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MULTISCALE MODEL. SIMUL . ccirclecopyrt2015 SIAM. Published by SIAM under the terms [610100]

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MULTISCALE MODEL. SIMUL . c/circlecopyrt2015 SIAM. Published by SIAM under the terms
Vol. 13, No. 3, pp. 1061–1105 of the Creative Commons 4.0 license
LOCALLY PERIODIC UNFOLDING METHOD AND TWO-SCALE
CONVERGENCE ON SURFACES OF LOCALLY PERIODIC
MICROSTRUCTURES∗
MARIYA PTASHNYK†
Abstract. In this paper we generalize the periodic unfolding method and the notion of two-
scale convergence on surfaces of periodic microstructures to locally periodic situations. The methodsthat we introduce allow us to consider a wide range of nonperiodic microstructures, especially to
derive macroscopic equations for problems posed in domains with perforations distributed nonperi-
odically. Using the methods of locally periodic two-scale convergence on oscillating surfaces and thelocally periodic boundary unfolding operator, we are able to analyze diﬀerential equations deﬁnedon boundaries of nonperiodic microstructures and consider nonhomogeneous Neumann conditions onthe boundaries of perforations, distributed nonperiodically.
Key words. unfolding method, two-scale convergence, locally periodic homogenization, non-
periodic microstructures, signalling processes
AMS subject classiﬁcations. 35B27, 35D30, 35Kxx
DOI.10.1137/140978405
1. Introduction. Many natural and man-made composite materials comprise
nonperiodic microscopic structures, e.g., ﬁbrous microstructures in heart muscles[23, 45], exoskeletons [27], industrial ﬁlters [49], or space-dependent perforations in
concrete [47]. An important special case of no nperiodic microstructures is that of the
so-called locally periodic (l-p) microstructures, where spatial changes are observed
on a scale smaller than the size of the domain under consideration, but larger than
the characteristic size of the microstructure. For many l-p microstructures spatialchanges cannot be represented by periodic functions depending on slow and fast vari-
ables, e.g.,plywood-likestructuresofgraduallyrotatedplanesofparallelalignedﬁbers
[12]. Thus, in these situations the standard two-scale convergence and periodic un-folding method cannot be applied. Hence, for a multiscale analysis of problems posed
in domains with nonperiodic perforations, in this paper we extend the periodic un-
folding method and two-scale convergence on oscillating surfaces to locally periodic
situations (seeDeﬁnitions 3.2–3.5). Thesegeneralizationsaremotivated bythe locally
periodic two-scale (l-t-s) convergence introduced in [46].
Two-scaleconvergenceonsurfacesofperiodicmicrostructureswasﬁrstintroduced
in [5, 41]. An extension of two-scale convergence associated with a ﬁxed periodic
Borel measure was considered in [53]. The unfolding operator maps functions deﬁnedon perforated domains, depending on small parameter ε, onto functions deﬁned on
the whole ﬁxed domain; see [19, 18] and ref erences therein. This helps to overcome
one of the diﬃculties of perforated domains, which is the use of extension operators.
Using the boundary unfolding operatorwe can proveconvergenceresults for nonlinear
equations posed on oscillating boundaries of microstructures [15, 18, 20, 34, 43]. Theunfolding method is also an eﬃcient tool to d erive error estimates; see, e.g., [28, 29,
30, 31, 44].
∗Received by the editors July 21, 2014; accepted for publication (in revised form) July 9, 2015;
published electronically September 30, 2015. This research was supported by EPSRC First Grant
“Multiscale modelling and analysis of mechanical properties of plant cells and tissues.”
http://www.siam.org/journals/mms/13-3/97840.html
†Division of Mathematics, University of Dundee, DD1 4HN, Scotland, UK (mptashnyk@maths.
dundee.ac.uk).
1061
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1062 MARIYA PTASHNYK
The main novelty of this paper is the derivation of new techniques for the multi-
scale analysis of nonlinear problems posed in domains with nonperiodic perforations
and on the surfaces of nonperiodic microstructures. The l-p unfolding operator allows
us to analyze nonlinear diﬀerential equations posed on domains with nonperiodic per-forations. The l-t-s convergence on oscillating surfaces and the l-p boundary unfold-
ing operator allow us to show strong convergence for sequences deﬁned on oscillating
boundaries of nonperiodic microstructures and to derive macroscopic equations for
nonlinear equations deﬁned on boundaries of nonperiodic microstructures. Until now,
this was not possible using existing methods.
The paper is structured as follows. First , in section 2, we present a mathematical
description of l-p microstructures and state the deﬁnition of an l-p approximation for
a function ψ∈C(
Ω;Cper(Yx)). In section 3 we introduce all of the main deﬁnitions
of the paper, i.e., the notion of an l-p unfolding operator, two-scale convergence for
sequences deﬁned on oscillating boundaries of l-p microstructures, and the l-p bound-
ary unfolding operator. The main results are summarized in section 4. The central
results of this paper are convergence results for sequences bounded in LpandW1,p,
withp∈(1,∞) (see Theorems 4.1–4.4). The proofs of the main results for the l-p
unfolding operator are presented in section 5. The properties of the decomposition
of aW1,p-function, with one part describing the macroscopic behavior and another
part of order ε, are shown in section 6. The proofs of the main results for the l-p
unfolding operator in perforated domains are given in section 7. The convergence
results for l-t-s convergence on oscillating surfaces and the l-p boundary unfolding
operator are proved in section 8. In section 9 we apply the l-p unfolding operator to
derive macroscopic problems f or microscopic models of signaling processes in cell tis-
sues comprising l-p microstru ctures. As examples of tissu es with l-p microstructures
we consider plant tissues, epithelial tissues, and nonperiodic ﬁbrous structure of heart
tissue. Finally, section 10 contains some concluding remarks.
There are some existing results on the homogenization of problems posed on l-p
media. The homogenization of a heat-conductivity problem deﬁned in domains with
nonperiodic microstructure consisting of spherical balls was studied in [13] using the
Murat–Tartar H-convergencemethod [40], and in [3] by applying the θ-2 convergence.
The nonperiodic distribution of balls is given by a C2-diﬀeomorphism θ, transforming
the centers of the balls. Estimates for a numerical approximationof this problem werederived in [50]. The notion of a Young measure was used in [37] to extend the concept
of periodic two-scale convergence and to deﬁne the so-called scale convergence .T h e
deﬁnition of scale convergence was motivated by the derivation of the Γ-limit for asequence of nonlinear energy functionals involving nonperiodic oscillations. Formal
asymptotic expansionsand the technique oftwo-scaleconvergencedeﬁned for periodic
test functions (see, e.g., [4, 42]) were used to derive macroscopic equations for models
posed on domains with l-p perforations, i.e., domains consisting of periodic cells with
smoothly changing perforations [9, 16, 17, 35, 36, 51]. The H-convergence method
[11, 12], the asymptotic expansion method [8], and the method of l-t-s convergence
[46] were applied to analyze microscopic models posed on domains consisting of non-
periodic ﬁbrous materials. The optimization of the elastic properties of a materialwith l-p microstructure was considered in [6, 7].
To illustrate the diﬀerence between the formulation of nonperiodic microstruc-
ture by using periodic functions and the l-p formulation of the problem, we consider
a plywood-like structure, given as the superposition of gradually rotated planes of
aligned parallel ﬁbers. We consider layers of cylindrical ﬁbers of radius εaorthogonal
to thex
3-axis and rotated around the x3-axis by an angle γ, constant in each layer
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LOCALLY PERIODIC UNFOLDING METHOD 1063
x
γx
εx Ω)
Ω εε
ε(x
xε
12
,
1ε
2r3
31
3ε
nWεεxx
xεε
Ωx
2r
123
Fig. 1.Schematic representation of slow rotating and fast rotating plywood-like structures.
and changing from one layer to another; see Figure 1. To describe the diﬀerence in
the material properties of ﬁbers and the interﬁber space with the help of a periodicfunction, we deﬁne a function
(1.1) A
ε(x)=A1˜η/parenleftbig
R(γ(x3))x/ε/parenrightbig
+A2/bracketleftBig
1−˜η/parenleftbig
R(γ(x3))x/ε/parenrightbig/bracketrightBig
,
whereA1,A2areconstant tensors and ˜ ηdenotes the characteristic functions of a ﬁber
of radius ain the direction of the x1-axis, i.e.,
(1.2) ˜ η(y)=/braceleftBigg
1f o r |ˆy−(1/2,1/2)|≤a,
0f o r |ˆy−(1/2,1/2)|>a ,
and extended ˆY-periodic to the whole R3, witha<1/2, ˆy=(y2,y3),Y=[ 0,1]3,a n d
ˆY=[ 0,1]2. The inverse of the rotation matrix around the x3-axis with rotation angle
αwith the x1-axis is deﬁned as
(1.3) R(α)=⎛
⎝cos(α) sin(α)0
−sin(α)c o s (α)0
00 1⎞
⎠,
andγ∈C1(R) is a given function such that 0 ≤γ(s)≤πfor alls∈R. Then,
considering, for example, an elliptic problem with a diﬀusion coeﬃcient or elasticity
tensor in the form (1.1) and using a change of variables ˜ x=R(γ(x3))x, we can apply
periodic homogenization techniques to derive corresponding macroscopic equations
(see [10, 11] for details). However, in the representation of the microscopic structureby (1.1), every point of a ﬁber is rotated diﬀerently, and the cylindrical structure of
the ﬁbers is deformed. Hence, A
εrepresent the properties of a material with a micro-
structure diﬀerent from the plywood-like s tructure, and for a co rrect representation
of a plywood-like structure, an l-p formulation of the microscopic problem is essential.
Also, applying periodic homogenization techniques, we obtain eﬀective macroscopic
coeﬃcients diﬀerent from the one obtain ed by using methods of l-p homogenization
(see [12, 46] for more details).
To deﬁne the characteristic function of the domain occupied by ﬁbers in a domain
with an l-p plywood-like structure, we divide R3into layers Lε
k=R2×((k−1)εr,kεr)
of height εrand perpendicular to the x3-axis, where k∈Zand 0<r<1. In
eachLε
kwe choose an arbitrary ﬁxed point xε
k∈Lε
k. Using the l-p approximation of
η∈C(
Ω,L∞
per(Yx)), withη(x,y)=˜η(R(x)y)f o rx∈Ωa n dy∈Yx,g i v e nb y
(Lεη)(x)=/summationdisplay
k∈Z˜η/parenleftbig
R(γ(xε
k,3))x/ε/parenrightbig
χLε
k(x)f o rx∈Ω,
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1064 MARIYA PTASHNYK
the characteristic function of the d omain occupied by ﬁbers is given by
(1.4) χΩε
f(x)=χΩ(x)(Lεη)(x).
Here ˜η∈L∞
per(Y)i sa si n( 1 . 2 ) ,a n d Yx=R−1(γ(x3))Y. For a microstructure
composed of fast rotating planes of parallel aligned ﬁbers (see Figure 1), we con-sider an approximation by an l-p plywood-like structure with shifted periodicity
D(x)Y=R
−1(x)W(x)Y; see [12, 46] for more details.
2. Locally periodic microstructures and locally periodic perforated do-
mains.In this sectionwe giveamathematical formulationofl-p microstructures. We
also deﬁne the approximation of functions, where the periodicity with respect to the
fast variableisdependent onthe slowvariable, byl-p functions, i.e., periodicin subdo-
mains smaller than the domain under considerationbut larger than the representativesize of the microstructure.
Let Ω⊂R
dbe a bounded Lipschitz domain. For each x∈Rdwe consider
a transformation matrix D(x)∈Rd×dand its inverse D−1(x) such that D,D−1∈
Lip(Rd;Rd×d)a n d0<D1≤|detD(x)|≤D2<∞for allx∈
Ω. We consider the
continuous family of parallelepipeds Yx=DxYon
Ω, where Y=( 0,1)dis the “unit
cell,” and denote Dx:=D(x)a n dD−1
x:=D−1(x).
Forε>0, in a manner similar to that of [13, 46], we consider the partition
covering of Ω by a family of open nonintersecting cubes {Ωε
n}1≤n≤Nεof sideεr, with
0<r<1:
Ω⊂Nε/uniondisplay
n=1
Ωεnand Ωε
n∩Ω/negationslash=∅.
For arbitrary chosen ﬁxed points xε
n,˜xε
n∈Ωε
n∩Ω we consider a covering of Ωε
nby
parallelepipeds εDxεnY:
Ωε
n⊂˜xε
n+/uniondisplay
ξ∈ΞεnεDxεn(
Y+ξ),where Ξε
n={ξ∈Zd:˜xε
n+εDxεn(Y+ξ)∩Ωε
n/negationslash=∅},
withDxεn=D(xε
n)a n d1≤n≤Nε.F o re a c h n=1,…,N ε,˜xε
nis a ﬁxed shift in the
representation of the microscopic structure of Ωε
n. Often we can consider ˜ xε
n=εDxεnξ
for some ξ∈Zd.
We consider the space C(
Ω;Cper(Yx)) given in a standard way; i.e., for any /tildewideψ∈
C(
Ω;Cper(Y)) the relation ψ(x,y)=/tildewideψ(x,D−1
xy) withx∈Ωa n dy∈Yxyields
ψ∈C(
Ω;Cper(Yx)). In the same way the spaces Lp(Ω;Cper(Yx)),Lp(Ω;Lq
per(Yx)),
andC(
Ω;Lq
per(Yx)), for 1≤p≤∞,1≤q<∞, are deﬁned.
To describe l-p microscopic properties of a composite material and to specify
test functions associated with the l-p microstructure of a material, as well as for the
deﬁnition of the l-t-s convergence, we shall consider an l-p approximation of functionswith space-dependentperiodicity, i.e., offunctions in C(
Ω;Cper(Yx)),Lp(Ω;Cper(Yx)),
orC(
Ω;Lq
per(Yx)). Locally periodic approximated functions are Yxεn-periodic in each
subdomain Ωε
n, withn=1,…,N ε, and are related to test functions associated with
the periodic structure of Ωε
n. Since the microscopic structure of Ωεnis represented by a
unionofperiodicitycells εYxεnshiftedbyaﬁxedpoint ˜ xε
n∈Ωε
n∩Ω, withn=1,…,N ε,
this shift is also reﬂected in the deﬁnition of the l-p approximation.
O f t e nc o e ﬃ c i e n t si nam i c r o s c o p i cm o d e lp o s e di nad o m a i nw i t ha nl – pm i c r o –
structure depend only on the microscopic fast variables x/εand the points xε
n,˜xε
n∈
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LOCALLY PERIODIC UNFOLDING METHOD 1065
r Ωnε
ε^^,m
Ωε
mΩnΩΩε
m
nε
εm
ΩεΩ
εxεΓ
K
1n
n
13
ε2
∗Yε
Fig. 2.Schematic representation of subdomains Ωε
nandˆΩε
n.
Ωε
n∩Ω, describing the periodic microstructurein each Ωε
n, withn=1,…,N ε,a n da r e
independent of the macroscopicslow variables x. To deﬁne such functions we shall in-
troduce the notion of an l-p approximation Lε
0of a function ψ∈C(
Ω;Cper(Yx)) (or in
Lp(Ω;Cper(Yx)),C(
Ω;Lq
per(Yx))). In each Ωε
nthe function Lε
0(ψ)i sYxεn-periodic and
depends only on the fast variables x/ε. This speciﬁc l-p approximation is important
for the derivationofmacroscopicequations for a microscopicproblemwith coeﬃcients
discontinuous with respect to the fast variables, since for ψ∈C(
Ω;Lp(Yx)) we have
thatLε
0(ψ) converges strongly l-t-s; see [46].
As an l-p approximation of ψwe name Lε:C(
Ω;Cper(Yx))→L∞(Ω) given by
(2.1) ( Lεψ)(x)=Nε/summationdisplay
n=1/tildewideψ/parenleftBigg
x,D−1
xεn(x−˜xε
n)
ε/parenrightBigg
χΩε
n(x)f o rx∈Ω.
We consider also the map Lε
0:C(
Ω;Cper(Yx))→L∞(Ω) deﬁned for x∈Ωa s
(Lε
0ψ)(x)=Nε/summationdisplay
n=1ψ/parenleftBig
xε
n,x−˜xε
n
ε/parenrightBig
χΩεn(x)=Nε/summationdisplay
n=1/tildewideψ/parenleftBigg
xε
n,D−1
xεn(x−˜xε
n)
ε/parenrightBigg
χΩεn(x).
If we choose ˜ xε
n=Dxεnεξfor some ξ∈Zd, then the periodicity of /tildewideψimplies
(Lεψ)(x)=Nε/summationdisplay
n=1/tildewideψ/parenleftBigg
x,D−1
xεnx
ε/parenrightBigg
χΩεn(x)a n d(Lε
0ψ)(x)=Nε/summationdisplay
n=1/tildewideψ/parenleftBigg
xε
n,D−1
xεnx
ε/parenrightBigg
χΩεn(x)
forx∈Ω.
In the following, we shall consider the case ˜ xε
n=εDxεnξ, withξ∈Zd. However,
all results hold for arbitrary chosen ˜ xε
n∈Ωε
nwithn=1,…,N ε; see [46]. In a similar
way we deﬁne LεψandLε
0ψforψinC(
Ω;Lq
per(Yx)) orLp(Ω;Cper(Yx)).
The l-p approximation reﬂects the micros copic properties of Ω, where in each Ωε
n
the microstructure is represented by a “unit cell” Yxεn=DxεnYfor an arbitrary ﬁxed
xε
n∈Ωε
n; see Figures 1 and 2.
In the context of admissible test functions in a weak formulation of partial diﬀer-
ential equations, we deﬁne a regular approximation of Lεψby
(Lε
ρψ)(x)=Nε/summationdisplay
n=1/tildewideψ/parenleftBigg
x,D−1
xεnx
ε/parenrightBigg
φΩεn(x)f o rx∈Ω,
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1066 MARIYA PTASHNYK
whereφΩεnare approximations of χΩεnsuch that φΩεn∈C∞
0(Ωεn)a n d
(2.2)Nε/summationdisplay
n=1|φΩεn−χΩεn|→0i nL2(Ω),||∇mφΩεn||L∞(Rd)≤Cε−ρmfor 0<r<ρ< 1;
see, e.g., [11, 13, 46]. In the deﬁnition of the l-p unfolding operator we shall use
subdomains of Ωε
ngiven by unit cells εDxεn(Y+ξ) that are completely included in
Ωε
n∩Ω( s e eF i g u r e2 ) :
ˆΩε=Nε/uniondisplay
n=1ˆΩε
n,withˆΩε
n=I n t/parenleftBigg/uniondisplay
ξ∈ˆΞεnεDxεn(
Y+ξ)/parenrightBigg
and Λε=Nε/uniondisplay
n=1Λε
n∩Ω, (2.3)
where Λε
n=Ωεn\ˆΩε
nandˆΞε
n={ξ∈Ξε
n:εDxε
n(Y+ξ)⊂(Ωε
n∩Ω)}.
As is known from the periodic case, the unfolding operator provides a power-
ful technique for the multiscale analysis of problems posed in perforated domains and
nonlinear equations deﬁned on oscillating surfaces of microstructures. Thus, the main
emphasis of this paper will be on the development of the unfolding method for do-mains with l-p perforations. Therefore, next we introduce perforated domains with
l-p changes in the distribution and in the shape of perforations.
We consider Y
0⊂Ywith a Lipschitz boundary Γ = ∂Y0and a matrix Kwith
K,K−1∈Lip(Rd;Rd×d), where 0 <K1≤|detK(x)|≤K2<∞,KxY0⊂Y,a n d
Y∗=Y\
Y0and/tildewideY∗
Kx=Y\Kx
Y0are connected for all x∈
Ω. Deﬁne Y∗
x,K=Dx/tildewideY∗
Kx
with the boundary Γ x=DxKxΓ, where Kx=K(x)a n dDx=D(x). Then, a domain
with l-p perforations is deﬁned as
Ω∗
ε,K=I n t/parenleftBiggNε/uniondisplay
n=1Ω∗,ε
n,K/parenrightBigg
∩Ω,where Ω∗,ε
n,K=/uniondisplay
ξ∈Ξ∗,ε
nεDxεn/parenleftBig
/tildewideY∗
Kxεn+ξ/parenrightBig
∪
Λ∗,ε
n.
Here Λ∗,ε
n=Ωεn\/uniontext
ξ∈Ξ∗,ε
nεDxεn(
Y+ξ), with Ξ∗,ε
n={ξ∈Ξε
n:εDxεn(Y+ξ)⊂Ωε
n},
/tildewideY∗
Kxεn=Y\Kxεn
Y0,an dKxεn=K(xε
n)forn=1,…,N εandxε
n∈ˆΩε
n. Theboundaries
of the l-p microstructure of Ω∗
ε,Kare denoted by
Γε=Nε/uniondisplay
n=1Γε
n,where Γε
n=/uniondisplay
ξ∈Ξ∗,ε
nεDxε
n(/tildewideΓKxεn+ξ)∩Ω with/tildewideΓKxεn=Kxε
nΓ.
Notice that changes in the microstructure of Ω∗
ε,Kare deﬁned by changes in the peri-
odicity given by D(x) and additional changes in the shape of perforations described
byK(x)f o rx∈Ω.
Along with plywood-like structures (see Figure 1), examples of l-p microstruc-
tures are, e.g., concrete materials with space-dependent perforations, and plant and
epithelial tissues; see Figure 3. In the deﬁnition of microstructure of concrete mate-rials with space-dependent perforations we have, e.g., D(x)=IandK(x)=ρ(x)I
for 0<ρ
1≤ρ(x)≤ρ2<∞such that K(x)
Y0⊂Y,w h e r e Idenotes the identity
matrix; see, e.g., [16, 51] and Figure 2. For plant or epithelial tissues additionally wehave space-dependent deformations of cells given by D(x)/negationslash=I;s e eF i g u r e3 .
Using the mathematical deﬁnition of general l-p microstructures, next we in-
troduce the deﬁnition of the l-p unfolding operator, mapping functions deﬁned on
ε-dependent domains to functions depending on two variables (i.e., a microscopic
variable and a macroscopic variable), but deﬁned on ﬁxed domains.
