Homogenization of parabolic problems with dynamical boundary conditions of reactive-diffusive type in perforated media
Abstract
This paper deals with the homogenization of the reaction-diffusion equations in a domain containing periodically distributed holes of size ε, with a dynamical boundary condition of reactive-diffusive type, i.e., we consider the following nonlinear boundary condition on the surface of the holes where denotes the Laplace–Beltrami operator on the surface of the holes, ν is the outward normal to the boundary, plays the role of a surface diffusion coefficient and g is the nonlinear term. We generalize our previous results established in the case of a dynamical boundary condition of pure-reactive type, i.e., with . We prove the convergence of the homogenization process to a nonlinear reaction-diffusion equation whose diffusion matrix takes into account the reactive-diffusive condition on the surface of the holes.
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Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ Esta es la versión aceptada del artículo publicado en: This is a accepted manuscript of a paper published in: Journal of Applied Mathematics and Mechanics (2020): 13/06/2020 DOI: https://doi.org/10.1002/zamm.202000088 Copyright: El acceso a la versión publicada del artículo puede requerir la suscripción de la revista. Access to the published version may require subscription. “This is the peer reviewed version of the following article: Homogenization of parabolic problems with dynamical boundary conditions of reactive-diffusive type in perforated media. Z Angew Math Mech. 2020, vol. 100, n.10, e202000088, which has been published in final form at https://doi.org/10.1002/zamm.202000088. This article may be used for noncommercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited."
Homogenization of parabolic problems with dynamical boundary conditions of reactive-di↵usive type in perforated media Mar´ıa ANGUIANO1 Abstract This paper deals with the homogenization of the reaction-di↵usion equations in a domain containing periodically distributed holes of size ", with a dynamical boundary condition of reactive-di↵usive type, i.e., we consider the following nonlinear boundary condition on the surface of the holes ru"·⌫+"@u" @t="u""g(u"), where denotes the Laplace-Beltrami operator on the surface of the holes, ⌫is the outward normal to the boundary, >0 plays the role of a surface di↵usion coefficient and gis the nonlinear term. We generalize our previous results (see [3]) established in the case of a dynamical boundary condition of pure-reactive type, i.e., with = 0. We prove the convergence of the homogenization process to a nonlinear reaction-di↵usion equation whose di↵usion matrix takes into account the reactive-di↵usive condition on the surface of the holes. AMS classification numbers: 35B27, 35K57 Keywords: Homogenization, perforated media, reaction-di↵usion systems, dynamical boundary conditions, surface di↵usion 1 Introduction and setting of the problem In the context of reaction-di↵usion equations, dynamical boundary conditions have been rigorously derived in Gal and Shomberg [11] based on first and second thermodynamical principles and their physical interpretation was also given in Goldstein [12]. It is worth emphasizing that the derivation in [11] obtains the dynamical boundary condition of reactive-di↵usive type both as a sufficient and necessary condition for thermodynamic processes which incorporate thermodynamic sources located along the boundary, and in which the second law plays a crucial role, while in [12] it has been introduced only as a sufficient condition. In particular, a dynamical boundary condition of reactive-di↵usive type accounts for (see [12, Section 3]) a heat source on the boundary that can depend on the heat flow along the boundary, the heat flux across the boundary and the temperature at the boundary. Consider the reaction-di↵usion equation, with dynamical boundary condition of reactive-di↵usive type, provides, in addition to classical bulk di↵usion, a di↵usion mechanism present along the boundary. A typical example in the theory of heat conduction (see [11] for more details) arises when a given body is in perfect thermal contact with a sufficiently thin metal sheet, possibly of di↵erent material and completely insulating the internal body from external contact, say, a well-stirred hot or cold fluid. In a recent article (see [3]) we addressed the problem of the homogenization of the reaction-di↵usion equations with a dynamical boundary condition of pure-reactive type in a domain perforated with holes. The present article is devoted to the generalization of that previous study to the case of a dynamical boundary condition of reactivedi↵usive type, i.e., we add to the dynamical boundary condition a Laplace-Beltrami correction term. Let us introduce the model we will be involved with in this article. 1Departamento de An´alisis Matem´atico. Facultad de Matem´aticas. Universidad de Sevilla, 41012 Sevilla (Spain) [email protected] 1
