www.elsevier.com/locate/anihpc
Semi-strong convergence of sequences satisfying a variational inequality
Marc Briane
a,∗, Gabriel Mokobodzki
b, François Murat
caCentre de Mathématiques, I.N.S.A. de Rennes & I.R.M.A.R., CS 14 315, 35043 Rennes cedex, France bÉquipe d’Analyse Fonctionnelle, Université Paris VI, Boîte courrier 186, 75252 Paris cedex 05, France cLaboratoire Jacques-Louis Lions, Université Paris VI, Boîte courrier 187, 75252 Paris cedex 05, France
Received 27 September 2005; received in revised form 27 November 2006; accepted 30 November 2006 Available online 28 December 2006
Abstract
In this paper we study the properties of any sequence(un)n1weakly converging to a nonnegative functionuinW01,p(Ω), p >1, and satisfying a variational inequality of type−div(an(·,∇un))fn, where(an)n1is a suitable sequence of monotone operators and(fn)n1is any strongly convergent sequence in the dual spaceW−1,p(Ω). We prove that the sequence(un− (1−ε)u)−strongly converges to 0 inW01,p(Ω)for anyε∈(0,1). We show by a counter-example that the result does not hold true ifε=0. A remarkable corollary of these strongε-convergences is that the sequence(un)n1satisfies, up to a subsequence, a kind of semi-strong convergence:(un)n1can be bounded from below by a sequence which converges to the same limitubut strongly inW01,p(Ω). We also give an example of a nonnegative weakly convergent sequence which does not satisfy this semi- strong convergence property and hence cannot satisfy any variational inequality of the previous type. Finally, in the linear case of a sequence of highly-oscillating matrices, we improve the strongε-convergences by replacing the arbitrary small constantε >0 by a sequence(εn)n1converging to 0.
©2006 Elsevier Masson SAS. All rights reserved.
Résumé
Dans cet article, on étudie les propriétés de toute suite(un)n1 qui converge faiblement dansW01,p(Ω),p >1, vers une fonction positiveuet qui satisfait une inégalité variationnelle du type−div(an(·,∇un))fn, où(an)n1est une suite convenable d’opérateurs monotones et où(fn)n1est une suite fortement convergente dans l’espace dualW−1,p(Ω). On montre que, pour toutε∈ ]0,1[, la suite(un−(1−ε)u)−converge fortement vers 0 dansW01,p(Ω), et, à l’aide d’un contre-exemple, que cette convergence n’est pas en général vraie lorsqueε=0. Un corollaire remarquable de cesε-convergences fortes est que la suite (un)n1satisfait, à une sous-suite près, une sorte de semi-convergence forte : la suite(un)n1peut en effet être minorée par une suite qui converge fortement dansW01,p(Ω)vers la même limiteu. On explicite aussi un exemple d’une suite positive et faiblement convergente qui ne vérifie pas la semi-convergence forte et qui par conséquent, n’est solution d’aucune inéquation variationnelle du type précédent. Enfin, dans le cas linéaire d’une suite fortement oscillante de matrices, on améliore laε-convergence en remplaçant la constante arbitraireε >0 par une suite(εn)n1convergeant vers 0.
©2006 Elsevier Masson SAS. All rights reserved.
* Corresponding author.
E-mail addresses:mbriane@insa-rennes.fr (M. Briane), murat@ann.jussieu.fr (F. Murat).
0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2006.11.004
MSC:35J60; 35B27 Keywords:???
Mots-clés :Homogenization ; Monotone operator ; Potential ; Variational inequality
1. Introduction
In [2] we proved the following result:
For any pair of sequences(Bn, Cn)n1of equi-coercive and bounded matrix-valued functions defined in a bounded open setΩofRd, for any pair of sequences(vn, wn)n1weakly converging to(v, w)inH01(Ω)2, and for any pair of strongly convergent sequences(hn, gn)n1inH−1(Ω)2, such that for anyn1,
−div(Bn∇vn)gn and −div(Bn∇vn)hn inD(Ω), (1.1)
we have the semi-continuity property
∀ψ∈C∞ Ω
, ψ0, lim inf
n→+∞
Ω
ψ Bn∇vn· ∇wn
Ω
ψ B∗∇v· ∇w, (1.2)
where B∗ is the H-limit (well-defined up to a subsequence) of the sequence (Bn)n1 in the sense of the H-convergence of Murat and Tartar [6].
