Error estimate for a homogenization problem involving the Laplace-Beltrami operator

TL;DR

This paper provides an error estimate and concentration result for heat conduction models in composite materials with microscopic periodic structures and thermally active membranes, advancing understanding of homogenization in such systems.

Contribution

It introduces a novel error estimate for a homogenization problem involving the Laplace-Beltrami operator in composite materials with membranes.

Findings

Established a concentration result for the model

Derived an explicit error estimate for the homogenization approximation

Enhanced understanding of heat conduction in complex composite structures

Abstract

In this paper we prove a concentration result and an error estimate for a model of heat conduction in composite materials having a microscopic structure arranged in a perodic array and thermally active membranes separating the heat conductive phases.

Equations

$$\Vert u_{\epsilon }-(u_0+\epsilon u_1){\Vert }_{L^2(0,T;H^1(\Omega ))}$$

$$\Vert u_{\epsilon }-u_0{\Vert }_{L^2({\Omega }_T)}$$

$${\nabla }^B\varphi =\nabla \varphi -(n\cdot \nabla \varphi )n$$

$$\begin{gathered} {\mathrm{div}}^B\mathbf{\Phi }=\mathrm{div}\mathbf{\Phi }-(n\cdot \nabla {\mathbf{\Phi }}_i)n_i-(\mathrm{div}n)(n\cdot \mathbf{\Phi }) \\ ={\mathrm{div}}^B\left(\mathbf{\Phi }-(n\cdot \mathbf{\Phi })n\right)=\mathrm{div}\left(\mathbf{\Phi }-(n\cdot \mathbf{\Phi })n\right), \end{gathered}$$

$${\Delta }^B\varphi ={\mathrm{div}}^B({\nabla }^B\varphi ),$$

$$\begin{gathered} {\Delta }^B\varphi =\Delta \varphi -n^t{\nabla }^2\varphi n-(n\cdot \nabla \varphi )\mathrm{div}n \\ =({\delta }_{ij}-n_i{n}_j){\partial }_{ij}^2\varphi -n_j{\partial }_j\varphi {\partial }_i{n}_i=(Id-n\otimes n{)}_{ij}{\partial }_{ij}^2\varphi -(n\cdot \nabla \varphi )\mathrm{div}n, \end{gathered}$$

$${\int }_S{\mathrm{div}}^B\mathbf{\Phi }\text{d}\sigma =0.$$

$${\lambda }^{\epsilon }={\lambda }_{\text{int}}\text{in Ωintε,}{\lambda }^{\epsilon }={\lambda }_{\text{out}}\text{in Ωoutε;}$$

$${\mu }^{\epsilon }={\mu }_{\text{int}}\text{in Ωintε,}{\mu }^{\epsilon }={\mu }_{\text{out}}\text{in Ωoutε.}$$

$${\mu }^{\epsilon }\frac{\partial u_{\epsilon }}{\partial t}-\mathrm{div}({\lambda }^{\epsilon }\nabla u_{\epsilon })$$

$$[u_{\epsilon }]$$

$$\epsilon \alpha \frac{\partial u_{\epsilon }}{\partial t}-\epsilon \beta {\Delta }^B{u}_{\epsilon }$$

$$u_{\epsilon }(x,t)$$

$$u_{\epsilon }(x,0)$$

$$[u_{\epsilon }]=u_{\epsilon }^{\text{out}}-u_{\epsilon }^{\text{int}},$$

$${\mathcal{X}}_0^{\epsilon }(\Omega ):=H_0^1(\Omega )\cap H_B^1({\Gamma }^{\epsilon }).$$