Mar´ıa Anguiano The geometrical setting. Let ⌦be a bounded connected open set in RN(N2), with smooth enough boundary @⌦. Let us introduce a set of periodically distributed holes. As a result, we obtain an open set ⌦", where "represents a small parameter related to the characteristic size of the holes. Let Y=[0,1]Nbe the representative cell in RNand Fan open subset of Ywith smooth enough boundary @F, such that ¯ F⇢Y. We denote Y⇤=Y\¯ F. For k2ZNand "2(0,1], each cell Yk,"="k+"Yis similar to the unit cell Yrescaled to size "and Fk,"="k+"Fis similar to Frescaled to size ". We denote Y⇤ k,"=Yk,"\¯ Fk,". We denote by F"the set of all the holes contained in ⌦, i.e. F"=[k2K{Fk,":¯ Fk,"⇢⌦}, where K:= {k2ZN:Yk,"\⌦6=;}. Let ⌦"=⌦\¯ F". By this construction, ⌦"is a periodically perforated domain with holes of the same size as the period. Remark that the holes do not intersect the boundary @⌦. Let @F"=[k2K{@Fk,":¯ Fk,"⇢⌦}.So @⌦"=@⌦[@F". Position of the problem. The prototype of the parabolic initial-boundary value problems that we consider in this article is 8 > > > > > > > < > > > > > > > : @u" @tu"+u"=0 in ⌦ "⇥(0,T), ru"·⌫+"@u" @t="u""g(u") on @F"⇥(0,T), u"=0,on @⌦⇥(0,T), u"(x, 0) = u0 "(x),for x2⌦", u"(x, 0) = 0 "(x),for x2@F", (1) where u"=u"(x, t), x2⌦",t2(0,T) and T>0. The first equation states the law of standard di↵usion in ⌦", = xdenotes the Laplacian operator with respect to the space variable and >0 is a given constant. The boundary equation (1)2is multiplied by "to compensate the growth of the surface by shrinking ", where the value of u"is assumed to be the trace of the function u"defined for x2⌦", denotes the Laplace-Beltrami operator on @F",⌫denotes the outward normal to @F", and >0 is a given constant. The term ru"·⌫ represents the interaction domain-boundary, while stands for a boundary di↵usion. We assume that the function g2C(R) is given, and satisfies that there exist constants q2, ↵1>0, ↵2>0,>0, and l>0, such that ↵1|s|qg(s)s↵2|s|q+,for all s2R, (2) (g(s)g(r)) (sr)l(sr)2,for all s, r 2R. (3) Finally, we also assume that u0 "2L2(⌦), 0 "2L2(@F"),(4) are given, and we suppose that |u0 "|2 ⌦"+"| 0 "|2 @F"C, (5) where Cis a positive constant, and we denote by |·|⌦"and |·|@F"the norm in L2(⌦") and L2(@F"), respectively. Depending of , two classes of boundary conditions are modeled by (1). For >0, we have boundary conditions of reactive-di↵usive type, and for = 0 the boundary conditions are purely reactive. In [3], we consider the homogenization of the problem (1) with = 0 and we obtain rigorously a nonlinear parabolic problem with zero Dirichlet boundary condition and with extra-terms coming from the influence of the dynamical boundary conditions as the homogenized model. Though the results of the present article are similar to those of [3], the generalization of their proof is not trivial. Some new technical results are required in order to carry out the machinery of [3]. Due to the presence of Laplace-Beltrami operator in the boundary condition, the variational formulation of the reaction-di↵usion equation is di↵erent that in [3]. We have to work in the space W=(v,0(v)) 2H1(⌦")⇥H1(@F") ,>0,(6) where 0denotes the trace operator v7! v|@⌦", and where we define by H1(@F") the completion of C1(@F") with respect to the induced norm by the inner product ((, ))@F":= Z@F" d+Z@F"r·r d,8, 2C1(@F"), 2
Mar´ıa Anguiano where rdenotes the tangential gradient on @F"and ddenotes the natural volume element on @F".The estimates of [3] did not allow to cover this case and new estimates are needed to deal with problem (1). In order to prove estimates in H2-norm, we have to combine estimates for general elliptic boundary value problems with interpolation properties of Sobolev spaces (see Lemma 4.2). On the other hand, in order to pass to the limit, as "!0, for the term which involves the tangential gradient rwe make use of a convergence result based on a technique introduced by Vanninathan [20] for the Steklov problem which transforms surface integrals into volume integrals. This convergence result can be used taking into account the estimates in