A diffusion equation for composite materials

Transcription

1 Electronic Journal of Differential Equations, Vol. 2000(2000), No. 15, pp ISSN: URL: or ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login: ftp) A diffusion equation for composite materials Mohamed El Hajji Abstract In this article, we study the asymptotic behavior of solutions to the diffusion equation with non-homogeneous Neumann boundary conditions. This equation models a composite material that occupies a perforated domain, in R N, with small holes whose sizes are measured by a number r. We examine the case when r < N/(N 2) with zero-average data around the holes, and the case when 0 r / = 0 with nonzero-average data. 1 Introduction As a model for a composite material occupying a perforated domain in R N, the diffusion equation with non-homogeneous boundary conditions has been the object of many studies. In particular, we are interested in the properties of the solution to the equation div (A u )=f in, (A (x) u ) n = h on, (1) u =0 on, where is the perforated domain obtained by extracting a set of holes S from, f and h are given functions, and A is an operator in the space M(α, β;) = { A [L (R N )] N 2 :(A(x)λ, λ) α λ 2, A(x)λ β λ λ R N,p.p x } which is defined for all real numbers α and β. When h L 2 ( ) and the domain has holes of size r, solutions to (1) have been studied by D. Cioranescu and P. Donato [3] for r = and A (x) =A( x ) with A M(α, β; ), and by C. Conca and P. Donato [5] for A = I and r. Using the concept of H 0 -convergence introduced by M. Briane et al. [2], P. Donato and M. El Hajji [6] showed convergence of solutions in not-necessarily 1991 Mathematics Subject Classifications: 31C40, 31C45, 60J50, 31C35, 31B35. Key words and phrases: Diffusion equation, composite material, asymptotic behavior, H 0 -convergence. c 2000 Southwest Texas State University and University of North Texas. Submitted October 14, Published February 22,

2 2 Diffusion equation for composite materials EJDE 2000/15 periodic domains. The H 0 -convergence is proven by showing strong convergence in H 1 () of the distribution, concentrated on the boundary of S,givenby νh,ϕ H 1 (), H0 1() = h,ϕ H 1/2 (), H 1/2 (), ϕ H0 1 (). (2) This method allows the study the asymptotic behavior of solutions to (1) for h L 2 ( )withr > N/N 2 (see [6]), and for perforated domains with double periodicity with h H 1/2 ( )andr / 0as 0 (see T. Levy [8]). In this article, we study the perforated domains with r < N/N 2,and perforated domains such that r / 0as 0. In these situation the distribution given by (2) does not converge strongly in H 1 (), and so the method described above can not be applied. In spite of this, we describe the asymptotic behavior of solutions to (1) using oscillating test functions. This method was introduced by L. Tartar [10] and has been used by many authors. 2 Statement of the main result Let be a bounded open set of R N, Y =[0,l 1 [.. [0,l N [ be a representative cell, and S be an open set of Y with smooth boundary S such that S Y. Let and r be terms of positive sequences such that r. Let c denote positive constants independent of. Wedenotebyτ(r S) the set of translations of r S of the form k 1 + r S with k Z N. Let k l =(k 1 l 1,.., k N l N ) represent the holes in R N. We assume that the holes τ(r S) do not intersect the boundary. If S is the set of the holes enclosed in, it follows that there exists a finite set K Z N such that S = r (k l + S). k K We set =\S, (3) and denote by χ the characteristic function of. Let V denote the Hilbert space V = { v H 1 ( ),v =0 } equipped with the H 1 -norm. Let A(y) =(a ij (y)) ij be a matrix such that A ( L ( R N)) N 2, A is Y-periodic, and there there exist α>0 such that N a ij (y) λ i λ j α λ 2, a.e. y in R N, λ R N. (4) i,j=1 We note that for every >0, A (x) =A( x ) a.e. x in RN. (5)

3 EJDE 2000/15 Mohamed El Hajji 3 In this paper, we study the system div(a u )=0 in, (A u ).n = h on, (6) u =0, on, where is given by (3), A is given by (5), and h is given by h (x) =h( x r ), (7) where h L 2 ( S) isy-periodic function. Set I h = 1 hdσ. (8) Y We examine the case where r < N/(N 2) with I h 0, and the case where o r / =0withI h = 0. The following result describes the asymptotic behavior of the solution to (6) in the two cases. Theorem 1 Let u be the solution of (6). Suppose that one of the following hypotheses is satisfied r I h 0 and 0 S N/(N 2) =0 if N>2, 0 2 (ln(/r )) 1 =0 if N =2, or r I h =0 and =0. (10) 0 Then, for every >0, there exists an extension operator P defined from V to H0 1 () satisfying P L ( V,H0 1()), (11) (P v) = v v V, (12) (P v) (L2 ()) C v N (L 2 (, v V )) N. (13) (9) such that and ( r ) N/2 P u u weakly in H0 1 (), for N>2 P [( r ) 1/2 (log ) 1/2 u ] u r weakly in H0 1 (), for N =2, where u is the solution of the problem div(a 0. u) =0 in, u =0 on,

