Homogenization of a parabolic equation in perforated domain with Neumann boundary condition

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1 Proc. Indin Acd. Sci. Mth. Sci. Vol. 112, No. 1, Februry 22, pp Printed in Indi Homogeniztion of prbolic eqution in perforted domin with Neumnn boundry condition A K NANDAKUMARAN nd M RAJESH Deprtment of Mthemtics, Indin Institute of Science, Bnglore 56 12, Indi Corresponding uthor E-mil: nnds@mth.iisc.ernet.in; rjesh@mth.iisc.ernet.in Abstrct. In this rticle, we study the homogeniztion of the fmily of prbolic equtions over periodiclly perforted domins,u div,u, u =fx,t in,t,,u, u.ν = on S,T, u = on,t, u x, = u x in. Here, = \ S is periodiclly perforted domin. We obtin the homogenized eqution nd corrector results. The homogeniztion of the equtions on fixed domin ws studied by the uthors [15]. The homogeniztion for fixed domin nd b,u bu hs been done by Jin [11]. Keywords. Homogeniztion; perforted domin; two-scle convergence; correctors. 1. Introduction Let be bounded domin in R n with smooth boundry. Let T > be constnt nd let > be smll prmeter which eventully tends to zero. Let Y =, 1 n nd Sclosed Y. We define periodiclly perforted domin s follows: First define I ={k Z n :k + S } nd, 1.1 S = k + S. 1.2 k I Set = \ S. 1.3 We ssume tht, i is smooth nd, ii is connected. Note tht in the definition of the holes S we hve tken them to hve no intersection with the externl boundry. This is only done to void techniclities nd the results of our pper will remin vlid without this ssumption cf. [2] for refinement of our rguments. 195

2 196 A K Nndkumrn nd M Rjesh We consider the following fmily of equtions with Neumnn conditions on the boundry of the holes,u div,u, u =fx,t in,t,,u, u.ν = on S,T, u = on,t, u x, = u x in, 1.4 where u is given function on nd f is given function on,t. For given, the Cuchy problem 1.4 will lso be denoted by P. It is known tht under suitble ssumptions on nd b cf. ssumptions A1 A4 below, tht the problem P hs solution u. Our im in this pper is to study the homogeniztion of the equtions P s i.e. to study the limiting behviour of u s nd obtin the limiting eqution stisfied by the limit. The theory of homogeniztion is well-developed re of reserch [5,12] nd we re not going into the detils of this theory. The cse when b is liner i.e. by,s = s hs been studied quite widely [4, 6 8, 1,16]. When b is not liner, the homogeniztion of the eqution in fixed domin ws studied by Jin [11] for by,s bs nd in the generl cse, by the uthors [15]. Assuming tht ũ re priori bounded in L,T where denotes the extension of u by zero in the holes, we show tht ũ converges, for subsequence, to solution of the homogenized problem u div Au, u = m fx,t in,t, u= on,t, ux, = in, 1.5 where b nd A hve been identified cf. eqs ; m denotes the volume frction of in Y, i.e. m = Y\S, the Lebesgue mesure of. We lso improve the wek convergence of ũ by obtining correctors. Such results re very useful in obtining numericl pproximtions nd they form n importnt spect of homogeniztion. The lyout of the pper is s follows: In 2, we give the wek formultion for the problem P. Then we stte our min results viz. Theorems 2.4 nd 2.6. In 3, we sketch some preliminry results nd recll fcts bout the two-scle convergence method. Finlly, in 4, we prove our min results in the frmework of two-scle convergence. 2. Assumptions nd min results For p>1, p. will denote the conjugte exponent p/p 1. Define the spce V ={v W 1,p : v = on }; V is its dul nd E. = L p,t;v. We define u E to be wek solution of P if it stisfies: b,u L,T;L 1,,u L p,t;v, 2.1 tht is T,u,ξx,t dt + T b,u b,u t ξ dx dt = 2.2