One of the key-ingredient of the proof (1.2) is given by the following auxiliary result: Under the same assumptions for the sequences(Cn)n1and(wn)n1, ifTε, forε >0, is a smoothε-approximation of the function(t →t−)onR such that
Tε(0)=0 and ∀t∈R,
|Tε(t )−t−|ε,
−1Tε(t )0, then we have the strong convergence
Tε(wn−w)−→0 inH01(Ω). (1.3)
Moreover, whenBn=B is independent ofnthe strong convergence (1.3) holds withTε(t )=t−. But in general, the sequence(wn−w)−does not strongly converge to 0 because of the oscillations ofBn like in the homogenization theory (see Remark 3.6 of [2]). The proofs of (1.3) and (1.2) are rather technical and need a fine result of potential theory.
The purpose of this study is to give another and simpler strong convergence of type (1.3) under the extra assumption of nonnegativity of the limit and in a nonlinear framework. This new approach has a surprising consequence on the behavior on the sequences satisfying a variational inequality such (1.1). The main result of the paper (see Theorem 2.1) is the following:
For any sequence (an)n1 of uniformly p-monotone, p >1, and uniformly bounded Carathéodory functions fromΩ×RdintoRd(see Section 2 for details), for any sequence(un)n1weakly converging tou0 inW01,p(Ω), and for any strongly convergent sequence(fn)n1inW−1,p(Ω),p:=pp−1>1, such that for anyn1,
−div
an(·,∇un)
fn inD(Ω), (1.4)
we have thesemi-strongconvergences
∀ε∈(0,1),
un−(1−ε)u−
−→0 strongly inW01,p(Ω). (1.5)
Contrary to (1.3) we do not consider in (1.5) anε-approximation of(t →t−). We keep this function but we have to introduce the shiftεuinside to obtain the strong convergence. The price to pay is to assume the nonnegativity of the weak limitu.
A noteworthy corollary of (1.5) (see Corollary 2.4) is that there exist a subsequence(uθ (n))n1and a sequence (vk)k1strongly converging touinW01,p(Ω), such that
∀nk, uθ (n)vk a.e. inΩ. (1.6)
Thanks to inequality (1.6) the qualifying “semi-strong” for convergence (1.5) takes its whole meaning.
The strong convergences (1.5) are in some sense optimal since we cannot takeε=0 ifandepends actually onn. We give a counter-example (see Proposition 3.1) showing that the oscillations ofanprevent from the strong convergence of(un−u)−. The assumptions cannot be relaxed anymore. On the one hand, it is easy to check that the nonnegativity ofuis a consequence of (1.5). On the other hand, the variational inequality (1.4) is a crucial assumption to obtain (1.5) and (1.6). Indeed, there exists a nonnegative sequenceunweakly converging inW01,p(Ω)which does not satisfy (1.6) and hence neither (1.5) (see Proposition 3.2). Such a sequence has the particular property to satisfy none variational inequality of type (1.4). This counter-example is quite general since it holds true provided thatpd.
If it is not possible to takeε=0 in the strong convergence (1.5), a natural question is to know if we can replace the arbitrary small but fixed constantε >0 in (1.5) by a positive sequence(εn)n1converging to 0. We prove that the answer is positive (see Theorem 4.1) in the case of any sequence of highly-oscillating matrices, i.e.an(x, ξ )= A(x/τn)ξ, where Ais a periodic matrix-valued function. In this linear framework we assume that the variational inequality (1.4) holds true and thatubelongs toW2,d(Ω). Then, there exists a positive sequenceεn, withτnεn1, such that
un−(1−εn)u−
−→0 strongly inW01,p(Ω). (1.7)
The paper is organized as follows: Section 2 is devoted to the results (1.5) and (1.6) and to their proof. In Section 3 we give two counter-examples showing the optimality of these results. In Section 4 we prove the strong convergence (1.7) in the case of a sequence of highly-oscillating matrices.