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LOCALLY PERIODIC UNFOLDING METHOD 1067
Fig. 3.Examples of l-p microstructures with local changes in the shape and the periodicity
of a microstructure. We observe changes in shape and size of cells in an epithelial tissue dueto maturation (left) and changes in the size of plant cells in a wood tissue (right). Reproducedwith permission from Anatomy & Physiology (http://anatomyandphysiologyi.com) (left) and from
W. Schoch, I. Heller, F. H. Schweingruber, and F. Kienast, 2004 [48] (right).
3. Deﬁnitions of lp unfolding operator and l-t-s convergence on oscil-
lating surfaces. The main idea of the two-scale convergence is to consider test func-
tions which comprise the information about the microstructure and the microscopic
properties of a composite material and of model equations. The same idea is used
in the deﬁnition of l-t-s by considering an l-p approximation of ψ∈Lq(Ω;Cper(Yx))
(reﬂecting the l-p properties of microscopic problems) as a test function.
Definition 3.1 ( see [46] ).Letuε∈Lp(Ω)for allε>0andp∈(1,+∞).W e
say the sequence {uε}converges l-t-s to u∈Lp(Ω;Lp(Yx))asε→0if/bardbluε/bardblLp(Ω)≤C
and for any ψ∈Lq(Ω;Cper(Yx))
lim
ε→0/integraldisplay
Ωuε(x)Lεψ(x)dx=/integraldisplay
Ω−/integraldisplay
Yxu(x,y)ψ(x,y)dydx,
whereLεis the l-p approximation of ψ, deﬁned as in (2.1),a n d1/p+1/q=1.
Remark. Notice that the deﬁnition of l-t-s and convergence results presented in
[46] forp= 2 are directly generalized to p∈(1,∞).
Motivated by the notion of the periodic unfolding operator and l-t-s convergence,
we deﬁne the l-p unfolding operator in the following way.
Definition 3.2. For any function ψthat is Lebesgue-measurable on Ω,t h el – p
unfolding operator Tε
Lis deﬁned as
Tε
L(ψ)(x,y)=Nε/summationdisplay
n=1ψ/parenleftbig
εDxεn/bracketleftbig
D−1
xεnx/ε/bracketrightbig
Y+εDxεny/parenrightbig
χˆΩεn(x)forx∈Ωandy∈Y.
The deﬁnition implies that Tε
L(ψ) is Lebesgue-measurable on Ω ×Yand is zero
forx∈Λε.
For perforated domains with local changes in the distribution of perforations, but
without additional changes in the shape of perforations, i.e., K=Iand
Ω∗
ε=I n t/parenleftBiggNε/uniondisplay
n=1Ω∗,ε
n/parenrightBigg
∩Ω,where Ω∗,ε
n=/uniondisplay
ξ∈Ξ∗,ε
nεDxε
n(
Y∗+ξ)∪
Λ∗,ε
n
andY∗=Y\
Y0, we deﬁne the l-p unfolding operator in the following way.
Definition 3.3. For any function ψthat is Lebesgue-measurable on Ω∗
ε,t h el – p
unfolding operator T∗,ε
Lis deﬁned as
T∗,ε
L(ψ)(x,y)=Nε/summationdisplay
n=1ψ/parenleftbig
εDxεn/bracketleftbig
D−1
xεnx/ε/bracketrightbig
Y+εDxεny/parenrightbig
χˆΩεn(x)forx∈Ωandy∈Y∗.
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1068 MARIYA PTASHNYK
The deﬁnition implies that T∗,ε
L(ψ) is Lebesgue-measurable on Ω ×Y∗and is zero
forx∈Λε.
In mathematical models posed in perforated domains we often have some pro-
cesses deﬁned on the surfaces of the microstructure (e.g., nonhomogeneous Neumannconditions or equations deﬁned on the boundaries of the microstructure). Therefore
it is important to have a notion of a convergence for sequences deﬁned on oscillating
surfaces of l-p microstructures. Applying the same idea as in the deﬁnition of l-t-s
convergence for sequences in L
p(Ω) (i.e., considering l-p approximations of functions
with space-dependent periodicity as test functions), we deﬁne the l-t-s convergenceon surfaces of l-p m icrostructures.
Definition 3.4. As e q u e n c e {u
ε}⊂Lp(Γε),w i t hp∈(1,+∞), is said to converge
l-t-s tou∈Lp(Ω;Lp(Γx))ifε/bardbluε/bardblp
Lp(Γε)≤Cand for any ψ∈C(
Ω;Cper(Yx))
lim
ε→0ε/integraldisplay
Γεuε(x)Lεψ(x)dσx=/integraldisplay
Ω1
|Yx|/integraldisplay
Γxu(x,y)ψ(x,y)dσydx,
whereLεis the l-p approximation of ψdeﬁned in (2.1).
Often, to show the strong convergence of a sequence deﬁned on oscillating bound-
aries of a microstructure, we need to map it to a sequence deﬁned on a ﬁxed domain.This can be achieved by using the boundary unfolding operator.
Definition 3.5. For any function ψthat is Lebesgue-measurable on Γ
ε,t h el – p
boundary unfolding operator Tb,ε
Lis deﬁned as
Tb,ε
L(ψ)(x,y)=Nε/summationdisplay
n=1ψ/parenleftbig
εDxεn/bracketleftbig
D−1
xεnx/ε/bracketrightbig
Y+εDxεnKxεny/parenrightbig
χˆΩεn(x)forx∈Ωandy∈Γ.
The deﬁnition implies that Tb,ε
L(ψ) is Lebesgue-measurable on Ω ×Γ and is zero
forx∈Λε.
The l-p boundary unfolding operator is a generalization of the periodic bound-
ary unfolding operator; see, e.g., [18, 20, 21, 43]. Similar to the periodic unfoldingoperator, the l-p unfolding operator maps functions deﬁned in domains depending on
ε(on Ω
∗
εor Γε) to functions deﬁned on ﬁxed domains (Ω ×Y∗or Ω×Γ). The l-p
microstructuresof domainsarereﬂectedinthe deﬁnitionofthe l-punfolding operator.
4. Main convergence results for the l-p unfolding operator and l-t-s
convergence on oscillating surfaces. In this section we summarize the main re-
sults of the paper. Similar to the periodic case [18, 21], we obtain compactness results
for the l-t-s convergence on oscillating boundaries, for the l-p unfolding operator, andfor the l-p boundary unfolding operator. We prove convergence results for sequences
bounded in L
p(Γε),H1(Ω), and H1(Ω∗
ε), respectively. The properties of the transfor-
mation matrices DandK, assumed in section 3, are used to prove the convergence
results stated in this section.
Theorem 4.1. For a sequence {wε}⊂Lp(Ω),w i t hp∈(1,+∞), satisfying
/bardblwε/bardblLp(Ω)+ε/bardbl∇wε/bardblLp(Ω)≤C,
there exist a subsequence (denoted again by {wε})a n dw∈Lp(Ω;W1,p
per(Yx))such that
Tε
L(wε)/arrowrighttophalfw(·,Dx·) weakly in Lp(Ω;W1,p(Y)),
εTε
L(∇wε)/arrowrighttophalfD−T
x∇yw(·,Dx·) weakly in Lp(Ω×Y).
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LOCALLY PERIODIC UNFOLDING METHOD 1069
Forauniformlyin εbounded sequencein W1,p(Ω), in additionweobtainthe weak
convergenceofthe unfolded sequenceofderivatives, important forthe homogenization
of equations comprising elliptic operators of second order.
Theorem 4.2. For a sequence {wε}⊂W1,p(Ω),w i t hp∈(1,+∞), that converges
weakly to winW1,p(Ω), there exist a subsequence (denoted again by {wε})a n da
function w1∈Lp(Ω;W1,p
per(Yx))such that
Tε
L(wε)/arrowrighttophalfw weakly in Lp(Ω;W1,p(Y)),
Tε
L(∇wε)(·,·)/arrowrighttophalf∇xw(·)+D−T
x∇yw1(·,Dx·)weakly in Lp(Ω×Y).
Twoofthemainadvantagesoftheunfoldingoperatorarethatithelpstoovercome
one of the diﬃculties of perforated domains—the use of extension operators—and itallows us to prove strong convergence for sequences deﬁned on boundaries of micro-
structures. Thus, next we formulate convergenceresults for the l-p unfolding operator
in perforated domains and the l-p boundary unfolding operator.
Theorem 4.3. For a sequence {w
ε}⊂W1,p(Ω∗
ε),w h e r ep∈(1,+∞), satisfying
(4.1) /bardblwε/bardblLp(Ω∗
ε)+ε/bardbl∇wε/bardblLp(Ω∗
ε)≤C,
there exist a subsequence (denoted again by {wε})a n dw∈Lp(Ω;W1,p
per(Y∗
x))such that
(4.2)T∗,ε
L(wε)/arrowrighttophalfw(·,Dx·) weakly in Lp(Ω;W1,p(Y∗)),
εT∗,ε
L(∇wε)/arrowrighttophalfD−T
x∇yw(·,Dx·)weakly in Lp(Ω×Y∗).
In the case when wεis bounded in Wp(Ω∗
ε) uniformly with respect to ε,w e
obtain weak convergence of T∗,ε
L(∇wε)i nLp(Ω×Y∗) and local strong convergence
ofT∗,ε
L(wε).
Theorem 4.4. For a sequence {wε}⊂W1,p(Ω∗
ε),w h e r ep∈(1,+∞), satisfying
/bardblwε/bardblW1,p(Ω∗ε)≤C,
there exist a subsequence (denoted again by {wε}) and functions w∈W1,p(Ω)and
w1∈Lp(Ω;W1,p
per(Y∗
x))such that
T∗,ε
L(wε)/arrowrighttophalfw weakly in Lp(Ω;W1,p(Y∗)),
T∗,ε
L(∇wε)/arrowrighttophalf∇w+D−T
x∇yw1(·,Dx·)weakly in Lp(Ω×Y∗),
T∗,ε
L(wε)→w strongly in Lp
loc(Ω;W1,p(Y∗)).
Notice that the weak limit of εT∗,ε
L(∇wε)a n dT∗,ε
L(∇wε) reﬂects the l-p micro-
structure of Ω∗
εand depends on the transformation matrix D.
For l-t-s convergence on oscillating surfaces of microstructures we have the fol-
lowing compactness result.
Theorem 4.5. For a sequence {wε}⊂Lp(Γε),w i t hp∈(1,+∞), satisfying
ε/bardblwε/bardblp
Lp(Γε)≤C,
there exist a subsequence (denoted again by {wε})a n dw∈Lp(Ω;Lp(Γx))such that
wε→wl-t-s.
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1070 MARIYA PTASHNYK
Similar to the periodic case [18, 21], we show the relation between the l-t-s con-
vergence on oscillating surfaces and the weak convergence of a sequence obtained by
applying the l-p boundary unfolding operator.
Theorem 4.6. Let{wε}⊂Lp(Γε)withε/bardblwε/bardblp
Lp(Γε)≤C,w h e r ep∈(1,+∞).
The following assertions are equivalent:
(i) wε→w l-t-s,w∈Lp(Ω;Lp(Γx)),
(ii)Tb,ε
L(wε)/arrowrighttophalfw(·,DxKx·)weakly in Lp(Ω×Γ).
Theorems 4.5 and 4.6 imply that for {wε}⊂Lp(Γε) withε/bardblwε/bardblp
Lp(Γε)≤Cwe
have the weak convergence of {Tb,ε
L(wε)}inLp(Ω×Γ), where p∈(1,+∞).
The deﬁnition of the l-p boundary unfolding operator and the relation between
the l-t-s convergence of sequences deﬁned on l-p oscillating boundaries and the l-p
boundary unfolding operator allow us to obtain homogenization results for equationsposed on the boundaries of l-p microstructures.
5. The l-p unfolding operator: Proofs of convergence results. First we
prove some properties of the l-p unfolding operator. Similar to the periodic case, weobtain that the l-p unfolding operator is linear and preserves strong convergence.
Lemma 5.1.
(i)Forφ∈L
p(Ω),w i t hp∈[1,+∞), it holds that
(5.1)1
|Y|/integraldisplay
Ω×YTε
L(φ)(x,y)dydx=/integraldisplay
Ωφ(x)dx−/integraldisplay
Λεφ(x)dx,
/integraldisplay
Ω×Y|Tε
L(φ)(x,y)|pdydx≤|Y|/integraldisplay
Ω|φ(x)|pdx.
(ii)Tε
L:Lp(Ω)→Lp(Ω×Y)is a linear continuous operator, where p∈[1,+∞).
(iii)Forφ∈Lp(Ω),w i t hp∈[1,+∞), we have strong convergence
Tε
L(φ)→φinLp(Ω×Y). (5.2)
(iv)Ifφε→φinLp(Ω),w i t hp∈[1,+∞),t h e nTε
L(φε)→φinLp(Ω×Y).
Proof. Using the deﬁnition of the l-p unfolding operator we obtain
/integraldisplay
Ω×Y|Tε
L(φ)(x,y)|pdydx=Nε/summationdisplay
n=1/summationdisplay
ξ∈ˆΞεnεd|DxεnY|/integraldisplay
Y|φ(Dxεn(εξ+εy))|pdy
=Nε/summationdisplay
n=1|Y|/summationdisplay
ξ∈ˆΞεn/integraldisplay
εDxεn(ξ+Y)|φ(x)|pdx=Nε/summationdisplay
n=1|Y|/integraldisplay
ˆΩε
n|φ(x)|pdx.(5.3)
Then the equality and estimate in (5.1) follow from the deﬁnition of Λεand the
properties of the covering of Ω by {Ωε
n}Nε
n=1.
The result in (ii) is ensured by the deﬁnition of the l-p unfolding operator and
inequality (5.1).
(iii) Using the fact that φ∈Lp(Ω) and|Λε|→0a sε→0 (ensured by the proper-
ties ofthe coveringof Ω by {Ωε
n}Nε
n=1and the deﬁnition of Λε) and applying Lebesgue’s
dominated convergence theorem ( see, e.g., [26]), we obtain that/integraltext
Λε|φ(x)|pdx→0a s
ε→0.
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LOCALLY PERIODIC UNFOLDING METHOD 1071
Then considering the approximation of Lp-functions by continuous functions and
using the deﬁnition of Tε
L, equality (5.3) and the estimate in (5.1) imply the conver-
gence stated in (iii).
(iv) The linearity of the l-p unfolding operator along with (5.1) and (5.2) yields
/bardblTε
L(φε)−φ/bardblLp(Ω×Y)≤|Y|1
p/bardblφε−φ/bardblLp(Ω)+/bardblTε
L(φ)−φ/bardblLp(Ω×Y)→0a sε→0.
Similartol-t-sconvergence,the averageofthe weaklimit ofthe unfolded sequence
withrespecttomicroscopicvar iablesisequaltotheweaklimitoftheoriginalsequence.
Lemma 5.2. For{wε}bounded in Lp(Ω),w i t hp∈(1,+∞), we have that
{Tε
L(wε)}is bounded in Lp(Ω×Y),a n di f
Tε
L(wε)/arrowrighttophalf˜wweakly in Lp(Ω×Y),
then
wε/arrowrighttophalf−/integraldisplay
Y˜wdyweakly in Lp(Ω).
Proof. The boundedness of {Tε
L(wε)}inLp(Ω×Y) follows directly from the
boundedness of {wε}inLp(Ω) and the estimate (5.1). For ψ∈Lq(Ω), 1/p+1/q=1 ,
using the deﬁnition of Tε
L(wε)w eh a v e
/integraldisplay
Ωwεψdx=1
|Y|/integraldisplay
Ω×YTε
L(wε)Tε
L(ψ)dydx+Aε,whereAε=/integraldisplay
Λεwεψdx.
For{wε}bounded in Lp(Ω) and ψ∈Lq(Ω), using the properties of the covering of
Ω and the deﬁnition of ˆΩεand Λε,w eo b t a i n Aε→0a sε→0. Then, the weak
convergence of Tε
L(wε) and the strong convergence of Tε
L(ψ), shown in Lemma 5.1,
imply
lim
ε→0/integraldisplay
Ωwε(x)ψ(x)dx=1
|Y|/integraldisplay
Ω/integraldisplay
Y˜w(x,y)ψ(x)dydx
for anyψ∈Lq(Ω).
For the periodic unfolding operator we have that Tε(ψ(·,·/ε))→ψinLq(Ω×Y)
forψ∈Lq(Ω,Cper(Y)). A similar result holds for the l-p unfolding operator and
ψ∈Lq(Ω,Cper(Yx)), but with ψ(·,·/ε) replaced by the l-p approximation Lεψ(·).
Lemma 5.3.
(i)Forψ∈Lq(Ω;Cper(Yx)),w i t hq∈[1,+∞), we have
Tε
L(Lεψ)→ψ(·,Dx·)strongly in Lq(Ω×Y).
(ii)Forψ∈C(
Ω;Lq
per(Yx)),w i t hq∈[1,+∞), we have
Tε
L(Lε
0ψ)→ψ(·,Dx·)strongly in Lq(Ω×Y).
Proof. (i) For ψ∈C(
Ω;Cper(Yx)) using the deﬁnition of LεandTε
L,w eo b t a i n
/integraldisplay
Ω×Y|Tε
L(Lεψ)|qdydx=Nε/summationdisplay
n=1/integraldisplay
ˆΩεn×Y/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideψ/parenleftBigg
εD
xε
n/bracketleftBigg
D−1
xεnx
ε/bracketrightBigg
Y+εDxε
ny,y/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleq
dydx,
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1072 MARIYA PTASHNYK
whereq∈[1,+∞)a n d/tildewideψ∈C(
Ω;Cper(Y)) such that ψ(x,y)=/tildewideψ(x,D−1
xy)f o rx∈Ω
andy∈Yx. Then, using the properties of the covering of Ωε
nbyεYξ
xεn=εDxεn(Y+ξ),
withξ∈Ξε
n, and considering ﬁxed points yξ∈Y+ξforξ∈ˆΞε
n,w eo b t a i n
/integraldisplay
Ω×Y|Tε
L(Lεψ)|qdydx=Nε/summationdisplay
n=1/summationdisplay
ξ∈ˆΞε
nεd|Yxε
n|/integraldisplay
Y|/tildewideψ(εDxε
n(ξ+yξ),y)|qdy+δ(ε),
where, due to the continuity of ψand the properties of the covering of Ω by {Ωε
n}Nε
n=1,
δ(ε)=Nε/summationdisplay
n=1/summationdisplay
ξ∈ˆΞεnεd|Yxεn|/integraldisplay
Y/parenleftBig
|/tildewideψ(εDxεn(ξ+yξ),y)|q−|/tildewideψ(εDxεn(ξ+y),y)|q/parenrightBig
dy→0
asε→0. Then, using the continuity of ψandDtogether with the relation between
ψand/tildewideψ,w eo b t a i n
lim
ε→0/integraldisplay
Ω×Y|Tε
L(Lεψ)|qdydx=/integraldisplay
Ω×Y|/tildewideψ(x,y)|qdydx=/integraldisplay
Ω×Y|ψ(x,Dxy)|qdydx.
Thecontinuityof ψwithrespectto xyieldsthepointwiseconvergenceof Tε
L(Lεψ)(x,y)
toψ(x,Dxy)a . e .i nΩ ×Y.
Considering an approximation of ψ∈Lq(Ω;Cper(Yx)) byψm∈C(
Ω;Cper(Yx))
and the convergences
lim
m→∞lim
ε→0/integraldisplay
Ω|Lεψm(x)−Lεψ(x)|qdx=0,
lim
m→∞lim
ε→0/integraldisplay
Ω/parenleftbig
|Lεψm(x)|q−|Lεψ(x)|q/parenrightbig
dx=0
(see [46, Lemma 3.4] for the proof) implies Tε
L(Lεψ)(·,·)→ψ(·,Dx·)i nLq(Ω×Y)
forψ∈Lq(Ω;Cper(Yx)).
(ii) Forψ∈C(
Ω;Lq
per(Yx)) we can provethe strong convergenceonly of Tε
L(Lε
0ψ).
Consider
lim
ε→0/integraldisplay
Ω×Y|Tε
L(Lε
0ψ)(x,y)|qdydx=|Y|lim
ε→0/bracketleftbigg/integraldisplay
Ω|Lε
0ψ(x)|qdx−/integraldisplay
Λε|Lε
0ψ(x)|qdx/bracketrightbigg
.
Then, using Lemma 3.4 in [46] along with the regularity of ψand the properties of
Λε,w eo b t a i n
|Y|lim
ε→0/integraldisplay
Ω|Lε
0ψ(x)|qdx=/integraldisplay
Ω×Y|ψ(x,Dxy)|qdydx, lim
ε→0/integraldisplay
Λε|Lε
0ψ(x)|qdx=0.
The continuity of ψwith respect to x∈Ω implies Tε
L(Lε
0ψ)(x,y)→ψ(x,Dxy)p o i n t –
wise a.e. in Ω ×Y.
Remark. Notice that for ψ∈C(
Ω;Lq
per(Yx)) we have the strong convergence
only ofTε
L(Lε
0ψ). However, this convergence resul t is suﬃcient for the derivation of
homogenization results, since the microscopic properties of the considered processes
or domains can be represented by coeﬃcients in the form BLε
0A, with some given
functions B∈L∞(Ω) and A∈C(
Ω;Lq
per(Yx)).
The strong convergence of Tε
L(Lεψ)f o rψ∈Lq(Ω;Cper(Yx)) is now used to show
the equivalence between the weak convergence of the l-p unfolded sequence and the
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LOCALLY PERIODIC UNFOLDING METHOD 1073
l-t-s convergence of the original sequence. Notice that Lq(Ω;Cper(Yx)) represents the
set of test functions admissible in the deﬁnition of the l-t-s convergence.
Lemma 5.4. Let{wε}be a bounded sequence in Lp(Ω),w h e r ep∈(1,+∞).T h e n
the following assertions are equivalent:
(i) wε→w l-t-s,w∈Lp(Ω;Lp(Yx)),
(ii)Tε
L(wε)(·,·)/arrowrighttophalfw(·,Dx·)weakly in Lp(Ω×Y).
Proof. [(ii)⇒(i)] Since {wε}is bounded in Lp(Ω), there exists (up to a subse-
quence) an l-t-s limit of wεasε→0. For an arbitrary ψ∈Lq(Ω;Cper(Yx)), the weak
convergence of Tε
L(wε) and the strong convergence of Tε
L(Lε(ψ)) ensure
lim
ε→0/integraldisplay
ΩwεLε(ψ)dx= lim
ε→0/bracketleftbigg/integraldisplay
Ω−/integraldisplay
YTε
L(wε)Tε
L(Lε(ψ))dydx+/integraldisplay
ΛεwεLε(ψ)dx/bracketrightbigg
=/integraldisplay
Ω−/integraldisplay
Yw(x,D(x)y)ψ(x,Dxy)dydx=/integraldisplay
Ω−/integraldisplay
Yxwψdydx.
