Nonlocal limits in the study of linear elliptic systems arising in periodic homogenization
Abstract
In the present paper, we obtain the two-scale limit system of a sequence of linear elliptic periodic problems with varying coefficients. We show that this system has not the same structure than the classical one, obtained when the coefficients are fixed. This is due to the apparition of nonlocal effects. Our results give an example showing that the homogenization of elliptic problems with varying coefficients, depending on one parameter, gives in general a nonlocal limit problem.
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Journal of Computational and Applied Mathematics 204 (2007) 3–9 www.elsevier.com/locate/cam Nonlocal limits in the study of linear elliptic systems arising in periodic homogenization夡 Carmen Calvo-Juradoa,∗, Juan Casado-Díazb aDepartamento de Matemáticas, Escuela Politécnica, Universidad de Extremadura, Avda. de la Universidad, s/n, 10071 Cáceres, Spain bDepartamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/Tarfia, s/n, 41012 Sevilla, Spain Received 15 July 2005; received in revised form 1 March 2006 Abstract In the present paper, we obtain the two-scale limit system of a sequence of linear elliptic periodic problems with varying coefficients. We show that this system has not the same structure than the classical one, obtained when the coefficients are fixed. This is due to the apparition of nonlocal effects. Our results give an example showing that the homogenization of elliptic problems with varying coefficients, depending on one parameter, gives in general a nonlocal limit problem. © 2006 Elsevier B.V. All rights reserved. MSC: 2000: 35J20 Keywords: Homogenization; Nonlocal problems; Elliptic equations 1. Introduction It is well known (see [10,8]) that given a bounded open subset ⊂RNand a sequence of matrices A∈L∞()N×N, which are uniformly elliptic and bounded, there exist a matrix A(homogenized matrix) in the same conditions that A, and a subsequence of , still denoted by , such that for every fwhich converges strongly in H−1()to a distribution f, and every uwhich converges in H1 0()and satisfies −div A∇u=fin , (1) the limit of uis a solution of the analogue equation, where Ais substituted by A, and fby f. The aim of the present paper is to show that the analogue of this result is not true when the matrices Ameasurably depend on a parameter, i.e., when at the place of (1), we have −divyA(x, y)∇yu(x, y) =f(x, y) in a.e.x∈, (2) where (,,)is a given space of measure. Indeed, for this type of problems the limit operator is in general nonlocal in y. This is due to the fact that the set of solutions of (2) is not compact in general in L2 (;L2()). 夡This work was partially supported by D.G.I., Spain. ∗Corresponding author. E-mail address: [email protected] (C. Calvo-Jurado). 0377-0427/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2006.04.023
4C. Calvo-Jurado, J. Casado-Díaz / Journal of Computational and Applied Mathematics 204 (2007) 3–9 The homogenization problem (2) appears, for example, in the study of the asymptotic behavior of partial differential problems with varying coefficients, which depend on an aleatory parameter x∈(stochastic homogenization problems). Another interesting situation is the study of some systems which usually appear in periodic homogenization. In this way, let us consider in the present paper the homogenization problem −div Ax, x ∇u−Gx, x =0in,u =0onj, (3) where as above is a bounded open subset of RN,Aand Gare, respectively, continuous matrices and vectorial functions in ×RN, which are periodic in the second variable, of period YN,Y=(−1 2,1 2). The matrices Aare uniformly elliptic and bounded. We remark that (3) is not a particular case of (1) because −div G(x, x )does not converges strongly, in general, in H−1()(it converges in W−1,∞()Nweakly-∗). When Ais a constant matrix A, the two-scale convergence theory of Nguetseng and Allaire (see e.g. [1,9]) shows that denoting by u0∈H1 0(), u1∈L2(;H1(Y N)/R)the solutions of −divxYN (A(x, y)(∇xu0(x) +∇yu1(x, y)) −G(x, y)) dy=0in, (4) −divy(A(x, y)(∇xu0(x) +∇yu1(x, y)) −G(x, y)) =0inRNa.e.x∈, (5) then the solutions of (3) converge weakly in H1 0()to u0, while ∇utwo-scale converges to ∇xu0+∇yu1(the idea is to approximate u(x) of the type u0(x) +u1(x, x/), which is better than to approximate ujust by u). When divyG(x, y) =0, Eq. (5) permits to calculate u1from u0and then, substituting in (4) we get the homogenized problem (see e.g. [1,9]) for u0. When this condition is not satisfied we can still calculate u1from u0and then to obtain an equation which only contains u0, but their coefficients depend on Gand thus, it is better to remain with the system (4), (5). In the present paper, given uthe solution of (3), let us search for the system satisfied by the limit of uin H1 0()and the two-scale limit of ∇u. Clearly, this must contain in particular the limit of a system like (4), (5), where the second equation has a structure similar to (2). As we announced above, we will obtain a nonlocal limit system (see Theorem 4 and Proposition 5). In particular there does not exist in general a matrix Asuch that the corresponding system is (4), (5), as it happens when Ais constant. Related results have been obtained in [4] for a different problem, the asymptotic behavior of thin structures. Other results, about the apparition of nonlocal terms in the homogenization of linear elliptic problems, can be found by using the theory of Dirichlet forms (see [7]), where it is assumed strong convergence in L2, which as we said above does not hold in our context. 2. Homogenization of periodic problems We take Y=(−1 2,1 2), and ⊂RNa bounded open subset of RN. L(L2(Y N)N,L 2(Y N)N)is the space of lineal continuous functions from L2(Y N)Ninto itself. As it is usual, we use the index to mean periodicity. For example, Lp (Y N)is the space of functions of Lp loc(RN) which are periodic, of period YN. For a sequence of matrices A∈C0(¯ ;C0 (Y N))N×N, such that there exist ,>0, with A(x, y)⩾min{||2,|A(x, y)|2}∀∈RN∀(x, y) ∈×RN, (6) let us study the homogenization of (3), with G∈C0(¯ ;C0 (Y N))N. Remark 1. We recall (see [8]) that (6) is equivalent to the existence of ,>0, such that A(x, y)⩾||2,|A(x, y)|⩽||∀∈RN∀(x, y) ∈×RN. (7) To study the asymptotic behavior of u, let us apply the Arbogast et al. method’s [2], strongly related to the two-scale theory (see [1,3,5,6,9]). For this purpose, we define :RN→ZNby the following rule: assuming RNdecomposed as the union of the cubes k+YN, with k∈ZN, then, for a.e. x∈RN,(x) gives the center kof the cube which contains x.
C. Calvo-Jurado, J. Casado-Díaz / Journal of Computational and Applied Mathematics 204 (2007) 3–9 5 We remark that if we decompose RNas the union of the cubes k+YN, with k∈ZN, then the center of the cube which contains xis (x/). For the proof of the following theorem we refer to [2,5,6]. Theorem 2. Consider a sequence uwhich is bounded in H1 0()and define ˆu∈L2(RN;H1(Y N)) by (uis extended by zero outside of ) ˆu(x, y) =u x +y,a.e. (x, y) ∈RN×YN. (8) then there exists a subsequence of ,still denoted by ,and there exist u0∈H1 0(),u1∈L2(;H1 (Y N)/R),such that uu in H1 0(), (9) 1 ∇yˆu∇xu0+∇yu1in L2(RN×YN). (10) Remark 3. For k∈ZN,ˆu(x, y) restricted to (k+YN)×YNdoes not depend on x, and as a function of y,itis obtained from uby using the change of variables y=(x −k)/which transforms the small cube k+YNon YN. Statement (10) is equivalent to ∇utwo-scale converges to ∇xu0+∇yu1. The homogenization of (3) is given by the following theorem. Its proof is based on the one of the classical result of F. Murat and L. Tartar for the compactness of the H-convergence [8]. Theorem 4. There exist a subsequence of ,still denoted by ,and an operator A∈L∞(;L(L2(Y N)N,L 2(Y N)N)) such that: For every ∈RN,every w∈H1(RN)and a.e. x∈,we have YN A(x)(+∇w) ·(+∇w)dy ⩾min ||2+YN|∇w|2dy,YN|A(x)(+∇w)|2dy. (11) For every G∈C0(¯ ;C0 (Y N))N,the solution uof (3) satisfies (9) and (10), where ˆuis given by (8) and u0∈H1 0(), u1∈L2(;H1 (Y N)/R)are the unique solutions of −divxYN (A(x)(∇xu0(x) +∇yu1(x, .)) −G) dy=0in , (12) −divy(A(x)(∇xu0(x) +∇yu1(x, .)) −G) =0in RNa.e. x∈. (13) Proof. We divide the proof in several steps: Step 1: For G∈C0(¯ ;C0 (Y N))N, the solutions uof (3) are bounded in H1 0(). So, by Theorem 2, up to a subsequence, there exist u0∈H1 0(),u1∈L2(;H1(Y N)/R)such that (9) and (10) hold. Moreover, using that ˆ A defined by (take an extension of Aoutside of ) ˆ A(x, y) =A x +y,ya.e. (x,y) ∈RN×YN(14) is bounded in L∞(×YN)N×Nand that (1/)∇yˆuis bounded in L2(×YN)N, we can also assume that there exists ∈L2(×YN)N, such that 1 ˆ A∇yˆuin L2(×YN)N. (15)