$$\begin{gathered} -{\int }_0^T{\int }_{\Omega }{\mu }^{\epsilon }{u}_{\epsilon }\frac{\partial \varphi }{\partial \tau }\text{d}x\text{d}\tau +{\int }_0^T{\int }_{\Omega }{\lambda }^{\epsilon }\nabla u_{\epsilon }\cdot \nabla \varphi \text{d}x\text{d}\tau -\epsilon \alpha {\int }_0^T{\int }_{{\Gamma }^{\epsilon }}{u}_{\epsilon }\frac{\partial \varphi }{\partial \tau }\text{d}\sigma \text{d}\tau \\ +\epsilon \beta {\int }_0^T{\int }_{{\Gamma }^{\epsilon }}{\nabla }^B{u}_{\epsilon }\cdot {\nabla }^B\varphi \text{d}\sigma \text{d}\tau ={\int }_{\Omega }{\mu }^{\epsilon }{\overline{u}}_0\varphi (x,0)\text{d}x+\epsilon \alpha {\int }_{{\Gamma }^{\epsilon }}{\overline{u}}_0\varphi (x,0)\text{d}\sigma , \end{gathered}$$

$$\epsilon \alpha \frac{\partial u_{\epsilon }}{\partial t}-\epsilon \beta \left(\Delta u_{\epsilon }-{\nu }_{\epsilon }^t{\nabla }^2{u}_{\epsilon }{\nu }_{\epsilon }-({\nu }_{\epsilon }\cdot \nabla u_{\epsilon })\mathrm{div}{\nu }_{\epsilon }\right)=[\lambda \nabla u_{\epsilon }\cdot {\nu }_{\epsilon }],\text{on Γε,}$$

$$\lambda ={\lambda }_{\text{int}}\text{in Eint,}\lambda ={\lambda }_{\text{out}}\text{in Eout;}$$

$$\mu ={\mu }_{\text{int}}\text{in Eint,}\mu ={\mu }_{\text{out}}\text{in Eout.}$$

$${\overline{u}}_0\in H_0^1(\Omega )\cap H^2(\Omega ).$$

$$\epsilon {\int }_{{\Gamma }^{\epsilon }}|{\overline{u}}_0{|}^2\text{d}\sigma \le \gamma ,\epsilon {\int }_{{\Gamma }^{\epsilon }}|{\nabla }^B{\overline{u}}_0{|}^2\text{d}\sigma \le \gamma ,$$

$$\begin{gathered} \frac{1}{2}{\int }_0^t{\int }_{\Omega }{\mu }^{\epsilon }\frac{\partial u_{\epsilon }^2}{\partial \tau }\text{d}x\text{d}\tau +{\int }_0^t{\int }_{\Omega }{\lambda }^{\epsilon }|\nabla u_{\epsilon }{|}^2\text{d}x\text{d}\tau + \\ \frac{\epsilon \alpha }{2}{\int }_0^t{\int }_{{\Gamma }^{\epsilon }}\frac{\partial u_{\epsilon }^2}{\partial \tau }\text{d}\sigma \text{d}\tau +\epsilon \beta {\int }_0^t{\int }_{{\Gamma }^{\epsilon }}|{\nabla }^B{u}_{\epsilon }{|}^2(x)\text{d}\sigma \text{d}\tau =0. \end{gathered}$$

$$\begin{gathered} \frac{1}{2}{\int }_{\Omega }{\mu }^{\epsilon }{u}_{\epsilon }^2(t)\text{d}x+{\int }_0^t{\int }_{\Omega }{\lambda }^{\epsilon }|\nabla u_{\epsilon }{|}^2\text{d}x\text{d}\tau +\frac{\epsilon \alpha }{2}{\int }_{{\Gamma }^{\epsilon }}{u}_{\epsilon }^2(t)\text{d}\sigma +\epsilon \beta {\int }_0^t{\int }_{{\Gamma }^{\epsilon }}|{\nabla }^B{u}_{\epsilon }{|}^2\text{d}\sigma \text{d}\tau = \\ \frac{1}{2}{\int }_{\Omega }{\mu }^{\epsilon }{\overline{u}}_0^2\text{d}x+\frac{\epsilon \alpha }{2}{\int }_{{\Gamma }^{\epsilon }}{\overline{u}}_0^2\text{d}\sigma . \end{gathered}$$