4 4 Diffusion equation for composite materials EJDE 2000/15 and the matrix A 0 =(a 0 ij ) ij has entries a 0 ij = 1 Y Y (a ji N k=1 and χ j is a Y -periodic function that satisfies a ki χ j y k dy), (14) div(a t (y j χ j )) = 0 in Y (15) Remark. One can replace the first equation of system (6) by div(a u )=f in, with ( r ) N/2 f f weakly in L 2 () if N>2, ( r ) 1 (ln(/r )) 1/2 f f weakly in L 2 () if N =2. Then u will be the solution of div(a 0. u) =f in, u =0 on. This approach has been used in [5] for the case A = I, in[3]whena=iand r =, and in [6] for the case where r > N/(N 2) using the H 0 -convergence and some arguments given by S. Kaizu in [7]. 3 Proof of the main result Observe first that S is admissible in, in the sense of the H 0 -convergence ([6, 4, 5, 10]). Then there exists an extension operator P satisfying (14). On the other hand, the matrix A 0 can be defined by A 0 λ = M Y ( t A w λ )= 1 t A w λdy, λ R N, Y Y where for every λ R, w λ is the solution of the problem div( t A w λ )=0 iny, (16) with w λ λy Y -periodic. For x R N,let wλ (x)=w λ( x ). (17) To simplify notation, let δ = { (r /) N/2 if N>2, (r /) 1 (ln(/r )) 1/2 if N =2. (18)

5 EJDE 2000/15 Mohamed El Hajji 5 Taking u as a test function in the variational formulation of (6), and using the classical techniques of a priori estimates, one can easily show the existence of two constants c and c independent of such that Hence, from (13), up to a subsequence, c (δ u ) L2 ( ) c. P (δ u ) u weakly in H 1 0 (). (19) Set now ξ = A [P (δ u )]. Then using (19), (11)-(13) and (4)-(5), one shows that ξ is bounded in L 2 (), and so up to a subsequence ξ ξ weakly in L 2 (). (20) Case where (9) is satisfied. Let φ D(). Then from the variational formulation of (6), one has χ ξ. φdx =δ h φdσ. (21) If N() denotes the number of the holes included in, one has then δ h φdσ φ L ()δ h( x ) dσ(x) k K (r (S+k)) r cδ N()r N 1 h dσ (22) From (18), one can write δ rn 1 N = and so in virtue of (9), { S cδ rn 1 N S 1/2 h L 2 ( S). ( rn 2 ) 1/2 N if N>2, [ 2 (ln(/r )) 1 ] 1/2 if N =2, r N 1 0 δ hence 0 δ h φdσ =0. On the other hand, it is easy to show that =0, (23) N χ 1 strongly in L p () p [1, [, hence χ ξ. φdx ξ. φdx. (24)

6 6 Diffusion equation for composite materials EJDE 2000/15 One can deduce that, as 0 in (21), ξ φdx =0, φ D(). Consequently div ξ =0 in. (25) It remains to identify the function ξ. Let wλ be the function defined by (16)- (17). Then w λ λx weakly in H1 (), and so L p () strong p <2 where 1/2 =1/2 1/N,withN 2. Let φ D(), by choosing φwλ as a test function in the variational formulation of (6), one has ξ (φwλ) dx = δ h φwλ dx. (26) To pass to the it as 0 in (26), we set ξ (φw λ) dx = J 1 + J 2, (27) where J1 = ξ φ.wλ dx and J2 = ξ wλ.φ dx. Using the results given by G. Stampacchia in [9] (see also [1]), one can deduce that w λ L (Y ), so This with convergence (20), gives J1 = χ wλ ξ φdx χ w λ λx strongly in L 2 (). (28) λxξ φdx as 0. (29) Now, we may write J2 = ξ wλφdx ξ wλφdx. S (30) One the one hand, ξ wλ φdx = t A wλ [P (δ u )φ]dx = t A wλ φp [(δ u ] dx t A w λ φp [(δ u ] dx

7 EJDE 2000/15 Mohamed El Hajji 7 because, from the definition of wλ, t A wλ [P (δ u )φ] dx =0. From the definition of A 0, t A wλ A0 λweakly in L 2 (). From (19), up to a subsequence, P (δ u ) u strongly inl 2 (), which implies t A wλ φp (δ u ) dx A 0 λu φdx. Hence ξ wλφdx A 0 λ φu dx. (31) On the other hand, ξ wλφdx c ξ L 2 () wλ L 2 (S ). (32) S Since ξ L2 () is bounded, ξ wλφdx c w λ L 2 (S ). (33) S Note that wλ 2 L 2 (S ) = ( wλ)(x) 2 dx S = ( wλ)(x) 2 dx k K r (S+k) = ( y w λ )( x k K r (S+k) ) 2 dy = N() N ( y w λ )(y) 2 dy r S c w λ 2 dy. r S Since r / 0andw λ H 1 (Y), it follows that w λ 2 dy 0. r S/ Using (33), one has S ξ w λφdx 0 as 0.

