IN this paper, we study the corrector of the homogenization

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1 , June 29 - July, 206, London, U.K. A Corrector Result for the Homogenization of a Class of Nonlinear PDE in Perforated Domains Bituin Cabarrubias Abstract This paper is devoted to the corrector of the homogenization of a quasilinear elliptic problem with oscillating coefficients in a periodically perforated domain. A nonlinear Robin condition is prescribed on the boundary of the holes, which depends on a real parameter γ. We suppose that the data satisfy some suitable assumptions to ensure the existence uniqueness of a weak solution of the problem. The periodic unfolding method is used to prove the result. Index Terms correctors, homogenization, nonlinear, unfolding method, quasilinear. I. INTRDUCTIN IN this paper, we study the corrector of the homogenization of a quasilinear elliptic problem with oscillating coefficients posed in a periodically perforated domain. We assume that the holes are of the same size as the period on the boundary of the holes, we prescribe a nonlinear Robin condition, which depend on a real parameter γ. Some suitable growth conditions are also assumed on the nonlinear boundary term while a weaker than a Lipschitz condition is prescribed on the quasilinear term. The assumptions used here are the same as those considered in [9] [6] see also [4] for the existence uniqueness of the weak solution of the problem to hold. The physical motivation is that, in several composites the thermal conductivity depends in a nonlinear way from the temperature itself like the case of a glass or wood, where the conductivity is nonlinearly increasing with the temperature, as well as ceramics, where it is decreasing, or aluminium semi-conductors, where the dependence is not even monotone see [], [2] [23] for details. n the other h, nonlinear Robin conditions appear in several physical situations like in some chemical reactions see for instance [7] or climatization see [24]. To prove the main result in this work, we apply the Periodic Unfolding Method PUM, a method of homogenization recently formed for the periodic case. It was first introduced in [] for fixed domains see also [2] for a general setting detailed proofs [20] for more simple approach, extended to perforated domains in [4] see also [5] for complete proofs [6] for more applications, to more general situations comprehensive presentation in [0] to time-dependent functions in [22]. Some nice properties of this method are: it only deals with the classical notion of convergences in L p -spaces; any function in the perforated domain is mapped to the unfolded function in a fixed domain; one do not need any extensions operators anymore when dealing with nonhomogeneous boundary conditions, like the case in this work. Manuscript received: March 2, 206. B. Cabarrubias is with the Institute of Mathematics, University of the Philippines Diliman, 0 Diliman, Quezon City, PHILIPPINES bituin@math.upd.edu.ph. The correctors for the homogenizations of a class of linear elliptic problem in a periodically perforated domain when the oscillating matrix field depends on a weakly converging sequence, a linear elliptic problem with Dirichlet condition in a fixed domain a linear elliptic problem with Robin boundary conditions in a perforated domain have been done in [8], [2] [5], respectively, via PUM. Some corrector results were also obtained by applying mainly some lemmas, for composites with imperfect interface in [2] see also the references therein. ne can also refer to [8] for a corrector result for H-converging parabolic problems with time-dependent coefficients via Tartar s oscillating test functions. The reader can also see [6] the references therein, for