$$\begin{gathered} \underset{t\in (0,T)}{\sup }{\int }_{\Omega }{u}_{\epsilon }^2(t)\text{d}x+{\int }_0^T{\int }_{\Omega }|\nabla u_{\epsilon }{|}^2\text{d}x\text{d}\tau \\ +\underset{t\in (0,T)}{\sup }\epsilon {\int }_{{\Gamma }^{\epsilon }}{u}_{\epsilon }^2(t)\text{d}\sigma +\epsilon {\int }_0^T{\int }_{{\Gamma }^{\epsilon }}|{\nabla }^B{u}_{\epsilon }{|}^2\text{d}\sigma \text{d}\tau \le \gamma , \end{gathered}$$

$$u_{\epsilon }=u_{\epsilon }(x,y,t)=u_0(x,y,t)+\epsilon u_1(x,y,t)+{\epsilon }^2{u}_2(x,y,t)+\dots .$$

$$\mathrm{div}=\frac{1}{\epsilon }{\mathrm{div}}_y+{\mathrm{div}}_x,\nabla =\frac{1}{\epsilon }{\nabla }_y+{\nabla }_x,$$

$$\nabla u_{\varepsilon}=\frac{1}{\varepsilon}\nabla_{y}u_{0}+\big{(}\nabla_{x}u_{0}+\nabla_{y}u_{1}\big{)}+\varepsilon\big{(}\nabla_{y}u_{2}+\nabla_{x}u_{1}\big{)}+\dots\,,$$

$$\Delta u_{\epsilon }=\frac{1}{{\epsilon }^2}{A}_0{u}_0+\frac{1}{\epsilon }(A_0{u}_1+A_1{u}_0)+(A_0{u}_2+A_1{u}_1+A_2{u}_0)+\dots ,$$

$$A_0={\Delta }_y,A_1={\mathrm{div}}_y{\nabla }_x+{\mathrm{div}}_x{\nabla }_y,A_2={\Delta }_x.$$

$${\Delta }^B{u}_{\epsilon }=\frac{1}{{\epsilon }^2}{A}_0^B{u}_0+\frac{1}{\epsilon }(A_0^B{u}_1+A_1^B{u}_0)+(A_0^B{u}_2+A_1^B{u}_1+A_2^B{u}_0)+\dots ,$$

$$\begin{gathered} A_0^B={\Delta }_y^B,A_2^B={\Delta }_x^B \\ {A}_1^B={\mathrm{div}}_x^B{\nabla }_y^B+{\mathrm{div}}_y^B{\nabla }_x^B=2(Id-\nu \otimes \nu {)}_{ij}{\partial }_{x_i{y}_j}^2-({\mathrm{div}}_y\nu )\nu \cdot {\nabla }_x. \end{gathered}$$

$${\mathcal{P}}_0[u_0]:\{\begin{array}{rl}& -\lambda {\Delta }_y{u}_0=0,\text{in Eint, Eout;}\\ & \beta {\Delta }_y^B{u}_0+[\lambda {\nabla }_y{u}_0\cdot \nu ]=0,\text{on Γ.}\end{array}$$

Full text

Keywords: Homogenization, Asymptotic expansion, Laplace-Beltrami operator, Heat conduction.

AMS-MSC: 35B27, 35Q79

Acknowledgments: We would like to thank R. Lipton and P. Bisegna for some helpful discussions. The first author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

1. Introduction

Heat and electrical conduction in composite materials has been widely investigated in the last years in the context of homogenization theory (see among others, e.g. [3, 4, 5, 6, 7, 10, 15, 17, 19, 21, 22, 24]). In this paper we will focus on the study of models of heat conduction in composite materials used for encapsulation of electronic devices. This topic is attracting increasing interest among researchers, both from the point of view of applications and also in a more mathematical setting. In our previous paper [13] (to which we refer for a more detailed physical description of the problem) a composite medium was taken into account, which was made of a hosting material with inclusions separated from their surroundings by a thermally active membrane.