Thus the whole sequence wεconverges l-t-s to w.
[(i)⇒(ii)] On the other hand, the boundedness of {wε}inLp(Ω) implies the
boundedness of {Tε
L(wε)}and (up to a subsequence) the weak convergence of Tε
L(wε)
inLp(Ω×Y). Ifwε→wl-t-s, then
lim
ε→0/integraldisplay
Ω−/integraldisplay
YTε
L(wε)Tε
L(Lε(ψ))dydx= lim
ε→0/bracketleftbigg/integraldisplay
ΩwεLε(ψ)dx−/integraldisplay
ΛεwεLε(ψ)dx/bracketrightbigg
=/integraldisplay
Ω−/integraldisplay
Yxwψdydx
forψ∈Lq(Ω;Cper(Yx)). Since Tε
L(Lε(ψ))(·,·)→ψ(·,Dx·)i nLq(Ω×Y), we obtain
the weak convergenceof the whole sequence Tε
L(wε)t ow(·,Dx·)i nLp(Ω×Y). Notice
that the boundedness of {wε}inLp(Ω) and the fact that |Λε|→0a sε→0i m p l y
/integraldisplay
Λε|wεLε(ψ)|dx≤C/parenleftbigg/integraldisplay
Λεsup
y∈Y|ψ(x,Dxy)|qdx/parenrightbigg1/q
→0a sε→0
forψ∈Lq(Ω;Cper(Yx)) and 1/p+1/q=1 .
Next, we prove the main convergence results for the l-p unfolding operator, i.e.,
convergence results for {Tε
L(wε)},{εTε
L(∇wε)},a n d{Tε
L(∇wε)}.
The deﬁnition of the l-p unfolding operator yields that for w∈W1,p(Ω)
(5.4) ∇yTε
L(w)=εNε/summationdisplay
n=1DT
xεnTε
L(∇w)χΩεn.
Due to the regularity of D, the boundedness of ε∇wεimplies the boundedness of
∇yTε
L(wε). Thus, assuming the boundedness of {ε∇wε}, we obtain convergence of
the derivatives with respect to the microscopic variables, but have no informationabout the macroscopic derivatives.
P r o o fo fT h e o r e m 4.1. The assumptions on {w
ε}, together with inequality (5.1),
equality(5.4),andregularityof D,ensurethat {Tε
L(wε)}isboundedin Lp(Ω;W1,p(Y)).
This implies that there exist a subsequence, denoted again by {wε}, and a func-
tion/tildewidew∈Lp(Ω;W1,p(Y)) such that Tε
L(wε)/arrowrighttophalf/tildewidewinLp(Ω;W1,p(Y)). We deﬁne
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1074 MARIYA PTASHNYK
w(x,y)=/tildewidew(x,D−1
xy) for a.a. x∈Ω,y∈Yx. Due to the regularity of D,w eh a v e
w∈Lp(Ω;W1,p(Yx)). Forφ∈C∞
0(Ω×Y), using the convergence of Tε
L(wε), we have
lim
ε→0/integraldisplay
Ω×YεTε
L(∇wε)φdydx=−lim
ε→0/integraldisplay
Ω×YTε
L(wε)Nε/summationdisplay
n=1divy(D−1
xε
nφ(x,y))χΩε
ndydx
=−/integraldisplay
Ω×Yw(x,Dxy)divy(D−1
xφ(x,y))dydx=/integraldisplay
Ω×YD−T
x∇yw(x,Dxy)φ(x,y)dydx.
Hence,εTε
L(∇wε)(·,·)/arrowrighttophalfD−T
x∇yw(·,Dx·)i nLp(Ω×Y)a sε→0. To show the
Yx-periodicity of w, i.e.,Y-periodicity of /tildewidew, we show ﬁrst the periodicity in the
ed-direction. Then, considering similar calculations in each ej-direction, with j=
1,…,d−1a n d{ej}j=1,…,dbeing the canonical basis of Rd, we obtain the Yx-
periodicity of w.F o rψ∈C∞
0(Ω×Y/prime)w ec o n s i d e r
I=/integraldisplay
Ω×Y/prime[Tε
L(wε)(x,(y/prime,1))−Tε
L(wε)(x,(y/prime,0))]ψ(x,y/prime)dy/primedx,
whereY/prime=( 0,1)d−1.F o rj=1,…,dwe deﬁne
/tildewideΩε,j
n=I n t/parenleftBigg/uniondisplay
ξ∈
Ξε,j
n,1εDxεn(
Y+ξ)/parenrightBigg
,/tildewideΛε,j
n,l=I n t/parenleftBigg/uniondisplay
ξ∈/tildewideΞε,j
n,lεDxεn(
Y+ξ)/parenrightBigg
forl=1,2,
where
Ξε,j
n,1={ξ∈ˆΞε
n:εDxεn(Y+ξ−ej)⊂ˆΩε
n},
Ξε,j
n={ξ∈ˆΞε
n:εDxεn(Y+ξ+ej)⊂
ˆΩε
nandεDxεn(Y+ξ−ej)⊂ˆΩε
n},a n d/tildewideΞε,j
n=ˆΞε
n\
Ξε,j
n.W ew r i t e /tildewideΞε,j
n=/tildewideΞε,j
n,1∪/tildewideΞε,j
n,2,
where/tildewideΞε,j
n,1corresponds to upper and /tildewideΞε,j
n,2corresponds to lower cells in ˆΩε
nin the
Dxε
nej-direction. Using the deﬁnition of Tε
Lwe can write
I=Nε/summationdisplay
n=1/integraldisplay
/tildewideΩε,d
n×Y/primeTε
L(wε)(x,y0)/bracketleftbig
ψ(x−εDxεned,y/prime)−ψ(x,y/prime)/bracketrightbig
dy/primedx
+Nε/summationdisplay
n=1/bracketleftbigg/integraldisplay
/tildewideΛε,d
n,1×Y/primeTε
L(wε)(x,y1)ψ(x,y/prime)dy/primedx−/integraldisplay
/tildewideΛε,d
n,2×Y/primeTε
L(wε)(x,y0)ψ(x,y/prime)dy/primedx/bracketrightbigg
,
wherey1=(y/prime,1) andy0=(y/prime,0). Using the continuity of ψ, the boundedness of the
trace ofTε
L(wε)i nLp(Ω×Y/prime), ensured by the assumptions on wε, and the fact that/summationtextNε
n=1|/tildewideΛε,d
n,l|≤Cε1−r→0a sε→0, withr∈[0,1) andl=1,2, we obtain that I→0
asε→0. Similar calculations for ej, withj=1,…,d−1, and the convergence of
the trace of Tε
L(wε)i nLp(Ω×Y/prime), ensured by the weak convergence of Tε
L(wε)i n
Lp(Ω;W1,p(Y)), imply the Yx-periodicity of w.
If/bardblwε/bardblW1,p(Ω)is bounded uniformly in ε, we have the weak convergence of wεin
W1,p(Ω) and of Tε
L(∇wε)i nLp(Ω×Y). Hence we have information about the macro-
scopic and microscopic gradients of limit functions. The proof of the convergenceresults for T
ε
L(∇wε) makes use of the Poincar´ e inequality for an auxiliary sequence.
For this purpose we deﬁne a local average operator Mε
L, i.e., an average of the un-
folded function with respect to the microscopic variables.
Definition 5.5. The local average operator Mε
L:Lp(Ω)→Lp(Ω),p∈[1,+∞],
is deﬁned as
(5.5)Mε
L(ψ)(x)=−/integraldisplay
YTε
L(ψ)(x,y)dy=Nε/summationdisplay
n=1−/integraldisplay
Yψ/parenleftbig
εDxεn/parenleftbig
[D−1
xεnx/ε]+y/parenrightbig/parenrightbig
dyχˆΩεn(x).
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LOCALLY PERIODIC UNFOLDING METHOD 1075
P r o o fo fT h e o r e m 4.2. The proof of the convergence of Tε
L(∇wε) follows ideas
similar to those in the case of the periodic unfolding operator. However, the proof of
the periodicity of the corrector w1involves new ideas and technical details.
The convergence of Tε
L(wε) follows from Lemma 5.2 and the fact that due to the
assumption on {wε}and regularity of D,w eh a v e
/bardbl∇yTε
L(wε)/bardblLp(Ω×Y)≤Cε→0a sε→0.
To show the convergence of Tε
L(∇wε) we consider a function Vε:Ω×Y→Rdeﬁned
as
(5.6) Vε=ε−1(Tε
L(wε)−Mε
L(wε)).
Then, the deﬁnition of Tε
LandMε
Limplies
∇yVε=1
ε∇yTε
L(wε)=Nε/summationdisplay
n=1DT
xε
nTε
L(∇wε)χΩε
n.
The boundedness of {wε}inW1,p(Ω), together with (5.1) and regularity assumptions
onD, implies that the sequence {∇yVε}is bounded in Lp(Ω×Y). Considering
−/integraldisplay
YVεdy=0 a n d −/integraldisplay
Yyε
c·∇wdy=0 w i t h yε
c=Nε/summationdisplay
n=1DxεnycχΩεn,
whereyc=(y1−1
2,…,y d−1
2)f o ry∈Y, and applying the Poincar´ e inequality to
Vε−yε
c·∇wyields
/bardblVε−yε
c·∇w/bardblLp(Ω×Y)≤C1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∇
yVε−Nε/summationdisplay
n=1DT
xε
n∇wχΩεn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(Ω×Y)≤C2.
Thus, there exist a subsequence (denoted again by {Vε−yε
c·∇w}) and a function
/tildewidew1∈Lp(Ω;W1,p(Y)) such that
(5.7) Vε−yε
c·∇w/arrowrighttophalf/tildewidew1weakly in Lp(Ω;W1,p(Y)).
Forφ∈W1,p(Ω) we have the following relation:
Tε
L(∇φ)(x,y)=ε−1Nε/summationdisplay
n=1D−T
xεn∇yTε
L(φ)(x,y)χΩε
n(x).
Then the convergence in (5.7) and the continuity of Dyield
(5.8)Tε
L(∇wε)=Nε/summationdisplay
n=1D−T
xεn∇yVεχΩεn/arrowrighttophalf∇w+D−T
x∇y/tildewidew1weakly in Lp(Ω×Y).
We show now that /tildewidew1(x,y)i sY-periodic. Then the function w1(x,y)=/tildewidew1(x,D−1
xy)
for a.a.x∈Ω,y∈Yxwill beYx-periodic. For ψ∈C∞
0(Ω×Y/prime)w ec o n s i d e r
/integraldisplay
Ω/integraldisplay
Y/prime/bracketleftbig
Vε(x,y1)−Vε(x,y0)/bracketrightbig
ψ(x,y/prime)dy/primedx=Nε/summationdisplay
n=1(I1,n+I2,n)
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1076 MARIYA PTASHNYK
with
I1,n=/integraldisplay
/tildewideΩε,d
n/integraldisplay
Y/primeTε
L(wε)(x,y0)1
ε/bracketleftbig
ψ(x−εDxεned,y/prime)−ψ(x,y/prime)/bracketrightbig
dy/primedx,
I2,n=1
ε/bracketleftbigg/integraldisplay
/tildewideΛε,d
n,1×Y/primeTε
L(wε)(x,y1)ψ(x,y/prime)dy/primedx−/integraldisplay
/tildewideΛε,d
n,2×Y/primeTε
L(wε)(x,y0)ψ(x,y/prime)dy/primedx/bracketrightbigg
=IU
2,n−IL
2,n,
wherey1,y0,/tildewideΩε,d
n,a n d/tildewideΛε,d
n,l, withl=1,2, are deﬁned in the proof of Theorem 4.1.
Then Lemma 5.1 and the strong convergence of {wε}inLp(Ω), ensured by the
boundedness of {wε}inW1,p(Ω), imply the strong convergence of {Tε
L(wε)}tow
inLp(Ω×Y). The boundedness of {∇yTε
L(wε)}(ensured by the boundedness of
{∇wε}) yields the weak convergence of {Tε
L(wε)}inLp(Ω;W1,p(Y)) to the same w.
Applying the trace theorem in W1,p(Y), we obtain that the trace of Tε
L(wε)o nΩ×Y/prime
converges weakly to winLp(Ω×Y/prime)a sε→0. This together with the regularity of
ψandDgives
lim
ε→0Nε/summationdisplay
n=1I1,n=−/integraldisplay
Ω/integraldisplay
Y/primew(x)Dd(x)·∇xψ(x,y/prime)dy/primedx,
whereDj(x)=(D1j(x),…,D dj(x))T, withj=1,…,d. Next we consider the inte-
grals over the upper cells IU
2,n1and over the lower cells IL
2,n2in neighboring Ωε
n1and
Ωε
n2(in theej-direction, with ej·Dxεn1ed/negationslash=0 ,j=1,…,d), i.e., for 1 ≤n1,n2≤Nε
such that Θ n1,2=(∂Ωε
n1∩∂Ωε
n2)∩{xj=c o n s t}/negationslash=∅,d i m ( Θ n1,2)=d−1, and
xε
n1,j<xεn
2,j, and write
IU
2,n1−IL
2,n2=1
ε/bracketleftbigg/integraldisplay
/tildewideΛε,d
n1,1×Y/primeTε
L(wε)(x,y0)ψdy/primedx−/integraldisplay
/tildewideΛε,d
n2,2×Y/primeTε
L(wε)(x,y0)ψdy/primedx/bracketrightbigg
+/integraldisplay
/tildewideΛε,d
n1,11
ε/bracketleftbigg/integraldisplay
Y/primeTε
L(wε)(x,y1)ψdy/prime−/integraldisplay
Y/primeTε
L(wε)(x,y0)ψdy/prime/bracketrightbigg
dx=I1,2
2,n+I1
2,n.
The second integral I1
2,ncan be rewritten as
I1
2,n=1
ε/integraldisplay
/tildewideΛε,d
n1,1×Y∂ydTε
L(wε)(x,y)ψ(x,y/prime)dydx=/integraldisplay
/tildewideΛε,d
n1,1×YDd(xε
n1)·Tε
L(∇wε)ψdydx.
Using the boundedness of {∇wε}inLp(Ω) and/summationtextNε
n1=1|/tildewideΛε,d
n1,1|≤Cε1−r, we conclude
that/summationtextNε
n=1I1
2,n→0a sε→0a n dr<1.
InI1,2
2,nwedistinguish between variationsin the Dxε
nej-direction, for1 ≤j≤d−1,
and in the Dxεned-direction. For an arbitrary ﬁxed xε
n1,2∈Θn1,2we deﬁne ˆDl
xεn1,2=
(D1(xε
n1,2),…,D d−1(xε
n1,2),Dd(xε
nl)), withl=1,2, and introduce
ˆΛε
nl=I n t/parenleftBigg/uniondisplay
ξ∈/tildewideΞε,l
n1,2εˆDl
xεn1,2(
Y+ξ)/parenrightBigg
forl=1,2,
where
/tildewideΞε,1
n1,2=/braceleftBig
ξ∈Zd:εˆD1
xε
n1,2(
Y+ξ+ed)∩Θn1,2/negationslash=∅andεˆD1
xε
n1,2(Y+ξ)⊂Ωε
n1/bracerightBig
,
/tildewideΞε,2
n1,2=/braceleftBig
ξ∈Zd:εˆD2
xεn1,2(
Y+ξ−ed)∩Θn1,2/negationslash=∅andεˆD2
xεn1,2(Y+ξ)⊂Ωε
n2/bracerightBig
.
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LOCALLY PERIODIC UNFOLDING METHOD 1077
Then each of the integrals in I1,2
2,nis rewritten as
1
ε/integraldisplay
/tildewideΛε,d
nl,l/integraldisplay
Y/primeTε
L(wε)(x,y0)ψdy/primedx=1
ε/integraldisplay
ˆΛεnl/integraldisplay
Y/primewε(εˆDl
xεn1,2([xl
D,n/ε]+y0))ψdy/primedx
+1
ε/bracketleftbigg/integraldisplay
/tildewideΛε,d
nl,l/integraldisplay
Y/primeTε
L(wε)(x,y0)ψdy/primedx−/integraldisplay
ˆΛεnl/integraldisplay
Y/primewε(εˆDl
xεn1,2([xl
D,n/ε]+y0))ψdy/primedx/bracketrightbigg
=J1
l,n+J2
l,n,
wherexl
D,n=(ˆDl
xεn1,2)−1xandl=1,2. Using the deﬁnition of ˆΛε
nlforl=1,2
and the fact that |/tildewideΞε,l
n1,2||ˆDl
xεn1,2|=Ij|Dd(xε
nl)·ej|, withDd(xε
nl)·ej/negationslash= 0 and some
Ij>0,j=1,…,d, and denoting |/tildewideΞε,1
n1,2|=Iε
n1,2yields
J1
1,n−J1
2,n=εdIε
n1,2/summationdisplay
i=1/integraldisplay
Y/integraldisplay
Y/prime1
ε/bracketleftBig
wε/parenleftbig
εˆD1
xε
n1,2(ξ1
i+y0)/parenrightbig
ψ(ε˜yi
n1,ξ,y/prime)
−wε/parenleftbig
εˆD2
xε
n1,2(ξ2
i+y0)/parenrightbig
ψ(ε˜yi
n2,ξ,y/prime)/bracketrightBig/vextendsingle/vextendsingleˆD1
xε
n1,2/vextendsingle/vextendsingledy/primed˜y
−εr−1/summationdisplay
ξ∈/tildewideΞε,2
n1,2/integraldisplay
ε(Y+ξ)/integraldisplay
Y/primewε/parenleftbig
εˆD2
xεn1,2/parenleftbig
[˜x/ε]+y0/parenrightbig/parenrightbig
ψdy/prime1
εr/bracketleftBig
d(ˆD2
xεn1,2˜x)−d(ˆD1
xεn1,2˜x)/bracketrightBig
,
where ˜yi
nl,ξ=ˆDl
xεn1,2(˜y+ξl
i)f o rl=1,2. The ﬁrst integral in the last equality can be
estimated by
Cεrd+(1−r)/bardblwε/bardblW1,p(Ω)/bardblψ/bardblC1
0(Ω×Y/prime).
Inthesecondintegralwehave adiscretederivativeinthe ej-direction, ej·Dd(xε
n1)/negationslash=0 ,
j=1,…,d, ofanintegraloveranevolvingdomainwith thevelocityvector Dd. Then,
using the fact that |Nε|≤Cε−drandxε
n1,j<xεn
2,j, together with the regularity of D
and the deﬁnition of ˆDl
xε
n1,2,w h e r el=1,2, yields
Nε/summationdisplay
n=1/parenleftbig
J1
1,n−J1
2,n/parenrightbig
→−/integraldisplay
Ω/integraldisplay
Y/primew(x)ψ(x,y/prime)divDd(x)dy/primedxasε→0.
ForJ2
1,n−J2
2,n, using the deﬁnition of /tildewideΛε,d
nl,landˆΛε
nl, withl=1,2, the regularity of D
andψ, the boundedness of {wε}inW1,p(Ω), along with the properties of the covering
of Ω by{Ωε
n}Nε
n=1,w eo b t a i n
Nε/summationdisplay
n=1|J2
1,n−J2
2,n|≤Cε1−rd−1/summationdisplay
j=1/bardbldivDj/bardblL∞(Ω)/bardblwε/bardblW1,p(Ω)/bardblψ/bardblC1
0(Ω×Y/prime)→0
asε→0f o rr∈[0,1). Combining the obtained results, we conclude that
Nε/summationdisplay
n=1(I1,n+I2,n)→−/integraldisplay
Ω×Y/prime/bracketleftbig
w(x)Dd(x)·∇xψ(x,y/prime)+w(x)ψ(x,y/prime)divDd(x)/bracketrightbig
dy/primedx
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1078 MARIYA PTASHNYK
asε→0. The deﬁnition of yc
ε·∇wimplies
(yε
c·∇w(x))(y/prime,1)−(yε
c·∇w(x))(y/prime,0) =Nε/summationdisplay
n=1Dd(xε
n)·∇w(x)χΩεn(x)
fory/prime∈Y/primeandx∈Ω. Taking the limit as ε→0 yields
lim
ε→0/integraldisplay
Ω×Y/prime/bracketleftbig
(yε
c·∇w)(y1)−(yε
c·∇w)(y0)/bracketrightbig
ψdy/primedx=/integraldisplay
Ω×Y/primeDd(x)·∇wψdy/primedx
=−/integraldisplay
Ω×Y/primew(x)/bracketleftbig
Dd(x)·∇xψ(x,y/prime)+divDd(x)ψ(x,y/prime)/bracketrightbig
dy/primedx.
Then, using the convergence of Vε−yc
ε·∇wto/tildewidew1inLp(Ω;W1,p(Y)), we obtain
/integraldisplay
Ω/integraldisplay
Y/prime[/tildewidew1(x,(y/prime,1))−/tildewidew1(x,(y/prime,0))]ψ(x,y/prime)dy/primedx= lim
ε→0/integraldisplay
Ω/integraldisplay
Y/prime/bracketleftbig
Vε(x,(y/prime,1))
−(yε
c·∇w)(x,(y/prime,1))−Vε(x,(y/prime,0))+(yε
c·∇w)(x,(y/prime,0))/bracketrightbig
ψ(x,y/prime)dy/primedx=0.
Carrying out similar calculations for yjwithj=1,…,d−1 yields the Y-periodicity
of/tildewidew1and, hence, the Yx-periodicity of w1, deﬁned by w1(x,y)=/tildewidew1(x,D−1
xy)f o r
x∈Ωa n dy∈DxY.
6. Micro-macro decomposition: The interpolation operator Qε
L.Similar
to the periodic case [19, 18], in the context of convergence results for the unfolding
method in perforated domains as well as for the derivation of error estimates [28,
29, 30, 31, 44], it is important to consider micro-macro decomposition of a function
inW1,pand to introduce an interpolation operator Qε
L. For any ϕ∈W1,p(Ω) we
consider the splitting ϕ=Qε
L(ϕ)+Rε
L(ϕ)a n ds h o wt h a t Qε
L(ϕ) has a behavior
similar to that of ϕ,w h e r e a s Rε
L(ϕ)i so fo r d e r ε.