3 Homogeniztion of prbolic eqution 197 for ll ξ E W 1,1,T;L with ξt = ; nd, T T,u,ξx,t dt +,u, u. ξx,tdx dt T = fx,tξx,tdx dt 2.3 for ll ξ E. Here.,. denotes the dulity brcket with respect to E,E. Note: For ny ξ L p,t;w 1,p, its restriction to cn be used s test function in 2.3 for the problem P. We mke the following ssumptions on nd b: A1 The function by, sis continuous in y nd s, Y -periodic in y nd non-decresing in s nd by, =. A2 There exists constnt θ> such tht for every δ nd R with <δ<r, there exists Cδ, R > such tht by, s 1 by, s 2 > Cδ, R s 1 s 2 θ 2.4 for ll y Y nd s 1,s 2 [ R,R] with δ< s 1. Remrk 2.1. The prototype for b is function of the form cy s k sgns for some positive rel number k nd continuous nd Y -periodic function, c., which is positive on Y. A3 The mpping y,µ,λ y,µ,λ defined from R n R R n to R n is mesurble nd Y -periodic in y nd continuous in µ, λ. Further, it is ssumed tht there exists positive constnts α, r such tht y,µ,λ.λ α λ p, 2.5 y, µ, λ 1 y,µ,λ 2 λ 1 λ 2 >, λ 1 λ y,µ,λ α 1 1+ µ p 1 + λ p 1, 2.7 y,µ 1,λ y,µ 2,λ α 1 µ 1 µ 2 r 1+ µ 1 p 1 r + µ 2 p 1 r + λ p 1 r. 2.8 A4 We ssume tht the dt, f L T. Under the bove ssumptions, it is known tht P dmits solution u cf. [3]. Remrk 2.2. From hereon, we mke the following simplifying ssumption to reduce the techniclities in the proof of the homogeniztion, viz. tht y,µ,λ is of the form y,λ. The sme results cn be crried over for generl cf. see [11,15] for suitble modifictions. A5 The strong monotonicity condition; y, λ 1 y,λ 2.λ 1 λ 2 α λ 1 λ 2 p. 2.9

4 198 A K Nndkumrn nd M Rjesh Remrk 2.3. We use the condition A5 only for obtining the corrector result. It will not be required for identifying the homogenized problem. We now stte our min theorems. Theorem 2.4. Let u be fmily of solutions of P. Assume tht there is constnt C>, such tht sup u L,T C. 2.1 Then there exists subsequence of, still denoted by, such tht for ll q with <q<, we hve ũ ustronglyinl q T 2.11 nd u solves, u div A u = m f in T, u = on,t, ux, = u in, 2.12 where b nd A re defined below by , m = Y\S nd T denotes the set,t. The functions b nd A re defined through, b s = by,sdy, 2.13 Aλ = y,λ + λ y dy, 2.14 where λ R n nd λ Wper 1,p Y solves the periodic boundry vlue problem y,λ + λ y. φydy = 2.15 Y \S for ll φ Wper 1,p Y. See [9] for further detils bout eq nd for the properties of A. Remrk 2.5. The ssumption 2.1 is true in specil cses see [13] nd it is resonble on physicl grounds see [11]. We see tht by extending u by zero in the holes we do not get functions in L p,t;w 1,p. However, u cn be shown to be bounded sequence in L p T nd hs wek limit. We will improve the wek convergence through corrector result. For this, we consider the solution U 1 x,t,y L p,t;wper 1,p Y of T y, x ux, t + y U 1 x, t, y. y x,t,ydy dx dt = for ll L p,t;w 1,p per Y we will see tht such U 1 exists. Let χ be the chrcteristic function of defined on Y nd extended Y -periodiclly to R n. Then, χx/ is the chrcteristic function of. The corrector result is s follows:

