YANG ZHAN-YING,SHU WAN,PAN ZHANG-PING AND PENG CHAO-QUAN

(Department of Mathematics,South-Central University for Nationalities,Wuhan,430074)

Abstract:We are concerned with a class of parabolic equations in periodically perforated domains with a homogeneous Neumann condition on the boundary of holes.By using the periodic unfolding method in perforated domains,we obtain the homogenization results under the conditions slightly weaker than those in the corresponding case considered by Nandakumaran and Rajesh(Nandakumaran A K,Rajesh M.Homogenization of a parabolic equation in perforated domain with Neumann boundary condition.Proc.Indian Acad.Sci.(Math.Sci.),2002,112(1):195–207).Moreover,these results generalize those obtained by Donato and Nabil(Donato P,Nabil A.Homogenization and correctors for the heat equation in perforated domains.Ricerche di Matematica L.2001,50:115–144).

Key words:parabolic equation,perforated domain,homogenization,periodic unfolding method

1 Introduction

In this paper,we study the homogenization for the following problem:where ??Rnis an open and bounded set,is a domain perforated by Sεwhich is a set of ε-periodic holes of size ε,and nεis the outward unit normal vector field de fined on ?Sε.The initial data u0and f belong to L∞(?)and L2(0,T;L2(?)),respectively.The matrix Aε(x)is of formwith A being a periodic,bounded and elliptic matrix field.We also assumewith ρ being a Y-periodic and coercive function in L∞(?).This problem is widely used to model many phenomena in the heat theory.

The study of this problem was initiated by the work of Spagnolo[1],in which he achieved the homogenization of problem(1.1)for fixed domains.Further investigations were made by Bensollssa et al.[2].The corresponding correctors were achieved in[3].Subsequently,much attention has been paid to such an investigation for some different cases,see[4]–[16]and the references therein.Recently,Meshkova and Suslina[17]studied the homogenization of the second initial boundary value problem for parabolic systems with rapidly oscillating coefficients.Chakib et al.[18]investigated the periodic homogenization of nonlinear parabolic problem,which is de fined in periodical domain and is nonlinear at the interface.Amar et al.[19]obtained the homogenization of a parabolic problem in a perforated domain with Robin-Neumann boundary conditions oscillating in time.In particular,for the case in periodically perforated domains,Donato and Nabil carried out a study of the homogenization and correctors for the standard linear case in[20]and for the semi-linear case in[6],respectively.Then,Donato and Yang[7]extended the results in[20]to the case with non-periodic coefficients.In[13],Nandakumaran and Rajesh studied the following nonlinear degenerate problem

To obtain the homogenization results,they imposed some extra condition onis uniformly bounded in L∞(0,T;L2(?))).The proof mainly depends on the two-scale convergence method.For the corresponding case in fixed domains,the homogenization and the correctors were provided in[12]and given for the casein[11].

In this paper,we are devoted to obtaining the homogenization of the problem(1.1)under the conditions slightly weaker than those in the corresponding case of[13].Moreover,our results generalize the work in[20].Our method mainly relies on the periodic unfolding method,which was originally introduced in[21](see also[22])and extended to perforated domains in[23](see[24]for more general situations).Recently,the unfolding technique was extended to the time-periodic case.Then it was further used to consider the homogenization problem for a parabolic equation oscillating both in space and time,with general independent scales.

Throughout this paper,we make the following assumptions:

for any λ ∈ Rnand a.e.in Y,where α,β ∈ R with 0 ＜ α ＜ β.For any ε＞ 0,we set

(A-2) Let ρ be a Y-periodic function in L∞(Y)and there existμ1,μ2∈ R such that 0＜ μ1≤ ρ≤ μ2a.e.in Y?.For any ε＞ 0,let

(A-3) We assume that u0∈ L∞(?)and f∈L2(0,T;L2(?)).

These assumptions ensure the existence and uniqueness of the solution of(1.1)(see[13]and[25]for instance).

Now,we state the homogenization results,in which we use some notations to be de fined in the next section.These results are derived under the loss of the uniform boundedness onMoreover,the unfolded formulation(see Theorem 3.1 in Section 3)is provided later for the proof of homogenization results.

Theorem 1.1Let uεbe the solution of problem(1.1).Then there exists a u∈L2(0,T;such that

Also,u is the unique solution of the following problem:

Further,we have the following precise convergence of fl ux:

Observe that here A0is the same constant positive de finite matrix obtained by Cioranescu and Paulin[26],for the Laplace problem in a perforated domain with a Neumann condition on the boundary of the holes.Moreover,we also give the precise convergence of fl ux.

This paper is organized as follows.In Section 2,we recall some de finitions and properties related to the unfolding method in perforated domains.Section 3 is devoted to the homogenization results.

2 Preliminaries

In this section,we mainly introduce some notations and recall some necessary results related to the unfolding operator in perforated domains.

2.1 Some Notations

Let ? ? Rnbe an open bounded set with Lipschitz boundary ??,and b=(b1,···,bn)be a basis in Rn.Set

Denote ε by the general term of a sequence of positive real numbers which converge to zero.Let S is a closed proper subset ofwith Lipschitz continuous boundary.We de fine the perforated domain

where

We make the following assumption:

This implies

where Sεis the subset of τε(εS)contained in ?.