2. Semi-strong convergence results
LetΩbe a bounded open set ofRd,d1, letp∈ (1,+∞[andp:=pp−1. Letabe a fixed function fromΩ×Rd intoRdwhich satisfies the following properties:
• ais a Carathéodory function, i.e.
a.e.x∈Ω, a(x,·)is continuous onRd,
∀ξ∈Rd, a(·, ξ )is measurable onΩ;
• ais coercive, i.e. there exists a positive constantαand a nonnegative functionγ inL1(Ω)such that a.e.x∈Ω,∀ξ∈Rd, a(x, ξ )·ξα|ξ|p−γ (x);
• ais strictly monotone, i.e.
a.e.x∈Ω,∀ξ=η∈Rd,
a(x, ξ )−a(x, η)
·(ξ−η) >0;
• ais bounded, i.e. there exists a positive constantβ and a nonnegative functionδinLp(Ω)such that a.e.x∈Ω,∀ξ∈Rd, a(x, ξ )β
|ξ| +δ(x)p−1
.
Let(an)n1be a sequence of functions fromΩ×Rd intoRd which satisfies the following properties:
(i) for anyn1,anis a Carathéodory function,
(ii) anis uniformly monotone with respect toa, i.e. for anyn1, a.e.x∈Ω,∀ξ, η∈Rd,
an(x, ξ )−an(x, η)
·(ξ−η)
a(x, ξ )−a(x, η)
·(ξ−η); (iii) anis uniformly bounded, i.e. there exists a positive constantβ such that, for anyn1,
a.e.x∈Ω,∀ξ∈Rd, an(x, ξ )β|ξ|p−1. Under the previous assumptions we have the following result:
Theorem 2.1.Let(un)n1be a sequence inW01,p(Ω)such that
un u weakly inW01,p(Ω). (2.1)
Assume that there exists a strongly convergent sequence(fn)n1inW−1,p(Ω)such that
−div
an(·,∇un)
fn inD(Ω). (2.2)
Then, we have the implication
u0a.e. inΩ ⇒
⎧⎪
⎨
⎪⎩
∀v∈W01,p(Ω),
0vua.e. inΩ and v < ue. in{u >0}, (un−v)−→0 strongly inW01,p(Ω)
(2.3) (a.e. for almost everywhere and e. for everywhere).
Remark 2.2.It is easy to check that the functionv:=(1−ε)usatisfies the requirements of (2.3) for anyε∈(0,1).
In this case, we obtain the equivalence
u0 a.e. inΩ ⇐⇒ ∀ε∈(0,1),
un−(1−ε)u−
→0 strongly inW01,p(Ω). (2.4) The implication (⇒) is an immediate consequence of (2.3) withv:=(1−ε)u. Conversely, assume that the right-hand side of (2.4) holds true. Then, the sequence(un−(1−ε)u)−weakly converges toεu−and strongly to 0 inW01,p(Ω).
Therefore, the uniqueness of the weak limit inW01,p(Ω)implies thatu−=0 a.e. inΩ, or equivalently,u0 a.e.
inΩ.
In the sequel, we will only focus on the strong convergence (2.4).
Remark 2.3.The variational inequality (2.2) implies the strong convergence of the negative part of the sequence (un−u), up to an arbitrary small shiftεu. In [2] we proved that the strong convergence holds withε=0, without assuming the nonnegativity ofu but assuming that an does not depend on n. In general, the sequence (un−u)− does not strongly converge to zero inW01,p(Ω), even ifu0 a.e. inΩ. This is due to the oscillations effects of the sequencean(see Proposition 3.1 below). Moreover, inequality (2.2) cannot be relaxed (see Proposition 3.2 below).
The previous semi-strong convergence result allows us to obtain a strong approximation from below of the se- quenceun:
Corollary 2.4.Let(un)n1be a sequence inW01,p(Ω)which satisfies assumptions(2.1)withu0a.e., and(2.2).
