On the asymptotic behaviour of elliptic problems with periodic data

Partial Differential Equations

On the asymptotic behaviour of elliptic problems with periodic data

Michel Chipot, Yitian Xie

University of Zürich, Angewandte Mathematik, Winterthurerstrasse, 190, CH-8057 Zürich, Switzerland Received 30 August 2004; accepted 7 September 2004

Available online 25 September 2004 Presented by Philippe G. Ciarlet

Abstract

We study the asymptotic behaviour of the solution of elliptic problems with periodic data when the size of the domain on which the problem is set becomes unbounded. To cite this article: M. Chipot, Y. Xie, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Résumé

Sur le comportement asymptotique de problèmes elliptiques à données périodiques. On s’intéresse au comportement asymptotique de la solution de problèmes elliptiques à données périodiques lorsque la taille de l’ouvert sur lequel le problème est posé devient infinie. Pour citer cet article : M. Chipot, Y. Xie, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Version française abrégée

SoitT un nombre positif. Pourn, k1 entiers on pose Ω_n=(−nT , nT )^k, Q=(0, T )^k.

On dit quef∈L¹_Loc(R^k)estT-périodique dans toutes les directions si l’on a f (x+T e_j)=f (x) p.p.x∈R^k, ∀j=1, . . . , k.

((e_j)désigne la base canonique deR^k). On considère alorsa_ij,i, j=1, . . . , k,a,f_i,i=0, . . . , k, des fonctions T-périodiques dans toutes les directions. SoitV_nun sous espace deH¹(Ω_n)tel que

V_nest fermé dansH¹(Ω_n), H₀¹(Ω_n)⊂V_n⊂H¹(Ω_n).

E-mail addresses: [email protected] (M. Chipot), [email protected] (Y. Xie).

1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.

doi:10.1016/j.crma.2004.09.007

On désigne parunla solution faible de







un∈Vn,

Ωn

aij(x)∂xiun∂xjv+a(x)unvdx=

Ωn

f0v+fi∂xivdx ∀v∈Vn, et paru_∞la solution de









u_∞∈H_per¹ (Q),

Q

aij(x)∂xiu_∞∂xjv+a(x)u_∞vdx=

Q

f0v+fi∂xivdx ∀v∈H_per¹ (Q),

oùH_per¹ (Q)est defini par

H_per¹ (Q)= {v∈H¹(Q)|vestT-périodique dans toutes les directions}. Sous les hypothèses usuelles d’ellipticité on se propose de montrer que

– lorsque 0a(x)Λ,a≡0, pour toutn₀>0 et toutr >0, il existe une constanteCindépendante dentelle que

|u_n−u_∞|H¹(Ω_n

0) C

n^r.

(Par souci de simplicité nous donnerons ici la preuve de ce résultat dans le cas où 0< λa(x)renvoyant le lecteur à [2] pour le cas général.)

– Lorsquea≡0,

Qf0dx=0,Vn=H₀¹(Ωn), et siu_∞est solution de (7) de moyenne nulle, alors il existe une constanteCtelle que (à une sous suite près)

un u_∞+C dansL^∞(R^k)faible∗.

Autrement dit, dans presque tous les cas (sauf quand a≡0,

Qf0≡0 où un est non borné), les données périodiques forcent la convergence sur tout compact vers une fonction périodique.

1. Introduction

LetT denote a positive number. Forn,kintegersn, k1 we set

Ωn=(−nT , nT )^k, Q=(0, T )^k. (1)

We say thatf ∈L¹loc(R^k)isT-periodic in all directions if it holds that

f (x+T ej)=f (x) a.e.x∈R^k, ∀j=1, . . . , k. (2) ((ej)denotes the canonical basis ofR^k). Consider then

aij, i, j=1, . . . , k, a, functions inL^∞(R^k),T-periodic in all directions, (3) fi, i=0, . . . , k, functions inL²(Q),T-periodic in all directions. (4) (We suppose of course that all the functions we use are defined in allR^k– extended eventually by periodicity.) Let us denote byVna subspace ofH¹(Ωn)such that

Vnis closed inH¹(Ωn), H₀¹(Ωn)⊂Vn⊂H¹(Ωn). (5)