Now we recall some notations related to the unfolding method in[22]and[24].Let

In what follows,we use the following notations:

?|D|denotes the Lebesgue measure of a measurable set D in Rn;

?Vεis de fined by

? c and C denote generic constants which do not depend upon ε;

?The notation Lp(O)will be used both for scalar and vector-valued functions de fined on the set O,since no ambiguity will arise.

2.2 Some Results on the Unfolding Operator in Perforated Domains

In this subsection,we recall some de finitions and properties related to the unfolding method in perforated domains,which are proved in[27]and needed in the sequel.We refer the reader to[24]and[27]for further properties and related comments.

Then for each x∈Rn,we have

De finition 2.1For p∈[1,+∞)and q∈[1,∞],let ? be inThe unfolding operatoris de fined as follows:

The following propositions contain some basic properties associated to the unfolding operators.

Proposition 2.1Let p∈[1,+∞)and q∈[1,∞].

(vii)For q∈[1,+∞],let ?εbe inand satisfy

then

Then for any η ∈ D(?),we have

Proposition 2.2(i)For p,q ∈ [1,∞),let{vε}be a sequence in Lq(0,T;Lp(?))such that

Then

(ii)For p∈(1,∞)and q∈(1,∞],let{vε}be a sequence insuch that

If

then we have

For q= ∞,the weak convergences above are replaced by the weak?convergences,respectively.

Finally,we finish offthis section with a convergence theorem which plays a key role in obtaining our homogenization result.Its proof is very similar to that of assertion(i)of Theorem 2.19 in[27](the only difference is that here we consider L2functions in time instead of L∞).

Proposition 2.3Suppose that{vε}is a sequence in L2(0,T;Vε)such that

3 Proof of the Homogenization Result

This section is devoted to the proof of Theorem 1.1,which is carried out in two steps.The first step is to obtain some suitable a priori estimates on uε,which is the content of Subsection 3.1.Combining these estimates with the unfolding method,we complete the proof of Theorem 1.1 in Subsection 3.2.

For every fixed ε,we know that the problem(1.1)has a unique solution uεwhose uniform a priori estimates will be given in the following subsection.

3.1 A Priori Estimates

Proposition 3.1Let uεbe the solution of the problem(1.1).Then

where C is a constant independent of ε.

Proof. Choose uεas a test function in the variational formulation(3.1)and integrating over(0,t),we have

By(A-1),(A-2)and the H¨older’s inequality,we get

Moreover,

Consequently,it follows that

By the Gronwall’s inequality and(A-3),we deduce

which implies(3.2)(i).This,together with(3.4),yields(3.2)(ii).

Now we turn to the proof of(3.2)(iii).Let v ∈ L2(0,T;Vε).From the variational formulation(3.1),we have

Together with(A-1),(A-3)and(3.2)(ii),we obtain

which gives(3.2)(iii).This completes the proof.

Based on this proposition,we have the following compactness result which will be used to verify the initial conditions in the homogenized problem in Section 3.

Corollary 3.1Let uεbe the solution of problem(1.1).There existssuch that up to a subsequence(still denoted by uε),

Proof. In view of Proposition 3.1,by Proposition 2.3,we get that(3.5)(i)holds true for a subsequence.Moreover,it follows from Proposition 2.1(vii)that

By the fact that u is independent of y,this gives

Combining this with(A-2)and(3.2)(i),we have

Together with(3.2)(iii),we obtain(3.5)(ii)due to the compactness result in[6](see also[20]).This completes the proof.

3.2 Proof of Theorem 1.1

In this subsection,we first state the unfolded formulation of Theorem 1.1(see Theorem 3.1 below),which is very important to the proof of Theorem 1.1.Then by the unfolding method,we simultaneously give the proofs of Theorem 1.1 and Theorem 3.1.

Theorem 3.1Let uεbe the solution of(1.1).Then there existsand awithsuch that

Moreover,we have

Proofs of Theorem 1.1 and Theorem 3.1In virtue of Proposition 3.1,Proposition 2.3 provides that(3.6)holds,at least for a subsequence(still denoted by ε).By Proposition 2.2(ii),we further get

Notice that u is independent of y.We get(3.6)(iii)from(3.9)(i).

Then

From Proposition 2.2,we obtain

Let φ∈D(0,T).By(3.6)and(3.10),we use Proposition 2.1 to get

Choosing vεφ as a test function in the variational formulation(3.1),passing to the limit and making use of(3.11),we get

This gives the equation in(3.7),due to the density of D(?)inand the density of

Then,following the lines in the proof of Theorem 3.1 in[17],we know convergence(3.9)(ii)yields convergence(1.7)and the equation in(3.7)implies that in(1.4).

Now we check the initial condition.From(3.5)(ii),we have

On the other hand,for any ? ∈ D(?),it follows from Proposition 2.1(vii)that

This yields

Owing to the uniqueness of the limit,we get u(x,0)=u0.

Standard arguments give the uniform ellipticity of A0(see[24]and[26]–[28]for instance)and the uniqueness of the solution of problem(1.4).Furthermore,by(3.8),we get the uniqueness of a pairwithsolving problem(3.7).This implies that each convergence in Theorem 3.1 holds for the whole sequence.The proofs of the two theorems are completed.

AcknowledgmentsThe first author would like to thank Professor Patrizia Donato for some valuable discussions on this subject.