5 Homogeniztion of prbolic eqution 199 Theorem 2.6. Let the functions u,u,u 1 be s bove. Assume further tht A5 holds nd u, U 1 re sufficiently smooth, i.e. tht they belong to C 1,T nd C,T;Cper 1 Y respectively. Then ũ u U 1 x,t, x strongly in L p T nd, x u χ x u+ y U 1 x,t, x strongly in L p T. Remrk 2.7. Note tht, in prticulr, the L p wek limit of u is m ux, t + U 1 S ν x,t,ydσy. 3. Some preliminry results As we closely follow the tretment in [15], mny results will only be sketched nd we refer the reder to [15] for more detils s nd when necessry. We first obtin priori bounds under the ssumption 2.1. From now on, C will denote generic positive constnt which is independent of. Lemm 3.1. Let u be fmily of solutions of P nd ssume tht 2.1 holds. Then, sup sup u L p,t C, 3.1 x, u L C, 3.2 p,t x sup,u E C. 3.3 Proof. Define the function B.,. : R n R R by By,s = by,ss As in [15] we deduce tht,u x, T B = B,u s by, τ dτ. 3.4 T dx + T dx+ nd from this we obtin B,u x, T dx + T fu dxdt, u. u dx dt, u. u dx dt C 3.5 by 2.1 nd the ssumptions on b nd. Then, 3.1 follows from 3.5 nd 2.5, s B is non-negtive, while 3.2 follows from 3.1 nd 2.7. The estimte 3.3 my be obtined from 3.1, 3.2 nd 2.3. Thus the lemm. We now show tht ũ u.e.,t, for subsequence nd for some u. This is the crucil prt of the present nlysis. This result is to be seen in the context of the estimte

6 2 A K Nndkumrn nd M Rjesh 3.1 where there is no time grdient estimte. Hence, we cnnot use ny redy-mde compct imbedding theorem to get the strong convergence of u to u in some L r T. However, we re ble to chieve this by dpting technique found in [3]. This is given in the following lemm. Lemm 3.2. Let u be s bove. Then, the sequence {ũ } > is reltively compct in L θ T, where θ is s in A2. As result, there is subsequence of u such tht, ũ u.e.in T. 3.6 Proof. Anlogous to Step 1, Lemm 3.3 in [15], we cn derive the following estimte T h h 1 b,u t + h b,u t u t + h u t dx dt C for some constnt C which is independent of nd h. Thus, s we hve ssumed in A1 tht by, =, we get T h h 1 b, ũ t + h b t, ũ ũ t + h Proceeding s in [15] we obtin the desired result. ũ t dx dt C. From Lemm 3.2 bove, the continuity of b nd the ssumption 2.1 we derive the following corollry. COROLLARY3.3 We hve, b x, ũ b x,u strongly in L q T q, <q<. Proof. By the priori bound 2.1, it is enough to consider the function b on Y [ M,M] for lrge M >. As b is continuous, it is uniformly continuous on Y [ M,M]. Therefore, given h >, there exists δ> such tht, by,s by,s <h, whenever y y + s s <δ. Now, since ũ u.e in T, by Egoroff s theorem, given h 1 >, there exists E T such tht its Lebesgue mesure me < h 1 nd ũ converges uniformly to u on T \ E, which we denote by E. Therefore, we cn find 1 > such tht ũ u,e <δ < Therefore, for < 1 we hve x b T, ũ b,u q dxdt x x = b E,ũ b,u q dxdt x x + b,ũ b,u q dx dt E h q m T + 2 q sup b q me h q m T + 2 q sup b q h 1.

7 Homogeniztion of prbolic eqution 21 This completes the proof s h nd h 1 cn be chosen rbitrrily smll. As consequence, we hve the following corollry. COROLLARY3.4 We hve the following convergences: b, ũ bu wekly in L q T, χ b, ũ b u wekly in L q T for q>1. Further, bu = b u. Proof. We note tht b, ũ = b, ũ b,u + b,u +bu by Corollry 3.3 nd the verging principle for periodic functions mentioned below. Similrly, χ b, ũ = χ b,ũ b,u + χ b,u +b u. From χ b, ũ = b, ũ, we redily obtin the lst of the conclusions in the corollry. The homogeniztion will be done in the frmework of two-scle convergence. So to conclude this section, we recll n verging principle which is t the bsis of this method, the definition of two-scle convergence nd few relted results which we will use in the sequel. For n exposition of this method nd its min fetures, see [14,1,15]. For fx,y L q loc Rn ; C per Y, the oscilltory function f, x converges wekly in L q loc Rn to Y fx,ydy, for ll q > 1. A similr result holds for fx,y L loc Rn ; C per Y, with wek convergence replced by wek-* convergence. DEFINITION3.5 Let 1 <q<. A sequence of functions v L q T is sid to two-scle converge to function v L q T Yif v ψ x,t, x dx dt vx,t,yψx,t,ydy dx dt T Y for ll ψ L q T ; C per Y. We write v 2 s v. T Theorem 3.6. Let 1 <q< nd let v be bounded sequence in L q T. Then there exists function v L q 2 s T Ysuch tht, up to subsequence, v vx,t,y.