Then, there exist a subsequence(uθ (n))n1and a sequence(vk)k1strongly converging touinW01,p(Ω), such that
∀nk, uθ (n)vk. (2.5)
Proof of Theorem 2.1. Assume thatu0 a.e. inΩ. Letvbe a function inW01,p(Ω)such that 0vua.e. inΩ andv < ueverywhere in{u >0}. SetEn:= {un−v <0}. By using successively the uniform monotonicity(ii)ofan, the variational inequality (2.2) and the strong convergence offntof inW−1,p(Ω), we have
0−
Ω
a(x,∇un)−a(x,∇v)
· ∇(un−v)−dx
=
Ω
a(x,∇un)−a(x,∇v)
· ∇(un−v)1Endx
Ω
an(x,∇un)−an(x,∇v)
· ∇(un−v)1Endx
= −
Ω
an(x,∇un)−an(x,∇v)
· ∇(un−v)−dx
Ω
an(x,∇v)· ∇(un−v)−dx−
fn, (un−v)−
W−1,p(Ω),W01,p(Ω)
=
Ω
an(x,∇v)· ∇(un−v)−dx−
f, (u−v)−
W−1,p(Ω),W01,p(Ω)+o(1)
=
Ω
an(x,∇v)· ∇(un−v)−dx+o(1) (since(u−v)−=0 a.e. inΩ)
= −
Ω
an(x,∇v)· ∇(un−v)1Endx+o(1). (2.6)
Moreover, using the boundedness (iii) ofan, the boundedness ofuninW01,p(Ω)and the Hölder inequality yield
Ω
an(x,∇v)· ∇(un−v)1Endx c
Ω
|∇v|p1Endx p1
. (2.7)
Since v0 a.e. in Ω, we have∇v1En = ∇v1En∩{v>0} a.e. in Ω. Let E be the subset ofΩ, of thex satisfying the pointwise convergenceun(x)→u(x)and the inequalityv(x)u(x). Up to a subsequence, still denotedn, the setΩ\Ehas a zero Lebesgue measure. Letx∈E. Assume by contradiction that there exists a subsequencensuch thatx∈En∩ {v >0}, for anyn1. Then, passing to the limit in the inequalityun(x) < v(x)yieldsu(x)v(x), and consequently,u(x)=v(x). Sincev(x) >0, we also haveu(x) >0 and thusv(x) < u(x)by the assumption onv, which establishes a contradiction. So, anyx∈Ebelongs to a finite number of setsEn∩ {v >0},n1. Therefore, the sequence1En∩{v>0}converges to 0 a.e. inΩ. The Lebesgue dominated convergence theorem thus implies that the right-hand side of (2.7) tends to 0. This, combined with estimate (2.6), implies that
Ω
a(x,∇un)−a(x,∇v)
· ∇(un−v)−dx −→
n→+∞0. (2.8)
Finally, following the first step of the proof of Theorem 2.19 in [2], we deduce from convergence (2.8) and the properties ofa, the strong convergence of (2.3). 2
Proof of Corollary 2.4. By the strong convergences (2.4) of Remark 2.2, for each integerk1, the sequence∇(un− (1−k−1)u)strongly converges to 0 inLp(Ω)d. Therefore, there exists a subsequenceθk(n)ofnsuch that
∀n1, ∇
uθk(n)−
1−k−1 u−
Lp(Ω) 1 2n.
We may also assume thatθk+1(n)is a subsequence ofθk(n), for anyk1. Then, by considering the diagonal extrac- tionθ (n):=θn(n), we obtain, for anynk, the equalityθn(n)=θk(nk)for somenkn, hence the estimate
∇
uθ (n)−
1−k−1 u−
Lp(Ω)=∇
uθk(nk)−
1−k−1 u−
Lp(Ω) 1 2nk 1
2n. (2.9)
In particular, thanks to the Poincaré inequality, for any k1, the series
nk(uθ (n)−(1−k−1)u)− converges inW01,p(Ω). We can thus define, for eachk1, the function
vk:=
1−k−1 u−
nk
uθ (n)−
1−k−1 u−
∈W01,p(Ω).
On the one hand, by virtue of (2.9) we have
∇vk− ∇uLp(Ω)1
k∇uLp(Ω)+
nk
1 2n −→
k→+∞0,
which implies that the sequencevk strongly converges touinW01,p(Ω). On the other hand, we have, for anynk, uθ (n)=
1−k−1 u+
uθ (n)−
1−k−1 u+
−
uθ (n)−
1−k−1 u−
1−k−1 u−
uθ (n)−
1−k−1 u−
vk, which yields (2.5) and concludes the proof. 2
3. Counter-examples
The first counter-example shows that in general one cannot takeε=0 in the semi-strong convergence (2.4) of Theorem 2.1.
Proposition 3.1.There exist a sequence(an)n1and a nonnegative sequence(un)n1which satisfy assumptions(2.1) and(2.2), such that(un−u)−does not strongly converge to0inW01,p(Ω).
The second counter-example provides a nonnegative and weakly convergent sequence inW01,p(Ω), for which the result of Corollary 2.4 and thus the one of Theorem 2.1, does not hold true.