We consider a continuous extension operator P:W1,p(Ω)→W1,p(Rd) satisfying
/bardblP(ϕ)/bardblW1,p(Rd)≤C/bardblϕ/bardblW1,p(Ω)for allφ∈W1,p(Ω),
where the constant Cdepends only on pand Ω; see, e.g., [26]. In the following we use
the same notation for a function in W1,p(Ω) and its continuous extension into Rd.
WeconsideraboundedLipschitzdomainΩ 1⊂Rd,suchthatΩ ⊂Ω1,dist(∂Ω,∂Ω1)
≥2εr,a n dΩ 1⊂/uniontextNε,1
n=1
Ωεn,w h e r eΩε
nas in section 2, and identify Nε,1withNε.
We consider Y=I n t (/uniontext
k∈{0,1}d(
Y+k)) and deﬁne
Ωε
Y=I n t/parenleftBiggNε/uniondisplay
n=1
Ωε
n,Y/parenrightBigg
,with Ωε
n,Y=I n t/parenleftBigg/uniondisplay
ξ∈Ξε
n,YεDxεn(
Y+ξ)/parenrightBigg
,
Λε
Y=Ω\Ωε
Y,
where Ξε
n,Y={ξ∈Ξε
n:εDxε
n(Y+ξ)⊂(Ωε
n∩Ω1)}.
In order to deﬁne an interpolation between two neighboring Ωε
nand Ωεmwe in-
troduceY−=I n t (/uniontext
k∈{0,1}d(
Y−k)).
For 1≤n≤Nεandm∈Zn={1≤m≤Nε:∂Ωε
n∩∂Ωε
m/negationslash=∅}we shall consider
unit cells near the corresponding neighboring parts of the boundaries ∂ˆΩε
nand∂ˆΩε
m,
respectively. For ξn∈¯Ξε
n,w h e r e¯Ξε
n={ξ∈ˆΞε
n:εDxεn(
Y+ξ)∩∂ˆΩε
n/negationslash=∅},w ec o n s i d e r
/tildewideΞε
n,m=/braceleftbig
ξm∈¯Ξε
m:εDxεn(Y+ξn)∩εDxεm(Y−+ξm)/negationslash=∅/bracerightbig
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LOCALLY PERIODIC UNFOLDING METHOD 1079
3
1
−2,3
−Ω
ε
n,Ωnε
mΩε
mmε
ξmΩmεΩξn
εΩξn
ξm
mDεxεnxD
εmxDεnxD
Fig. 4.Schematic diagram of the covering of ΩbyΩε
n,o fDxεnYandDxεmY−,a n do ft h e
interpolation points ξnandξmforQε
LandQ∗,ε
L.
and
ˆKn={k∈{0,1}d:ξn+k∈¯Ξε
n},ˆK−
m={k∈{0,1}d:ξm−k∈¯Ξε
m}.
One of the important parts in the deﬁnition of Qε
Lis to deﬁne an interpolation
between neighboring Ωε
nand Ωεm. For two neighboring Ωεnand Ωεmwe consider trian-
gular interpolations between such vertices of εDxεn(Y+ξn)a n dεDxεm(Y+ξm)t h a t
are lying on ∂Ωε
n,Yand∂Ωε
m,Y, respectively.
Definition 6.1. The operator Qε
L:Lp(Ω)→W1,∞(Ω),f o rp∈[1,+∞],i s
deﬁned by
(6.1) Qε
L(ϕ)(εξ)=−/integraldisplay
Yϕ(Dxεn(εξ+εy))dyforξ∈Ξε
nand1≤n≤Nε,
and forx∈Ωε
n,Y∩Ωwe deﬁne Qε
L(ϕ)(x)as theQ1-interpolant of Qε
L(ϕ)(εξ)at the
vertices of ε[D−1
xεnx/ε]Y+εY,w h e r e1≤n≤Nε.
Forx∈Λε
Ywe deﬁne Qε
L(ϕ)(x)as a triangular Q1-interpolant of the values of
Qε
L(ϕ)(εξ)atξn+knandξmsuch that ξn∈¯Ξε
n,ξm∈/tildewideΞε
n,mform∈Zn,a n dkn∈ˆKn,
where1≤n≤NεandΩε
n∩Ω/negationslash=∅orΩε
m∩Ω/negationslash=∅.
The vertices of εDxεn(Y+ξn+kn)a n dεDxεm(Y+ξm)f o rξn∈¯Ξε
n,ξm∈/tildewideΞε
n,m,
andkn∈ˆKn, in the deﬁnition of Qε
L,b e l o n gt o ∂Ωε
n,Yand∂Ωε
m,Y; see Figure 4.
ForQε
L(ϕ)a n dRε
L(ϕ)=ϕ−Qε
L(ϕ) we have the following estimates.
Lemma 6.2. For every ϕ∈W1,p(Ω),w h e r ep∈[1,+∞), we have
(6.2)/bardblQε
L(ϕ)/bardblLp(Ω)≤C/bardblϕ/bardblLp(Ω), /bardblRε
L(ϕ)/bardblLp(Ω)≤Cε/bardbl∇ϕ/bardblLp(Ω),
/bardbl∇Qε
L(ϕ)/bardblLp(Ω)+/bardbl∇Rε
L(ϕ)/bardblLp(Ω)≤C/bardbl∇ϕ/bardblLp(Ω),
where the constant Cis independent of εand depends only on Y,D,a n dd= dim(Ω) .
Proof. Similar to the periodic case [19], we use the fact that the space of Q1-
interpolantsis a ﬁnite-dimensionalspace ofdimension2dand all normsareequivalent.
Then, for ξ∈Ξε
n,Y,w h e r en=1,…,N ε,w eo b t a i n
/bardblQε
L(ϕ)/bardblp
Lp(εDxεn(ξ+Y))≤C1εd/summationdisplay
k∈{0,1}d/vextendsingle/vextendsingleQε
L(ϕ)(εξ+εk)/vextendsingle/vextendsinglep.(6.3)
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1080 MARIYA PTASHNYK
Forξn∈¯Ξε
nand triangular elements ωε
ξn,mbetween Ωε
n,Yand Ωε
m,Y,m∈Zn,i th o l d s
that
/bardblQε
L(ϕ)/bardblp
Lp(ωε
ξn,m)≤C2εd/summationdisplay
k∈ˆKn,m∈Zn/summationdisplay
ξm∈/tildewideΞε
n,m/bracketleftBig/vextendsingle/vextendsingleQε
L(ϕ)(εξn+εk)/vextendsingle/vextendsinglep+/vextendsingle/vextendsingleQε
L(ϕ)(εξm)/vextendsingle/vextendsinglep/bracketrightBig
,
where|Zn|≤2dand|/tildewideΞε
n,m|≤22dfor every n=1,…,N ε.T h u s ,f o rΛε
Yit holds that
/bardblQε
L(ϕ)/bardblp
Lp(Λε
Y)
≤C3εdNε/summationdisplay
n=1/summationdisplay
ξn∈¯Ξεn,k∈ˆKn/summationdisplay
m∈Zn,ξm∈/tildewideΞεn,m/bracketleftBig/vextendsingle/vextendsingleQε
L(ϕ)(εξn+εk)/vextendsingle/vextendsinglep+/vextendsingle/vextendsingleQε
L(ϕ)(εξm)/vextendsingle/vextendsinglep/bracketrightBig
.(6.4)
From the deﬁnition of Qε
Lit follows that
|Qε
L(ϕ)(εξ)|p≤−/integraldisplay
Y|ϕ(εDxεn(ξ+y))|pdy=1
εd|DxεnY|/integraldisplay
εDxεn(ξ+Y)|ϕ(x)|pdx
forξ∈Ξε
nandn=1,…,N ε. Then, using (6.3) and (6.4) implies
/bardblQε
L(ϕ)/bardblp
Lp(εDxεn(ξ+Y))≤C4/summationdisplay
k∈{0,1}d/integraldisplay
εDxεn(ξ+k+Y)|ϕ(x)|pdx(6.5)
forξ∈Ξε
n,Yandn=1,…,N ε, and in Λε
Ywe have
/bardblQε
L(ϕ)/bardblp
Lp(Λε
Y)≤C5Nε/summationdisplay
n=1/summationdisplay
m∈Zn/summationdisplay
j=n,m/summationdisplay
ξ∈¯Ξε
j/integraldisplay
εDxε
j(ξ+Y)|ϕ(x)|pdx. (6.6)
Summing up in (6.5) over ξ∈Ξε
n,Yandn=1,…,N ε, and adding (6.6), we obtain
t h ee s t i m a t ef o rt h e Lp-norm of Qε
L(ϕ), as stated in the lemma.
From the deﬁnition of Q1-interpolants we obtain that for ξ∈Ξε
n,Y
(6.7)/bardbl∇Qε
L(φ)/bardblp
Lp(εDxεn(ξ+Y))≤Cεd−p/summationdisplay
k∈{0,1}d|Qε
L(φ)(εξ+εk)−Qε
L(φ)(εξ)|p.
For the triangular regions ωε
ξn,mbetween neighboring Ωεn,Yand Ωε
m,Ywe have
/bardbl∇Qε
L(φ)/bardblp
Lp(ωε
ξn,m)≤Cεd−p/summationdisplay
m∈Zn
ξm∈/tildewideΞε
n,m/summationdisplay
kn∈ˆKn,km∈ˆK−
m/bracketleftBig
|Qε
L(φ)(ε(ξn+kn))−Qε
L(φ)(εξn)|p
+|Qε
L(φ)(ε(ξn+kn))−Qε
L(φ)(ε(ξm−km))|p+|Qε
L(φ)(ε(ξm−km))−Qε
L(φ)(εξm)|p/bracketrightBig
.
Forφ∈W1,p(DxεnY)( a n dW1,p(DxεnY),W1,p(DxεnY−)), using the regularity of D
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LOCALLY PERIODIC UNFOLDING METHOD 1081
and the Poincar´ e inequality, we obtain
/vextenddouble/vextenddouble/vextenddouble/vextenddoubleφ−−/integraldisplay
DxεnYφdy/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(DxεnY)≤C/bardbl∇yφ/bardblLp(DxεnY),
/vextenddouble/vextenddouble/vextenddouble/vextenddoubleφ−−/integraldisplay
DxεnYφdy/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(DxεnY)≤C/bardbl∇yφ/bardblLp(DxεnY), (6.8)
/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/integraldisplay
DxεnYφdy−−/integraldisplay
DxεnYφdy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/integraldisplay
Dxεn(Y+k)φdy−−/integraldisplay
DxεnYφdy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
≤C/bardbl∇yφ/bardblp
Lp(DxεnY),
/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/integraldisplay
DxεnYφdy−−/integraldisplay
DxεnY−φdy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/integraldisplay
Dxεn(Y−k)φdy−−/integraldisplay
DxεnY−φdy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
≤C/bardbl∇yφ/bardblp
Lp(DxεnY−),
where 1≤n≤Nε,k∈{0,1}d, and the constant Cdepends on Dand is independent
ofεandn. Using a scaling argument, we obtain for every ξ∈Ξε
n
/vextenddouble/vextenddouble/vextenddouble/vextenddoubleφ−−/integraldisplay
εDxεn(ξ+Y)φdx/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(εDxεn(ξ+Y))≤Cε/bardbl∇φ/bardblLp(εDxεn(ξ+Y)). (6.9)
Hence, for ξ∈Ξε
n,Yandk∈{0,1}das well as for ξj∈¯Ξε
j, withj=n,mandkn∈ˆKn,
km∈ˆK−
m,u s i n gas c a l i n ga r g u m e n ti n( 6 . 8 ) ,w eh a v e
|Qε
L(ϕ)(εξ+εk)−Qε
L(ϕ)(εξ)|p=/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/integraldisplay
Y+kϕ(εDxε
n(ξ+y))dy−−/integraldisplay
Yϕ(εDxε
n(ξ+y))dy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
≤Cεp−d/bardbl∇ϕ/bardblp
Lp(εDxεn(ξ+Y)),
|Qε
L(ϕ)(εξn+εkn)−Qε
L(ϕ)(εξn)|p≤Cεp−d/bardbl∇ϕ/bardblp
Lp(εDxεn(ξn+Y)),
|Qε
L(ϕ)(εξm−εkm)−Qε
L(ϕ)(εξm)|p≤Cεp−d/bardbl∇ϕ/bardblp
Lp(εDxεm(ξm+Y−)),(6.10)
whereCdepends on Dand is independent of ε,n,a n dm.
Forξn∈¯Ξε
n,ξm∈/tildewideΞε
n,mandkn∈ˆKn,km∈ˆK−
m,u s i n gt h a t εDxεm(ξm+Y−)∩
εDxεn(ξn+Y)/negationslash=∅and applying the inequalities (6.8) with a connected domain
/tildewideYξn=/uniondisplay
m∈Zn,ξm∈/tildewideΞεn,m/uniondisplay
k∈{0,1}dDxεm(ξm+Y−+k)∪Dxεn(ξn+Y−k),
instead of YandY−, together with a scaling argument, yields
|Qε
L(ϕ)(εξn+εkn)−Qε
L(ϕ)(εξm−εkm)|p≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/integraldisplay
Dxεn(ξn+Y+kn)ϕ(εy)dy−−/integraldisplay
/tildewideYξnϕ(εy)dy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/integraldisplay
Dxεm(ξm+Y−km)ϕ(εy)dy−−/integraldisplay
/tildewideYξnϕ(εy)dy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
≤Cεp−d/bardbl∇ϕ/bardblp
Lp(ε/tildewideYξn),(6.11)
whereCdepends on Dand is independent of ε,n,a n dm. Thus, using (6.11) and
the last two estimates in (6.10), we obtain
/bardbl∇Qε
L(ϕ)/bardblp
Lp(Λε
Y)≤C1Nε/summationdisplay
n=1/summationdisplay
ξn∈¯Ξε
n
m∈Zn/summationdisplay
ξm∈/tildewideΞεn,m/bardbl∇ϕ/bardblp
Lp(ε/tildewideYξn)≤C2/bardbl∇ϕ/bardblp
Lp(Ω).(6.12)
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1082 MARIYA PTASHNYK
Applying (6.10) in (6.7), summing up over ξ∈Ξε
n,Yandn=1,…,N ε,a n dc o m –
bining with the estimate for /bardbl∇Qε
L(ϕ)/bardblLp(Λε
Y)in (6.12), we obtain the estimate for
/bardbl∇Qε
L(ϕ)/bardblLp(Ω)in terms of /bardbl∇ϕ/bardblLp(Ω), as stated in the lemma.
To show the estimates for Rε
L(ϕ)ﬁ r s tw ec o n s i d e r
/bardblϕ−Qε
L(ϕ)/bardblLp(εDxεn(ξ+Y))≤/bardblϕ−Qε
L(ϕ)(εξ)/bardblLp(εDxεn(ξ+Y))
+/bardblQε
L(ϕ)(εξ)−Qε
L(ϕ)/bardblLp(εDxεn(ξ+Y))
forξ∈Ξε
n,Y. Using the deﬁnition of Qε
Land (6.9), we obtain
/bardblϕ−Qε
L(ϕ)(εξ)/bardblLp(εDxεn(ξ+Y))≤Cε/bardbl∇ϕ/bardblLp(εDxεn(ξ+Y))forξ∈Ξε
n,Y.
The deﬁnition of Qε
L(ϕ) and the properties of Q1-interpolants along with (6.10) imply
/bardblQε
L(ϕ)−Qε
L(ϕ)(εξ)/bardblLp(εDxεn(ξ+Y))≤Cε/bardbl∇ϕ/bardblLp(εDxεn(ξ+Y))forξ∈Ξε
n,Y.
For triangular elements ωε
ξn,m⊂Λε
Ywithξn∈¯Ξε
nandξm∈/tildewideΞε
n,mwe have ωε
ξn,m⊂
ε/tildewideYξn. Then, the second inequality in (6.8) with /tildewideYξnand a scaling argument yield
/bardblϕ−Qε
L(ϕ)(εξn)/bardblLp(ωε
ξn,m)≤/bardblϕ−Qε
L(ϕ)(εξn)/bardblLp(ε/tildewideYξn)≤Cε/bardbl∇ϕ/bardblLp(ε/tildewideYξn),
whereas (6.10) and (6.11) together with the properties of Q1-interpolants ensure
/bardblQε
L(ϕ)−Qε
L(ϕ)(εξn)/bardblLp(ωε
ξn,m)≤Cε/bardbl∇ϕ/bardblLp(ε/tildewideYξn).
Thus, combining the estimates from above, we obtain
/bardblRε
L(ϕ)/bardblLp(Ω)≤Nε/summationdisplay
n=1/bardblϕ−Qε
L(ϕ)/bardblLp(Ωε
n)≤Nε/summationdisplay
n=1/summationdisplay
ξ∈Ξε
n,Y/bardblϕ−Qε
L(ϕ)/bardblLp(εDxεn(ξ+Y))
+Nε/summationdisplay
n=1/summationdisplay
ξn∈¯Ξεn,m∈Zn/summationdisplay
ξm∈/tildewideΞε
n,m/bardblϕ−Qε
L(ϕ)/bardblLp(ωε
ξn,m)≤Cε/bardbl∇ϕ/bardblLp(Ω).
Then the estimate for ∇Qε
L(ϕ) and the deﬁnition of Rε
L(ϕ) yield the estimate for
∇Rε
L(ϕ).
To show convergence results for sequences obtained by applying the l-p unfolding
operator to sequences of functions deﬁned on l-p perforated domains, we have tointroduce the interpolation operator Q
∗,ε
Lfor functions in Lp(Ω∗
ε). We deﬁne
ˆΩ∗
ε=I n t/parenleftBiggNε/uniondisplay
n=1ˆΩ∗,ε
n/parenrightBigg
,Λ∗
ε=Ω∗ε\ˆΩ∗
ε,whereˆΩ∗,ε
n=/uniondisplay
ξ∈ˆΞεnεDxε
n(
Y∗+ξ),
and
Ω∗
ε,Y=I n t/parenleftBiggNε/uniondisplay
n=1
Ω∗,ε
n,Y/parenrightBigg
,Λ∗
ε,Y=Ω∗
ε\Ω∗
ε,Y,where Ω∗,ε
n,Y=I n t/parenleftBigg/uniondisplay
ξ∈Ξε
n,YεDxεn(
Y∗+ξ)/parenrightBigg
,
with Ω instead of Ω 1in the deﬁnition of Ξε
n,Y,a sw e l la s /tildewideΩ∗
ε=Ω∗ε∩/tildewideΩε,w h e r e/tildewideΩεis
deﬁned as
(6.13) /tildewideΩε=/braceleftBig
x∈Ω: d i s t ( x,∂Ω)>4εmax
x∈∂Ωdiam(D(x)Y)/bracerightBig
.
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LOCALLY PERIODIC UNFOLDING METHOD 1083
We also consider Y∗=I n t (/uniontext
k∈{0,1}d(
Y∗+k)) andY∗,−=I n t (/uniontext
k∈{0,1}d(
Y∗−k)).
Similar to Qε
L, in the deﬁnition of the interpolation operator Q∗,ε
Lwe shall dis-
tinguish between Ωε
Yand ΛεY∩/tildewideΩε.F o rx∈Ωε
Ywe can consider Q1-interpolation
between vertices of the corresponding unit cells, whereas for x∈Λε
Y∩/tildewideΩεwe consider
triangular Q1-interpolation between vertices of unit cells in two neighboring Ωε
nand
Ωε
m. This approach ensures that Q∗,ε
L(φ) is continuous in /tildewideΩε.
Definition 6.3. The operator Q∗,ε
L:Lp(Ω∗
ε)→W1,∞(/tildewideΩε),f o rp∈[1,+∞],i s
deﬁned by
(6.14) Q∗,ε
L(φ)(εξ)=−/integraldisplay
Y∗φ(Dxεn(εξ+εy))dyforξ∈ˆΞε
nandn=1,…,N ε,
and forx∈Ωε
n,Y∩/tildewideΩεwe deﬁne Q∗,ε
L(φ)(x)as theQ1-interpolant of the values of
Q∗,ε
L(φ)(εξ)at vertices of ε[D−1
xεnx/ε]Y+εY,w h e r e1≤n≤Nε.
Forx∈Λε
Y∩/tildewideΩεwe deﬁne Q∗,ε
L(φ)(x)as a triangular Q1-interpolant of the values
ofQ∗,ε
L(φ)(εξ)atξn+knandξmsuch that ξn∈¯Ξ∗,ε
n={ξ∈¯Ξε
n:εDxε
n(Y+ξ)∩/tildewideΩε/2/negationslash=
∅},ξm∈/tildewideΞε
n,mform∈Zn,a n dkn∈ˆKn,w h e r e1≤n≤Nε;s e eF i g u r e 4.
In a similar way as for Qε
L(φ)a n dRε
L(φ) we obtain estimates for Q∗,ε
L(φ)a n d
R∗,ε
L(φ)=φ−Q∗,ε
L(φ).
Lemma 6.4. For every φ∈W1,p(Ω∗
ε),w h e r ep∈[1,+∞), we have
/bardblQ∗,ε
L(φ)/bardblLp(/tildewideΩε)≤C/bardblφ/bardblLp(Ω∗ε), /bardbl∇Q∗,ε
L(φ)/bardblLp(/tildewideΩε)≤C/bardbl∇φ/bardblLp(Ω∗ε),
/bardblR∗,ε
L(φ)/bardblLp(/tildewideΩ∗
ε)≤Cε/bardbl∇φ/bardblLp(Ω∗
ε), /bardbl∇R∗,ε
L(φ)/bardblLp(/tildewideΩ∗
ε)≤C/bardbl∇φ/bardblLp(Ω∗
ε),
where the constant Cis independent of ε.
Proof. The proof for the ﬁrst estimate follows along the same lines as the proof of
the corresponding estimate in Lemma 6.2. To show the estimates for ∇Q∗,ε
L(φ)a n d
R∗,ε
L(φ) we have to estimate the diﬀerences Q∗,ε
L(φ)(εξ)−Q∗,ε
L(φ)(εξ+k)f o rξ∈Ξε
n,Y
andk∈{0,1}d,a n dQ∗,ε
L(φ)(εξn+εkn)−Q∗,ε
L(φ)(εξm−εkm)f o rξn∈¯Ξ∗,ε
nand
ξm∈/tildewideΞε
n,m, withm∈Zn,a n dkn∈ˆKn,km∈ˆK−
m,w h e r e1 ≤n≤Nε.
