
a very high diffusion in the y-direction makes our analysis and results different from these other papers.
We write H1(U) for the space H1(U) with the equivalent norm
(cid:15)
(cid:13)∂w(cid:13)2 1(cid:13)∂w(cid:13)2
(cid:107)w(cid:107)2 =(cid:107)w(cid:107)2 +(cid:13) (cid:13) + (cid:13) (cid:13)
H(cid:15)1(U) L2(U) (cid:13)∂x1(cid:13)L2(U) (cid:15)2(cid:13)∂x2(cid:13)L2(U)
given by the inner product
(cid:90) (cid:110) ∂φ ∂ϕ 1 ∂φ ∂ϕ (cid:111)
(φ,ϕ) = φϕ+ + dx dx
H(cid:15)1(U) U ∂x1∂x1 (cid:15)2∂x2∂x2 1 2
where U is an arbitrary open set of R2, which may depend also on (cid:15).
The variational formulation of (2.4) is find u(cid:15) ∈H1(Ω(cid:15)) such that
(cid:90) (cid:110)∂u(cid:15) ∂ϕ 1 ∂u(cid:15) ∂ϕ (cid:111) (cid:90)
+ +u(cid:15)ϕ dx dx = f(cid:15)ϕdx dx , ∀ϕ∈H1(Ω(cid:15)). (2.7)
∂x ∂x (cid:15)2∂x ∂x 1 2 1 2
Ω(cid:15) 1 1 2 2 Ω(cid:15)
Remark that the solutions u(cid:15) satisfy an uniform a priori estimate on (cid:15). In fact, taking ϕ=u(cid:15) in expression
(2.7), we obtain
(cid:13)∂u(cid:15)(cid:13)2 1(cid:13)∂u(cid:15)(cid:13)2
(cid:13) (cid:13) + (cid:13) (cid:13) +(cid:107)u(cid:15)(cid:107)2 ≤(cid:107)f(cid:15)(cid:107) (cid:107)u(cid:15)(cid:107) . (2.8)
(cid:13)∂x1(cid:13)L2(Ω(cid:15)) (cid:15)2(cid:13)∂x2(cid:13)L2(Ω(cid:15)) L2(Ω(cid:15)) L2(Ω(cid:15)) L2(Ω(cid:15))
Consequently,
(cid:13)∂u(cid:15)(cid:13) 1(cid:13)∂u(cid:15)(cid:13)
(cid:107)u(cid:15)(cid:107) ,(cid:13) (cid:13) and (cid:13) (cid:13) ≤(cid:107)f(cid:15)(cid:107) ∀(cid:15)>0. (2.9)
L2(Ω(cid:15)) (cid:13)∂x1(cid:13)L2(Ω(cid:15)) (cid:15)(cid:13)∂x2(cid:13)L2(Ω(cid:15)) L2(Ω(cid:15))
Hence, it follows from (2.5) that there exists C >0, independent of (cid:15)>0, such that
(cid:13)∂u(cid:15)(cid:13) 1(cid:13)∂u(cid:15)(cid:13)
(cid:107)u(cid:15)(cid:107) ,(cid:13) (cid:13) and (cid:13) (cid:13) ≤C. (2.10)
L2(Ω(cid:15)) (cid:13)∂x1(cid:13)L2(Ω(cid:15)) (cid:15)(cid:13)∂x2(cid:13)L2(Ω(cid:15))
Observe that the solutions u(cid:15) of (2.4) can also be characterized as the minimum of a functional. That is,
u(cid:15) ∈H1(Ω(cid:15)) is the solution of (2.4) if only if
V (u(cid:15))= min V (ϕ), (2.11)
(cid:15) (cid:15)
ϕ∈H1(Ω(cid:15))
where
V :H1(Ω(cid:15))(cid:55)→R
(cid:15)
1(cid:90) (cid:110) ∂ϕ 2 1 ∂ϕ 2 (cid:111) (cid:90)
V (ϕ)= + +ϕ2 dx dx − f(cid:15)ϕdx dx .