8 8 Diffusion equation for composite materials EJDE 2000/15 This, with (30) and convergence (31) imply that J2 A 0 λ φu dx. (34) Next we pass to the it in the right hand of (26). With the same argument as in (22), δ h φwλ dσ cδ N()r N 1 hw λ dσ S Since we have shown that from (9) one deduces that 0 cδ rn 1 N w λ L2 ( S) S 1/2 h L2 ( S). rn 1 δ N =0, δ h φw λ dσ 0. Finally, by passing to the it as 0 in (26), and using (32) and (34) one obtains λxξ φdx A 0 λ φu dx =0, hence, from (25) it follows that ξλφdx = A 0 λ uφ dx φ D(), λ R N, i.e., ξ = A 0 u. Case where (10) is satisfied. Let φ D(). Then from the variational formulation of (6), χ ξ. φdx=δ h φdσ. (35) The arguments used the proof of (24) can be applied here to obtain χ ξ. φdx ξ. φdx. To pass to the it in the right-hand side of (35), we introduce N as the solution to div N =0 ins, N.n = h on S.

9 EJDE 2000/15 Mohamed El Hajji 9 The existence of N is assured by the hypothesis I h =0. SetN (x)=n( x k ), r for x in (Y \ r S) k.then h φdσ = φ.n dx, S hence δ h φdσ δ φ L 2 (S ) N L 2 (S ). Note that N L2 (S ) c( r )N/2 N L2 (S), so δ h φdσ cδ ( r )N/2 φ L1 (S ). (36) Since δ ( r { 1 if N>2 )N/2 = (ln(/r )) 1/2 if N =2, it follows from (36), when N =2,that 0 δ h φdσ =0. For N>2, one has δ h φdσ c φ L2 (S ). Since χ 1 strongly in L p (), for all p [1, [ andφ D(), one deduces that (1 χ ) φ 2 dx 0. Hence, by passing to the it as 0 in (35), one obtains ξ. φdx=0, then div ξ =0 in. Let wλ be the function defined by (16)-(17) and φ D(). As in the previous case, by using φwλ as a test function in the variational formulation of (6), one has ξ. (φwλ) dx = δ h φwλ dσ. From (27), (29) and (34), one has ξ. (φwλ) dx λx. φdx A 0 λ. φu dx. (37)

10 10 Diffusion equation for composite materials EJDE 2000/15 Nowweshowthat δ h φwλ dσ 0. (38) One has δ h φwλ dσ = δ (φwλ).n dx S δ φ.wλ.n dσ + δ φ. wλ.n dσ S S δ N L2 (S ) φwλ L 2 (S ) + δ N L2 (S ) φ wλ L 2 (S ) cδ ( r { )N/2 φwλ L 2 (S ) + φ wλ } L 2 (S ) cδ ( r { )N/2 φ L () wλ L 2 (S ) + φ L () wλ } L 2 (S ). cδ ( r { )N/2 wλ L 2 (S ) + wλ } L 2 (S ). Note that wλ 2 L 2 (S ) = wλ 2 dx = wλ 2 dx S k K r (S+k) N() N w λ 2 dx c w λ 2 dx. r r S S Since w λ H 1 (S) andr / 0, w λ 2 dx =0. 0 r S Hence wλ L 2 (S ) 0. On the other hand, one has wλ L 2 (S ) c. Finally, as 0 δ ( r )N/2 =0, one deduces (38). This and (37) completes the proof, using the same arguments as in the previous case. Acknowledgments The author would like to thank Professor Patrizia Donato for her help on this work. References [1] Brezis H. Analyse fonctionnelle, théorie et applications. Masson, 1992.

11 EJDE 2000/15 Mohamed El Hajji 11 [2] Briane M., A. Damlamian & Donato P. H-convergence in perforated domains, volume 13. Nonlinear partial Differential Equations & Their Applications, Collège de France seminar, H. Brézis & J.-L. Lions eds., Longman, New York, à paraitre. [3] Cioranescu D.& Donato P. Homogénéisation du problème de Neumann non homogène dans des ouverts perforés. Asymptotic Analysis, 1: , [4] Cioranescu D. & Saint Jean Paulin J. Homogenization in open sets with holes. Journal of Mathematical Analysis & Applications, 71: , [5] Conca C. & Donato P. Non-homogeneous Neumann problems in domains with small holes. Modélisation Mathématiques et Analyse Numérique, 4(22): , [6] Donato P. & El Hajji M. An application of the H 0 -convergence to some nonhomogeneous Neumann problems. GAKUTO International Series, Math. Sc. and Appl., Gakkotosho, Tokyo, Japan, 9: , [7] Kaizu S. The Poisson equation with nonautonomous semilinear doundary conditions in domains with many tiny holes. Japan. Soc. for Industr. & Appl. Math., 22: , [8] Levy T. Filtration in a porous fissured rock: influence of the fissures connexity. Eur. J. Mech, B/Fluids, 4(9): , [9] Stampacchia G. Equations elliptiques du second ordre à coefficients discontinues. Presses Univ.Montreal, [10] Tartar L. Cours Peccot. Collège de France, Mohamed El Hajji Université de Rouen, UFR des Sciences UPRES-A (Labo de Math.) Mont Saint Aignan, France Mohamed.Elhajji@univ-rouen.fr