the corrector of some wave equation with discontinuous coefficients in time. For the correctors for linear Dirichlet problems with simultaneously varying operators domains obtained by using some special test function depending on the varying matrices domains, see [9]. The reader is also referred to [3] the references therein, for the correctors for the homogenizations of the wave heat equations to [7] for a corrector result for the wave equation with high oscillating periodic coefficients. Let us also mention here, that the existence uniqueness of a solution of the problem as well as the homogenization were already studied in [4] [5], respectively. This paper is organized as follows: Section 2 gives the geometric setting of the problem as well as the assumptions on the data to ensure existence uniqueness of the weak solution of our problem; Section 3 contains a short discussion on PUM together with the operators the corresponding properties that we need to prove the main result; homogenization results for the problem obtained in [5], for which the corrector result is based, are also recalled in Section 4; the main results for this paper, which completes the study of the asymptotic bahavior in [5], are presented in Section 5. II. SETTING F THE PRBLEM Let us recall the geometric framework for the perforated domain see e.g. [4]. Let b = b, b 2,..., b N be a basis of R N the set of reference periods Y a subset of R N such that, R N = Y + k j b j = Y + ξ, ξ G where G = { k Z N ξ R N ξ = j= } k i b i, k,..., k N Z N, Ξ = {ξ G, ξ + Y }. ISSN: Print; ISSN: nline

2 , June 29 - July, 206, London, U.K. For z R N, we let [z] Y = k j b j, be the unique integer j= combination of periods such that z [z] Y is in Y, {z} Y = z [z] Y. That is, z = {z} Y + [z] Y, z R N. Fig. 2. The perforated domain its boundary = Γ 0 Γ. Fig.. The numbers {z} Y [z] Y. The set Y = 0, N is called the reference cell, that is, { } Y = y R N y = y i b i, y,..., y N Y. Let {} be a positive sequence converging to zero for each positive, one can write { x } [ x ] x = +, Y Y for all x R N. Denote a bounded open set on R N by. Let also S, a compact subset of Y, be the reference hole suppose that S has a Lipschitz continuous boundary with a finite number of connected components. Let us also define Y = Y \S the perforated reference cell. The perforated domain is then given by = \ S, where S = ξ + S. ξ G As introduced in [0] see also [2], we set = interior ξ + Ȳ Λ = \, ξ Ξ that is, is the interior of the largest union of ξ + Ȳ cells fully contained in, Λ contains the parts from the ξ + Ȳ cells that intersects the boundary. The corresponding perforated sets are then, given by, = \ S Λ = \. The boundary of the perforated domain is, = Γ 0 Γ, where Γ = S Γ 0 = \ Γ. Thus, Γ is the boundary of the set of holes contained in. In the perforated domain in Figure 2 below, the dark perforated part is the set the boundary of the holes contained is Γ while the remaining part is the set Λ, the boundary of the holes contained being the Γ 0. Lastly, let α, β R with 0 < α < β, denote by Mα, β, Y the set of N N matrix fields satisfying A = a ij i,j N L Y N N, Ayλ, λ α λ 2 Ayλ β λ, for all λ R N a.e. in Y. The goal of this paper is to provide a corrector result for the homogenization of the following quasilinear elliptic problem P: diva x, u u = f in, u = 0 on Γ 0, A x, u u n + γ τ xhu = g x on Γ, where n is the unit exterior normal to γ is a real parameter, with γ. Let us denote by M the mean value of an integrable function on, given by M Φ = Φydy, Φ L. We also set x τ x = τ, { g x if M S g = 0, g x = g x if M S g 0, x A x, t = A, t, for every x, t R N R. Suppose that the data satisfy the following assumptions: A. f, g, τ are functions such that for every, i f L 2, ii g is a Y -periodic function in L 2 Γ, iii τ is a positive Y -periodic function in L S; A2. h is a function from R to R such that i h is an increasing function in C R such that h0 = 0, ii there exists a constant C > 0 an exponent q with q if N = 2 q N N 2 if N > 2 such that s R, h s C + s q ; ISSN: Print; ISSN: nline