Such a situation is consistent with many physical applications in which a material must be modified in a way such that its thermal conductivity is enhanced while preserving other material properties e.g. ductility. This is, as stated above, the case of polymer encapsulation of electronic devices as well as, just to make an example, engine coolants. Specifically, in the first case, ductility of the material is required to fill the voids and the interstices among the electrical components by applying a moderate pressure. Polymers and rubbers have this property but they do not display a satisfactory heat dissipation which, on the other hand, can be attained by adding highly conductive nanoparticles. In some situations, these nanoparticles are enclosed in a membrane separating them from the surrounding medium. It is therefore only natural to investigate the influence of these membranes on the overall conductivity of the composite medium under different assumptions on the thermal behaviour of these interfaces. The case of perfect or imperfect thermal contact, though interesting from the point of view of applications, is mathematically well known, for this reason we focused on the case in which the membrane is thermally active, e.g. a tangential heat diffusion takes place. In [13] a macroscopic model was deduced, via the unfolding homogenization technique, assuming the periodicity of the microscopic structure, whose characteristic length is described by a small parameter $\varepsilon$. We make use of a sensible mathematical description of the behavior of the interfaces which are modeled by means of the Laplace-Beltrami operator (see, e.g. [1, 14]).

In this paper we complete the research started in [13] providing an “error estimate” which enables us to evaluate the rate of convergence, with respect to $\varepsilon\to 0$, of the solution $u_{\varepsilon}$ of the microscopic (physical) problem to the solution $u_{0}$ of the macroscopic one. More precisely, we prove

[TABLE]

for a proper constant $\gamma>0$ independent of $\varepsilon$, where $u_{1}$ is the so called first corrector and it is defined in (3.13).

To obtain this estimate we follow the classical approach given by the asymptotic expansions due to Bensoussan-Lions-Papanicolaou [16] which, under extra-regularity assumptions, gives an $H^{1}$-estimate for this error. The knowledge of the rate of convergence is a crucial tool for numerical applications. Moreover, we prove the symmetry and the strict positivity of the matrix describing the diffusivity of the macroscopic (homogenized) material. This last result is crucial to guarantee the well-posedness of the parabolic limit equation.

Though the results proved in this paper are along the same lines of other ones obtained in the framework of the homogenization theory, nevertheless they are of some mathematical interest due to the presence of the Laplace-Beltrami operator, which makes the computations a bit tricky.

The paper is organized as follows. In Section 2 we recall the definition and some properties of the tangential operators (gradient, divergence, Laplace-Beltrami operator), we state our geometrical setting and present our model. In Section 3, after having proved some energy inequalities, we follow the formal approach by Bensoussan-Lions-Papanicolau in order to introduce the cell functions and to guess the limit equation, proving the ellipticity of its principal part (see Theorem 3.1). Finally, in Section 4 taking advantage of the asymptotic expansions obtained in Section 3, we provide the error estimate (see Theorem 4.1).

2. Preliminaries

2.1. Tangential derivatives

Let $\phi$ be a ${\mathcal{C}}^{2}$-function, $\mathbf{\Phi}$ be a ${\mathcal{C}}^{2}$-vector function and $S$ a smooth surface with normal unit vector $n$. We recall that the tangential gradient of $\phi$ is given by

[TABLE]

and the tangential divergence of $\mathbf{\Phi}$ is given by

[TABLE]

where, taking into account the smoothness of $S$, the normal vector $n$ can be naturally defined in a small neighborhood of $S$ as $\frac{\nabla d}{|\nabla d|}$, where $d$ is the signed distance from $S$. Moreover, we define the Laplace-Beltrami operator as

[TABLE]

so that, by (2.1) and (2.2), we get that the Laplace-Beltrami operator can be written as

[TABLE]

where $\nabla^{2}\phi$ stands for the Hessian matrix of $\phi$. Finally, we recall that on a regular surface $S$ with no boundary (i.e. when $\partial S=\emptyset$) we have

[TABLE]

2.2. Geometrical setting

The typical periodic geometrical setting is displayed in Figure 1. Here we give, for the sake of clarity, its detailed formal definition.