As in the proof of Lemma 6.2, by considering the estimate (6.7), applying the
Poincar´e inequality, and using the estimates similar to (6.8) and (6.10), with Y∗and
Y∗instead of YandY,w eo b t a i n
(6.15)/vextendsingle/vextendsingleQ∗,ε
L(φ)(εξ)−Q∗,ε
L(φ)(εξ+k)/vextendsingle/vextendsinglep≤Cεp−d/bardbl∇φ/bardblp
Lp(εDxεn(Y∗+ξ)),
/bardbl∇Q∗,ε
L(φ)/bardblLp(εDxεn(Y+ξ))≤C/bardbl∇φ/bardblLp(εDxεn(Y∗+ξ)),
/bardblφ−Q∗,ε
L(φ)/bardblLp(εDxεn(Y∗+ξ))≤/bardblφ−Q∗,ε
L(φ)(εξ)/bardblLp(εDxεn(Y∗+ξ))
+/bardblQ∗,ε
L(φ)−Q∗,ε
L(φ)(εξ)/bardblLp(εDxεn(Y+ξ))≤Cε/bardbl∇φ/bardblLp(εDxεn(Y∗+ξ))
forξ∈Ξε
n,Yandn=1,…,N ε.F o rξn∈¯Ξ∗,ε
nandξm∈/tildewideΞε
n,m, withm∈Zn,w e
consider εDxε
j(Y0+ξ)f o rs u c h εDxε
j(Y+ξ), withξ∈ˆΞε
j, that have possible nonempty
intersections with a triangular element ωε
ξn,mbetween neighboring Ω∗,ε
n,Yand Ω∗,ε
m,Y,
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1084 MARIYA PTASHNYK
i.e.,
ˆY0
ξn=/uniondisplay
k+
n∈/tildewideKn
k∈/tildewideK−
n/uniondisplay
l∈Zn,ξl∈/tildewideΞε,+
l,n
kl∈/tildewideKlDxεn(
Y0+ξn+k+
n−k)∪Dxε
l(
Y0+ξl+kl),
ˆY0,−
ξn=/uniondisplay
m∈Zn,ξm∈/tildewideΞε
n,m
k−
m∈/tildewideK−
m,k∈/tildewideKm/uniondisplay
l∈Zm,ξl∈/tildewideΞε,+
l,m
kl∈/tildewideKlDxεm(
Y0+ξm−k−
m+k)∪Dxε
l(
Y0+ξl+kl),
ˆY0,+
ξn=/uniondisplay
m∈Zn,ξm∈/tildewideΞεn,m/uniondisplay
s∈Zm,ξs∈/tildewideΞεm,s/uniondisplay
k∈/tildewideK−
sDxεs(
Y0+ξs−k),
where/tildewideK−
n={k∈{0,1}d:ξn−k∈ˆΞε
n},/tildewideKm={k∈{0,1}d:ξm+k∈ˆΞε
m},a n d
/tildewideΞε,+
l,n=/braceleftbig
ξl∈¯Ξε
l:εDxε
l(Y+ξl)∩εDxεn(Y−+ξn)/negationslash=∅/bracerightbig
, assemble a set of such cells
εDxεn(Y+ξ)a n dεDxεm(Y+ξ) that have possible nonempty intersections with ωε
ξn,m,
i.e.,
ˆYξn=/uniondisplay
m∈Zn,ξm∈/tildewideΞεn,m/uniondisplay
k∈{0,1}dDxε
m(
Y−+ξm+k)∪Dxε
n(
Y+ξn−k),
and deﬁne /tildewideY∗
ξn=I n t (ˆYξn\(ˆY0
ξn∪ˆY0,−
ξn∪ˆY0,+
ξn)). We have that /tildewideY∗
ξnis connected and
ε/tildewideY∗
ξn⊂Ω∗
εfor allξn∈¯Ξ∗,ε
nandn=1,…,N ε. Applying the Poincar´ e inequality in
/tildewideY∗
ξnand using the regularity of Dyields
(6.16)/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/integraldisplay
Dxεn(Y∗+ξn+kn)φ(y)dy−−/integraldisplay
/tildewideY∗
ξnφ(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
≤C/integraldisplay
/tildewideY∗
ξn|∇yφ(y)|pdy,
/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/integraldisplay
Dxεm(Y∗+ξm−km)φ(y)dy−−/integraldisplay
/tildewideY∗
ξnφ(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
≤C/integraldisplay
/tildewideY∗
ξn|∇yφ(y)|pdy,
/vextenddouble/vextenddouble/vextenddouble/vextenddoubleφ−−/integraldisplay
Dxεn(Y∗+ξn)φ(y)dy/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lp(/tildewideY∗
ξn)≤C/bardbl∇yφ/bardblLp(/tildewideY∗
ξn)
forξn∈¯Ξ∗,ε
n,ξm∈/tildewideΞε
n,m, withm∈Zn,a n dkn∈ˆKn,km∈ˆK−
m, where the constant
Cdepends on Dand is independent of ε,n,a n dm. Then, using a scaling argument
in (6.16) implies
/vextendsingle/vextendsingleQ∗,ε
L(φ)(εξn+εkn)−Q∗,ε
L(φ)(εξn)/vextendsingle/vextendsinglep+/vextendsingle/vextendsingleQ∗,ε
L(φ)(εξm−εkm)−Q∗,ε
L(φ)(εξm)/vextendsingle/vextendsinglep
+/vextendsingle/vextendsingleQ∗,ε
L(φ)(εξn+εkn)−Q∗,ε
L(φ)(εξm−εkm)/vextendsingle/vextendsinglep≤Cεp−d/bardbl∇φ/bardblp
Lp(ε/tildewideY∗
ξn)(6.17)
forξn∈¯Ξ∗,ε
n,ξm∈/tildewideΞε
n,m, withm∈Zn,a n dkn∈ˆKn,km∈ˆK−
m. Hence, taking into
account that |Zn|≤2dand|/tildewideΞε
n,m|≤22d,w eo b t a i n
(6.18) /bardbl∇Q∗,ε
L(φ)/bardblp
Lp(Λε
Y∩/tildewideΩε)≤C1Nε/summationdisplay
n=1/summationdisplay
ξn∈¯Ξ∗,ε
n/bardbl∇φ/bardblp
Lp(ε/tildewideY∗
ξn)≤C2/bardbl∇φ/bardblp
Lp(Ω∗ε).
Applying a scaling argument in (6.16) and using the properties of Q1-interpolants and
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LOCALLY PERIODIC UNFOLDING METHOD 1085
the estimate (6.17) yields
(6.19)/bardblφ−Q∗,ε
L(φ)/bardblLp(Λ∗
ε,Y∩/tildewideΩε)≤Nε/summationdisplay
n=1/summationdisplay
ξn∈¯Ξ∗,ε
n/bracketleftBig/vextenddouble/vextenddoubleφ−Q∗,ε
L(φ)(εξn)/vextenddouble/vextenddouble
Lp(ε/tildewideY∗
ξn)
+/summationdisplay
m∈Zn,ξm∈/tildewideΞε
n,m/vextenddouble/vextenddoubleQ∗,ε
L(φ)(εξn)−Q∗,ε
L(φ)/vextenddouble/vextenddouble
Lp(ωε
ξn,m)/bracketrightBig
≤Cε/bardbl∇φ/bardblLp(Ω∗
ε).
Summing in (6.15) over Ξε
n,Yand 1≤n≤Nε, adding (6.18) or (6.1 9), respectively,
andusing the deﬁnition of R∗,ε
L(φ), we obtain the estimates stated in the lemma.
7. The l-p unfolding operator in perforated domains: Proofs of con-
vergence results. In this section we prove convergence results for the l-p unfolding
operator in domains with l-p perforations. First, we show some properties of the l-p
unfolding operator in perforated domains.
Lemma 7.1.
(i)T∗,ε
Lis linear and continuous from Lp(Ω∗
ε)toLp(Ω×Y∗),w h e r ep∈[1,+∞),
and
/bardblT∗,ε
L(w)/bardblLp(Ω×Y∗)≤|Y|1/p/bardblw/bardblLp(Ω∗ε).
(ii)Forw∈Lp(Ω),w i t hp∈[1,+∞),T∗,ε
L(w)→wstrongly in Lp(Ω×Y∗).
(iii)Letwε∈Lp(Ω∗
ε),w i t hp∈(1,+∞), such that /bardblwε/bardblLp(Ω∗ε)≤C.I f
T∗,ε
L(wε)/arrowrighttophalfˆwweakly in Lp(Ω×Y∗),
then
/tildewidewε/arrowrighttophalf1
|Y|/integraldisplay
Y∗ˆwdyweakly in Lp(Ω).
(iv)Forw∈Lp(Ω;Cper(Y∗
x))we have T∗,ε
L(Lεw)→w(·,Dx·)inLp(Ω×Y∗),
wherep∈[1,+∞).
(v)Forw∈C(
Ω;Lp
per(Y∗
x))we have T∗,ε
L(Lε
0w)→w(·,Dx·)inLp(Ω×Y∗),
wherep∈[1,+∞).
By/tildewidewwe denote the extension of wby zero from Ω∗
εinto Ω.
Sketch of the proof . The proof of (i) follows directly from the deﬁnition of T∗,ε
L
and by using calculations similar to those in the proof of Lemma 5.1.
Forwk∈C∞
0(Ω) the convergence in (ii) results from the deﬁnition of T∗,ε
L,t h e
properties of the covering of Ω∗
εby Ω∗,ε
n, and the following simple calculations:
lim
ε→0/integraldisplay
Ω×Y∗|T∗,ε
L(wk)|pdydx= lim
ε→0/bracketleftBiggNε/summationdisplay
n=1|ˆΩε
n||Y∗||wk(xε
n)|p+δε/bracketrightBigg
=/integraldisplay
Ω×Y∗|wk(x)|pdydx.
We used the fact that |Λε|→0a sε→0 and, due to the continuity of wk,w eh a v e
δε=Nε/summationdisplay
n=1/summationdisplay
ξ∈ˆΞεn|Y|/integraldisplay
εDxεn(ξ+Y∗)|wk(x)−wk(xε
n)|pdx→0a s ε→0.
The approximation of w∈Lp(Ω) by{wk}⊂C∞
0(Ω) and the estimate for the norm
ofT∗,ε
L(w−wk) in (i) yield the convergence for w∈Lp(Ω).
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1086 MARIYA PTASHNYK
The proof of the convergence in (iii) is similar to the proof of Lemma 5.2 and the
corresponding result for the periodic unfolding operator.
The proof of (iv) follows along the same lines as the proof of the corresponding
result for Tε
Lin Lemma 5.3. In a way similar to that in [46, Lemma 3.4], we obtain
that
lim
ε→0/integraldisplay
Ω∗ε|Lε
0(w)(x)|pdx=/integraldisplay
Ω1
|Yx|/integraldisplay
Y∗x|w(x,y)|pdydx=/integraldisplay
Ω1
|Y|/integraldisplay
Y∗|w(x,Dx˜y)|pd˜ydx,
lim
ε→0/integraldisplay
Λ∗ε|Lε
0(w)(x)|pdx=0.
Then, the last two convergence re sults together with the equality
lim
ε→0/integraldisplay
Ω×Y∗|T∗,ε
L(Lε
0w)|pdydx=|Y|lim
ε→0/bracketleftbigg/integraldisplay
Ω∗ε|Lε
0w|pdx−/integraldisplay
Λ∗ε|Lε
0w|pdx/bracketrightbigg
and the continuity of wwith respect to ximply the convergence result stated in
(v).
Similar to Tε
Lwe have∇yT∗,ε
L(w)=ε/summationtextNε
n=1DT
xεnT∗,ε
L(∇w)χΩεnforw∈W1,p(Ω∗
ε).
Using the deﬁnition and properties of T∗,ε
L, we prove convergenceresults for T∗,ε
L(wε),
εT∗,ε
L(∇wε), andT∗,ε
L(∇wε). We start with the proof of Theorem 4.3. Here the proof
of the weak convergence follows the same steps as for Tε
Lin Theorem 4.1, whereas the
periodicity of the limit-function is shown in a diﬀerent way.
Proof of Theorem 4.3. The boundedness of {T∗,ε
L(wε)}and{∇yT∗,ε
L(wε)},e n –
sured by (4.1) and the regularity of D, imply the weak convergencesin (4.2). To show
the periodicity of wwe consider for φ∈C∞
0(Ω×Y∗)a n dj=1,…,d
/integraldisplay
Ω×Y∗T∗,ε
L(wε)(x,˜y+ej)φd˜ydx=/integraldisplay
Ω×Y∗Nε/summationdisplay
n=1T∗,ε
L(wε)φ(x−εDxε
nej,˜y)χ/tildewideΩε,j
nd˜ydx
+Nε/summationdisplay
n=1/integraldisplay
/tildewideΛε,j
n,1×Y∗T∗,ε
L(wε)(x,˜y+ej)φd˜ydx,
where/tildewideΩε,j
nand/tildewideΛε,j
n,l, withl=1,2, are deﬁned in the proofof Theorem 4.1 in section 5.
Considering the weak convergence of T∗,ε
L(wε)a l o n gw i t h/summationtextNε
n=1|/tildewideΛε,j
n,l|≤Cε1−r,f o r
l=1,2, and taking the limit as ε→0 implies
/integraldisplay
Ω×Y∗w(x,Dx(˜y+ej))φ(x,˜y)d˜ydx=/integraldisplay
Ω×Y∗w(x,Dx˜y)φ(x,˜y)d˜ydx
for allφ∈C∞
0(Ω×Y∗)a n dj=1,…,d. Thus, we obtain that wisYx-periodic.
Similar to the periodic case, we use the micro-macro decomposition of a function
φ∈W1,p(Ω∗
ε), i.e.,φ=Q∗,ε
L(φ)+R∗,ε
L(φ), toshowtheweakconvergenceof T∗,ε
L(∇wε).
Here we use the fact that for {wε}bounded in W1,p(Ω∗
ε), the sequence {Q∗,ε
L(wε)}is
bounded in W1,p(G) for every relatively compact open subset G⊂Ω.
Notice that for wε∈W1,p(Ω∗
ε) the function Q∗,ε
L(wε) is deﬁned on /tildewideΩε.T h u s ,w e
canapply Tε
LtoQ∗,ε
L(wε)andusetheconvergenceresultsforthel-punfoldingoperator
Tε
L(shown in Theorems 4.1 and 4.2) to prove the weak convergence of Tε
L(Q∗,ε
L(wε)∼)
andTε
L([∇Q∗,ε
L(wε)]∼), where∼denotes an extension by zero from /tildewideΩεto Ω.
Lemma 7.2. Let/bardblwε/bardblW1,p(Ω∗
ε)≤C,w h e r ep∈(1,+∞). Then there exist a
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LOCALLY PERIODIC UNFOLDING METHOD 1087
subsequence (denoted again by {wε})a n daf u n c t i o n w∈W1,p(Ω)such that
Tε
L(Q∗,ε
L(wε)∼)→w strongly in Lp
loc(Ω;W1,p(Y)),
Tε
L(Q∗,ε
L(wε)∼)/arrowrighttophalfw weakly in Lp(Ω;W1,p(Y)),
Tε
L([∇Q∗,ε
L(wε)]∼)/arrowrighttophalf∇wweakly in Lp(Ω×Y).
Proof. Similar to the periodic case [18], the estimates for Q∗,ε
Lin Lemma 6.4
ensure that there exists a function w∈W1,p(Ω) such that, up to a subsequence,
Q∗,ε
L(wε)∼→w strongly in Lp
loc(Ω) and weakly in Lp(Ω),
[∇Q∗,ε
L(wε)]∼/arrowrighttophalf∇wweakly in Lp(Ω).
Then, the ﬁrst two convergencesstated in th e lemma follow directly from the estimate
/bardbl∇yTε
L(Q∗,ε
L(wε)∼)/bardblLp(Ω×Y)≤C1ε/bardbl[∇Q∗,ε
L(wε)]∼/bardblLp(Ω)≤Cε, and convergence re-
sults for Tε
Lin Lemmas 5.1 and 5.2 and Theorem 4.1. To prove the ﬁnal convergence
stated in the lemma, we observe that Q∗,ε
L(wε)|Gis uniformly bounded in W1,p(G),
whereG⊂Ω is a relatively compact open set; see Lemma 6.4. Then, by Theorem 4.2
there exists ˆ w1,G∈Lp(G;W1,p
per(Yx)) such that
Tε
L(∇Q∗,ε
L(wε)|G)/arrowrighttophalf∇w+D−T
x∇yˆw1,G(·,Dx·)w e a k l y i n Lp(G×Y).
The deﬁnition of Qε
Limplies that ˆ w1,Gis a polynomial in yof degree less than or
equal to one with respect to each variable y1,…,y d.T h u s ,t h e Yx-periodicity of ˆ w1,G
yields that it is constant with respect to yand
Tε
L([∇Q∗,ε
L(wε)]∼)/arrowrighttophalf∇wweakly in Lp
loc(Ω;Lp(Y)).
Theboundednessof[ ∇Q∗,ε
L(wε)]∼inLp(Ω)impliestheboundednessof Tε
L([∇Q∗,ε
L(wε)]∼)
inLp(Ω×Y), and we obtain the last convergence stated in the lemma.
ForR∗,ε
L(wε)=wε−Q∗,ε
L(wε) we have the following convergence results.
Lemma 7.3. Consider a sequence {wε}⊂W1,p(Ω∗
ε),w i t hp∈(1,+∞), satisfying
/bardbl∇wε/bardblLp(Ω∗ε)≤C. Then, there exist a subsequence (denoted again by {wε})a n da
function w1∈Lp(Ω;W1,p
per(Y∗
x))such that
ε−1T∗,ε
L(R∗,ε
L(wε)∼)/arrowrighttophalfw1(·,Dx·) weakly in Lp(Ω;W1,p(Y∗)),
T∗,ε
L(R∗,ε
L(wε)∼)→0 strongly in Lp(Ω;W1,p(Y∗)),
T∗,ε
L([∇R∗,ε
L(wε)]∼)/arrowrighttophalfD−T
x∇yw1(·,Dx·)weakly in Lp(Ω×Y∗),(7.1)
where∼denotes the extension by zero from /tildewideΩ∗
εtoΩ∗
ε.
Proof. The estimates in Lemma 6.4 imply that ε−1T∗,ε
L(R∗,ε
L(wε)∼) is bounded in
Lp(Ω;W1,p(Y∗)) and there exist /tildewidew1∈Lp(Ω;W1,p(Y∗)) andw1(x,y)=/tildewidew1(x,D−1
xy)
forx∈Ω,y∈Y∗
x,w h e r eY∗
x=D(x)Y∗, such that the convergences in (7.1) are
satisﬁed. To show that w1isYx-periodic, we consider the restriction of ε−1R∗,ε
L(wε)
onG∗
ε, which belongs to W1,p(G∗
ε). HereG∗
ε=G∩Ω∗
ε,a n dG⊂Ω is a relatively
compact open subset of Ω. Using Lemma 6.4 we obtain
/bardblε−1R∗,ε
L(wε)/bardblLp(G∗ε)+ε/bardblε−1∇R∗,ε
L(wε)/bardblLp(G∗ε)≤C.
Applying Theorem 4.3 to ε−1R∗,ε
L(wε)|G∗εyieldsw1|G×Y∗x∈Lp(G;W1,p
per(Y∗
x)). Since
Gcan be chosen arbitrarily we obtain that w1∈Lp(Ω;W1,p
per(Y∗
x)).
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1088 MARIYA PTASHNYK
Combining the convergence results from above, we obtain directly the main con-
vergence theorem for the l-p unfolding operator in l-p perforated domains.
Proof of Theorem 4.4. Similar to the periodic case, the convergence results stated
in Theorem 4.4 follow directly from the fact that wε=Q∗,ε
L(wε)+R∗,ε
L(wε)a n df r o m
the convergence results in Lemmas 7.2 and 7.3.
Remark. InthedeﬁnitionofΩ∗
εweassumedthattherearenoperforationsinlayers
(Ω∗,ε
n\
ˆΩ∗,ε
n)∩/tildewideΩε/2, with/tildewideΩε/2={x∈Ω:d i s t (x,∂Ω)>2εmaxx∈∂Ωdiam(D(x)Y)}
and 1≤n≤Nε. In the proofs of convergenceres ults only local estimates for Q∗,ε
L(wε)
andR∗,ε
L(wε) are used; thus no restrictionson the distribution ofperforations near ∂Ω
are needed. For the macroscopic descriptio n of microscopic processes this assumption
is not restrictive since |/uniontextNε
n=1(Ω∗,ε
n\
ˆΩ∗,ε
n)∩Ω|≤Cε1−r→0a sε→0,r<1. If we
allow perforations in layers between two neighboring ˆΩ∗,ε
nandˆΩ∗,ε
min/tildewideΩε/2, then using
thatY∗=Y\
Y0is connected, the tran sformation matrix Dis Lipschitz continuous,
and dist(/tildewideΩε/2,∂Ω)>0, it is possible to construct an extension of wε∈W1,p(Ω∗
ε)
from (Ω∗,ε
n\
ˆΩ∗,ε
n)∩/tildewideΩε/2to (Ωε
n\
ˆΩεn)∩/tildewideΩε/2such that the W1,p-norm of the extension
is controlled by the W1,p-norm of the original function, uniform in ε, and apply
Lemmas 7.2 and 7.3 and Theorem 4.4 to the sequence of extended functions.
8. Two-scale convergence on oscillating surfaces and the l-p boundary
unfolding operator. To derive macroscopic equations for the microscopic problems
posed on boundaries of l-p microstructures or with nonhomogeneous Neumann con-
ditions on boundaries of l-p microstructures, we have to show convergence properties
for sequences deﬁned on oscillating surfaces . To show the compactness result for l-t-s
convergence on oscillating surfaces (see Theorem 4.5), we ﬁrst prove the convergence
of theLp(Γε)-norm of the l-p approximation of ψ∈C(
Ω;Cper(Yx)).
Lemma 8.1. Forψ∈C(
Ω;Cper(Yx))andp∈[1,+∞), we have that
lim
ε→0ε/integraldisplay
Γε|Lεψ(x)|pdσx=/integraldisplay
Ω1
|Yx|/integraldisplay
Γx|ψ(x,y)|pdσydx.