H2-norm. Several technical results are merely quoted, and we refer [3] for their proof. We present here a new result concerning the local problem, which involves the orthogonal projection (denoted by P), the tangential gradient (denoted by r) and the tangential divergence (denoted by div) on the boundary of the unit cell. More precisely, using the so-called energy method introduced by Tartar [19] and considered by many authors (see, for instance, Cioranescu and Donato [5]), we prove the following: Theorem 1.1 (Main Theorem).Under the assumptions (2)–(3) and (5), assume that g2C1(R), the exponent qsatisfies that 2q<+1if N=2 and 2q2N2 N2if N>2,(7) and (u0 ", 0 ")2W\(Lq(⌦")⇥Lq(@F")). Let (u", ")be the unique solution of the problem (1), where "(t)= 0(u"(t)) a.e. t2(0,T]. Then, as "!0, we have ˜u"(t)!u(t)strongly in L2(⌦),8t2[0,T], where ˜ ·denotes the extension to ⌦⇥(0,T)and uis the unique solution of the following problem 8 > > > > < > > > > : ✓|Y⇤| |Y|+|@F| |Y|◆@u @tdiv (Qru)+|Y⇤| |Y|u+|@F| |Y|g(u)=0,in ⌦⇥(0,T), u(x, 0) = u0(x),for x2⌦, u=0,on @⌦⇥(0,T). (8) The homogenized matrix Q=((qi,j )),1i, j N, which is symmetric and positive-definite, is given by qi,j =1 |Y|✓ZY⇤ (ei+rywi)·(ej+rywj)dy +Z@F (Pei+rwi)·(Pej+rwj)d(y)◆,(9) where wi2Hper/R,1iN, is the unique solution of the cell problem 8 > > > < > > > : divy(ei+rywi)=0,in Y⇤, (ei+rywi)·⌫=div(Pei+rwi),on @F, wiis Yperiodic. (10) Here, eiis the ielement of the canonical basis in RNand Hper is the space of functions from H:= {v2H1(Y⇤): v|@F2H1(@F)}which are Y-periodic. Remark 1.2. Note that in the case =0(i.e., in the absence of a surface di↵usion coefficient), the homogenized equation (8) is exactly the equation obtained in [3]. Remark 1.3. An example of a function g2C1(R), satisfying (2)-(3), is a odd degree polynomial, g(s)= 2k+1 X j=0 cjsj, where c2k+1 >0.Thetypicalcaseisg(s)=s3s. 3
Mar´ıa Anguiano The homogenization of problems which involve the Laplace-Beltrami operator has been considered in recent articles. In particular, in the context of periodic homogenization based on the periodic unfolding method, in [13] Graf and Peter extend the existing convergence results for the boundary periodic unfolding operator to gradients defined on manifolds. These results are then used to homogenize a system of five coupled reaction-di↵usion equations, three of which include di↵usion described by the Laplace-Beltrami operator and four of which consider a particular nonlinearity. In [1], Amar and Gianni state a new property of the unfolding operator regarding the unfolded tangential gradient. This property is used to homogenize a di↵erential system of linear equations in two disjoint conductive phases with a linear dynamical boundary condition which involves the Laplace-Beltrami operator in the separating interface. An error estimate for this model, under extra regularity assumptions on the data, can be found in Amar and Gianni [2]. More recently, in [9], Gahn derives some general two-scale compactness results for coupled bulk-surface problems and applies these results to an elliptic problem with a non-dynamical boundary condition, which involves the Laplace-Beltrami operator, in a multi-component domain. However, to our knowledge, there does not seem to be in the literature any study on the homogenization of parabolic models associated with nonlinear dynamical boundary conditions, which involves the Laplace-Beltrami operator, in a periodically perforated domain, as we consider in this article. The article is organized as follows. In Section 2, we introduce suitable functions spaces for our considerations. Especially, we consider some fundamentals from di↵erential geometry as the tangential gradient and the tangential divergence. To prove the main result, in Section 3 we prove the existence and uniqueness of solution of (1), a priori estimates are established in Section 4 and some compactness results are proved in Section 5. Finally, the proof of Theorem 1.1 is established in Section 6. 