3 , June 29 - July, 206, London, U.K. A3. A : Y R R N 2, is a matrix field satisfying the following conditions: i A is a Caratheodory function Y periodic for every t ii for every t R, A, t Mα, β, Y, iii there exists a function ω : R R such that a. ω is continuous, nondecreasing ωt > 0 for all t > 0, b. Ay, t Ay, t ω t t a.e. y Y, for t t R, c. for any r > 0, lim s 0 + r s dt ωt = +. Now, for p [, +, we define V p = { φ W,p φ = 0 on Γ } 0 V = V 2, which is a Banach space for the norm u V p = u L p u V p. The variational formulation VF of problem P is then given by Find u V such that A x, u u v dx + γ τ xhu v dσ x Γ = fv dx + g xv dσ x, v V. Γ Under assumptions A A3, the existence uniqueness of a solution for problem VF has been proved in [4]. III. A SHRT REVIEW F THE UNFLDING METHD In this section, we briefly recall the main definitions properties of the unfolding operators under PUM, that we need. DEFINITIN. For any Lebesgue-measurable function φ on, the unfolding operator T is a function from L p to L p Y, is defined by [ x ] T φ + y, a.e. in φx, y = Y, Y 0, a.e. in Λ Y. PRPSITIN 2. [0], [2], [4], [5] Let p [, +. T is linear continuous. 2 T φψ = T φt ψ for every φ, ψ L p. 3 For w L p, T w w strongly in L p Y. 4 For all φ L one has φx dx = φx dx = Λ Y T φx, y dx dy. φx dx 5 y T φx, y = T x φx, y for every x, y in R N Y. 6 Let φ L p such that Then φ φ strongly in L p. T φ φ strongly in L p Y. DEFINITIN 3. For p [, + ], the averaging operator U : L p Y L p is defined as U Φx = [ x ] { x } Y + z, dz, Y Y a.e. for x, a.e. for x Λ. ne has Y Φ U Φ Y U Φx = 0, = U Φ, when Φ belongs to L p Y. Some of the properties of the averaging operator are given in the next two propositions. PRPSITIN 4. [0], [2], [4], [5] Let p [, + [. U is linear continuous. 2 U is almost a left inverse of T on. 3 For any φ in L p, φ U φ L p 0. 4 Let w be in L p. Then the following are equivalent: i T w ŵ strongly in L p Y w p dx 0, Λ ii w U ŵ Lp 0. PRPSITIN 5. [0] For p [, +, suppose that ρ is in L p θ in L ; L p Y. Then the product U ρu θ belongs to L p U ρ θ U ρu θ 0 strongly in L p. Let us now define the boundary unfolding operator. Here, we assume that p ], + [ that S has a finite number of connected components. DEFINITIN 6. For any Lebesgue-measurable function ϕ on S, the boundary unfolding operator is defined by T b ϕx, y = { [ x ] ϕ Y + y, a.e. in S, 0 a.e. in Λ S. PRPSITIN 7. [0], [5] Let p [, [. Then T b is a linear operator. 2 T b φψ = T b φt b ψ for every φ, ψ L p S. 3 Let φ L p S be a Y -periodic function. Set φ x = φ x. Then T b φ x, y = φy. 4 For all φ L S, the integration formula is given by φx dσ x = T b φx, y dxdσ y. S Γ ISSN: Print; ISSN: nline

4 , June 29 - July, 206, London, U.K. 5 Let φ L p S. Then T b φ φ strongly in L p S. IV. SME KNWN RESULTS In this section, we recall the homogenization results as obtained in [5], on which the corrector result will be based upon. THEREM 8. Under assumptions A A3, let u be the unique solution of VF γ. Then, there exists u 0, û in H0 L 2 ; HperY with M Y û = 0, such that: i T u u 0 strongly in L 2 ; H Y, ii T u u 0 + y û weakly in L 2 Y, iiit b hu hu 0 weakly in L t ; W t,t S. Case. If γ =, the couple u 0, û is the unique solution in the space H0 L 2 ; HperY with M Y û = 0, of the limit equation Ay, u 0 u 0 + y û φx + y Ψx, y dx dy Y + S M S τ hu 0 φ dx = Y fφ dx + S M S g φ dx for all φ H0 Ψ L 2 ; HperY. Case 2. If γ >, the couple u 0, û is the unique solution in the space H0 L 2 ; HperY with