Proof. The deﬁnition of the l-p approximation Lεimplies
ε/integraldisplay
Γε|Lεψ|pdσx=εNε/summationdisplay
n=1/summationdisplay
ξ∈ˆΞεn/integraldisplay
εΓξ
xn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideψ/parenleftbigg
x,D
−1
xεnx
ε/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideψ/parenleftbigg
x
ε
n,D−1
xεnx
ε/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
dσx
+εNε/summationdisplay
n=1/bracketleftBigg/summationdisplay
ξ∈ˆΞεn/integraldisplay
εΓξ
xn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideψ/parenleftbigg
xε
n,D−1
xε
nx
ε/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
dσx+/summationdisplay
ξ∈/tildewideΞεn/integraldisplay
εΓξ
xn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideψ/parenleftbigg
x,D−1
xε
nx
ε/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglep
χΩεnχΩdσx/bracketrightBigg
=I1+I2+I3,
where/tildewideΞε
n=Ξεn\ˆΞε
nand Γξx
n=Dxεn(ξ+/tildewideΓKxεn). Then, the continuity of ψ,t h e
properties of Ωε
n, and the inequality ||a|p−|b|p|≤p|a−b|(|a|p−1+|b|p−1)i m p l y
I1→0a sε→0. Using the properties of the covering of Ω by {Ωε
n}Nε
n=1,w eo b t a i n
|I3|≤Cε−rdsup
1≤n≤Nεεd|/tildewideΞε
n||Dxε
n/tildewideΓKxεn|≤Cε1−r→0a sε→0f o r 0≤r<1.
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LOCALLY PERIODIC UNFOLDING METHOD 1089
Considering the proper ties of the covering of ˆΩε
nbyDxεn(Y+ξ), where ξ∈ˆΞε
nand
1≤n≤Nε,a n dt h e Y-periodicity of /tildewideψ, the second integral can be rewritten as
I2=Nε/summationdisplay
n=1εd|ˆΞε
n|/integraldisplay
Dxεn/tildewideΓKxεn|/tildewideψ(xε
n,D−1
xεny)|pdσy=Nε/summationdisplay
n=1|ˆΩε
n|
|Yxεn|/integraldisplay
Dxεn/tildewideΓKxεn|ψ(xε
n,y)|pdσy.
Then, the regularity assumptions on ψ,D,a n dK, the deﬁnition of ˆΩε
n,a n dt h e
properties of the covering of Ω by {Ωε
n}Nε
n=1imply the convergence result stated in the
lemma.
Similar to l-t-s convergence and two-scale convergence for sequences deﬁned on
surfaces of periodic microstructures, the convergence of l-p approximations (shown in
Lemma 8.1) and the Riesz representation theorem ensure the compactness result for
sequences {wε}⊂Lp(Γε) withε/bardblwε/bardblp
Lp(Γε)≤C.
Proof of Theorem 4.5. The Banach space C(
Ω;Cper(Yx)) is separable and dense
inLp(Ω;Lp(Γx)). Thus, using the convergence result in Lemma 8.1, the Riesz rep-
resentation theorem, and arguments similar to those in [46, Theorem 3.2], we ob-
tain l-t-s convergence of {wε}⊂Lp(Γε)t ow∈Lp(Ω;Lp
per(Γx)), as stated in the
theorem.
Using the structure of Ω∗,ε
n,Kand the covering properties of Ω∗
ε,Kby{Ω∗,ε
n,K}Nε
n=1,
we can derive the trace inequalities for functions deﬁned on Γε. Applying ﬁrst the
trace inequality in Y∗,ξ
xεn,K=Dxεn(/tildewideY∗
Kxεn+ξ), withξ∈ˆΞε
n, yields
/bardblu/bardblp
Lp(Dxεn(/tildewideΓKxεn+ξ))≤μΓ/bracketleftbigg
/bardblu/bardblp
Lp(Y∗,ξ
xεn,K)+/bardbl∇u/bardblp
Lp(Y∗,ξ
xεn,K)/bracketrightbigg
,
/bardblu/bardblp
Lp(Dxεn(/tildewideΓKxεn+ξ))≤μΓ/bracketleftBigg
/bardblu/bardblp
Lp(Y∗,ξ
xεn,K)+/integraldisplay
Y∗,ξ
xεn,K/integraldisplay
Y∗,ξ
xεn,K|u(y1)−u(y2)|p
|y1−y2|d+βpdy1dy2/bracketrightBigg
foru∈W1,p(Y∗,ξ
xε
n,K)o ru∈Wβ,p(Y∗,ξ
xε
n,K), for 1/2<β<1, respectively, where the
constant μΓdepends only on D,K,a n dY∗; see, e.g., [26, 52]. Then, scaling by εand
summing up over ξ∈ˆΞε
nand 1≤n≤Nεimplies the estimates
ε/bardblu/bardblp
Lp(ˆΓε)≤μΓ/bracketleftBig
/bardblu/bardblp
Lp(Ω∗
ε,K)+εp/bardbl∇u/bardblp
Lp(Ω∗
ε,K)/bracketrightBig
(8.1)
foru∈W1,p(Ω∗
ε,K),p∈[1,+∞),
ε/bardblu/bardblp
Lp(ˆΓε)≤μΓ/bracketleftBigg
/bardblu/bardblp
Lp(Ω∗
ε,K)+εβp/integraldisplay
Ω∗
ε,K/integraldisplay
Ω∗ε,K|u(x1)−u(x2)|p
|x1−x2|d+βpdx1dx2/bracketrightBigg
(8.2)
foru∈Wβ,p(Ω∗
ε,K)w i t h1 /2<β<1,p∈[1,+∞),
where the constant μΓdepends on D,K,a n dY∗and is independent of ε,a n d
ˆΓε=Nε/uniondisplay
n=1ˆΓε
nwithˆΓε
n=/uniondisplay
ξ∈ˆΞε
nεDxεn(/tildewideΓKxεn+ξ).
Since Γ xεnis given by a linear transformation of Γ, for a parametrization y=y(w)
of Γ, where w∈Rd−1,w eo b t a i nb y x(w)=εDxε
nKxεny(w) the parametrization
ofεΓxεn. We consider for Γ that dσy=√
gdwwithw∈Rd−1,a n df o rΓε
xεnwe
havedσn
x=εd−1√
gxεndw,w h e r eg=d e t (gij),gxεn=d e t (gxεn,ij), and (gij), (gxεn,ij)
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1090 MARIYA PTASHNYK
are the corresponding ﬁrst fundamental forms (metrics). We have also/integraltext
Γεdσε
x=/summationtextNε
n=1/integraltext
Γεndσn
xand Γx=D(x)K(x)Γ withdσx=/radicalbig
g(x)dw.
Using the deﬁnition of the l-p boundary unfolding operator, the trace inequality
(8.1), and the assumptions on DandK, we show the following properties of Tb,ε
L.
Lemma 8.2. Forψ∈W1,p(Ω∗
ε,K),w i t hp∈[1,+∞), we have
(i)/integraldisplay
Ω×ΓNε/summationdisplay
n=1√
gxεn
√
g|Yxεn||Tb,ε
L(ψ)(x,y)|pχΩεndσydx=ε/integraldisplay
ˆΓε|ψ(x)|pdσε
x,
(ii)/integraldisplay
Ω×Γ|Tb,ε
L(ψ)(x,y)|pdσydx=εNε/summationdisplay
n=1/integraldisplay
ˆΓε
n√
g|Yxεn|
√
gxεn|ψ(x)|pdσn
x≤Cε/integraldisplay
ˆΓε|ψ(x)|pdσε
x,
(iii)/bardblTb,ε
L(ψ)/bardblLp(Ω×Γ)≤C/parenleftBig
/bardblψ/bardblLp(Ω∗
ε,K)+ε/bardbl∇ψ/bardblLp(Ω∗
ε,K)/parenrightBig
,
where the constant Cdepends on DandKand is independent of ε.
Proof. Equality (i) follows directly from the deﬁnition of Tb,ε
L, i.e.,
/integraldisplay
Ω×ΓNε/summationdisplay
n=1√
gxεn
√
g|Yxεn||Tb,ε
L(ψ)|pχΩεndσydx
=Nε/summationdisplay
n=1/summationdisplay
ξ∈ˆΞεnεd/integraldisplay
Γ√
gxεn
√
g|ψ(εDxεn(ξ+Kxεny))|pdσy=ε/integraldisplay
ˆΓε|ψ(x)|pdσε
x.
Similar calculations and the regularity assumptions on DandKimply the equality
and the estimate in (ii). The estimate in (iii) is ensured by (ii) and (8.1).
Remark. Due to the second estimate in Lemma 8.2 and the assumptions on D
andK, theboundednessof ε/bardblwε/bardblp
Lp(ˆΓε)impliesthe boundednessof /bardblTb,ε
L(wε)/bardblp
Lp(Ω×Γ)
and, hence, the weak convergence of Tb,ε
L(wε)i nLp(Ω×Γ).
Applyingthepropertiesofthel-pboundaryunfoldingoperatorshowninLemma8.2,
we prove the relation between the l-t-s convergence on oscillating boundaries and the
l-p boundary unfolding operator.
P r o o fo fT h e o r e m 4.6. Usingthedeﬁnitionof Tb,ε
Landconsidering ψ∈C(
Ω;Cper(Yx))
together with the corresponding /tildewideψ∈C(
Ω;Cper(Y)) yields
/integraldisplay
Ω/integraldisplay
ΓNε/summationdisplay
n=1√
gxεn
√
g|Yxε
n|Tb,ε
L(wε)/tildewideψ(x,Kxεny)χΩεndσydx
=Nε/summationdisplay
n=1/summationdisplay
ξ∈ˆΞεnε/integraldisplay
εΓξ
xεnwε(z)/tildewideψ/parenleftBig
z,D−1
xεnz
ε/parenrightBig
dσn
z
+Nε/summationdisplay
n=1/summationdisplay
ξ∈ˆΞεnε1−d1
|Yxεn|/integraldisplay
εΓξ
xεnwε(z)/integraldisplay
εYξ
xεn/bracketleftBig
/tildewideψ/parenleftBig
x,D−1
xεnz
ε/parenrightBig
−/tildewideψ/parenleftBig
z,D−1
xεnz
ε/parenrightBig/bracketrightBig
dxdσn
z,
where Γξ
xεn=Dxεn(/tildewideΓKxεn+ξ)a n dYξ
xεn=Dxεn(Y+ξ). The continuity of ψand the
boundedness of ε/bardblwε/bardblp
Lp(Γε)ensure the convergence of the last integral to zero as
ε→0. Consider ﬁrst that wε→wl-t-s. The assumption on wε, i.e.,ε/bardblwε/bardblp
Lp(Γε)≤
C, withp∈(1,+∞), ensures that, up to a subsequence, Tb,ε
L(wε)/arrowrighttophalfˆwweakly in
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LOCALLY PERIODIC UNFOLDING METHOD 1091
Lp(Ω×Γ). Using the continuity of ψ,D,a n dK,a l o n gw i t h |Γε\ˆΓε|→0a sε→0,
yields
(8.3)/integraldisplay
Ω/integraldisplay
Γ√
gx
|Yx|√
gˆw(x,y)/tildewideψ(x,Kxy)dσydx
= lim
ε→0/integraldisplay
Ω/integraldisplay
ΓNε/summationdisplay
n=1√
gxεn
|Yxεn|√
gTb,ε
L(wε)/tildewideψ(x,Kxεny)χΩεndσydx
= lim
ε→0ε/integraldisplay
Γεwε(x)Lε(ψ)dσε
x=/integraldisplay
Ω1
|Yx|/integraldisplay
Γxw(x,y)ψ(x,y)dσydx
for allψ∈C(
Ω;Cper(Yx)). Applying the coordinate transformation in the integral
on the right-hand side yields ˆ w(x,y)=w(x,DxKxy) for a.a. x∈Ω,y∈Γ, and the
whole sequence {Tb,ε
L(wε)}converges to w(·,DxKx·).
Consider Tb,ε
L(wε)/arrowrighttophalfw(·,DxKx·)i nLp(Ω×Γ). The boundedness of ε/bardblwε/bardblp
Lp(Γε)
impliesthat, uptoasubsequence, wε→ˆwl-t-sand ˆ w∈Lp(Ω;Lp(Γx)). Interchanging
ˆwandwin (8.3), we obtain that the whole sequence wεl-t-s converges to w.
For functions in Wβ,p(Ω), with p∈(1,+∞)a n d1/2<β≤1, or for sequences
deﬁned on oscillating boundaries and converging in the Lp(Γε)-norm, scaled by ε1/p,
we obtain the strong convergence of the corresponding unfolded sequences.
Lemma 8.3. Foru∈Wβ,p(Ω),w i t hp∈(1,+∞)and1/2<β≤1, we have
(8.4) Tb,ε
L(u)→ustrongly in Lp(Ω×Γ).
If for{vε}⊂Lp(Γε)and some v∈C(
Ω;Cper(Yx))we haveε/bardblvε−Lεv/bardblp
Lp(Γε)→0as
ε→0,t h e n
(8.5) Tb,ε
L(vε)→v(·,DxKx·)strongly in Lp(Ω×Γ).
Proof. For an approximation of ubyuk∈C1(
Ω) we can write
/integraldisplay
Ω×Γ|Tb,ε
L(uk)|pdσydx=Nε/summationdisplay
n=1/integraldisplay
Ω×Γ/vextendsingle/vextendsingleuk/parenleftbig
εDxεn/bracketleftbig
D−1
xεnx/ε/bracketrightbig
Y+εDxεnKxεny/parenrightbig/vextendsingle/vextendsinglepχˆΩεndσydx
=Nε/summationdisplay
n=1/summationdisplay
ξ∈ˆΞε
nεd|Yxε
n|/integraldisplay
Γ|uk(εDxε
n(ξ+Kxε
ny))|pdσy=Nε/summationdisplay
n=1/summationdisplay
ξ∈ˆΞε
n|εYxε
n||Γ||uk(˜xε
n,ξ)|p+δε
for some ﬁxed ˜ xε
n,ξ∈εDxε
n(Kxε
nΓ+ξ), where, due to the continuity of uk,w eh a v e
δε=Nε/summationdisplay
n=1/summationdisplay
ξ∈ˆΞε
nεd|Dxε
nY|/integraldisplay
Γ|uk(εDxε
n(ξ+Kxε
ny))−uk(˜xε
n,ξ)|pdσy→0a sε→0.
The properties of the covering of Ω by {Ωε
n}Nε
n=1and|Ω\ˆΩε|→0a sε→0i m p l y
lim
ε→0Nε/summationdisplay
n=1/summationdisplay
ξ∈ˆΞεnεd|DxεnY||Γ||uk(˜xε
n,ξ)|p=/integraldisplay
Ω/integraldisplay
Γ|uk(x)|pdσydx.
Then, the density of C1(
Ω) inWβ,p(Ω), relation (ii) in Lemma 8.2, and the trace
estimates (8.1) and (8.2) ensure the convergence result for u∈Wβ,p(Ω).
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1092 MARIYA PTASHNYK
Fig. 5.Left: Images of laminar cleavage planes in longitudinal-radial (A)and circumferential-
radial(B)tissue sections from basal and apical measurement sites in anterior left ventricular
free wall. Republished with permission of The American Physiological Society, from K. D. Costa,Y. Takayana, A. D. McCulloch, and J. W. Covell, Laminar ﬁber architecture and three-dimensional
systolic mechanics in canine ventricular myocardium , Am. J. Physiol. Heart Circ. Physiol., 276
(1999), pp.H595–H607 [23] . Right: Cardiac muscle ﬁber orientations vary continuously through
the left ventricular wall from a negative angle at the epicardium to positive values toward the endo-cardium. Republished with permission of Taylor and Francis Group, from A. D. McCulloch, Cardiac
biomechanics , in The Biomedical Engineering Handbook, 2nd ed., J.D. Bronzino, ed., CRC Press,
Boca Raton, FL, 2000 [39]; permission conveyed through Copyright Clearance Center, Inc.
To show the convergence in (8.5) we consider
/bardblTb,ε
L(vε)−v(·,DxKx·)/bardblLp(Ω×Γ)≤/bardbl Tb,ε
L(vε)−Tb,ε
L(Lεv)/bardblLp(Ω×Γ)
+/bardblTb,ε
L(Lεv)−v(·,DxKx·)/bardblLp(Ω×Γ).
Then, estimate (ii) in Lemma 8.2, the regularity of v,D,a n dK, and the convergence
lim
ε→0/integraldisplay
Ω×Γ|Tb,ε
L(Lεv)|pdσydx= lim
ε→0Nε/summationdisplay
n=1|εYxεn|/summationdisplay
ξ∈ˆΞε
n/integraldisplay
Γ|/tildewidev(εDxεn(ξ+Kxεny),Kxεny)|pdσy
=/integraldisplay
Ω/integraldisplay
Γ|v(x,DxKxy)|pdσydx,
where/tildewidev(x,y)=v(x,Dxy)f o rx∈Ωa n dy∈Y, yield (8.5).
The results in Lemma 8.3 will ensure the strong convergence of coeﬃcients in
equations deﬁned on oscillating boundaries and are used in the derivation of macro-scopic problems for microscopic equations d eﬁned on surfaces of l-p microstructures.
9. Homogenization of a model for a signaling process in a tissue with
l-p distribution of cells. In this section we apply the methods of the l-p unfolding
operator and l-t-s convergence on oscillating surfaces to derive macroscopic equations
for microscopic models posed in domains with l-p perforations. We consider a gener-alization of the model for an intercellular signaling process presented in [34] to tissues
with l-p microstructures. As examples for tissues with space-dependent changes in
the size and shape of cells, we consider epithelial and plant cell tissues; see Figure 3.
As an example of a tissue which has a plywood-like structure we consider the cardiac
muscle tissue of the left ventricular wall; see Figure 5.
The microstructure of cardiac muscle is described in the same way as a plywood-
like structure considered in the introduction, where D(x)=R
−1(γ(x3)) and the
rotation matrix Ris as deﬁned in the introduction. Additionally we assume that the
radius of ﬁbers may change locally; i.e., K(x)Y0⊂Y, with
K(x)=/parenleftbigg10T
0ρ(x)I2/parenrightbigg
,
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LOCALLY PERIODIC UNFOLDING METHOD 1093
Y0={(y1,y2,y3)∈Y:|y2|2+|y3|2<a2},0<a<1/2,Y=(−1/2,1/2)3,a n d
ρ∈C1(
Ω) with 0 <ρ1≤ρ(x)a<1/2 for all x∈
Ω. Then, for the plywood-like
s t r u c t u r ew eh a v e Dxεn=R−1(γ(xε
n,3)),/tildewideY∗
Kx=Y\K(x)
Y0,Y∗
x,K=R−1(γ(x3))/tildewideY∗
Kx,
and the characteristic function of ﬁbers is given by
χΩε
f(x)=χΩ(x)Nε/summationdisplay
n=1˜η(xε
n,R(γ(xε
n,3))x/ε)χΩε
n,
where
˜η(x,y)=/braceleftBigg
1f o r |ˆK(x)−1ˆy|≤a,
0e l s e w h e r e ,
andisextended ˆY-periodicallytothewholeof R3.H e r eˆy=(y2,y3),ˆY=(−1/2,1/2)2,
andˆK(x)=ρ(x)I2,w h e r e I2denotes the identity matrix in R2×2.
In the case of an epithelial tissue consider Yx=D(x)Y, with, e.g.,
D(x)=/parenleftbigg
I20
0Tκ(x)/parenrightbigg
,
whereκ∈C1(
Ω) and 0 <κ1≤κ(x)<1 for all x∈Ω deﬁnes a compression of
cells in the x3-direction. The changes in the size and shape of cells can be deﬁned
by the boundaries of the microstructure Γ x=S(x)Γ⊂Yx=DxYfor allx∈
Ωa n d
S∈Lip(
Ω;R3×3). Then, in the deﬁnition of the intercellular space Ω∗
ε,Kwe have
Y∗
x,K=D(x)/tildewideY∗
Kx=D(x)(Y\K(x)
Y0), where K(x)=D(x)−1S(x).
We deﬁne the intercellular space in a tissue as
Ω∗
ε,K=I n t/parenleftBiggNε/uniondisplay
n=1
Ω∗,ε
n,K/parenrightBigg
∩Ω,where Ω∗,ε
n,K=Ωε
n\/uniondisplay
ξ∈Ξ∗,ε
nDxεn(Kxεn
Y0+ξ),
ˆΩ∗
ε,K=I n t/parenleftBiggNε/uniondisplay
n=1
ˆΩ∗,ε
n,K/parenrightBigg
,ˆΩ∗,ε
n,K=I n t/parenleftBigg/uniondisplay
ξ∈ˆΞε
nεDxεn(
/tildewideY∗
Kxεn+ξ)/parenrightBigg
,Λ∗
ε,K=Ω∗ε,K\ˆΩ∗
ε,K.
In the model for a signaling process in a cell tissue we consider diﬀusion of signal-
ing molecules lεin the intercellular space and their interactions with free and bound
receptors rε
fandrε
blocated on cell surfaces. The microscopic model reads as
(9.1)∂tlε−div(Aε(x)∇lε)=Fε(x,lε)−dε
l(x)lεin (0,T)×Ω∗
ε,K,
Aε(x)∇lε·n=ε/bracketleftbig
βε(x)rε
b−αε(x)lεrε
f/bracketrightbig
on (0,T)×Γε,
Aε(x)∇lε·n=0 o n( 0 ,T)×(∂Ω∩∂Ω∗
ε,K),
lε(0,x)=l0(x)i n Ω∗
ε,K,
where the dynamics in the con centrations of free and bo und receptors on cell surfaces
are determined by two ordinary diﬀerential equations:
(9.2)∂trε
f=pε(x,rε
b)−αε(x)lεrε
f+βε(x)rε
b−dε
f(x)rε
fon (0,T)×Γε,
∂trε
b= αε(x)lεrε
f−βε(x)rε
b−dε
b(x)rε
bon (0,T)×Γε,
rε
f(0,x)=rε
f0(x),rε
b(0,x)=rε
b0(x)o n Γε.