2 Functional setting Notation. We denote by (·,·)⌦"(respectively, (·,·)@F") the inner product in L2(⌦")(respectively,inL2(@F")), and by |·|⌦"(respectively, |·|@F") the associated norm. We also denote by (·,·)⌦"the inner product in (L2(⌦"))N. If r6= 2, we will also denote by (·,·)⌦"(respectively, (·,·)@F") the duality product between Lr0(⌦") and Lr(⌦") (respectively, the duality product between Lr0(@F") and Lr(@F")). We will denote by |·|r,⌦"(respectively |·|r,@F") the norm in Lr(⌦")(respectivelyinLr(@F")). We denote by (·,·)⌦the inner product in L2(⌦), and by |·|⌦the associated norm. If r6= 2, we will also denote by (·,·)⌦the duality product between Lr0(⌦) and Lr(⌦). We will denote by |·|r,⌦the norm in Lr(⌦). By k·k⌦"we denote the norm in H1(⌦"), which is associated to the inner product ((u, v))⌦":= (u, v)⌦"+(ru, rv)⌦",8u, v 2H1(⌦"), and by ||·||⌦",T we denote the norm in L2(0,T;H1(⌦")). By k·k⌦we denote the norm in H1(⌦), by ||·||⌦,T we denote the norm in L2(0,T;H1(⌦)) and, if r6= 2, we denote by |·|r,⌦,T the norm in Lr(0,T;Lr(⌦)). We denote by 0the trace operator u7! u|@⌦", which belongs to L(H1(⌦"),H1/2(@⌦")). We introduce, for any s>1, the space Hs(⌦"), which is naturally embedded in H1(⌦"), and it is a Hilbert space equipped with the norm inherited, which we denote by ||·||Hs(⌦"). Moreover, we denote by Hr @⌦(⌦") and Hr @⌦(@⌦"), for r0, the standard Sobolev spaces which are closed subspaces of Hr(⌦") and Hr(@⌦"), respectively, and the subscript @⌦means that, respectively, traces or functions in @⌦", vanish on this part of the boundary of ⌦", i.e. Hr @⌦(⌦")={v2Hr(⌦"):0(v) = 0 on @⌦}, 4
Mar´ıa Anguiano and Hr @⌦(@⌦")={v2Hr(@⌦"):v= 0 on @⌦}. Analogously, for r2, we denote Lr @⌦(@⌦"):={v2Lr(@⌦"):v= 0 on @⌦}. Let us notice that, in fact, we can consider the given 0 "as an element of L2 @⌦(@⌦"). Let us consider the space Hq:= Lq(⌦")⇥Lq @⌦(@⌦"), 8q2, with the natural inner product ((v,),(w,'))Hq=(v,w)⌦"+"(,')@F",which in particular induces the norm |(·,·)|Hqgiven by |(v,)|q Hq=|v|q q,⌦"+"||q q,@F",(v,)2Hq. For the sake of clarity, we shall omit to write explicitly the index qif q= 2, so we denote by Hthe Hilbert space H:= L2(⌦")⇥L2 @⌦(@⌦"). For functions u2H1 @⌦(⌦") which satisfy u2L2 @⌦(⌦"), we have Z⌦" uvdx=Z⌦"ru·rvdx +Z@F"ru·⌫vd(x),8v2H1 @⌦(⌦"). Tangential gradient and Laplace-Beltrami operator. We recall here, for the reader’s convenience, some well-known facts on the tangential gradient rand the Laplace-Beltrami operator . We refer to Sokolowski and Zolesio [17] for more details and proofs. Let Sbe a smooth surface with normal unit vector ⌫. For every v2(L2(S))N, we can define an element Pv2(L2(S))Nsuch that Pv·⌫= 0 a.e. on S,whereP(y) for y2Sis the orthogonal projection on the tangent space at y2S, i.e., it holds that P(y)v(y)=v(y)(v(y)·⌫(y)) ⌫(y) for a.e. y2S. Let 2C1(S), there exist a tubular neighborhood Uof Sand an extension ˜ 2C1(U) of . We define the tangential gradient of on Sby r:= Pr˜ =r˜ (r˜ ·⌫)⌫on S. We emphasize that this definition is independent of the chosen extension of . Let 2(C1(S))N, then there exists an extension ˜ 2(C1(U))N(Uas above a suitable neighborhood of S) and we define the tangential divergence of on Sby div:=r·:=r·˜ D˜ ⌫·⌫on S, where D˜ is the Jacobi-matrix of ˜ . Now, we consider the surface @F". First, an equivalent definition of the Sobolev space H1(@F") on @F"is given. We introduce the inner product ((, ))@F":= (, )@F"+(r,r )@F",8, 2C1(@F"),0, and denote by ||·||@F"the induced norm. The Sobolev space H1(@F") is the closure of the space C1(@F")with respect to the norm induced by the inner product. Therefore, the space C1(@F") is dense by definition in the space H1(@F"). An equivalent definition of H1(@F") can be given via local coordinates or distributional meaning, see, for instance, Strichartz [18]. We denote by ||·||@F",T the norm in L2(0,T;H1(@F")). 5
Mar´ıa Anguiano By definition, for every 2H1(@F")thereexistsr2L2(@F")withr·⌫= 0 a.e. on @F", the tangential gradient in the distributional sense. We introduce, for any s>1, the space Hs(@F"), which is naturally embedded in H1(@F"), equipped with the norm inherited, which we denote by ||·||Hs(@F"). For all 2H1(@F") and v2(C1(@F"))Nsuch that v·⌫= 0 a.e. on @F", we have the Stokes formula (see [17, Proposition 2.58]) Z@F"r ·vd=Z@F" divvd.(11) Let h2H2(@F"), then we have rh2H1(@F") such that rh·⌫= 0 a.e. on @F". The Laplace-Beltrami operator on @F"is defined as follows h=div (rh)8h2H2(@F"). Hence h2L2(@F"), and from (11) it follows that the element h2L2(@F") is uniquely determined by the integral identity Z@F" h d=Z@F"rh·r d8 2H1(@F").(12) If 2H1(@F"), then there exists (see [17, Chapter 2, Section 2.20]) an element #2H3/2(⌦"), the extension of , and #|@F"= ,furthermore r#·⌫= 0 on @F".(13) Therefore r#=r on @F". It should be noted that on the right-hand side of (12) there is the scalar product of vector fields rhand r tangent to @F". On the other hand, if is a smooth function defined in an open neighbourhood of @F"in ⌦, then (see [17, Chapter 2, Section 2.20]) rh·(r |@F")=rh·r because of (r ·⌫⌫)·rh=0. Hence, if is the restriction to @F"of a given function defined in ⌦, then Z@F" h d=Z@F"rh·r d8 2H2(⌦).