8 22 A K Nndkumrn nd M Rjesh The following theorem [1] is useful in obtining the limit of the product of two two-scle convergent sequences. Let 1 <q<. Theorem 3.7. Let q>1. Let v be sequence in L q T nd w be sequence in L q 2 s 2 s T such tht v v nd w w. Further, ssume tht the sequence w stisfies w q x, t dx dt wx,t,y q dy dx dt. 3.8 T T Y Then, v w dx dt vx,t,ywx,t,ydy dx dt. 3.9 T T Y Remrk 3.8. A sequence w which two-scle converges nd stisfies 3.8 is sid to be strongly two-scle convergent. Exmples of strongly two-scle convergent sequences re {ψx, t, x } for ny ψ L q pery ; C T. We now stte result for perforted domins the proof of which my be found in [1]. Theorem 3.9. such tht Let q>1. Let v L q,tnd let v L q,tbe sup v L q,t C, sup v L q,t C. 3.1 Then there exist v L q,t;w 1,q nd V 1 L q T ; Wper 1,q Y such tht, up to subsequence 2 s ṽ vx, t, 2 s v χy x vx, t + y V 1 x, t, y Proofs of the min Theorems We first obtin the two-scle homogenized eqution. From the estimtes 3.1 nd 3.2 nd Theorems 3.8 nd 3.5, there exists U 1 L p T ; Wper 1,p Y nd L p T Y such tht, up to subsequence, 2 s u χy x ux, t + y U 1 x, t, y, 4.1 χ, 2 s u x,t,y. 4.2 Theorem 4.1. The limits u, stisfy the following two-scle homogenized problem T u, φ dt + x, t, y. x φ + y x,t,ydy dx dt T = fφdxdt 4.3 T for ll φ C T nd C T ; Cper Y.

9 Homogeniztion of prbolic eqution 23 Proof. We pss to the limit in 2.3 in the presence of pproprite test functions. Let φ C T nd C T ; Cper Y. Set, φ = φx,t + x,t, x. We my use these s test functions in 2.3 s the restrictions of φ to re in E nd we note T T,u,φ dt + x φ+ y x,t, x χ dx dt + o1=, u T χ fx,tφx,tdx dt+o Now we pss to the limit in the bove identity. We my hndle the first term s follows. Observing tht φ., T = = φ.,, we hve from 2.2, T T,u,φ dt = b,u t φ dxdt T x = χ b,ũ t φdxdt+o1 T b u t φ dx dt T χ = T u, φ dx dt, where the convergence in the bove follows from Corollry 3.4. For pssing to the limit in the other two terms in 4.4 we observe tht T, u. x φ + y x,t, x dx dt T =, u. x φ + y x,t, x dx dt Y x, t, y. x φ + y x,t,ydy dx dt, where the convergence in the lst step follows from Theorem 3.6. For this, one observes from Remrk 3.7 tht x φ + y,t, x is strongly two-scle convergent to x φ + y x,t,y. Further, observe tht since χ x x, u is zero in the holes, we hve x,t,yto be zero in S nd the integrl over Y cn be replced by tht over. Finlly, note tht T χ fx,tφx,tdx dt = T T Y χyfx,tφx,tdy dx dt m fφdxdt. Thus, from the bove we deduce tht the limit of the equtions 2.3 with φ s test functions is nothing but 4.3. This is referred to s the two-scle homogenized problem.