(We refer the reader to [5,1] for the notation used in this Note.) We denote byunthe weak solution to







u_n∈V_n,

Ωn

aij(x)∂xiun∂xjv+a(x)unvdx=

Ωn

f0v+fi∂xivdx ∀v∈Vn, (6) and byu_∞the solution to









u_∞∈H_per¹ (Q),

Q

aij(x)∂xiu_∞∂xjv+a(x)u_∞vdx=

Q

f0v+fi∂xivdx ∀v∈H_per¹ (Q), (7) whereH_per¹ (Q)is defined by

H_per¹ (Q)= {v∈H¹(Q)|visT-periodic in all directions}. (8) Under the usual ellipticity conditions described below it is clear that when 0a,a≡0 then both (6), (7) admit a unique solution. Then we would like to show thatu_nconverges towardsu_∞.

In the case wherea≡0, that we call the degenerate case, (6) possesses a unique solution whenVn=H₀¹(Ωn).

Moreover, if we impose tou_∞to be of average of 0 then (7) admits also a unique solution. Then, roughly speaking, whenf₀is of average 0, we will show that up to a constantu_nconverges towardsu_∞.

2. The nondegenerate case

In this section we assume that for some positive constantλ,Λit holds that

0< λa(x)Λ a.e.x∈R^k, (9)

λ|ξ|²a_ij(x)ξ_iξ_j a.e.x∈R^k,∀ξ∈R^k. (10)

((10) is the usual ellipticity condition – there is no loss of generality in assumingλto be the same in (9) and (10)).

Clearly, under the above assumptions there exists a unique solution to (6) and also to (7). Moreover we have:

Theorem 2.1. For anyn₀>0 and any exponentr >0 there exists a constantCindependent ofnsuch that

|u_n−u_∞|_H¹_(Ωn

0) C

n^r. (11)

In the above estimate – as in the following – we set

|u|H¹(Ω)=

Ω

|∇u|²+u² dx 1/2

. (12)

In order to prove our theorem, we will need some lemmas. First Lemma 2.2 (Estimate ofun). It holds that

|un|²_H1(Ωn)2^k

λ²|f|²2,Qn^k (13)

where we have set

|f|²_2,Q=

Q

|f|²dx=

Q

k i=0

f_i²dx. (14)

Proof of the Lemma. We takev=unas test function in (6). Using (9), (10) and the Cauchy–Schwarz inequality it comes after some easy computations

λ|u_n|²_H1(Ωn)

Ωn

a_ij(x)∂x_iu_n∂x_ju_n+au²_ndx=

Ωn

f₀u_n+f_i∂x_iu_ndx

Ωn

|f|²dx 1/2

|u_n|_H¹_(Ωn)

(|f|²=_k

1=0f_i^k). (14) follows easily due to the periodicity off =(f₀, . . . , f_k). 2

Next we will use the following result well known in homogenization theory (see [4] for an idea of the proof).

Lemma 2.3. The solutionu_∞of (7) satisfies

−∂x_j

a_ij(x)∂x_iu_∞

+a(x)u_∞=f₀−∂x_if_i inD(R^k). (15)

Proof of Theorem 2.1. Letbe the piecewise continuous affine function such that =1 on

−1 2,1

2

, =0 outside(−1,1), is affine on

−1,−1 2

, 1

2,1

. (16)

It is clear that it holds that

||2. (17)

Moreover, for anyn1n (u_n−u_∞)

k i=1

² xi

n₁T

:=(u_n−u_∞)Π² (18)

is a test function for (6). It follows that it holds – see Lemma 2.3

Ωn1

a_ij∂x_i(u_n−u_∞)∂x_j

(u_n−u_∞)Π² +a(u_n−u_∞)²Π²dx=0. (19) This leads to

Ωn1

a_ij∂x_i(u_n−u_∞)∂x_j(u_n−u_∞)Π²+a(u_n−u_∞)²Π²dx= −2

Ωn1

a_ij∂x_i(u_n−u_∞)∂x_jΠ (u_n−u_∞)Π.

Since∂x_jΠ=_n¹

1T(_n^xj

1T)

i=j(_n^xi

1T)we derive easily λ

Ωn1

∇(u_n−u_∞)²+(u_n−u_∞)² Π²dx C n1

Ωn1

∇(u_n−u_∞)|u_n−u_∞|Πdx

whereC=C(T , aij)is independent ofn. Using Cauchy–Schwarz inequality we obtain λ

Ωn1

{|∇(u_n−u_∞)|²+(u_n−u_∞)²}Π²dx C n₁

Ωn1

∇(u_n−u_∞)²Π²dx 1/2

Ωn1

(u_n−u_∞)²dx 1/2

.