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1094 MARIYA PTASHNYK
The coeﬃcients Aε,αε,βε,dε
jand the functions Fε(·,ξ),pε(·,ξ),rε
i0are deﬁned as
Aε(x)=Lε
0(A(x,y)),Fε(x,ξ)=Lε
0(F(x,y,ξ)),pε(x,ξ)=Lε
0(p(x,y,ξ)),
αε(x)=Lε
0(α(x,y)),βε(x)=Lε
0(β(x,y)),dε
j(x)=Lε
0(dj(x,y)),
rε
i0(x)=Lε(ri0(x,y)),j =l,f,b, i =f,b,
forx∈Ω,y∈Yx,a n dξ∈R,w h e r eA(x,·),α(x,·),β(x,·),dj(x,·),p(x,·,ξ),F(x,·,ξ),
andri0(x,·)a r eYx-periodic functions. We assume also that αε(x)=0a n d βε(x)=0
forx∈Λε. The last assumption is not restrictive, since the domain Λεis very small
compared to the size of the whole domain Ω and |Λε|≤Cε1−r→0a sε→0f o r
0≤r<1.
Here,Aε:Ω→Rdenotes the diﬀusion coeﬃcient for signaling molecules (li-
gands),Fε:Ω×R→Rmodels the production of new ligands, pε:Ω→Rdescribes
the productionof new free receptors, dε
j:Ω→R,j=l,f,b, denotes the ratesof decay
of ligands, free receptors, and bo und receptors, respectively, βε:Ω→Ri st h er a t eo f
dissociation of bound receptors, and αε:Ω→Ris the rate of binding of ligands to
free receptors.
Assumption 9.1.
•A∈C(
Ω;L∞
per(Yx)) is symmetric with ( A(x,y)ξ,ξ)≥d0|ξ|2ford0>0,
ξ∈Rd,x∈Ω, and a.a. y∈Yx.
•F(·,·,ξ)∈C(
Ω;L∞
per(Yx)) is Lipschitz continuous in ξwithξ≥−κ,f o rs o m e
κ>0, uniformly in ( x,y)a n dF(x,y,ξ)≥0f o rξ≥0,x∈Ω, andy∈Yx.
•p(·,·,ξ)∈C(
Ω;Cper(Yx)) is Lipschitz continuous in ξwithξ≥−κ,f o rs o m e
κ>0, uniformly in ( x,y) and nonnegative for nonnegative ξ.
•Coeﬃcients α,β,d j∈C(
Ω;Cper(Yx)) are nonnegative, j=l,f,b.
•Initial conditions l0∈H1(Ω)∩L∞(Ω),rj0∈C(
Ω;Cper(Yx)) are nonnegative,
j=f,b.
Notice that these assumptions are satisﬁ ed by the physical processes described
by our model, since for most signaling processes in biological tissues we have that
A=c o n s t , F(x,y,ξ)=μ1ξ/(μ2+μ3ξ), andp(x,y,ξ)=κ1ξ/(κ2+κ3ξ) with some
nonnegative constants μiandκifori=1,2,3, and the coeﬃcients α,β,a n ddj, with
j=l,f,b, can be chosen as constant or as some smooth functions.
We shall use the following notation: ˆΓε
T=( 0,T)×ˆΓε,Γε
T=( 0,T)×Γε,ΩT=
(0,T)×Ω, ΓT=( 0,T)×Γ, and Γ x,T=( 0,T)×Γx.F o rv∈Lp(0,σ;Lq(A)),
w∈Lp/prime(0,σ;Lq/prime(A)) we denote /angbracketleftv,w/angbracketrightA,σ=/integraltextσ
0/integraltext
Avwdxdt.
Definition 9.1. A weak solution of the problem (9.1)–(9.2) are functions (lε,rε
f,rε
b)
such that lε∈L2(0,T;H1(Ω∗
ε,K))∩H1(0,T;L2(Ω∗
ε,K)),rε
j∈H1(0,T;L2(Γε))∩
L∞(Γε
T)forj=f,b, satisfying (9.1)i nt h ew e a kf o r m
(9.3)/angbracketleft∂tlε,φ/angbracketrightΩ∗
ε,K,T+/angbracketleftAε(x)∇lε,∇φ/angbracketrightΩ∗
ε,K,T=/angbracketleftFε(x,lε)−dε
l(x)lε,φ/angbracketrightΩ∗
ε,K,T
+ε/angbracketleftβε(x)rε
b−αε(x)lεrε
f,φ/angbracketrightΓε,T
for allφ∈L2(0,T;H1(Ω∗
ε,K));e q u a t i o n s (9.2)are satisﬁed a.e. on Γε
T;a n dlε(t,·)→
l0(·)inL2(Ω∗
ε,K),rε
j(t,·)→rε
j0(·)inL2(Γε)ast→0.
In a similar way as in [15, 34] we obtain the existence and uniqueness results and
a priori estimates for a weak solution of the microscopic problem (9.1)–(9.2).
Lemma 9.2. Under Assumption 9.1there exists a unique nonnegative weak so-
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LOCALLY PERIODIC UNFOLDING METHOD 1095
lution of the microscopic problem (9.1)–(9.2) satisfying a priori estimates
/bardbllε/bardblL∞(0,T;L2(Ω∗
ε,K))+/bardbl∇lε/bardblL∞(0,T;L2(Ω∗
ε,K))+/bardbl∂tlε/bardblL2((0,T)×Ω∗
ε,K)≤C,
ε1/2/bardbllε/bardblL2(ˆΓε
T)+/bardblrε
j/bardblL∞(0,T;L∞(Γε))+ε1/2/bardbl∂trε
j/bardblL2(Γε
T)≤C,(9.4)
withj=f,b, where the constant Cis independent of ε. Additionally, we have that
(9.5)/bardbl(lε−MeBt)+/bardblL∞(0,T;L2(Ω∗
ε,K))+/bardbl∇(lε−MeBt)+/bardblL2((0,T)×Ω∗
ε,K)≤Cε1/2,
wherev+=m a x{0,v},M≥supΩl0(x),B=B(F,β,p),a n dCis independent of ε.
Sketch of the proof . To prove the existence of a sol ution of the microscopic model
we show the existence of a ﬁxed point of an operator Bdeﬁned on L2(0,T;Hς(Ω∗
ε,K)),
with 1/2<ς<1, bylε
n=B(lε
n−1) given as a solution of (9.1)–(9.2) with lε
n−1in
(9.2) and in the nonlinear function Fε(x,lε) instead of lε
n. For a given nonnegative
lε
n−1∈L2(0,T;Hς(Ω∗
ε,K)) there exists a nonnegative solution ( rε
f,n,rε
b,n) of (9.2).
Then, the nonnegativity of solutions, the equality
∂t(rε
f,n+rε
b,n)=pε(x,rε
b,n)−dε
b(x)rε
b,n−dε
f(x)rε
f,n,
and the Lipschitz continuity of pensure the boundedness of rε
f,nandrε
b,n. Considering
lε,−
n=m i n{0,lε
n}as a test function in (9.3) and using the nonnegativity of rε
f,n,rε
b,n
and the initial data, we obtain the nonnegativity of lε
n. Applying Galerkin’s method
and using a prioriestimates similarto those in (9.4), we obtain the existence of a weak
nonnegative solution lε
n∈H1(0,T;L2(Ω∗
ε,K))∩L2(0,T;H1(Ω∗
ε,K)). The compactness
of the embedding H1(0,T;L2(Ω∗
ε,K))∩L2(0,T;H1(Ω∗
ε,K))⊂L2(0,T;Hς(Ω∗
ε,K)) and
Schauder’s theorem imply the existence of a ﬁxed point lεofB. Notice that the
strong convergence of lε
ninL2(Γε
T), asn→∞, implies the strong convergence of rε
j,n,
j=f,b. Taking lε
nand∂tlε
nas test functions in (9.3) and using the trace estimate
(8.1), we obtain a priori estimates for lε
nand∂tlε
n. Testing (9.2) by ∂trε
f,nand∂trε
b,n,
respectively, yields the estimates for the time derivatives. Then, using the lower
semicontinuity of a norm, we obtain the a priori estimates (9.4) for lε,rε
f,a n drε
b.
Especially for the derivation of a priori estimates for ∂tlεwe consider
ε/integraldisplay
Γε(βεrε
b−αεrε
flε)∂tlεdσx=εd
dt/integraldisplay
Γεβεrε
blεdσx−ε/integraldisplay
Γεβε∂trε
blεdσx
−ε
2d
dt/integraldisplay
Γεαεrε
f|lε|2dσx+ε
2/integraldisplay
Γεαε∂trε
f|lε|2dσx.
Using the equation for ∂trε
f, the last integral can be rewritten as
ε
2/integraldisplay
Γεαε/parenleftbig
pε(x,rε
b)−αεlεrε
f+βεrε
b−dε
frε
f/parenrightbig
|lε|2dσx.
Applying the trace estimate (8.1) and using the assumptions on αεandβε,a l o n gw i t h
the nonnegativity of lεandrε
j, the boundedness of rε
j, uniform in ε, and the estimate
ε/bardbl∂trε
b/bardbl2
L2(Γε
T)≤C,w eo b t a i n
ε/integraldisplayτ
0/integraldisplay
Γε(βεrε
b−αεrε
flε/parenrightbig
∂tlεdσxdt≤C1/bracketleftbig
/bardbllε(τ)/bardbl2
L2(Ω∗
ε,K)+ε2/bardbl∇lε(τ)/bardbl2
L2(Ω∗
ε,K)/bracketrightbig
+C2/bracketleftbig
/bardbllε/bardbl2
L2((0,τ)×Ω∗
ε,K)+ε2/bardbl∇lε/bardbl2
L2((0,τ)×Ω∗
ε,K)/bracketrightbig
+C3
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1096 MARIYA PTASHNYK
forτ∈(0,T]. Standard arguments pertaining to the diﬀerence of two solutions lε
1−lε
2,
rε
j,1−rε
j,2, withj=f,b, imply the uniqueness of a weak solution of the microscopic
problem (9.1)–(9.2). In particular, the nonnegativity of αε,rε
j,a n dlεalong with the
boundedness of rε
j,w h e r ej=f,b, ensures that
∂t/bardblrε
f,1−rε
f,2/bardbl2
L2(Γε)≤C/parenleftBigg/summationdisplay
j=f,b/bardblrε
j,1−rε
j,2/bardbl2
L2(Γε)+/bardbllε
1−lε
2/bardbl2
L2(ˆΓε)/parenrightBigg
. (9.6)
Testing the diﬀerence of the equations for rε
f,1+rε
b,1andrε
f,2+rε
b,2byrε
f,1+rε
b,1−
rε
f,2−rε
b,2yields
/bardblrε
b,1(τ)−rε
b,2(τ)/bardbl2
L2(Γε)≤C/integraldisplayτ
0/summationdisplay
j=f,b/bardblrε
j,1−rε
j,2/bardbl2
L2(Γε)+/bardbllε
1−lε
2/bardbl2
L2(ˆΓε)dt. (9.7)
Applying Gronwall’s lemma yields the estimate for /bardblrε
j,1(τ)−rε
j,2(τ)/bardbl2
L2(Γε), with
τ∈(0,T]a n dj=f,b,i nt e r m so f /bardbllε
1−lε
2/bardbl2
L2(ˆΓε
τ).T a k i n g( lε−S)+as a test function
in (9.3) and using the boundedness of rε
j,w eo b t a i n
/bardbl(lε−S)+/bardblL∞(0,T;L2(Ω∗
ε,K))+/bardbl∇(lε−S)+/bardblL2((0,T)×Ω∗
ε,K)≤2S/parenleftbigg/integraldisplayT
0|Ω∗,S
ε,K(t)|dt/parenrightbigg1
2
,
whereS≥max{supΩl0(x),supΩ×Yx|β(x,y)|,supΩ×Yx|α(x,y)|,/bardblrε
j/bardblL∞(Γε
T)}and
Ω∗,S
ε,K(t)={x∈Ω∗
ε,K:lε(t,x)>S}for a.a.t∈(0,T). Then, applying Theorem II.6.1
in [33] yields the boundedness of lεfor every ﬁxed ε>0. Considering (9.3) for lε
1and
lε
2, we obtainthe estimate for /bardbllε
1−lε
2/bardblL2(0,τ;H1(Ω∗
ε,K))in termsof ε1/2/bardblrε
j,1−rε
j,2/bardblL2(Γετ),
withj=f,bandτ∈(0,T]. Then, using the estimates for /bardblrε
j,1(τ)−rε
j,2(τ)/bardblL2(Γε)in
(9.6) and (9.7) yields that rε
j,1=rε
j,2a.e. in Γε
T,w h e r ej=f,b, and hence lε
1=lε
2a.e.
in (0,T)×Ω∗
ε,K.
To show (9.5), we consider ( lε−MeBt)+as a test function in (9.3). Using the
boundedness of rε
j, uniform in ε, and the trace estimate (8.1), we obtain for τ∈(0,T)
/bardbl(lε(τ)−MeBτ)+/bardbl2
L2(Ω∗
ε,K)+/bardbl∇(lε−MeBt)+/bardbl2
L2((0,τ)×Ω∗
ε,K)
≤C1/bardbl(lε−MeBt)+/bardbl2
L2((0,τ)×Ω∗
ε,K)+C2ε,
whereM≥supΩl0(x),MB≥/parenleftbig
supΩ×Yx|F(x,y,0)|+μΓsupΩ×Yxβ(x,y)/bardblrε
b/bardblL∞(ˆΓε
T)/parenrightbig
,
withμΓas in (8.1). Then, applying Gronwall’s lemma in the last inequality yields
(9.5).
Notice that in the case of a perforated domain where the periodicity and the
shape of perforations vary in space, i.e., K/negationslash=I, we cannot apply the l-p unfolding
operator to functions deﬁned on Ω∗
ε,Kdirectly. To overcome this problem we consider
a local extension of a function from ˆΩ∗,ε
n,KtoˆΩε
nand then apply the l-p unfolding
operator Tε
L, determined for functions deﬁned on ˆΩε. Applying the assumptions on
themicrostructureofΩ∗
ε,Kconsideredhere, i.e., Kx
Y0⊂Yoraﬁbrousmicrostructure,
we obtain the following lemma.
Lemma 9.3. Forxε
n∈ˆΩε
n,w h e r e1≤n≤Nε,a n du∈W1,p(Y∗
xεn,K),w i t h
p∈(1,+∞), there exists an extension ˆu∈W1,p(Yxεn)such that
/bardblˆu/bardblLp(Yxεn)≤μ/bardblu/bardblLp(Y∗
xεn,K),/bardbl∇ˆu/bardblLp(Yxεn)≤μ/bardbl∇u/bardblLp(Y∗
xεn,K), (9.8)
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LOCALLY PERIODIC UNFOLDING METHOD 1097
whereμdepends on Y,Y0,D,a n dKand is independent of εandn.
Foru∈W1,p(Ω∗
ε,K)we have an extension ˆu∈W1,p(ˆΩε)fromˆΩ∗
ε,KtoˆΩεsuch
that
/bardblˆu/bardblLp(ˆΩε)≤μ/bardblu/bardblLp(ˆΩ∗
ε,K),/bardbl∇ˆu/bardblLp(ˆΩε)≤μ/bardbl∇u/bardblLp(ˆΩ∗
ε,K), (9.9)
whereμdepends on Y,Y0,D,a n dKand is independent of ε.
Sketch of the proof . The proof follows along the same lines as in the periodic
case; see, e.g., [14, 22, 32]. The only diﬀer ence is that the extension depends on the
Lipschitz continuity of KandDand the uniform boundedness from above and below
of|detK(x)|and|detD(x)|. To show (9.9), we ﬁrst consider an extension from
Dxεn(/tildewideY∗
Kxεn+ξ)t oDxεn(Y+ξ) satisfying the estimates (9.8), where ξ∈ˆΞε
n. Then,
scaling by εand summing up over ξ∈ˆΞε
nandn=1,…,N εimplies the estimates
(9.9).
Remark. Notice that the deﬁnition of Ω∗
ε,Kimplies that there are no perforations
in/parenleftbig
Ω∗,ε
n,K\
ˆΩ∗,ε
n,K/parenrightbig
∩/tildewideΩε/2,with/tildewideΩε/2={x∈Ω: d i s t ( x,∂Ω)>2εmaxx∈∂Ωdiam(D(x)Y)}.
Also in the case of a plywood-like structure the ﬁbers are orthogonal to the bound-
aries of Ωε
n, and near ∂Ωε
nwe need to extend lεonly in the directions parallel to ∂Ωε
n.
Thus, applying Lemma 9.3 we can extend lεfrom Ω∗,ε
n,KintoˆΩε
n∪/parenleftbig
Ωε
n∩/tildewideΩε/2/parenrightbig
for
n=1,…,N ε.
Theorem 9.4. A sequence of solutions of the microscopic problem (9.1)–(9.2)
converges to a solution (l,rf,rb)withl∈L2(0,T;H1(Ω))∩H1(0,T;L2(Ω))andrj∈
H1(0,T;L2(Ω;L2(Γx)))∩L∞(ΩT;L∞(Γx))of the macroscopic equations
(9.10)|Y∗
x,K|
|Yx|∂tl−div(A(x)∇l)=1
|Yx|/integraldisplay
Y∗
x,KF(x,y,l)dy
+1
|Yx|/integraldisplay
Γx(β(x,y)rb−α(x,y)rfl)dσyinΩT,
A(x)∇l·n=0 on∂Ω,
∂trf=p(x,y,rb)−α(x,y)lrf+β(x,y)rb−df(x,y)rffory∈Γx,
∂trb= α(x,y)lrf−β(x,y)rb−db(x,y)rbfory∈Γx,
and for(t,x)∈ΩT,w h e r eY∗
x,K=Dx(Y\KxY0)and the macroscopic diﬀusion matrix
is deﬁned as
Aij(x)=1
|Yx|/integraldisplay
Y∗
x,K/bracketleftbig
Aij(x,y)+(A(x,y)∇yωj(x,y))i/bracketrightbig
dyforx∈Ω,
fori,j=1,…,d,w i t h
(9.11)divy(A(x,y)(∇yωj+ej)) = 0 inY∗
x,K,
A(x,y)(∇yωj+ej)·n=0 onΓx,ωjisYx-periodic.
We have that ˆlε→linL2(ΩT),∂tlε/arrowrighttophalf∂tland∂trε
j/arrowrighttophalf∂trjl-t-s,rε
j→rjstrongly
l-t-s,j=f,b,a n d
∇lε/arrowrighttophalf∇l+∇yl1 l-t-s,l1∈L2(ΩT;H1
per(Y∗
x,K)),
lim
ε→0/angbracketleftAε∇lε,∇lε/angbracketrightΩ∗
ε,K,T=/angbracketleft|Yx|−1A(x,y)(∇l+∇yl1),∇l+∇yl1/angbracketrightΩT,Y∗
x,K,
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1098 MARIYA PTASHNYK
wherel1(t,x,y)=/summationtextd
j=1∂l
∂xj(t,x)ωj(x,y).H e r eˆφdenotes the extension as in Lemma
9.3from(0,T)×Ω∗
ε,Kto(0,T)×(/tildewideΩε/2∪Ω∗
ε,K)and then by zero to ΩT.
Proof. Applying Lemma 9.3, we can extend lεfrom Ω∗
ε,KintoˆΩε∪Λ∗
ε,K.W e
shall use the same notation for original functions and their extensions. The a prioriestimates in Lemma 9.2 imply
(9.12) /bardbll
ε/bardblL2(0,T;H1(ˆΩε∪Λ∗
ε,K))+/bardbl∂tlε/bardblL2((0,T)×(ˆΩε∪Λ∗
ε,K))≤C,
where the constant Cdepends on DandKand is independent of ε. Then the
sequences {lε},{∇lε},a n d{∂tlε}are deﬁned on ˆΩε, and we can determine Tε
L(lε),
Tε
L(∇lε), and∂tTε
L(lε). The properties of Tε
Ltogether with (9.12) ensure that
/bardblTε
L(lε)/bardblL2(ΩT×Y)+/bardblTε
L(∇lε)/bardblL2(ΩT×Y)+/bardbl∂tTε
L(lε)/bardblL2(ΩT×Y)≤C.
The aprioriestimatesin Lemma9.2yieldthe estimatesforthe l-pboundaryunfolding
operator
/bardblTε,b
L(lε)/bardblL2(ΩT×Γ)+/bardblTε,b
L(rε
f)/bardblH1(0,T;L2(Ω×Γ))+/bardblTε,b
L(rε
b)/bardblH1(0,T;L2(Ω×Γ))≤C.
Notice that due to the assumptions on Ω∗
ε,Kwe have that /tildewideΩε/2⊂ˆΩε∪Λ∗
ε,K.
Then, the convergence results in Theorems 4.2, 4.4, 4.5, and 4.6 imply that there
existsubsequences(denotedagainby lε,rε
f,rε
b)andthefunctions l∈L2(0,T;H1(Ω))∩
H1(0,T;L2(Ω)),l1∈L2(ΩT;H1
per(Yx)), andrj∈H1(0,T;L2(Ω;L2(Γx))) such that
Tε
L(lε)/arrowrighttophalfl weakly in L2(ΩT;H1(Y)),
Tε
L(lε)→l strongly in L2(0,T;L2
loc(Ω;H1(Y))),
∂tTε
L(lε)/arrowrighttophalf∂tl weakly in L2(ΩT×Y),
Tε
L(∇lε)/arrowrighttophalf∇l+D−T
x∇˜yl1(·,Dx·)w e a k l y i n L2(ΩT×Y),
Tb,ε
L(lε)/arrowrighttophalfl weakly in L2(ΩT×Γ),
Tb,ε
L(lε)→l strongly in L2(0,T;L2
loc(Ω;L2(Γ))),
rε
j/arrowrighttophalfrj,∂trε
j/arrowrighttophalf∂rj l-t-s,rj,∂trj∈L2(ΩT;L2(Γx)),
Tb,ε
L(rε
j)/arrowrighttophalfrj(·,DxKx·)w e a k l y i n L2(ΩT×Γ),
∂tTb,ε
L(rε
j)/arrowrighttophalf∂trj(·,DxKx·)w e a k l y i n L2(ΩT×Γ),j=f,b.(9.13)
Notice that for lεwe have a priori estimates only in L2(0,T;H1(ˆΩε∪Λ∗
ε,K))
and not in L2(0,T;H1(Ω)) and cannot apply the convergence results in Theorem 4.2
directly. However, using /bardbllε/bardblL2(0,T;H1(/tildewideΩε/2))+/bardbl∂tlε/bardblL2((0,T)×/tildewideΩε/2)≤C, ensured by
(9.12), applying Lemmas 7.2 and 7.3 to Qε
L(lε)a n dRε
L(lε), respectively, and consid-
ering the proof of Theorem 4.4, we obtain the convergences for Tε
L(lε),∂tTε
L(lε), and
Tε
L(∇lε) in (9.13). Lemma 5.4 implies that ∇lε/arrowrighttophalf∇l+∇yl1l-t-s and ∂tlε/arrowrighttophalf∂tl
l-t-s. The local strong convergence of Tε
L(lε) together with the estimate /bardbl(lε−
MeBt)+/bardblL∞(0,T;L2(Ω∗
ε,K))≤Cε1/2, shown in Lemma 9.2, yields the strong conver-
gence of ˆlεinL2(ΩT).