M Y û = 0, of the limit equation Ay, u 0 u 0 + y û φx + y Ψx, y dx dy Y = Y fφ dx + S M S g φ dx for all φ H0 Ψ L 2 ; HperY. CRLLARY 9. Under assumptions A A3, let u be the unique solution of VF. Then, if γ, ũ Y u 0 weakly in L 2, where denotes the extension by 0 to. If γ =, the function u 0 is the unique solution of the limit problem diva 0 u u 0 + S M Sτhu 0 = Y f + S M Sg in u 0 = 0 on. If γ >, the function u 0 is the unique solution of the problem diva 0 u u 0 = Y f + S M Sg in u 0 = 0 on. The homogenized matrix field A 0 t is given by A 0 tλ = Ay, t w λ t, ydy, λ R N, Y ISSN: Print; ISSN: nline where w λ y, t = χ λ y, t + λ y a.e. in Y χ λ, t is, for every t, the solution of the cell problem diva, t χ λ, t = diva, tλ in Y, A, t χ λ, t n = 0 on S, χ λ, t Y periodic, M Y χ λ, t = 0. REMARK 0. ne has see e.g. [3] for the details of the computation, ûx, y = χ ei y, u 0 x u 0 x, i where u 0 is the one given in Theorem 8. V. MAIN RESULT First we recall a classical result that we need to prove the main result given in Theorem 4. LEMMA. [0] Let {D } be a sequence of n n matrix fields in Mα, β, for some open set, such that D D a.e. on, or more generally, in measure in. If the sequence {ζ } converges weakly to ζ in [L 2 ] N, then lim inf D ζ ζ dx Dζζ dx. Furthermore, if lim sup D ζ ζ dx Dζζ dx, then Dζζ dx = lim D ζ ζ dx, ζ ζ strongly in [L 2 ] N. Next, we present a proposition which is also essential in proving the corrector result. PRPSITIN 2. Let γ suppose M S g 0. Under the assumptions in Theorem 8, one has T u u 0 + y û strongly in L 2 Y, 2 lim u 2 dx = 0. 3 Λ Proof: We prove the case γ =. When γ >, the proof is similar but the the nonlinear term in the boundary approaches 0 at the limit. By applying PUM, fu dx = Y T ft u dx dy, so that from Proposition 2-3 Theorem 8 convergence i, lim fu dx = lim T ft u dx dy Y = fu 0 dx dy. Y

5 , June 29 - July, 206, London, U.K. Since f u 0 are just functions of x, one gets lim fu dx = Y fu 0 dx. 4 Also, from Proposition 3.5 of [0], g xu dσ x = S M Sg Γ Moreover, Γ = τ xhu u dσ x S u 0 dx. 5 τyt b hu T u dx dσ y, from Proposition 7. Thus, one obtains from Theorem 8 i iii that, lim τyt b hu T u dx dσ y S = S M Sτ hu 0 u 0 dx, which yields, lim Γ Now, using Lemma with τ xhu u dσ x = S M Sτ hu 0 u 0 dx. D = T A x, u ζ = T u, together with 4-6, one obtains Ay, u 0 u 0 + y û u 0 + y û dx dy Y lim inf T A x, u u u dx dy Y lim sup lim sup = lim sup = Y Y T A x, u u u dx dy A x, u u u dx γ fv dx + g xv dσ x Γ τ xhu v dσ x Γ fu 0 dx + S M Sg S M Sτ hu 0 u 0 dx = Ay, u 0 u 0 + y û Y u 0 + y û dx dy, ISSN: Print; ISSN: nline u 0 dx 6 by choosing φ = u 0 Ψ = û in the limit equation for Case in Theorem 8. Thus, lim T A x, u u u dx dy Y = Y Ay, u 0 u 0 + y û u 0 + y û dx dy which implies, using Lemma, the convergence given in 2. n the other h, from Proposition 2-4 the ellipticity of A, α u 2 dx A x, u u u dx Λ Λ = A x, u u u dx Y T A x, u T u T u dx dy. This, together with 7 gives the second convergence in 3. REMARK 3. From the computations above, we have the following convergence of the energy: A x, u u u dx lim = Y Ay, u 0 u 0 + y û u 0 + y û dx dy. Let us now have the main result in this paper, the corrector result. THEREM 4. Let γ suppose M S g 0. Under the assumptions in Theorem 8, we have u u 0 + U y χ ei y, u 0 xu u0 i L 2 converges to 0. Proof: From Proposition 2, Proposition 4-4, one has u U u 0 + y û L 2 0. This together with Proposition 4-,3 gives u u 0 U u 0 + U u 0 U y û L2 u U u 0 U y û L2 + u 0 U u 0 L 2 0. Thus, from, Proposition 5 the computations above, one obtains u u 0 U y χ ei y, u 0 x u 0 i +U y χ ei y, u 0 x u 0 i 7