(14) The space W.We now introduce, as anticipated in the introduction, the space Wgiven in (6) (see [10, Subsection 2.2] for more details). Let V @⌦,0, be the completion of C1(⌦") in the norm ||u||2 V @⌦:= Z⌦"|u(x)|2+|ru(x)|2dx +"Z@F"|u(x)|2+|ru(x)|2d(x). Note that for any f2V @⌦,wehavef2H1 @⌦(⌦") so that f@F"makes sense in the trace sense. The space V @⌦is topologically isomorphic to H1(⌦")⇥H1 @⌦(@⌦")if>0, and V0 @⌦=H1 @⌦(⌦"). For all 0, we define the linear space W:= (v,0(v)) : v2V @⌦ . We emphasize that Wis not a product space as V @⌦. Clearly, W⇢Hdensely since the trace operator acting on function H1(⌦") and into H1/2(@⌦") is bounded and onto, and Wis a Hilbert space with respect to the inner product inherited from V @⌦,0. Thus, by definition we can identify W=(v,0(v)) 2H1(⌦")⇥H1 @⌦(@⌦") ,for each >0, where k(v,0(v))k2 W:= kvk2 ⌦"+"k0(v)k2 @F", and W0=n(v,0(v)) 2H1(⌦")⇥H1/2 @⌦(@⌦")o. 6
Mar´ıa Anguiano 3 Existence and uniqueness of solution Along this paper, we shall denote by Cdi↵erent constants which are independent of ". We state in this section a result on the existence and uniqueness of solution of problem (1). First, we observe that it is easy to see from (2) that there exists a constant C>0 such that |g(s)|C⇣1+|s|q1⌘,for all s2R. (15) Definition 3.1. A weak solution of (1) is a pair of functions (u", "), satisfying u"2C([0,T]; L2(⌦")), "2C([0,T]; L2 @⌦(@⌦")),for all T>0,(16) u"2L2(0,T;H1(⌦")),for all T>0,(17) "2L2(0,T;H1 @⌦(@⌦")) \Lq(0,T;Lq @⌦(@⌦")),for all T>0,(18) 0(u"(t)) = "(t),a.e. t2(0,T],(19) 8 > > > < > > > : d dt(u"(t),v)⌦"+"d dt( "(t), 0(v))@F"+(ru"(t),rv)⌦"+(u"(t),v)⌦" +"(r "(t),r0(v))@F"+"(g( "(t)), 0(v))@F"=0 in D0(0,T),forallv2H1(⌦")such that 0(v)2H1 @⌦(@⌦")\Lq @⌦(@⌦"), (20) u"(0) = u0 ",and "(0) = 0 ".(21) We have the following result. Theorem 3.2. Under the assumptions (2)–(3) and (4), there exists a unique solution (u", ")of the problem (1). Moreover, this solution satisfies the energy equality 1 2 d dt |(u"(t), "(t))|2 H+|ru"(t)|2 ⌦"+|u"(t)|2 ⌦"+"|r "(t)|2 @F"+"(g( "(t)), "(t))@F"=0,(22) a.e. t2(0,T). Proof. On the space Wwe define a continuous symmetric linear operator A:W!W0 , given by hA((v,0(v))),(w,0(w))i=(rv,rw)⌦"+(v,w)⌦"+"(r0(v),r0(w))@F", (23) for all (v,0(v)), (w,0(w)) 2W. We observe that Ais coercive. In fact, for all (v,0(v)) 2W,wehave hA((v,0(v))) ,(v,0(v))i+|(v,0(v))|2 Hmin {1,}kvk2 ⌦"+"|r0(v)|2 @F"+|v|2 ⌦"+"|0(v)|2 @F" min {1,}k(v,0(v))k2 W. Let us denote V1=W,A 1=A,V 2=L2(⌦")⇥Lq @⌦(@⌦"),A 2(v,)=(0,"g()). From (15) one deduces that A2:V2!V0 2. With this notation, and denoting V=V1\V2,p 1=2,p 2=q, ~u"=(u", "), one has that (16)–(21) is equivalent to ~u"2C([0,T]; H),~u "2 2 \ i=1 Lpi(0,T;Vi),for all T>0,(24) 7
Mar´ıa Anguiano (~u")0(t)+ 2 X i=1 Ai(~u"(t)) = 0 in D0(0,T;V0),(25) ~u"(0) = (u0 ", 0 ").(26) Applying a slight modification of [14, Chapter 2,Theorem 1.4], it is not difficult to see that problem (24)–(26) has a unique solution. Moreover, ~u"satisfies the energy equality 1 2 d dt|~u"(t)|2 H+ 2 X i=1hAi(~u"(t)),~u "(t)ii= 0 a.e. t2(0,T), where h·,·iidenotes the duality product between V0 iand Vi. This last equality turns out to be just (22). 4 A priori estimates In this section we obtain some energy estimates for the solution of (1). By (22) and taking into account (2), we have d dt |(u"(t), "(t))|2 H+2min{1,}ku"(t)k2 ⌦"+2"|r "(t)|2 @F"+2↵1"| "(t)|q q,@F"2"|@F"|,(27) where |@F"|denotes the measure of @F". Observe that the number of holes is given by N(")= |⌦| (2")N(1 + o(1)) , then using the change of variable y=x ",d(y)="(N1)d(x), we can deduce |@F"|=N(")|@Fk,"|=N(")"N1|@F|C ".(28) Let us denote G(s):=Zs 0 g(r)dr. Then, there exist positive constants e↵1,e↵2,and e such that e↵1|s|qe G(s)e↵2|s|q+e 8s2R.(29) We observe that the linear term u"in the boundary condition is coercive, so that this term is of no real significance to the energy estimates and only enhances the regularity of the solution. Lemma 4.1. Under the assumptions (2)–(3) and (5), assume that g2C 1(R). Then, for any initial condition (u0 ", 0 ")2W\Hq, there exists a constant Cindependent of ", such that the solution (u", ")of the problem (1) satisfies ku"k⌦",T C, sup t2[0,T ]ku"(t)k⌦"C, ku0 "k⌦",T C, p"k0(u0 ")k@F",T C, (30) p"| "(t)|@F"C, |u0 "(t)|⌦"C, p"|0(u0 "(t))|@F"C, (31) for all t2(0,T). 8