Let us introduce a periodic open subset $E$ of $\mathbb{R}^{N}$, so that $E+z=E$ for all $z\in\mathbb{Z}^{N}$. We employ the notation $Y=(0,1)^{N}$, and $E_{\textup{int}}=E\cap Y$, $E_{\textup{out}}=Y\setminus\overline{E}$, $\varGamma=\partial E\cap\overline{Y}$. As a simplifying assumption, we stipulate that $|\varGamma\cap\partial Y|_{N-1}=0$.

Let $\varOmega$ be an open connected bounded subset of $\mathbb{R}^{N}$; for all $\varepsilon>0$ define $\varOmega_{\textup{int}}^{\varepsilon}=\varOmega\cap\varepsilon E$, $\varOmega_{\textup{out}}^{\varepsilon}=\varOmega\setminus\overline{\varepsilon E}$, so that $\varOmega=\varOmega_{\textup{int}}^{\varepsilon}\cup\varOmega_{\textup{out}}^{\varepsilon}\cup\varGamma^{\varepsilon}$, where $\varOmega_{\textup{int}}^{\varepsilon}$ and $\varOmega_{\textup{out}}^{\varepsilon}$ are two disjoint open subsets of $\varOmega$, and $\varGamma^{\varepsilon}=\partial\varOmega_{\textup{int}}^{\varepsilon}\cap\varOmega=\partial\varOmega_{\textup{out}}^{\varepsilon}\cap\varOmega$. The region $\varOmega_{\textup{out}}^{\varepsilon}$ [respectively, $\varOmega_{\textup{int}}^{\varepsilon}$] corresponds to the outer phase [respectively, the inclusions], while $\varGamma^{\varepsilon}$ is the interface. We assume also that $\varOmega$ and $E$ have regular boundary and we stipulate that $\operatorname{dist}(\varGamma^{\varepsilon},\partial\varOmega)\geq\gamma_{0}\varepsilon$, for a suitable $\gamma_{0}>0$. To this purpose, for each $\varepsilon$, we are ready to remove the inclusions in all the cells which are not completely contained in $\varOmega$ (see Figure 1). This assumption is in accordance with our previous papers (see [6, 7, 8, 9, 10, 11]) and maybe it can be dropped as in [2, 18]; nevertheless we will not pursue this line of investigation in this paper.

Moreover, let $\nu$ denote the normal unit vector to $\varGamma$ pointing into $E_{\textup{out}}$, extended by periodicity to the whole of $\mathbb{R}^{N}$, so that $\nu_{\varepsilon}(x)=\nu(x/\varepsilon)$ denotes the normal unit vector to $\varGamma^{\varepsilon}$ pointing into $\varOmega_{\textup{out}}^{\varepsilon}$.

Finally, given $T>0$, we denote by $\varOmega_{T}=\varOmega\times(0,T)$. More in general, for any spatial domain $G$, we denote by $G_{T}=G\times(0,T)$.

2.3. Position of the problem

Let $\mu^{\varepsilon},\lambda^{\varepsilon}:\varOmega\to\mathbb{R}$ be defined as

[TABLE]

For every $\varepsilon>0$, we consider the problem for $u_{\varepsilon}(x,t)$ given by

[TABLE]

[TABLE]

where we denote

[TABLE]

and the same notation is employed also for other quantities. We assume that all the constants $\mu_{\textup{int}},\mu_{\textup{out}},\lambda_{\textup{int}},\lambda_{\textup{out}},\alpha,\beta$, involved in equations (2.6) and (2.8) are strictly positive.