Proposition 3.2.Assume thatpd. Then, there exists a nonnegative weakly convergent sequence inW01,p(Ω), such that inequality(2.5)is satisfied by none of its subsequences.
Remark 3.3.Forp > d, the situation is completely different. Indeed, letΩ be a smooth bounded open subset ofRd and let(un)n1be a sequence which weakly converges touinW1,p(Ω). Then, by the Morrey embedding theorem there exists a subsequence(uθ (n))n1which converges uniformly touinΩ, and thus satisfying
δk:=sup
nk
uθ (n)−uL∞(Ω) −→
k→+∞0.
Therefore, the sequences (uθ (n))n1 and (vk :=u−δk)k1 satisfy inequality (2.5) without any assumption of type (2.2).
Proof of Proposition 3.1. The dimension isd:=1 andΩ:=(0,1). For each integern1, letρn be the function defined in(0,1)by
ρn(x):=
1
2 ifx∈ [nk,kn+2n1),
3
2 ifx∈ [nk+2n1,k+n1) fork∈ {0, . . . , n−1}, and letunbe the solution of
− ρn−1un
=1 in(0,1), un(0)=un(1)=0.
The sequenceunclearly satisfies the assumptions (2.1) and (2.2) of Theorem 2.1 in the linear case. The weak limit of uninH01((0,1))isu(x):=12x(1−x).
An easy but rather long computation yields for anyx∈ [pn,pn+2n1],p∈ {0, . . . , n−1}, un(x)−u(x)= − p
8n2+1 4
x−p n
x+p
n −1 + 1
8n x 0
ρn(t )dt.
Therefore, ifx <12 andx∈ [pn +4n1,pn +2n1], thenx+pn −1<2x−1<0, hence un(x)−u(x) 1
16n(2x−1)+ 3
16nx= 1
16n(5x−1).
In particular, we have {un−u <0} ⊃
0,1 5
∩ n−1
k=0
k n+ 1
4n,k n+ 1
2n
,
which implies lim inf
n→+∞
{un−u <0} ∩
0,1 5
1
20. (3.1)
On the other hand, we have un(x)−u(x)2
=
ρn(x)−11 2−x
+ 1
8nρn(x) 2
1
4 1
2−x 2
+O 1
n
,
which combined with estimate (3.1) yields lim inf
n→+∞
1/4 0
un−u2
1{un−u<0}dx 1 43× 1
20>0.
Therefore,(un−u)−does not strongly converge to 0 inH01((0,1)). 2
Proof of Proposition 3.2. Let Y :=(−12,12)d. Denote byBr the ball of radiusr >0, centered at the origin. Let R∈(0,12)and(Rn)n1be a sequence in(0, R)converging to 0. LetVn, forn1, be the unique solution inW#1,p(Y ) (the set of theY-periodic functions inWloc1,p(Rd)) of
⎧⎪
⎨
⎪⎩ div
|∇Vn|p−2∇Vn
=0 inBR\BRn, Vn=1 inY\BR,
Vn=0 inBRn.
(3.2) Let(εn)n1be a positive sequence converging to 0. We consider theεn-rescaled function defined by
ˆ
vn(x):=Vn x
εn
, forx∈Ω. (3.3)
The functionvˆn(x)was introduced in [3] (forp=2) to obtain a capacitary effect in homogenization. The sequence (vˆn)n1satisfies the following result:
Lemma 3.4.Assume thatpd and set Rn:=
εp/(dn −p) ifp < d, exp
−εp/(1n −p)
ifp=d. (3.4)
Then, we have ˆ
vn1 weakly inW1,p(Ω). (3.5)
Setωn:= {ˆvn=0} ∩Ω. Then, there exists a positive constantCsuch that the following estimate holds
∀v∈W01,p(Ω), 1
|BRn|
ωn
v−
Ω
v
C∇vLp(Ω). (3.6)
Let us prove that the result of Proposition 3.2 is satisfied under the assumptions of Lemma 3.4. Letϕbe a nonneg- ative and nonzero function inCc∞(Ω)and considerun:=ϕvˆnforn1. The sequenceunis nonnegative and by (3.5) weakly converges toϕinW01,p(Ω). Assume by contradiction that there exists a subsequence, still denotedun, and a sequencevkstrongly converging toϕinW01,p(Ω), such that inequality (2.5) holds. Thanks to estimate (3.6) we have, for anynk,