Mar´ıa Anguiano for all r2[2,2?). We separate the cases N>2 and N= 2. Case 1: N>2.Since 2? ¯q=(N2)(q2) + N N2=q1+ 2 N2>q1, there exists r2[2,2?) such that r ¯qq1 and |g(s)|C⇣1+|s|q1⌘C⇣1+|s|r ¯q⌘. Then, applying Theorem 2.4 in [8] for G(x, v)=g(v), t=¯qand r2[2,2?) such that r ¯qq1, we have that the map v2Lr(⌦)7! g(v)2L¯q(⌦) is continuous in the strong topologies. Then, taking into account (68), we get (63). Finally, we prove (64). First, we observe that it is easy to see from (2) that there exists a constant C>0 such that |g0(s)|C⇣1+|s|q2⌘. Then, we get Z⌦ @g @xi (˜u"(t)) ¯q dx CZ⌦⇣1+|˜u"(t)|(q2)¯q⌘ @˜u"(t) @xi ¯q dx (69) C 1+✓Z⌦|˜u"(t)|(q2)¯qdx◆1/!✓Z⌦|r˜u"(t)|¯q⌘dx◆1/⌘ , where we took and ⌘such that ¯q⌘= 2, 1/+1/⌘= 1 and (q2)¯q=2 ?. Note that from here we get ¯q=2N (N2)(q2) + N. Observe that ¯q>1. Indeed, q2N2 N2=N N2+1<N N2+2)(N2)(q2) + N<2N)2N (N2)(q2) + N>1. Then, we have Z⌦ @g @xi (˜u"(t)) ¯q dx C⇣1+|˜u"|2?/ 2?,⌦⌘|r˜u"|2/⌘ ⌦, and taking into account the continuous embedding H1 0(⌦)⇢L2?(⌦) and (55), we get |rg(˜u"(t))|¯q,⌦C. (70) Then, from (63) and (70), we can deduce (64). Case 2: N=2.We consider s2[2q2,+1) and ¯q=2s 2(q2) + s.(71) Since s ¯q=2(q2) + s 2=q1+s2 2q1, 15
Mar´ıa Anguiano we have |g(s)|C⇣1+|s|q1⌘C⇣1+|s|s ¯q⌘. Then, applying Theorem 2.4 in [8] for G(x, v)=g(v), t=¯qand r=s, we have that the map v2Ls(⌦)7! g(v)2L¯q(⌦) is continuous in the strong topologies. Then, taking into account (68), we get (63). Finally, we prove (64). In (69) we took and ⌘such that ¯q⌘= 2, 1/+1/⌘= 1 and (q2)¯q=s. Note that from here we get ¯qgiven by (71). Observe that ¯q2(1,2). Indeed, taking into account that 1 ¯q=q2 s+1 2, we can deduce 2q2s<+1)0<1 s1 2q2)1 2<1 ¯qq2 2q2+1 2)2(2q2) 2(q2) + 2q2¯q<2, and using that 2(2q2) 2(q2)+2q2>1, we have that ¯q2(1,2). Then, we have Z⌦ @g @xi (˜u"(t)) ¯q dx C⇣1+|˜u"|s/ s,⌦⌘|r˜u"|2/⌘ ⌦, and taking into account the continuous embedding H1 0(⌦)⇢Ls(⌦) and (55), we get |rg(˜u"(t))|¯q,⌦C. (72) Then, from (63) and (72), we can deduce (64). Because we have the linear term u"in the boundary condition, in order to pass to the limit in the integral which involves this term, we need the following result. Proposition 5.2. Under the assumptions in Lemma 4.2, there exists a function ⇠2L2(0,T;H1(⌦)) such that for all T>0, ˜ ⇠"*⇠ weakly in H1(⌦),8t2[0,T],(73) where ˜ ⇠"is given by (58). Proof. From the estimate (43) and (58), we have ||˜ ⇠"||⌦C. Then, we see that the sequence {˜ ⇠"}is bounded in H1(⌦), and hence, up to a subsequence and by (62), we can deduce (73). 6 Homogenized model: proof of the main Theorem In this section, we identify the homogenized model. We multiply system (1) by a test function v2D(⌦), integrating by parts and taking into account (14) and (58), we have d dt ✓Z⌦ ⌦"˜u"(t)vdx◆+"d dt ✓Z@F" 0(u"(t))vd(x)◆+Z⌦ ˜ ⇠"·rvdx +Z⌦ ⌦"˜u"(t)vdx +"Z@F"r0(u"(t)) ·rvd(x)+"Z@F" g(0(u"(t)))vd(x)=0, in D0(0,T). 16
Mar´ıa Anguiano We consider '2C1 c([0,T]) such that '(T) = 0 and '(0) 6= 0. Multiplying by 'and integrating between 0 and T,wehave '(0) ✓Z⌦ ⌦"˜u"(0)vdx◆ZT 0 d dt'(t)✓Z⌦ ⌦"˜u"(t)vdx◆dt "'(0) ✓Z@F" 0(u"(0))vd(x)◆"ZT 0 d dt'(t)✓Z@F" 0(u"(t))vd(x)◆dt +ZT 0 '(t)Z⌦ ˜ ⇠"·rvdxdt +ZT 0 '(t)Z⌦ ⌦"˜u"(t)vdxdt (74) +"ZT 0 '(t)Z@F"r0(u"(t)) ·rvd(x)dt +"ZT 0 '(t)Z@F" g(0(u"(t)))vd(x)dt =0. For the sake of clarity, we split the proof in three parts. Firstly, we pass to the limit, as "!0, in (74) in order to get the limit equation satisfied by u. Secondly we identify ⇠making use of the solutions of the cell-problems (10), and finally we prove that uis uniquely determined. Step 1. In order to pass to the limit, as "!0, we reason as in [3, Theorem 6.1] for all the terms except the term which involves the tangential gradient r. Exactly, for the integrals on ⌦we only require to use Proposition 5.1 and the convergence (57) and for the integrals on the boundary of the holes we make use of a convergence result based on a technique introduced by Vanninathan [20] for the Steklov problem which transforms surface integrals into volume integrals, which was already used as a main tool to homogenize the non homogeneous Neumann problem for the elliptic case by Cioranescu and Donato [5]. For the term which involves the tangential gradient, we also use this technique together with Proposition 5.2. By Definition 3.2 in Cioranescu and Donato [5], let us introduce, for any h2Ls0(@F), 1 s01, the linear form µ" hon W1,s 0(⌦) defined by hµ" h,'i="Z@F" h⇣x "⌘'(x)d(x),8'2W1,s 0(⌦), with 1/s +1/s0= 1. It is proved in Lemma 3.3 in Cioranescu and Donato [5] that µ" h!µhstrongly in (W1,s 0(⌦))0,(75) where hµh,'i=µhZ⌦ '(x)dx,with µh=1 |Y|Z@F h(y)d(y). In the particular case in which h2L1(@F) or even when his constant, we have µ" h!µhstrongly in W1,1(⌦). We denote by µ" 1the above introduced measure in the particular case in which h= 1. Notice that in this case µhbecomes µ1=|@F|/|Y|. For the term which involves the tangential gradient, we proceed as follows. Taking into account (13), there exists an element #"2H3/2(⌦"), the extension of 0(u"(t)), such that r#"=r0(u"(t)) on @F".Then,we can deduce "Z@F"r0(u"(t)) ·rvd(x)="Z@F"r#"·rvd(x)=hµ" 1,˜ ⇠"·rvi, 17