Since problem (2.6)–(2.10) is not standard, in order to define a proper notion of weak solution, we will need to introduce some suitable function spaces. To this purpose and for later use, we will denote by $H^{1}_{B}(\varGamma^{\varepsilon})$ the space of Lebesgue measurable functions $u:\varGamma^{\varepsilon}\to\mathbb{R}$ such that $u\in L^{2}(\varGamma^{\varepsilon})$, $\nabla^{B}u\in L^{2}(\varGamma^{\varepsilon})$. Let us also set

[TABLE]

Definition 2.1**.**

We say that u_{\varepsilon}\in L^{2}\big{(}0,T;{\mathcal{X}}^{\varepsilon}_{0}(\varOmega)\big{)} is a weak solution of problem (2.6)–(2.10) if

[TABLE]

for every test function $\phi\in\mathcal{C}^{\infty}(\varOmega_{T})$ such that $\phi$ has compact support in $\varOmega$ for every $t\in(0,T)$ and $\phi(\cdot,T)=0$ in $\varOmega$. ∎

If $u_{\varepsilon}$ is smooth, by (2.4) it follows that equation (2.8) can be written in the form

[TABLE]

where, as in (2.4), $\nabla^{2}u_{\varepsilon}$ stands for the Hessian matrix of $u_{\varepsilon}$. By [12], for every $\varepsilon>0$, problem (2.6)–(2.10) admits a unique solution u_{\varepsilon}\in L^{2}\big{(}0,T;{\mathcal{X}}^{\varepsilon}_{0}(\varOmega)\big{)}\cap\mathcal{C}^{0}\big{(}[0,T];L^{2}(\varOmega)\cap L^{2}(\varGamma^{\varepsilon})\big{)}, if $\overline{u}_{0}\in H^{1}_{0}(\varOmega)$.

Finally, it will be useful in the sequel to define also $\mu,\lambda:Y\to\mathbb{R}$ as

[TABLE]

[TABLE]

3. Homogenization of the microscopic problem

In the following, we will assume that the initial data satisfies

[TABLE]

By the trace inequality (see [13, Proposition 1] and [6, proof of Lemma 7.1]) we get that $\overline{u}_{0}$ satisfies

[TABLE]

where $\gamma>0$ is independent of $\varepsilon$. Notice that, for our purposes, it should be enough to assume that $\overline{u}_{0}\in H^{1}_{0}(\varOmega)$ and satisfies (3.2), but we prefer to assume (3.1) since it is reasonable to choose $\overline{u}_{0}$ not depending on $\varepsilon$.

We are interested in understanding the limiting behaviour of the heat potential $u_{\varepsilon}$ when $\varepsilon\to 0$; this leads us to look at the homogenization limit of problem (2.6)–(2.10).

To this purpose, we first obtain some energy estimates for the heat potential $u_{\varepsilon}$. Multiplying (2.6) by $u_{\varepsilon}$ and integrating, formally, by parts, we obtain

[TABLE]

Then, evaluating the time integral and taking into account the initial condition (2.10), we obtain, for all $0<t<T$,

[TABLE]

By (3.2) the right hand side of (3.4) is stable as $\varepsilon\to 0$, hence

[TABLE]

where $\gamma$ is a constant independent of $\varepsilon$.

Notice that inequality (3.5) implies that there exists a function $u$ belonging to L^{2}\big{(}0,T;H^{1}_{0}(\varOmega)\big{)} such that, up to a subsequence, $u_{\varepsilon}\rightharpoonup u$, weakly in L^{2}\big{(}0,T;H^{1}_{0}(\varOmega)\big{)}. It will be our purpose to characterize the limit function $u$.

3.1. The two-scale expansion

We summarize here, to establish the notation, some well-known asymptotic expansions needed in the two-scale method (see, e.g., [16], [23]), when applied to stationary or evolutive problems involving second order partial differential equations. Introduce the microscopic variables $y\in Y$, $y=x/\varepsilon$ and assume

[TABLE]

Note that $u_{0}$, $u_{1}$, $u_{2}$ are periodic in $y$, and $u_{1}$, $u_{2}$ are assumed to have zero integral average over $Y$. Recalling that

[TABLE]

we compute

[TABLE]

and

[TABLE]

where

[TABLE]

Moreover, recalling (2.3) and taking into account that the normal vector $\nu_{\varepsilon}$ depends only on the microscopic variable, we obtain also

[TABLE]

where

[TABLE]

Substituting in (2.6)–(2.10) the expansion (3.6), and using (3.7)–(3.12), one readily obtains, by matching corresponding powers of $\varepsilon$, that $u_{0}$ solves $[u_{0}]=0$ on $\varGamma$, and

[TABLE]

By the equality

[TABLE]

we obtain that $u_{0}$ is independent of $y$, i.e., $u_{0}=u_{0}(x,t)$.