To derive macroscopic equations for lε,w ec o n s i d e r ψε(x)=ψ1(x)+εLε
ρ(ψ2)(x)
withψ1∈C1(
Ω) andψ2∈C1
0(Ω;C1
per(Yx)) as a test function in (9.3). Applying the
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LOCALLY PERIODIC UNFOLDING METHOD 1099
l-p unfolding operator and the l-p boundary unfolding operator implies
1
|Y|/bracketleftBig
/angbracketleftTε
L(χε
Ω∗
ε,K)∂tTε
L(lε),Tε
L(ψε)/angbracketrightΩT×Y+/angbracketleftTε
L(χε
Ω∗
ε,K)Tε
L(Aε)Tε
L(∇lε),Tε
L(∇ψε)/angbracketrightΩT×Y/bracketrightBig
=|Y|−1/angbracketleftTε
L(χε
Ω∗
ε,K)ˆFε(x,˜y,Tε
L(lε)),Tε
L(ψε)/angbracketrightΩT×Y
+/angbracketleftBiggNε/summationdisplay
n=1√
gxε
n
√
g|Yxεn|/bracketleftbig
Tb,ε
L(βε)Tb,ε
L(rε
b)−Tb,ε
L(αε)Tb,ε
L(lε)Tb,ε
L(rε
f)/bracketrightbig
χΩεn,Tb,ε
L(ψε)/angbracketrightBigg
ΩT×Γ
−/angbracketleft∂tlε,ψε/angbracketrightΛ∗
ε,K,T−/angbracketleftAε(x)∇lε,∇ψε/angbracketrightΛ∗
ε,K,T+/angbracketleftFε(x,lε),ψε/angbracketrightΛ∗
ε,K,T,
whereˆFε(x,˜y,Tε
L(lε)) =/summationtextNε
n=1F(xε
n,Dxεn˜y,Tε
L(lε))χˆΩεn(x)f o r˜y∈Y,x∈Ω, and
χε
Ω∗
ε,K=Lε
0(χY∗
x,K). HereχY∗
x,Kis the characteristicfunction of Y∗
x,K=Dx(Y\KxY0),
extended Yx-periodically to Rd. We notice that ˆFε(x,˜y,ξ)=Tε
L(Lε
0(F(x,y,ξ))).
Applying Lemma 5.3 yields Tε
L(χε
Ω∗
ε,K)(x,˜y)→χY∗
x,K(x,Dx˜y),Tε
L(Aε)(x,˜y)→
A(x,Dx˜y), andˆFε(x,˜y,l)→F(x,Dx˜y,l)i nLp(ΩT×Y)f o rp∈(1,+∞)a sε→0.
Lemma 8.3 ensures that Tb,ε
L(φε)(x,ˆy)→φ(x,DxKxˆy)i nLp(Ω×Γ) asε→0, where
φε(x)=βε(x),αε(x), ordε
j(x)a n dφ(x,y)=α(x,y),β(x,y), ordj(x,y), withj=
f,b,l, respectively.
For an arbitrary δ>0w ec o n s i d e rΩδ={x∈Ω:d i s t (x,∂Ω)>δ}and rewrite
the boundary integral in the form
/angbracketleftBiggNε/summationdisplay
n=1√
gxεn
√
g|Yxεn|Tb,ε
L(αε)Tb,ε
L(lε)Tb,ε
L(rε
f)χΩε
n,Tb,ε
L(ψε)/angbracketrightBigg
Ωδ×ΓT
+/angbracketleftBiggNε/summationdisplay
n=1√
gxεn
√
g|Yxε
n|Tb,ε
L(αε)Tb,ε
L(lε)Tb,ε
L(rε
f)χΩεn,Tb,ε
L(ψε)/angbracketrightBigg
(Ω\Ωδ)×ΓT=I1+I2.
Usingtheaprioriestimatesfor lεandrε
j,theweakconvergenceof Tε
L(lε)inL2(ΩT;H1(Y)),
and the strong convergence in L2(0,T;L2
loc(Ω;H1(Y))), we obtain
lim
δ→0lim
ε→0I1=/angbracketleftbigg√
gx
√
g|Yx|α(x,DxKxˆy)rf(x,DxKxˆy)l(x),ψ1(x)/angbracketrightbigg
ΩT×Γ,
lim
δ→0lim
ε→0I2=0.(9.14)
To obtain (9.14), we also use the strong convergenceand boundedness of Tb,ε
L(αε), the
weak convergence and boundedness of Tb,ε
L(rε
f), the regularity of DandK,a n dt h e
strong convergence of Tb,ε
L(ψε). Similar arguments, along with the Lipschitz continu-
ity ofFand the strong convergence of ˆFε(x,˜y,l)a n dTε
L(χε
Ω∗
ε,K)=Tε
L(Lε
0(χY∗
x,K)),
ensure that
/angbracketleftTε
L(χε
Ω∗
ε,K)ˆFε(x,˜y,Tε
L(lε)),Tε
L(ψε)/angbracketrightΩT×Y→/angbracketleftχY∗
x,K(x,Dx˜y)F(x,Dx˜y,l),ψ1/angbracketrightΩT×Y
asε→0a n dδ→0. Using the convergence results (9.13), the strong convergence
ofTε
L(ψε)a n dTε
L(∇ψε), and the fact that |Λ∗
ε,K|→0a sε→0, taking the limit
asε→0, and considering the transformation of variables y=Dx˜yfor ˜y∈Yand
y=DxKxˆyfor ˆy∈Γ yield
/angbracketleft|Yx|−1l,ψ1/angbracketrightY∗
x,K×ΩT+/angbracketleft|Yx|−1A(x,y)(∇l+∇yl1),∇ψ1+∇yψ2/angbracketrightY∗
x,K×ΩT
+/angbracketleft|Yx|−1/bracketleftbig
α(x,y)rfl−β(x,y)rb/bracketrightbig
,ψ1/angbracketrightΓx×ΩT=/angbracketleft|Yx|−1F(x,y,l),ψ1/angbracketrightY∗
x,K×ΩT.
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1100 MARIYA PTASHNYK
Considering ψ1(t,x)=0f o r( t,x)∈ΩT,w eob t ai n l1(t,x,y)=/summationtextd
j=1∂xjl(t,x)ωj(x,y),
whereωjare solutions of the unit cell problems (9.11). Choosing ψ2(t,x,y)=0f o r
x∈ΩTandy∈Yximplies the macroscopicequation for l. Applying the l-p boundary
unfolding operator to the equations on Γε,w eo b t a i n
∂tTb,ε
L(rε
f)=ˆpε(x,ˆy,Tb,ε
L(rε
b))−Tb,ε
L(αε)Tb,ε
L(lε)Tb,ε
L(rε
f)
+Tb,ε
L(βε)Tb,ε
L(rε
b)−Tb,ε
L(dε
f)Tb,ε
L(rε
f),
∂tTb,ε
L(rε
b)=Tb,ε
L(αε)Tb,ε
L(lε)Tb,ε
L(rε
f)−Tb,ε
L(βε)Tb,ε
L(rε
b)−Tb,ε
L(dε
b)Tb,ε
L(rε
b)(9.15)
in ΩT×Γ, where ˆ pε(x,ˆy,Tb,ε
L(rε
b)) =/summationtextNε
n=1p(xε
n,DxεnKxεnˆy,Tb,ε
L(rε
b))χˆΩεn(x)f o rˆy∈Γ
andx∈Ω. In order to pass to the limit in the nonlinear function ˆ pε(x,ˆy,Tb,ε
L(rε
b)),
we have to show the strong convergence of Tb,ε
L(rε
b). We consider the diﬀerence of
the equations for Tb,εk
L(rεk
f)a n dTb,εm
L(rεm
f) and use Tb,εk
L(rεk
f)−Tb,εm
L(rεm
f)a sat e s t
function. Applying the Lipschitz continuity of palong with the strong convergence of
Tb,ε
L(αε),Tb,ε
L(βε), andTb,ε
L(dε
j), and the nonnegativity of lεandαεyields
d
dt/bardblTb,εk
L(rεk
f)−Tb,εm
L(rεm
f)/bardbl2
L2(Ω×Γ)≤C/bracketleftBigg/summationdisplay
j=f,b/bardblTb,εk
L(rεk
j)−Tb,εm
L(rεm
j)/bardbl2
L2(Ω×Γ)
+/bardblTb,εk
L(lεk)−Tb,εm
L(lεm)/bardbl2
L2(Ωδ×Γ)+δ1
2/bardblTb,εk
L(lεk)−Tb,εm
L(lεm)/bardblL2((Ω\Ωδ)×Γ)
+σ(εk,εm)/bracketrightBigg
,
whereσ(εk,εm)→0a sεk,εm→0. Considering the sum of the equations for
Tb,εk
L(rεk
j)−Tb,εm
L(rεm
j), withj=f,b,u s i n g/summationtext
j=f,b/parenleftbig
Tb,εk
L(rεk
j)−Tb,εm
L(rεm
j)/parenrightbig
as
a test function, and applying the Lipschitz continuity of pimply
/bardblTb,εk
L(rεk
b)−Tb,εm
L(rεm
b)/bardbl2
L2(Ω×Γ)≤C1/integraldisplayτ
0/bardblTb,εk
L(lεk)−Tb,εm
L(lεm)/bardbl2
L2(Ωδ×Γ)dt
+C2/integraldisplayτ
0/summationdisplay
j=f,b/bardblTb,εk
L(rεk
j)−Tb,εm
L(rεm
j)/bardbl2
L2(Ω×Γ)dt+σ(εk,εm)+C3δ1
2.
Using the a priori estimates for lεand the local strong convergence of Tb,ε
L(lε), col-
lecting the estimates from above, and applying the Gronwall inequality, we obtain
/bardblTb,εk
L(rεk
j)(τ)−Tb,εm
L(rεm
j)(τ)/bardblL2(Ω×Γ)≤C/parenleftbig
σ(εk,εm)+δ1
4/parenrightbig
forj=f,b,
whereσ(εk,εm)→0a sεk,εm→0a n dδ>0 is arbitrary. Thus, we conclude that
{Tb,ε
L(rε
j)},f o rj=f,b, are Cauchy sequences in L2(ΩT×Γ). Using the strong con-
vergence of Tb,ε
L(rε
b) and the Lipschitz continuity of p,w eo b t a i nˆ pε(x,ˆy,Tb,ε
L(rε
b))/arrowrighttophalf
p(x,DxKxˆy,rb)i nL2(ΩT×Γ). Then, passing in the weak formulation of (9.15) to the
limit asε→0 implies the macroscopic equations (9.10) for rfandrb. This concludes
the proof of the convergence up to subsequences. The strong convergence of Tb,ε
L(rε
j)
together with the estimates in Lemma 8.2, the boundedness of rε
j, withj=f,b,a n d
the regularity of DandKensures the strong l-t-s convergence of rε
j; i.e.,
lim
ε→0ε/bardblrε
j/bardbl2
L2(Γε
T)=/integraldisplay
ΩT1
|Yx|/integraldisplay
Γx|rj(t,x,y)|2dσxdxdt forj=f,b.
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LOCALLY PERIODIC UNFOLDING METHOD 1101
The nonnegativity of lεandrε
jand the uniform boundedness of rε
j, withj=
f,b(see Lemma 9.2), along with the weak convergence of Tε
L(rε
j)a n dlεensure the
nonnegativity of rjandland the boundedness of rj(t,x,y) for a.a. ( t,x)∈ΩTand
y∈Γx. Considering ( l−M1eM2t)+as a test function in the weak formulation of the
macroscopic model (9.10) and using the boundedness of rfandrb,w eo b t a i n
/bardbl(l−M1eM2t)+/bardblL∞(0,T;L2(Ω))+/bardbl∇(l−M1eM2t)+/bardblL2(ΩT)≤0.
Hence, 0 ≤l(t,x)≤M1eM2Tfora.a.(t,x)∈ΩT,whereM1≥supΩl0(x)andM1M2≥
(/bardblF(x,y,0)/bardblL∞(Ω;L∞(Yx))+|Y∗
x,K|−1/bardblβ(x,y)/bardblL∞(Ω;L∞(Yx))/bardblrb/bardblL∞(Ω;L1(Γx))).
Considering equations for the diﬀerence of two solutions of (9.10), taking l1−l2,
rf,1−rf,2,a n drb,1−rb,2as test functions in the weak formulation of the macroscopic
problem, and using the Lipschitz continuity of Fandpalong with boundedness of
rjandl, we obtain uniqueness of a weak solution of the problem (9.10). Thus, we
have that the entire sequence of weak solutions ( lε,rε
f,rε
b) of the microscopic problem
(9.1)–(9.2) convergences to the weak solut ion of the macroscopic equations (9.10).
Applying the lower semicontinuity of a norm, the ellipticity of A, and the strong
convergence of Tε
L(Aε)a n dTε
L(χε
Ω∗
ε,K)i nLp(ΩT×Y) for any p∈(1,+∞) yields
/angbracketleft|Yx|−1A(x,y)(∇l+∇yl1),∇l+∇yl1/angbracketrightΩT,Y∗
x,K
≤liminf
ε→0|Y|−1/angbracketleftTε
L(Aε)Tε
L(χε
Ω∗
ε,K)Tε
L(∇lε),Tε
L(χε
Ω∗
ε,K)Tε
L(∇lε)/angbracketrightΩT,Y
≤limsup
ε→0|Y|−1/angbracketleftTε
L(Aε)Tε
L(χε
Ω∗
ε,K)Tε
L(∇lε),Tε
L(χε
Ω∗
ε,K)Tε
L(∇lε)/angbracketrightΩT,Y
≤limsup
ε→0/angbracketleftAε∇lε,∇lε/angbracketrightΩ∗
ε,K,T= limsup
ε→0[I1+I2+I3],
where
I1=|Y|−1/angbracketleftbigˆFε(x,˜y,Tε
L(lε))−∂tTε
L(lε),Tε
L(lε)/angbracketrightbig
ΩT,Y,
I2=/integraldisplay
ΩT×ΓNε/summationdisplay
n=1√
gxεn
√
g|Yxε
n|/bracketleftBig
Tb,ε
L(βε)Tb,ε
L(rε
b)−Tb,ε
L(αε)Tb,ε
L(lεrε
f)/bracketrightBig
Tb,ε
L(lε)χΩε
ndσydxdt,
I3=/angbracketleftFε(x,lε)−∂tlε,lε/angbracketrightΛ∗
ε,K,T.
Using the estimates in Lemma 9.2, together with 0 ≤lε≤M+(lε−M)+and the
deﬁnition of Λ∗
ε,K, we obtain lim ε→0I3=0 .
Considering the strong convergence Tb,ε
L(rε
j), withj=f,b, and the local strong
convergence of Tε
L(lε)a n dTb,ε
L(lε), together with (9.5), taking las a test function in
(9.3), and using the fact that l1is a solution of the unit cell problem yields
lim
ε→0[I1+I2]=/angbracketleft|Yx|−1A(x,y)(∇l+∇yl1),∇l+∇yl1/angbracketrightΩT,Y∗
x,K.
Hence, we conclude the convergence of the energy
(9.16) lim
ε→0/angbracketleftAε∇lε,∇lε/angbracketrightΩ∗
ε,K,T=/angbracketleft|Yx|−1A(x,y)(∇l+∇yl1),∇l+∇yl1/angbracketrightΩT,Y∗
x,K,
as well as
lim
ε→0|Y|−1/angbracketleftTε
L(Aε)Tε
L(χε
Ω∗
ε,K)Tε
L(∇lε),Tε
L(∇lε)/angbracketrightΩT,Y
=/angbracketleft|Yx|−1A(x,y)(∇l+∇yl1),∇l+∇yl1/angbracketrightΩT,Y∗
x,K.
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1102 MARIYA PTASHNYK
This implies also the strong convergence of the unfolded gradient
(9.17)Tε
L(χΩ∗
ε,K)Tε
L(∇lε)→χY∗
x,K(Dx·)(∇l+D−T
x∇˜yl1(·,Dx·)) inL2(ΩT×Y).
To show the strong convergence in (9.17), we consider
/angbracketleftbig
Tε
L(Aε)Tε
L(χε
Ω∗
ε,K)(Tε
L(∇lε)−∇l−D−T
x∇˜yl1),Tε
L(∇lε)−∇l−D−T
x∇˜yl1/angbracketrightbig
ΩT×Y
=/angbracketleftbig
Tε
L(Aε)Tε
L(χε
Ω∗
ε,K)Tε
L(∇lε),Tε
L(∇lε)/angbracketrightbig
ΩT×Y
−/angbracketleftbig
Tε
L(Aε)Tε
L(χε
Ω∗
ε,K)Tε
L(∇lε),∇l+D−T
x∇˜yl1/angbracketrightbig
ΩT×Y
−/angbracketleftbig
Tε
L(Aε)Tε
L(χε
Ω∗
ε,K)(∇l+D−T
x∇yl1),Tε
L(∇lε)/angbracketrightbig
ΩT×Y
+/angbracketleftbig
Tε
L(Aε)Tε
L(χε
Ω∗
ε,K)(∇l+D−T
x∇yl1),∇l+D−T
x∇˜yl1/angbracketrightbig
ΩT×Y.
Applying the strong convergence of Tε
L(Aε)a n dTε
L(χε
Ω∗
ε,K) along with the weak con-
vergence of Tε
L(∇lε), the convergence of the energy (9.16), and the uniform ellipticity
ofA(x,y) implies the convergence (9.17).
Remark. Since in Ω∗
ε,Kwe have spatial changes both in the periodicity of the
microstructure and in the shape of perforations, the l-p unfolding operator T∗,ε
Lis not
deﬁned on Ω∗
ε,Kdirectly, and in the derivation of the macroscopic equations we use
a local extension of lεfromˆΩ∗,ε
KtoˆΩε. The local extension allows us to apply the
l-p unfolding operator Tε
Ltolε. If we have changes only in the periodicity and no
additional changes in the shape of perforations, then we can apply the l-p unfolding
operator deﬁned in a perforated domain Ω∗
εdirectly, without cons idering an extension
fromˆΩ∗
εtoˆΩε, and derive macroscopic equations in the same way as in the proof of
Theorem 9.4.
10. Discussions. The macroscopic model (9.10) derived from the microscopic
description of a signaling process in a doma in with l-p perforations reﬂects spatial
changes in the microscopic structure of a cell tissue. The eﬀective coeﬃcients of the
macroscopic model describe the impact of changes in the microstructure on the move-ment (diﬀusion) of signaling molecules (ligands) and on interactions between ligands
and receptors in a biological tissue. The mul tiscale analysis also allows us to consider
the inﬂuence of nonhomogeneous distribution of receptors in a cell membrane, as wellas nonhomogeneous membrane properties (e.g., cells with top-bottom and front-back
polarities) on the signaling process. The dependence of the coeﬃcients on the macro-
scopic variables represents the diﬀerence in the signaling properties of cells depending
on the size and/or position. For example, the changes in the size and shape of cells in
epithelium tissues are caused by the maturation process and, hence, cells of diﬀerentage may show diﬀerent activity in a signaling process. Expanding the microscopic
model by including equations for cell biomechanics and using the proposed multiscale
analysis techniques, we can also consider the impact of mechanical properties of abiological tissue with a nonperiodic microstructure on signaling processes.
Techniques of l-p homogenization allow us to consider a wider range of com-
posite and perforated materials than the methods of periodic homogenization allow.The structures of macroscopic equations obtained for microscopic problems posed in
domains with periodic and l-p microstructures are similar. If we consider the mi-
croscopic model (9.1)–(9.2) in a domain with periodic microstructure, i.e., D(x)=I
andK(x)=I,w h e r e Idenotes the identity matrix, then the macroscopic equations
(9.10) with D(x)=IandK(x)=Icorrespond to the macroscopic equations ob-
tained in [34] by considering the periodic distribution of cells and applying methods
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LOCALLY PERIODIC UNFOLDING METHOD 1103
of periodic homogenization. For some l-p microstructures, e.g., domains consisting of
periodic cells with smoothly changing perforations, it is possible to derive the same
macroscopic equations by applying periodic and l-p homogenization techniques; see,
e.g., [36, 37, 46]. However, as mentioned in the introduction, for the microscopic de-scription and homogenization of processes deﬁned in domains with, e.g., plywood-like
microstructures or on oscillating surfaces of l-p microstructures the techniques of l-p
homogenization are essential. Notice that methods of l-p homogenization are applied
to analyze microscopic problems posed in domains with nonperiodic but deterministic
microstructures, in contrast to stochastic homogenization techniques used to derivemacroscopic equations for problems posed in domains with random microstructures.
The corrector function l
1and the macroscopic diﬀusion coeﬃcient in the macro-
scopic problem (9.10) are determined by solutions of the unit cell problems (9.11),w h i c hd e pe n do nt h em a croscopicvariables x. This dependence correspondsto spatial
changes in the structure of the microscopic domains. To compute solutions of the unit
cell problems (9.11) (and hence the eﬀectiv e macroscopic coeﬃcients and the correc-
torl
1) numerically, approaches from the two-scale ﬁnite element method [38] or the
heterogeneous multiscale method [1, 2, 24, 25] can be applied. Using heterogeneousmultiscale methods, one would have to compute the solutions of (9.11) only at the
grid points of a discretization of the macroscopic domain, which requires much lower
spatial resolution than computing the microscopic problem on the scale of a singlecell. A similar approach can be applied for numerical simulations of the ordinary
diﬀerential equations determining the dy namics of receptor densities, which depend
on the macroscopic xand the microscopic yvariables as parameters.
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