Mar´ıa Anguiano where ˜ ⇠"is given by (58). Note that using (75) with s= 2 and taking into account (73), we can deduce, for "!0, "Z@F"r0(u"(t)) ·rvd(x)=hµ" 1,˜ ⇠"·rvi!µ1Z⌦ ⇠·rvdx =|@F| |Y|Z⌦ ⇠·rvdx, 8v2D(⌦), which integrating in time and using Lebesgue’s Dominated Convergence Theorem, gives "ZT 0 '(t)Z@F"r0(u"(t)) ·rvd(x)dt !|@F| |Y|ZT 0 '(t)✓Z⌦ ⇠·rvdx◆dt. (76) Therefore, using the proof of the main Theorem in [3] and (76), we pass to the limit, as "!0, in (74), and we obtain '(0) ✓|Y⇤| |Y|+|@F| |Y|◆✓Z⌦ u(0)vdx◆✓|Y⇤| |Y|+|@F| |Y|◆ZT 0 d dt'(t)✓Z⌦ u(t)vdx◆dt +ZT 0 '(t)Z⌦ ⇠·rvdxdt +|Y⇤| |Y|ZT 0 '(t)Z⌦ u(t)vdxdt +|@F| |Y|ZT 0 '(t)Z⌦ ⇠·rvdxdt +|@F| |Y|ZT 0 '(t)Z⌦ g(u(t))vdxdt =0. Hence, ⇠verifies ✓|Y⇤| |Y|+|@F| |Y|◆@u @t✓1+|@F| |Y|◆div⇠+|Y⇤| |Y|u+|@F| |Y|g(u)=0,in ⌦⇥(0,T).(77) Step 2. It remains now to identify ⇠. We shall make use of the solutions of the cell problems (10). For any fixed i=1,...,N, let us define i"(x)="⇣wi⇣x "⌘+yi⌘8x2⌦",(78) where y=x/". By periodicity ˜ i"*x iweakly in H1(⌦), where ˜ ·denotes the extension to ⌦given by Cioranescu and Saint Jean Paulin [7]. Then, by Rellich-Kondrachov Theorem, we can deduce ˜ i"!xistrongly in L2(⌦).(79) Let r i"be the gradient of i"in ⌦". Denote by ] r i"the extension by zero of r i"inside the holes. From (78), we have ] r i"=^ ry(wi+yi)=^ rywi(y)+eiY⇤, and taking into account [4, Corollary 2.10], we have ] r i"*1 |Y|ZY⇤ (ei+rywi(y)) dy weakly in L2(⌦).(80) Due to that wi2Hper \R(see [9, Theorem 4.1]), let r0( i") be the tangential gradient of 0( i") on @F" and we denote by µ" hthe above introduced linear form in the particular case in which h⇣x "⌘=r0( i"(x)). From (78), we have r0( i")=Pei+rwi(y), 18
Mar´ıa Anguiano where Peiis defined on @Fand the tangential gradient of wiis given by rwi:= Pry˜wi=ry˜wi(ry˜wi·⌫)⌫on @F, where ˜wiis an extension of wi. In this case, µhbecomes µh=1 |Y|Z@F (Pei+rwi(y)) d(y). Then, using (75), we obtain "Z@F"r0( i"(x))'(x)d(x)=hµ" h,'i!hµh,'i=µhZ⌦ '(x)dx, 8'2W1,s 0(⌦).(81) On the other hand, it is not difficult to see that i"satisfies (div (r i")=0,in ⌦", r i"·⌫="div(r i"),on @F". (82) Let v2D(⌦). Multiplying the first equation in (82) by vu", integrating by parts over ⌦"and taking into account (14), we get "Z@F"r0( i")·rv 0(u")d(x)"Z@F"r0( i")·r0(u")vd(x) (83) =Z⌦"r i"·rvu "dx +Z⌦"r i"·ru"vdx. On the other hand, we multiply system (1) by the test function v i", integrating by parts over ⌦"and taking into account (14), we obtain d dt ✓Z⌦ ⌦"˜u"v˜ i"dx◆+"d dt ✓Z@F" 0(u")v0( i")d(x)◆+Z⌦"ru"·rv i"dx +Z⌦"ru"·r i"vdx +Z⌦ ⌦"˜u"v˜ i"dx +"Z@F"r0(u")·rv 0( i")d(x)+"Z@F"r0(u")·r0( i")vd(x) +"Z@F" g(0(u"))v0( i")d(x)=0, in D0(0,T). Using (83), we have d dt ✓Z⌦ ⌦"˜u"v˜ i"dx◆+"d dt ✓Z@F" 0(u")v0( i")d(x)◆+Z⌦"ru"·rv i"dx Z⌦"r i"·rvu "dx "Z@F"r0( i")·rv 0(u")d(x)+Z⌦ ⌦"˜u"v˜ i"dx +"Z@F"r0(u")·rv 0( i")d(x) +"Z@F" g(0(u"))v0( i")d(x)=0, in D0(0,T). 19
Mar´ıa Anguiano We consider '2C1 c([0,T]) such that '(T) = 0 and '(0) 6= 0. Multiplying by 'and integrating between 0 and T,wehave '(0) ✓Z⌦ ⌦"˜u"(0)v˜ i"dx◆ZT 0 d dt'(t)✓Z⌦ ⌦"˜u"(t)v˜ i"dx◆dt (84) "'(0) ✓Z@F" 0(u"(0))v0( i")d(x)◆"ZT 0 d dt'(t)✓Z@F" 0(u"(t))v0( i")d(x)◆dt +ZT 0 '(t)Z⌦ ˜ ⇠"·rv˜ i"dxdt ZT 0 '(t)Z⌦ ] r i"·rv˜u"dxdt "ZT 0 '(t)Z@F"r0( i")·rv 0(u")d(x)dt +ZT 0 '(t)Z⌦ ⌦"˜u"v˜ i"dxdt +"ZT 0 '(t)Z@F"r0(u")·rv 0( i")d(x)dt +"ZT 0 '(t)Z@F" g(0(u"))v0( i")d(x)dt =0. Now, we have to pass to the limit, as "!0. We will focus on the terms which involve the gradient and the tangential gradient. Taking into account (79), we reason as in [3, Theorem 6.1] for the others terms. Firstly, using (62), (79) and Lebesgue’s Dominated Convergence Theorem, we have ZT 0 '(t)Z⌦ ˜ ⇠"·rv˜ i"dxdt !ZT 0 '(t)Z⌦ ⇠·rvx idxdt, and by (60), (80) and Lebesgue’s Dominated Convergence Theorem, we obtain ZT 0 '(t)Z⌦ ] r i"·rv˜u"dxdt !1 |Y|ZT 0 '(t)Z⌦✓ZY⇤ (ei+rywi)dy◆·rv udxdt. On the other hand, using (59) and (81), we can deduce "Z@F"r0( i")·rv 0(u")d(x)! |Y|Z⌦✓Z@F (Pei+rwi)d(y)◆·rv udx, which integrating in time and by Lebesgue’s Dominated Convergence Theorem, we obtain "ZT 0 '(t)Z@F"r0( i")·rv 0(u")d(x)dt ! |Y|ZT 0 '(t)Z⌦✓Z@F (Pei+rwi)d(y)◆·rv udxdt. Similarly to the proof of (76) together with (79), we have "ZT 0 '(t)Z@F"r0(u")·rv 0( i")d(x)dt !|@F| |Y|ZT 0 '(t)Z⌦ ⇠·rvx idxdt. 20