6 , June 29 - July, 206, London, U.K. U y χ ei y, u 0 xu u0 i L2 U y χ ei y, u 0 x u 0 i U y χ ei y, u 0 xu u0 i L2 + u u 0 U y χ ei y, u 0 x u 0 i 0, L 2 which yields the desired result. Remark: The case M S g = 0 has been recently done in [8]. [8] A. Dall Aglio F. Murat, A corrector result for H-converging parabolic problems with time-dependent coefficients, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4 e série, tome 25, n 0-2, pp , 997. [9] G. Dal Maso F. Murat, Asymptotic behavior correctors for linear Dirichlet problem with simultaneously varying operators domains, Ann. I. H. Poincaré, AN 2, pp , [20] A. Damalamian, An elementary introduction to periodic unfolding, Gakuto International Series, Math.Sci. Appl. Vol. 24, pp. 9-36, [2] P. Donato, Some corrector results for composites with imperfect interface, Rendiconti di Matematica, Serie VII, Vol 26, pp , [22] P. Donato Z. Yang, The Periodic unfolding method for the wave equation in domains with holes, Advances in Mathematical Sciences Applications, 222, pp , 202. [23] Encyclopedia of Material Science Engineering, edited by Michael B. Bever, Pergamon Press, xford, New York, tome 3 988, p. 209, tome 7 986, p , p. 4950, p [24] C. Timofte, n the homogenization of a climatization problem, Stud. Univ. Babes-Bolyai Math., 52 2, pp. 7-25, REFERENCES [] S. Bendib, Homogenization of some class of nonlinear problem with Fourier condition in the holes Homogénéisation d une classe de problèmes non linéaires avec des conditions de Fourier dans des ouverts perforés, Thèse, Institut National Polytechnique de Lorraine, [2] S. Bendib R.L. Tcheugoué Tébou, Homogenization of a class of nonlinear problem in perforated domains Homogénéisation d une classe de problèmes non linéaires dans des domaines perforés, C.R. Acad. Sci. Paris, t. 328, Serie, pp , 999. [3] S. Brahim-tsmane, G.A. Francfort, F. Murat, Correctors for the homogenization of the wave heat equations, J. Math. Pures Appl., 7, pp , 992. [4] B. Cabarrubias P. Donato, Existence uniqueness for a quasilinear elliptic problem with nonlinear Robin conditions, Carpathian J. Math. Vol 27, No.2, pp , 20. [5] B. Cabarrubias P. Donato, Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions, Applicable Analysis, DI:0.080/ , 20. [6] J. Casado-Diaz, et al., Homogenization corrector for the wave equation with discontinuous coefficients in time, J. Math. Anal. Appl., 379, pp , 20. [7] J. Casado-Diaz, et al., A corrector result for the wave equation with hgh oscillating periodic coefficients, Advances in Differential Equations Applicatgions SEMA SIMAI Springer Series 4, pp , Springer International Publishing Switzerl 204, DI: 0.007/ [8] I. Chourabi P. Donato, Homogenization correctors of a class of elliptic problems in perforated domains, Asymptotic Analysis, 92-2, pp. -43, 205. [9] M. Chipot, Elliptic Equations: An Introductory Course. Germany: Birkhäuser Verlag AG, [0] D. Cioranescu, A. Damlamian, P. Donato, G. Griso R. Zaki, The Periodic Unfolding Method in Domains With Holes, SIAM J. Math. Anal., Vol. 44, No. 2, pp , 202. [] D. Cioranescu, A. Damlamian G. Griso, Periodic unfolding homogenization, C.R. Acad. Sci. Paris, Ser. I 335, pp , [2] D. Cioranescu, A. Damlamian G. Griso, The periodic unfolding homogenization, SIAM J. Math. Anal., Vol. 40, No. 4, pp , [3] D. Cioranescu P. Donato, An Introduction to Homogenization, xford University Press, 999. [4] D. Cioranescu, P. Donato R. Zaki, The Periodic Unfolding Robin problems in perforated domains, C.R.A.S. Paris, Série, 342, pp , [5] D. Cioranescu, P. Donato R. Zaki, The Periodic Unfolding Method in Perforated Domains,Portugal Math 634, pp , [6] D. Cioranescu, P. Donato R. Zaki, Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptotic Analysis 53, pp , [7] C. Conca, J.I. Diaz, A. Linan C. Timofte, Homogenization in Chemical Reactive Flows, Electronic Journal of Differential Equations, Vol. 40, pp. -22, ISSN: Print; ISSN: nline