(cid:15) 2 ∂x (cid:15)2∂x 1 2 1 2
Ω(cid:15) 1 2 Ω(cid:15)
It follows from (2.7) with ϕ=u(cid:15) that
1(cid:90)
V (u(cid:15))=− f(cid:15)u(cid:15)dx dx .
(cid:15) 2 1 2
Ω(cid:15)
Hence, due to (2.5) and (2.9) we obtain
1
|V (u(cid:15))|≤ (cid:107)f(cid:15)(cid:107) (cid:107)u(cid:15)(cid:107) ≤C2 (2.12)
(cid:15) 2 L2(Ω(cid:15)) L2(Ω(cid:15))
Important tools for the analysis are appropriate extension operators for functions defined in the sets Ω(cid:15).
Wewillobtainsuchoperatorandwewillbeabletoconstructitevenformoregeneraldomainsthattheones
like Ω(cid:15).
Hence, let us consider the following open sets:
O ={(x ,x )∈R2 |x ∈I and 0<x <G }
1 2 1 2 1
O(cid:15) ={(x ,x )∈R2 |x ∈I and 0<x <G (x )}
1 2 1 2 (cid:15) 1
HOMOGENIZATION IN THIN DOMAINS 7
where I ⊂R is an open interval, G :I (cid:55)→R is a piecewise C1-function satisfying 0<G ≤G (x )≤G for
(cid:15) 0 (cid:15) 1 1
all x ∈ I and (cid:15) > 0 and there exists 0 = ξ < ξ < ... < ξ < ξ = 1 such that G is C1 in the intervals
0 1 N−1 N (cid:15)
(ξ ,ξ ). Let us define
i i+1
η((cid:15))= sup {|G(cid:48)(x)|} (2.13)
(cid:15)
x∈I\ξ0,...,ξN
and assume η((cid:15))<+∞ for fixed (cid:15), although in general η((cid:15))→+∞ as (cid:15)→0.
Also, we denote by
Oˆ =O\∪N {(ξ ,x ):G <x <G }.
i=1 i 2 0 2 1
Notice that O(cid:15) ⊂O.
Lemma 2.1. With the notation above, there exists an extension operator
P ∈L(Lp(O(cid:15)),Lp(Oˆ))∩L(W1,p(O(cid:15)),W1,p(Oˆ))∩L(W1,p(O(cid:15)),W1,p(Oˆ))
(cid:15) ∂l ∂l
(where W1,p is the set of functions in W1,p which are zero in the lateral boundary ∂ ) and a constant K
∂l l
independent of (cid:15) and p such that
(cid:107)P ϕ(cid:107) ≤K(cid:107)ϕ(cid:107)
(cid:15) Lp(Oˆ) Lp(O(cid:15))
(cid:13)∂P ϕ(cid:13) (cid:110)(cid:13) ∂ϕ (cid:13) (cid:13) ∂ϕ (cid:13) (cid:111)
(cid:13) (cid:15) (cid:13) ≤K (cid:13) (cid:13) +η((cid:15))(cid:13) (cid:13)
(cid:13) ∂x1 (cid:13)Lp(Oˆ) (cid:13)∂x1(cid:13)Lp(O(cid:15)) (cid:13)∂x2(cid:13)Lp(O(cid:15)) (2.14)
(cid:13)∂P ϕ(cid:13) (cid:13) ∂ϕ (cid:13)
(cid:13) (cid:15) (cid:13) ≤K(cid:13) (cid:13)
(cid:13) ∂x2 (cid:13)Lp(Oˆ) (cid:13)∂x2(cid:13)Lp(O(cid:15))
for all ϕ∈W1,p(O(cid:15)) where 1≤p≤∞ and η((cid:15)) is defined in (2.13).