10 24 A K Nndkumrn nd M Rjesh Remrk 4.2. We note from the two-scle homogenized problem 4.3, by setting tht u div x,t,ydy =m f in,t 4.5 nd by setting φ we get T x,t,y. y x,t,ydy dx dt = 4.6 Y \S for ll C T ; Cper Y. In view of 4.5 nd 4.6, it is enough to show tht x,t,y=y, x ux, t + y U 1 x,t,yto conclude Theorem 2.4. Proof of Theorem 2.4. Note tht 2.11 follows from the convergence 3.6 nd the ssumption 2.1. We now justify 2.12, for which it is enough to show tht x,t,y equls y, x ux, t + y U 1 x,t,y, by the preceding remrk. Let ξ L p,t;w 1,p W 1,1,T;L with ξ., T = = ξ.,. This my be used s test function in 2.2 nd we hve T T,u,ξx,t dt + b,u t ξdxdt =. This cn be rewritten s T T x R tb,u,ξ dt+ χ b,ũ t ξdxdt=. 4.7 Observe tht sup R tb x,u L p,t ;W 1,p C by 3.3 where R is the restriction opertor from L p,t;w 1,p to E. Thus for some w L p,t;w 1,p nd for subsequence R tb,u w wekly * in L p,t;w 1,p. Thus, pssing to the limit in 4.7 we get T T w, ξ + b u t ξ dx dt = for ll ξ s bove. So, u = w in the sense of distributions. We now show tht x,t,y=y, x ux, t + y U 1 x,t,y. Let φ, be s in the proof of Theorem 4.1. Let λ> nd ψ C T ; Cper Y n. Set η x, t =. x φx,t + y x,t, x + λψ x,t, x. By 2.6 we note tht T, u,η. u η dx dt. 4.8

11 We hve T Homogeniztion of prbolic eqution 25 x T, u. u = = T T x T,u,u dt + T u, u + Y\S m fudxdt x,t,y. x u dydxdt, fu dxdt where the lst step follows from 4.5. Also, since η nd,η re strongly two-scle convergent, Theorem 3.6 gives T T Y\ S x T x,η. u dxdt =,η. u dxdt y, x φ + y +λψ. x u + y U 1 dydxdt. Similrly, nd, T T T T = x T,η.η dx dt = χ,η.η dx dt y, x φ + y +λψ. x φ + y +λψ dy dx dt x T, u.η dx dt = T χ, u.η dx dt x, t, y. x φ + y +λψ dy dx dt x, t, y. x φ + λψ dy dx dt. The equlity in the lst step follows from 4.6. Thus T T lim, u,η. u η = T x u φ λψ y, x φ + y +λψ Y \S x u φ + y U 1 λψ. x,t,y Letting φ u nd U 1 in the bove nd using the continuity of we get T y, x u + y U 1 +λψ x, t, y.λψ dy dx dt Y \S

12 26 A K Nndkumrn nd M Rjesh for ll λ nd ψ. Dividing the bove inequlity by λ nd letting λ we get, T y, x u + y U 1 x, t, y.ψ dy dx dt Y \S for ll ψ. From this we conclude tht x,t,y=y, x u + y U 1. Proof of Theorem 2.6. Set η x, t =. x ux, t + y U 1 x,t, x. From 2.9 it follows tht T, u,η. u η dx, dt T α u η p dx dt, tht is, α T T, u,η. u χ η dx dt ũ χ [ x ux, t + y U 1 x,t, x ] p dx dt. As we hve ssumed tht u, U 1 re smooth, we my proceed s in the proof of Theorem 2.4 to obtin T α lim sup T lim This completes the proof. ũ χ x ux, t + y U 1 x,t, x p dx dt, u,η. u χ η dx dt =. Remrk 4.3. A weker form of the corrector result with no smoothness ssumptions cn lso be obtined. See [15] for the corresponding formultion. Acknowledgement The second uthor would like to thnk the Ntionl Bord of Higher Mthemtics, Indi for finncil support. He would lso like to thnk the orgnizers for their invittion to give tlk t the Workshop on Spectrl nd Inverse Spectrl Theories for Schrödinger Opertors held t Go, where he presented this work.

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