Thus – for some constantCindependent ofn–

Ωn1

∇(un−u_∞)²+(un−u_∞)² Π²dx C n²₁

Ωn1

∇(un−u_∞)²+(un−u_∞)²dx.

Due to the definition ofΠ this implies

|un−u_∞|H¹(Ω_n1/2) C

n₁|un−u_∞|H¹(Ωn1) ∀n1n.

Choosingn₁=₂pⁿ−1 and iterating the above formula leads to

|u_n−u_∞|H¹(Ω_n/2p) C

n^p|u_n−u_∞|H¹(Ω_n) C n^p−k/2

(cf. Lemma 2.2). Choosing₂ⁿpn0,p−^k₂> rthe result follows. 2

Remark 1. The method allows us to deal with more general periodic data (cf. [2]) also the parabolic analogue could be considered, as well as nonlinear versions – cf. [3,1]. We can extend the results to more generalΩ_nthan (1).

Note also that our convergence result does not depend on our boundary conditions on∂Ω_n. It is also valid when (9) is replaced by

0a(x)Λ a.e.x∈R^k, a≡0, (20)

(see [2] for details).

3. The degenerate case In this section we assume that

a≡0. (21)

Moreover we consider only the case of homogeneous boundary conditions that is to say the case

V_n=H₀¹(Ω_n). (22)

Then it is easy to see that, in order to preventu_nsolution of (6) to be unbounded, one has to assume

Q

f₀(x)dx=0. (23)

Under this assumption the second hand side of (7) defines a continuous linear form on H_per¹ (Q)=

v∈H_per¹ (Q)

Q

vdx=0

(24) and by the Lax–Milgram theorem there exists a uniqueu_∞solution to









u_∞∈ H_per¹ (Q),

Q

a_ij(x)∂_xiu_∞∂_xjvdx=

Q

f₀v+f_i∂_xivdx ∀v∈ H_per¹ (Q). (25) Moreover we have:

Theorem 3.1. Under the above assumptions, if in addition

u_∞∈L^∞(Q), (26)

there exists a subsequence ofu_nthat we still label bynsuch that

u_n u_∞+C inL^∞(R^k) weak∗ (27)

(unis supposed to be extended by 0 outsideΩn,Cdenotes some constant).

Proof. By (15), (6) we remark thatun−u_∞satisfies −∂_xj

a_ij(x)∂_xi(u_n−u_∞) =0 inΩ_n, un−u_∞= −u_∞ on∂Ωn.

By the maximum principle,un−u_∞is uniformly bounded inR^k and there isv_∞∈L^∞(R^k)such that – up to a subsequence:

un−u_∞ v_∞ inL^∞(R^k)weak∗.

The above convergence takes also place inD(R^k)and thus it holds that

−∂_xj

a_ij(x)∂_xiv_∞

=0 inD(R^k).

By the Liouville theorem (see [6,7] for references) it follows that v_∞=Cst

which completes the proof. 2

Remark 2. In the case of dimension 1 or in the case whereaij are constants, one can remove the assumption (23) and show that the whole sequenceunsatisfies

un→u_∞+C inD(R^k) whereCcan be determined, see [2].

In the case of dimension 1 and for n, δ_n∈(0,1), if Ω_n=

−(n+ n)T , (n+δ_n)T

it can happen – when _n, δ_nhave no limits – that the sequenceu_nhas no limit.

Acknowledgements

The work of both authors has been supported by the Swiss National Fond under the contract # 20-103300/1. We are very grateful to this institution.

References

[1] M. Chipot,Goes to Plus Infinity, Birkhäuser, 2002.

[2] M. Chipot, Y. Xie, in preparation.

[3] M. Chipot, Y. Xie, in preparation.

[4] D. Cioranescu, P. Donato, An Introduction to Homogenization, Oxford Lecture Series, vol. 17, Oxford University Press, 1999.

[5] R. Dautray, J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, 1988.

[6] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983.

[7] M. Meier, Liouville Theorems for nonlinear elliptic equations and systems, Manuscripta Math. 29 (1979) 207–228.