Mar´ıa Anguiano Therefore, when we pass to the limit in (84), we obtain '(0) ✓|Y⇤| |Y|+|@F| |Y|◆✓Z⌦ u(0)vxidx◆✓|Y⇤| |Y|+|@F| |Y|◆ZT 0 d dt'(t)✓Z⌦ u(t)vxidx◆dt +ZT 0 '(t)Z⌦ ⇠·rvx idxdt 1 |Y|ZT 0 '(t)Z⌦✓ZY⇤ (ei+rywi)dy◆·rv udxdt |Y|ZT 0 '(t)Z⌦✓Z@F (Pei+rwi)d(y)◆·rv udxdt +|Y⇤| |Y|ZT 0 '(t)Z⌦ uvxidxdt +|@F| |Y|ZT 0 '(t)Z⌦ ⇠·rvx idxdt +|@F| |Y|ZT 0 '(t)Z⌦ g(u(t))vxidxdt =0. Using Green’s formula and equation (77), we have ZT 0 '(t)Z⌦ ⇠·rxivdxdt +1 |Y|ZT 0 '(t)Z⌦✓ZY⇤ (ei+rywi)dy◆·ruvdxdt + |Y|ZT 0 '(t)Z⌦✓Z@F (Pei+rwi)d(y)◆·ruvdxdt|@F| |Y|ZT 0 '(t)Z⌦ ⇠·rxivdxdt =0. The above equality holds true for any v2D(⌦) and '2C1 c([0,T]). This implies that ✓1+|@F| |Y|◆⇠·rxi+1 |Y|✓ZY⇤ (ei+rywi)dy◆·ru+ |Y|✓Z@F (Pei+rwi)d(y)◆·ru=0, in ⌦⇥(0,T).We conclude that ✓1+|@F| |Y|◆div⇠=div(Qru),(85) where Q=((qij)), 1 i, j N, is given by qij =1 |Y|✓ZY⇤ (ei+rywi)·ejdy +Z@F (Pei+rwi)·Pejd(y)◆. Observe that if we multiply system (10) by the test function wj, integrating by parts over Y⇤, we obtain ZY⇤ (ei+rywi)·rywjdy +Z@F (Pei+rwi)·rwjd(y)=0, then we conclude that qij is given by (9). Step 3. Finally, thanks to (77) and (85), we observe that usatisfies the first equation in (8). A weak solution of (8) is any function u, satisfying u2C([0,T]; L2(⌦)),for all T>0, u2L2(0,T;H1 0(⌦)) \Lq(0,T;Lq(⌦)),for all T>0, ✓|Y⇤| |Y|+|@F| |Y|◆d dt(u(t),v)+(Qru(t),rv)+|Y⇤| |Y|(u(t),v)+|@F| |Y|(g(u(t)),v)=0,in D0(0,T), for all v2H1 0(⌦)\Lq(⌦), and u(0) = u0. Due to that the homogenized matrix Qis positive-definite (see [9, Theorem 4.1]), applying a slight modification of [14, Chapter 2,Theorem 1.4], we obtain that the problem (8) has a unique solution, and therefore Theorem 1.1 is proved. 21
Mar´ıa Anguiano Remark 6.1. It is worth remaking that if we consider a nonlinear term f(u")in the first equation in (1) which satisfies the same assumptions as g, we obtain Theorem 1.1 with an additional term |Y⇤| |Y|f(u)in the first equation in (8). References [1] M. Amar, R. Gianni, Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices, Discrete and Continuous Dynamical Systems Series B, 23, No. 4 (2018) 1739-1756. [2] M. Amar, R. Gianni, Error estimate for a homogenization problem involving the Laplace-Beltrami operator, Mathematics and Mechanics of Complex Systems, 6, No. 1 (2018) 41-59. [3] M. Anguiano, Existence, uniqueness and homogenization of nonlinear parabolic problems with dynamical boundary conditions in perforated media, Mediterr. J. Math. (2020) 17:18. [4] D. Cioranescu, A. Damlamian, G. Griso, The periodic Unfolding Method in Homogenization, SIAM Journal on Mathematical Analysis, 40, No. 4 (2008) 1585-1620. [5] D. Cioranescu, P. Donato, Homog´en´eisation du probl`eme de Neumann non homog`ene dans des ouverts perfores, Asymptotic Analysis, 1, (1988) 115-138. [6] D. Cioranescu, P. Donato, An Introduction to Homogenization, Oxford Lectures Series in Mathematics and its Applications, 17, New York, 1999. [7] D. Cioranescu, J. Saint Jean Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl., 71, (1979) 590-607. [8] C. Conca, J.I. D´ıaz, A. Li˜n´an, C. Timofte, Homogenization in chemical reactive flows, Electronic Journal of Di↵erential Equations. 40 (2004), 1-22. [9] M. Gahn, Multi-scale modeling of processes in porous media-coupling reaction-di↵usion processes in the solid and the fluid phase and on the separating interfaces, Discrete and Continuous Dynamical Systems Series B, 24, No. 12 (2019) 6511-6531. [10] C.G. Gal, The role of surface di↵usion in dynamic boundary conditions: where do we stand?, Milan J. Math. 83, (2015) 237-278. [11] C.G. Gal, J. Shomberg, Coleman-Gurtin type equations with dynamic boundary conditions, Phys. D. 292, (2015) 29-45. [12] G.R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Di↵erential Equations, 11, No. 4 (2006) 457-480. [13] I. Graf, M.A. Peter, Di↵usion on surfaces and the boundary periodic unfolding operator with an application to carcinogenesis in human cells, SIAM J. Math. Analysis, 46, No. 4 (2014) 3025-3049. [14] J.L. Lions, Quelques M´ethodes de R´esolution des Probl`emes aux Limites Non lin`eaires, Dunod, 1969. [15] J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer- Verlag Berlin Heidelberg, 1972. [16] J.C. Robinson, Infinite-dimensional dynamical systems. Cambridge University Press, 2001. [17] J. Sokolowski, J.-P. Zolesio, Introduction to Shape Optimization, Springer-Verlag, 1992. [18] R.S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, Journal of Functional Analysis, 52, (1983) 48-79. 22
Mar´ıa Anguiano [19] L. Tartar, Probl`emes d’homog´en´eisation dans les ´equations aux d´eriv´ees partielles, Cours Peccot Coll`ege de France, 1977. [20] M. Vanninathan, Homogenization of eigenvalues problems in perforated domains, Proc. Indian Acad. of Sciences, 90, (1981) 239-271. 23