Proof. Observe first that the set O0 = (0,1)×(0,G ) ⊂ O(cid:15) for all (cid:15). Hence, if we have that G ≤ 2G ,
0 1 0
which implies that G (x )≤2G , we can define the operator:
(cid:15) 1 0
(cid:26) ϕ(x ,x ) (x ,x )∈O(cid:15)
(P ϕ)(x ,x )= 1 2 1 2
(cid:15) 1 2 ϕ(x ,2G (x )−x ) (x ,x )∈O/O(cid:15).
1 (cid:15) 1 2 1 2
Observe that this operator is obtained through a “reflection” procedure in the x direction along the
2
oscillating boundary. It is straigth forward to check that this operator satisfies (2.14).
If we are in the case where G > 2G , we will need to extend first the function ϕ in the direction of
1 0 |O0
negative x , with a finite number of successive reflections. That is, if ϕ is defined in O(cid:15) then we extend ϕ
2 0 0
to the set (0,1)×(−G ,0) with the reflecting along the line x =0. This produces the function
0 2
(cid:26)
ϕ (x ,x ) (x ,x ) with 0<x <G (x )
ϕ (x ,x )= 0 1 2 1 2 2 (cid:15) 1
1 1 2 ϕ (x ,−x ) (x ,x ) with −G <x ≤0.
0 1 2 1 2 0 2
We can continue producing these reflections inductively as follows
(cid:26)
ϕ (x ,x ) (x ,x ) with −(n−1)G <x ≤G (x )
ϕ (x ,x )= n−1 1 2 1 2 0 2 (cid:15) 1
n 1 2 ϕ (x ,−x −2(n−1)G ) (x ,x ) with −nG <x ≤−(n−1)G
n−1 1 2 0 1 2 0 2 0
Choosing n large enough so that nG > G , constructing ϕ and applying to ϕ the procedure by
0 1 n n
reflection in the x direction along the oscillating boundary, we obtain the extension operator P which
2 (cid:15)
satisfies (2.14). (cid:3)
Remark 2.2. 1)Thisoperatorpreservesperiodicityinthex variable. Thatis, ifthefunctionϕ isperiodic
1 (cid:15)
in x , then the extended function P ϕ is also periodic in x .
1 (cid:15) (cid:15) 1
2) This result can be applied to the case G (x) = G(x) independent of (cid:15). In particular, we can apply the
(cid:15)
extension operator to the basic cell.
Now we are in contidion to state our main result. We consider the general case, that is, the domain Ω(cid:15) is
given as
Ω(cid:15) ={(x ,x )∈R2 |x ∈I, 0<x <G (x )}.
1 2 1 2 (cid:15) 1
8 J.M.ARRIETAANDM.C.PEREIRA
where the function G (·) satisfies hypothesis (H). Recall that we denote by Ω=(0,1)×(0,G ) and
(cid:15) 1
Ωˆ =Ω\∪N {(ξ ,x ):G <x <G },
i=1 i 2 0 2 1
where the points ξ ,ξ ,...,ξ are given by (H).
0 1 N
Theorem 2.3. Let u(cid:15) be the solution of (2.4) with f(cid:15) ∈L2(Ω(cid:15)) satisfying (cid:107)f(cid:15)(cid:107) ≤C for some positive
L2(Ω(cid:15))
constant C independent of (cid:15) > 0. Assume that the function fˆ(cid:15)(x) = (cid:82)G(cid:15)(x)f(x,y)dy satisfies that fˆ(cid:15) (cid:42) fˆ,
0
w-L2(0,1).