1
|BRn|
ωn
vk−
Ω
vk
1
|BRn|
ωn
ϕ−
Ω
ϕ
+C∇vk− ∇ϕLp(Ω). Moreover, the regularity ofϕand the asymptotic|ωn| ∼ |Ω||BRn|imply that
n→+∞lim 1
|BRn|
ωn
ϕ=
Ω
ϕ,
which combined with the strong convergence ofvktoϕinW01,p(Ω), gives 1
|BRn|
ωn
vk−
Ω
vk
on(1)+ok(1), (3.7)
where on(1)(respectively ok(1)) denotes a sequence converging to 0 asn→ +∞(respectivelyk→ +∞). Then, by using inequality (2.5) and the fact thatun=0 inωn, we deduce from (3.7) that
Ω
vk 1
|BRn|
ωn
un+on(1)+ok(1)=on(1)+ok(1). (3.8)
Therefore, passing successively to the limitsn→ +∞andk→ +∞in (3.8), implies
Ω
ϕ0,
which yields the contradiction. 2
Proof of Lemma 3.4. Proof of(3.5)The functionVndefined by (3.2) is radial in the setBR\BRn. More precisely, we have, for anyr∈(Rn, R),
Vn(r):=
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩ 1+ r
p−d p−1 −R
p−d p−1
R
p−d p−1 −R
p−d p−1
n
ifp < d,
1+ lnr−lnR lnR−lnRn
ifp=d,
hence there exists a positive constantcd,pindependent ofnsuch that ∇Vn(r)p
Lp(Y )=
⎧⎨
⎩cd,p R
p−d p−1
n −R
p−d p−11−p
∼cd,pRdn−p ifp < d, cd,p(lnR−lnRn)1−p∼cd,p|lnRn|1−p ifp=d.
This estimate, combined with the choice (3.4) of Rn, implies that the sequence vˆn defined by (3.3) is bounded inW1,p(Ω). Moreover, sinceVn=1 in the setY\BR, the weak limit ofvˆnis 1, which yields (3.5).
Proof of(3.6).Denote bySr the sphere centered at the origin and of radiusr >0. LetV ∈C1(Y )and letVbe the function defined in spherical coordinates byV (r, ξ ) :=V (y), wherey=rξ withr >0 andξ∈S1. By starting from the equality
V (R, ξ ) −V (R n, ξ )= R Rn
∂V
∂r(r, ξ )dr
and by using the Hölder inequality, we obtain the inequality V (R, ξ )−V (R n, ξ )αn
R Rn
∂V
∂r(r, ξ )
prd−1dr 1
p
,
where αn:=
⎧⎪
⎨
⎪⎩ p−1
p−d(R
p−d p−1 −R
p−d p−1
n )
p1
ifp < d, [lnR−lnRn]p1 ifp=d.
(3.9) Then, integrating the previous inequality with respect toξ∈S1 and using the Hölder inequality with respect to the integral inξ, imply
−
SRn
V − −
SR
V
cαn∇VLp(Y ), (3.10)
where−
denotes the average-value andcis a positive constant. On the other hand, using a scaling of orderRnin the Poincaré–Wirtinger type inequality
−
BR
W− −
SR
W
c∇WLp(Y ), withW (y):=V (Rny), implies that
−
BRn
V− −
SRn
V cR
p−d p
n ∇VLp(RnY )cR
p−d p
n ∇VLp(Y ). (3.11)
The following Poincaré–Wirtinger type inequality also holds true:
−
Y
V − −
SR
V
c∇VLp(Y ). (3.12)
Then, combining estimate (3.10) with (3.11) and (3.12), we get −
BRn
V− −
Y
V c
αn+R
p−d p
n +1
∇VLp(Y ), (3.13)
wherecis a positive constant independent of the functionV. Letvbe a function inW01,p(Ω), extended by 0 inRd\Ω. Then, putting, forκ∈Zd, the functionV (y):=v(κ+εny)in estimate (3.13) and summing overκ∈Zd, it follows
1
|BRn|
ωn
v−
Ω
v c
εnαn+εnR
p−d p
n +εn
∇vLp(Ω). (3.14)
Moreover, by the definition (3.9) of αn and the choice (3.4) ofRn, the sequencesεnαn andεnR
p−d p
n are bounded.