6C. Calvo-Jurado, J. Casado-Díaz / Journal of Computational and Applied Mathematics 204 (2007) 3–9 For 0,1∈C1 0(),∈C1 (Y ), we take v(x) =0(x) +(x/)1(x) as test function in (3). Decomposing RNas the union of the cubes k+YN,k∈ZNand using in each cube the change of variables y=(x −k)/,weget 0=lim →0Ax, x ∇u−Gx, x ·∇vdx =lim →0×YNˆ A 1 ∇yˆu−G·(∇x0(x) +1(x)∇y(y)) dxdy =×YN (−G) ·(∇x0(x) +1(x)∇y(y)) dxdy. Since 0,1,are arbitrary, this means that satisfies the equations −divxY ((x, y) −G(x, y)) dy=0a.e.in , (16) −divy((x, y) −G(x, y)) =0inRNa.e.x∈. (17) For ∈C1 0(), we take uas test function in (3), using then the Rellich–Kondrachov compactness theorem, (16) and (17), we get lim →0 Ax, x ∇u·∇udx =−lim →0Ax, x ∇u·∇u−Gx, x ·(∇u+u∇)dx =−lim →0×YNˆ A 1 ∇yˆu·∇xu0−G·1 ∇yˆu+u0∇xdxdy =−×YN (·∇xu0−G·((∇xu0+∇yu1)+u0∇x)) dxdy =×YN ·(∇xu0+∇yu1)dxdy. (18) On the other hand, from (6) and the lower semicontinuity of the weak convergence, we have lim →0 Ax, x ∇u·∇udx⩾lim inf →0 min |∇u|2,|Ax, x ∇u|2dx =lim inf →0×YN min 1 ∇yˆu 2 , 1 ˆ A∇yˆu 2dxdy ⩾×YN min{|∇xu0+∇yu1|2,||2}dxdy. Thus, from (18) and arbitrary, for a.e. x∈we get YN (∇xu0+∇yu1)dy⩾YN min{|∇xu0+∇yu1|2,||2}dy. (19) Step 2: We consider a countable subset Dof C1 0()×C1 0(;C1 (Y N)) which is dense in H1 0()×L2(;H1 (Y N)/R). By a diagonal argument, there exists a subsequence of , still denoted by (this will be the subsequence which appears in the statement of Theorem 4) such that for every (w0,w 1)∈Span(D), the solution uof (3), with G=∇xw0+∇yw1,
C. Calvo-Jurado, J. Casado-Díaz / Journal of Computational and Applied Mathematics 204 (2007) 3–9 7 is such that there exist u0∈H1 0(),u1∈L2(;H1 (Y N)/R),∈L2(×YN)Nsuch that (9), (10), (15)–(19) hold. From (19), (16) and (17), we have |∇xu0|2dx+×YN|∇yu1|2dxdy ⩽×YN|∇xu0+∇yu1|2dxdy ⩽×YN ·(∇xu0+∇yu1)dxdy =×YN (∇xw0+∇yw1)·(∇xu0+∇yu1)dxdy(20) and thus, we deduce (u0,u 1)H1 0()×L2(;H1 (Y N)/R)⩽(w0,w 1)H1 0()×L2(;H1 (Y N)/R). (21) The two first lines of (20) also show (u0,u 1)H1 0()×L2(;H1 (Y N)/R)⩽L2(×YN)N, (22) while from (16) and (17), we have |∇xw0|2dx+×YN|∇yw1|2dxdy=×YN ·(∇xw0+∇yw1)dxdy. (23) So, taking into account (19) and (22) we get (w0,w 1)H1 0()×L2(;H1 (Y N)/R)⩽1 2(u0,u 1)H1 0()×L2(;H1 (Y N)/R). (24) Following (21) and (24) we can extend the linear application (w0,w 1)∈D→ (u0,u 1), to a linear application Q on H1 0()×L2(;H1 (Y N)/R)which still satisfies (21), (24). From Lax–Milgram’s theorem, Qis bijective and has a continuous inverse. From (19) and (21) we can also extend the linear application (w0,w 1)∈D→ to a continuous application Rfrom H1 0()×L2(;H1 (Y N)/R)into L2(×YN)N. In particular, we can define S=RQ−1: H1 0()×L2(;H1 (Y N)/R)→L2(×YN)N, which extends the application (u0,u 1)∈QD → . Now, given an increasing sequence of compact subsets Knof such that n∈NKn=,K0=∅, we consider n∈C1 0(), such that n(x) =x, for every x∈Kn. Then, we define A∈L∞(;L(L2(Y N)N,L 2(Y N)N)) by A(x)(g) =SYN g(y) dy·n,w , (25) ∀g∈L2(Y N)N, a.e. x∈Kn\Kn−1, with w∈H1 (Y N)/R,−w=−div gin RN. From (19), Asatisfies (11). Step 3: Let us take G∈C0(¯ ;C0 (Y N))N, and define uas the solution of (3) (with the subsequence of given by the previous step). By Theorem 2, and Step 1, there exist a subsequence of , still denoted by , and u0∈H1 0(), u1∈L2(;H1 (Y N)/R),∈L2(×YN)Nsuch that (9), (10), (15) hold. Let us prove that u0,u 1satisfy (12), (13), and then, by uniqueness that there is not necessary to extract any subsequence. Applying Step 1 to Greplaced by G−∇xw0−∇yw1,(w0,w 1)∈Span(D), we deduce from (19) 2YN|−R(w0,w 1)|2dy⩽YN|∇xu0+∇yu1−Q(w0,w 1)|dya.e.in .