Then, there exists uˆ∈H1(Ωˆ), such that, if P is the extension operator constructed in Lemma 2.1, then
(cid:15)
P u(cid:15) (cid:42)uˆ w−H1(Ωˆ)
(cid:15)
where uˆ(x ,x ) depends only on the first variable, that is, uˆ(x ,x )=u(x ), and u is the unique solution of
1 2 1 2 1
the Neumann problem
(cid:90) (cid:110) (cid:111) (cid:90)
r(x)u (x)ϕ (x)+p(x)u(x)ϕ(x) dx= fˆ(x)ϕ(x)dx (2.15)
x x
I I
for all ϕ∈H1(I), where
1 (cid:90) (cid:110) ∂X(x) (cid:111)
r(x)= 1− (y ,y ) dy dy
L ∂y 1 2 1 2
Y∗(x) 1 (2.16)
|Y∗(x)|
p(x)=
L
and X(x) is the unique solution of problem
 −∆X(x)=0 in Y∗(x)
 ∂∂XX∂(N(xx)) =0 on B2(x)
=N on B (x) (2.17)
∂N 1 1
X(x)(0(cid:90),y2)=XX(x()x)d(yL1,dyy22)=on0B0(x)
Y∗(x)
in the representative cell Y∗(x) given by
Y∗(x)={(y ,y )∈R2 : 0<y <L, 0<y <G(x,y )}. (2.18)
1 2 1 2 1
B (x) is the lateral boundary, B (x) is the upper boundary and B (x) is the lower boundary of ∂Y∗(x) for
0 1 2
all x∈I.
Remark 2.4. i) If the function r(x) is a continuous function, then, the integral formulation (2.15) is the
weak formulation of problem (1.7) with f(x)=fˆ(x)/p(x)
ii) If initially the function f(cid:15)(x,y) = f (x), then it is not difficult to see that fˆ(cid:15)(x) = G (x)f (x) and
0 (cid:15) 0
G (x)(cid:42) |Y∗(x)| ≡p(x) as (cid:15)→0, and therefore, fˆ(x)=p(x)f (x).
(cid:15) L 0
3. The piecewise periodic case
In this section we find the limit of the sequence {u(cid:15)} given by the Neumann problem (2.4) as (cid:15) goes
(cid:15)>0
to zero for the case where the oscillating boundary is piecewise periodic.
SoletusconsiderthatthefamilyofdomainsΩ(cid:15) satisfies(H)andmoreverthefunctionGisindependentof
thefirstvariableineachofthedomains(ξ ,ξ )×R. Thatis,thereexist0=ξ <ξ <...<ξ <ξ =1
i−1 i 0 1 N−1 N
sothatthefunctionGfrom(2.2)satisfiesthat G(x,y)=G (y)forx∈I =(ξ ,ξ )andG (y+L)=G (y)
i i i−1 i i i
HOMOGENIZATION IN THIN DOMAINS 9
for all y ∈ R. Moreover, the function G (·) is C1 for all i = 1,2,...N, there exists 0 < G < G such that
i 0 1
0<G ≤G (·)≤G for all i=1,...,N; and the domain is given by
0 i 1
(cid:110) (cid:111)
Ω(cid:15) = (x,y):ξ <x<ξ ,0<y <G (x/(cid:15)),i=1,...,N ∪
i−1 i i
(3.1)
(cid:110) (cid:111)
∪N−1 (ξ ,y),0<y <min{G (ξ /(cid:15)),G (ξ /(cid:15))}
i=1 i i−1 i i i
(see Figure 2). Denote also by Ω the open rectangle Ω=I×(0,G ) and by
1
Ωˆ =Ω\∪N {(ξ ,x ):G <x <G }.
i=1 i 2 0 2 1
Observe that for Ω(cid:15) and Ωˆ we have the extension operator P constructed in Lemma 2.1.