Therefore, (3.14) yields estimate (3.7). 2
4. The case of highly-oscillating linear operators
We restrict ourselves to a sequence of linear operators defined by highly-oscillating matrix-valued functions in a bounded open setΩofRd,d1.
LetY :=(0,1)d, letAbe aY-periodic matrix-valued function onRdand letα, βbe two positive constants such that
a.e.y∈Rd,∀ξ∈Rd, A(y)ξ·ξα|ξ|2 and A(y)−1ξ·ξβ−1|ξ|2.
Let(τn)n1be a positive sequence converging to 0 and let(An)n1be the sequence of oscillating matrices defined by
An(x):=A x
τn
a.e.x∈Ω. (4.1)
Let (e1, . . . , ed) be the canonical basis ofRd. By [1] we know that An H-converges, in the sense of Murat and Tartar [6], to the constant matrixA∗defined by
A∗ei:=
Y
A(y)
ei− ∇χi(y)
dy, fori∈ {1, . . . , d}, (4.2)
whereχi is the unique function inH#1(Y ), with zero average-value inY, solution of div(Aei−A∇χi)=0 inD
Rd
. (4.3)
Moreover, for any sequenceunconverging touweakly inH01(Ω)such that div(An∇un)is compact inH−1(Ω), we define the so-called corrector
¯
un:=u−τn d i=1
χi x
τn
∂u
∂xi
. (4.4)
Indeed, ifuis smooth enough the sequenceu¯nstrongly converges touinHloc1 (Ω).
In this framework, Theorem 2.1 can be improved in the following way:
Theorem 4.1.Let Ω be a bounded open set of Rd,d1, with a Lipschitz boundary. Let(un)n1 be a sequence weakly converging touinH01(Ω), such that
u0a.e. inΩ and u∈W2,d∨2(Ω), (4.5)
whered∨2denotes the maximum betweendand2. Assume that there exists a sequence(fn)n1strongly converging inH−1(Ω), such that
−div(An∇un)fn inD(Ω). (4.6)
Then, there exists a positive sequence(εn)n1converging to0such that un−(1−εn)u−
−→0 strongly inH01(Ω). (4.7)
Proof of Theorem 4.1. First, we need to modify the corrector (4.4) by introducing truncatures and a cut-off function
¯
un:=u−τn d i=1
ψn(x)Tkn(χi) x
τn
Tkn
∂u
∂xi
, (4.8)
whereTk, fork∈N, is the function defined byTk(t ):=max(−k,min(k, t )), fort∈R,(kn)n1 is a sequence of positive integers which tends to+∞, and(ψn)n1is a sequence of functions inC01(Ω)satisfying, for anyn1,
⎧⎨
⎩
0ψn1 inΩ
ψn(x)=1 if dist(x, ∂Ω) > ηn, whereηn→0,
|∇ψn|cη−n1 inΩ.
Such a sequenceψnexists sinceΩis regular. So, the functionu¯nbelongs toH01(Ω).
The proof is then divided in two steps:
First step:(un− ¯un)−strongly converges to 0 inH01(Ω).
We get rid of the cut-off functionψnby introducing the new function
˜
un:=u−τn d i=1
Tkn(χi) x
τn
Tkn ∂u
∂xi
. (4.9)
We have
∇ ˜un− ∇ ¯un=τn d i=1
∇ψn(x)Tkn(χi) x
τn
Tkn ∂u
∂xi
+ d i=1
ψn(x)−1
∇Tkn(χi) x
τn
Tkn ∂u
∂xi
+τn d
i=1
ψn(x)−1 Tkn(χi)
x τn
∇
Tkn ∂u
∂xi
. (4.10)
Sinceχi∈W#1,p(Y ), for somep >2, by the Meyers theorem [5], and since∇u∈L
2p
p−2(Ω)dby (4.5) and the Sobolev embedding theorem, the first term of the right-hand side of (4.10) is O(τnηn−1)inL2(Ω)-norm by the Hölder inequal- ity. Similarly, by the Hölder inequality, the second term is O(ηγn)inL2(Ω)-norm, for anyγ < p2p−2. Finally, since
∇2u∈Ld∨2(Ω)d×dby (4.5), the last term of (4.10) is O(τnkn)inL2(Ω)-norm. Then, choosingknandδnsuch that
n→+∞lim τn
kn+ηn−1
=0,