8C. Calvo-Jurado, J. Casado-Díaz / Journal of Computational and Applied Mathematics 204 (2007) 3–9 By density this holds for every (w0,w 1)∈H1 0()×L2(;H1 (Y N)/R). Taking (w0,w 1)=Q−1(·n,∇yw), with ∈RN,w∈H1 (Y N)/R)we get 2YN|−A(x)(+∇yw)|2dy⩽YN|∇xu0−+∇y(u1−w)|2dy a.e. in Kn, and then in . Thus, =A(x)(∇xu0(x) +∇yu1)a.e. in , which by (16) and (17) proves (12), (13). The question now is if effectively there is some example with a nonlocal term. If N=1, it is possible to show that the homogenized problem of (3) is local. For N=2, the following result gives an example where a nonlocal term appears. Proposition 5. Let be ∈C0 (Y ) a sequence which converges almost everywhere to the function =l∈Z(l−(1/2),l), and it is such that 0⩽⩽1in R.Given A1=10 01 2,A 2=10 03 2, we define the sequence of matrices A∈C0(R2;C0 (Y 2))2×2by A(x, y) =A1(x1/√)+A2(1−(x1/√)), x=(x1,x 2)∈,y=(y1,y 2)∈R2. Then,for every bounded open set ⊂R2and every G∈C0(¯ ;C0 (Y 2))2,the solution uof (3) is such that (9), (10) are satisfied,where u0∈H1 0(),u1∈L2(;H1 (Y 2)/R)are the solutions of −u0=−divxY2 G(x, y) dyin , (26) −yu1+1 4 j jy2 Rju1 jy2=−divyGin R2a.e. in , (27) where R:L2 (Y 2)→L2 (Y 2)is the nonlocal operator given by R(z) =jw/jy2,for every z∈L2 (Y 2),with w∈ H1 (Y 2)/R,−w=−jz/jy2in R2. Proof. An easy application of the two-scale convergence theory shows that (9) and (10) hold, where taking A∈ L∞(Y )2×2as A1(−1/2,0)+A2(0,1/2), there exist ˆu0∈L2(,H1 (Y )/R),ˆu1∈L2(;L2 (Y ;H1 (Y 2)/R)) such that (u0,ˆu0,ˆu1)satisfy the variational problem ×Y×Y2A(t) ∇xu0+dˆu0 dte1+∇yˆu1−G∇xv0+dˆv0 dte1+∇yˆv1dxdtdy=0 ∀(v0,ˆv0,ˆv1)∈H1 0()×L2(,H1 (Y )/R)×L2(;L2 (Y ;H1 (Y 2)/R)), where e1is the first vector of the usual basis of R2and u1is given by u1(x, y) =Yˆu1(x,t,y)dta.e. (x,y) ∈×RN. (28) Taking v0=ˆv0=0, we deduce ˆu1(x,t,y)=1(x, y)(−1 2,0)(t) +2(x, y)(0,1 2)(t), with i∈L2(;H1 (Y 2)/R), −divy(Ai∇yi−G) =0 and RN, a.e. x∈,i=1,2. From (28), we have u1=(1+2)/2 and then, using A1+A2=2I,weget−yi=−divyAj∇yu1and RN, for a.e. x∈, where i, j ∈{1,2},i = j. Denoting by Υ∈L2(;H1 (Y 2)/R)the solution of −yΥ=−j2u1/jy2 2in RN, for a.e. x∈,we have that i=u1+(−1)i+1Υ/2, i=1,2, and then, using the equations satisfied by i, we conclude that u1satisfies (27). Taking in the variational equation v0=ˆv1=0, we now get ˆu0=0, and then using ˆv0=ˆv1=0 we conclude that u0 is the solution of (26).
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