(cid:15)
Figure 2. Piecewise periodic domain Ω(cid:15)
We can show,
Theorem 3.1. Assume that f(cid:15) ∈ L2(Ω(cid:15)) satisfies (2.5) and the function fˆ(cid:15)(x) = (cid:82)G(cid:15)(x)f(x,y)dy satisfies
0
that fˆ(cid:15) (cid:42) fˆ, w-L2(0,1). Let u(cid:15) be the unique solution of (2.4). Then, there exists uˆ ∈ H1(Ω) such that if
P is the extension operator constructed in Lemma 2.1, we have
(cid:15)
P u(cid:15) −(cid:15)→→0 uˆ w−H1(Ωˆ), s−L2(Ωˆ)
(cid:15)
where uˆ(x ,x )=u(x ) in Ωˆ and u(·) is the unique weak solution of the Neumann problem
1 2 1
(cid:90) (cid:110) (cid:111) (cid:90)
r(x)u (x)ϕ (x)+p(x)u(x)ϕ(x) dx= fˆ(x)ϕ(x)dx (3.2)
x x
I I
for all ϕ∈H1(I), where p(x) and r(x) are piecewise constant functions defined as follows: p(x)=p for all
i
x∈(ξ ,ξ ) where
i−1 i
|Y∗|
p = i , i=1,...,N (3.3)
i L
where Y∗ is the basic cell for x∈(ξ ,ξ ), that is
i i−1 i
Y∗ ={(y ,y )∈R2 : 0<y <L, 0<y <G (y )} (3.4)
i 1 2 1 2 i 1
and r(x)=r for all x∈(ξ ,ξ ) where
i i−1 i
1 (cid:90) (cid:110) ∂X (cid:111)
r = 1− i(y ,y ) dy dy
i L ∂y 1 2 1 2
Y∗ 1
i
10 J.M.ARRIETAANDM.C.PEREIRA
and the function X is the unique solution of
i
 −∆X =0 in Y∗
i i
 ∂∂∂∂XNXNii==N01oonnBB2i1i (3.5)
Xi(0,(cid:90)y2)X=iXdyi(1Ldy,2y2=) o0n B0i
Y∗
i
Remark 3.2. If we define f (x)=fˆ(x)/p(x), then problem (3.2) is equivalent to the following:
0
−q u (x)+u(x)=f (x) x∈(ξ ,ξ ) (3.6)
i xx 0 i−1 i
for i=1,...,N, where q =r /p , satisfying the following boundary conditions
i i i
(cid:40)
u (ξ )=u (ξ )=0
x 0 x N
(3.7)
q u (ξ −)−q u (ξ +)=0 i=1,...,N −1.
i x i i+1 x i
Here, u (ξ ±) denote the right(left)-hand side limits of u at ξ .
x i x i
Proof. Let us consider the family of representative cell Y∗, i=1,2...,N, defined by
i
Y∗ ={(y ,y )∈R2 : 0<y <L and 0<y <G (y )}
i 1 2 1 2 i 1
andletχ betheircharacteristicfunction. Weextendeachχ periodicallyonthevariabley ∈Randdenote
i i 1
this extension again by χ , for i=1,...,N.
i
If we denote by χ(cid:15) the characteristic function of the set Ω(cid:15) = {(x,y) : ξ < x < ξ , 0 < y < G (x/(cid:15)) =
i i i−1 i i
G(x,x/(cid:15))}, we easily see that
x
χ(cid:15)(x ,x )=χ ( 1,x ), for (x ,x )∈Ω(cid:15). (3.8)
i 1 2 i (cid:15) 2 1 2 i
Let us also denote by Ωi the rectangle Ωi ={(x,y):ξ <x<ξ ,0<y <G }, for i=1,...,N.
i−1 i 1
Let us also consider the following families of isomorphisms T(cid:15) :A(cid:15) (cid:55)→Y given by
k k
x −(cid:15)kL
T(cid:15)(x ,x )=( 1 ,x ) (3.9)
k 1 2 (cid:15) 2
where
A(cid:15) ={(x ,x )∈R2 |(cid:15)kL≤x <(cid:15)L(k+1) and 0<x <G }
k 1 2 1 2 1
Y =(0,L)×(0,G )
1
with k ∈N.
Let us consider the following auxiliary problem given by
 −∆X =0 in Y∗
∂Xi =−∂∂XNi i=G0(cid:48)i(yo1n)B2ii on Bi
∂N (cid:112)1+G(cid:48)(y )2 1 (3.10)
 Xi(0(cid:90),y2)X=iXdyi1(Ldiy,2y21=) o0n B0i
Y∗
i
where Bi is the lateral boundary, Bi is the upper boundary and Bi is the lower boundary of ∂Y∗.
0 1 2 i