Moreover, $u_{1}$ satisfies $[u_{1}]=0$ on $\varGamma$, and

[TABLE]

Following a classical approach, we introduce the factorization

[TABLE]

for a vector function $\chi:Y\to\mathbb{R}^{N}$, whose components $\chi_{h}$ satisfy

[TABLE]

The functions $\chi_{h}$ are also required to be periodic in $Y$, with zero integral average on $Y$ (here, $\boldsymbol{e}_{h}$ denotes the $h$ vector of the canonical basis of $\mathbb{R}^{N}$). We note that [12] assures existence and uniqueness of the cell functions $\chi_{h}\in\mathcal{C}^{\infty}_{\#}(Y)$, for $h=1,\dots,N$ (here and in the following, the subscript $\#$ denotes the $Y$-periodicity).

Finally, $u_{2}$ solves $[u_{2}]=0$ on $\varGamma$, and

[TABLE]

The limiting equation for $u_{0}$ is finally obtained as a compatibility condition for $\boldsymbol{\mathcal{P}}_{2}[u_{2}]$, and amounts to

[TABLE]

We replace now the factorization (3.13) in the previous equality and we take into account that

[TABLE]

[TABLE]

where (3.23) follows from (2.5), since $\varGamma$ has no boundary. Hence, we obtain for the homogenized solution $u_{0}$ the parabolic equation

[TABLE]

where

[TABLE]

Clearly, equation (3.24) must be complemented with a boundary and an initial condition which are $u_{0}=0$ on $\partial\varOmega\times(0,T)$ and $u_{0}(x,0)=\overline{u}_{0}(x)$ in $\varOmega$, respectively, as follows from the microscopic problem (2.6)–(2.10). Indeed, by (3.5) we obtain that $\{u_{\varepsilon}\}$ converges weakly in L^{2}\big{(}0,T;H^{1}_{0}(\varOmega)\big{)}, which implies the weak convergence of the trace on $\partial\varOmega$, while the initial data is already included in the weak formulation of the problem.

Theorem 3.1**.**

The matrix $\lambda_{0}I+A^{hom}$ is symmetric and positive definite.

Proof.

We first prove the symmetry. By (2.1), we have

[TABLE]

then, taking into account (3.14)–(3.16), we obtain

[TABLE]

From (3.25) and (3.27), we can rewrite

[TABLE]

which gives the symmetry of the matrix $A^{hom}$ and hence the symmetry of the whole matrix $\lambda_{0}I+A^{hom}$.

Let us now prove that it is also positive definite. Firstly, we observe that, using (3.26) and (3.27), we obtain

[TABLE]

Then, we can rewrite

[TABLE]

Finally, setting $\lambda_{min}=\min(\lambda_{\textup{int}},\lambda_{\textup{out}})$ and using Jensen’s inequality, we obtain

[TABLE]

where we have denoted by $n=(n_{1},\dots,n_{N})$ the outward unit normal to $\partial Y$. Moreover, we remark that the last integral vanishes because of the periodicity of the cell function $\chi_{h}$.

This proves that the homogenized matrix is positive definite and concludes the theorem. ∎

Remark 3.2*.*

We note that the homogenized matrix is positive definite independently of the strict positivity of $\beta$.

Once proved Theorem 3.1, the existence of a unique solution for equation (3.24) complemented with suitable initial and boundary conditions is standard. The next proposition state the regularity of this solution, which is a property needed in order to obtain the error estimate.

Proposition 3.3**.**

Assume that $\overline{u}_{0}\in\mathcal{C}^{\infty}_{c}(\varOmega)$ (i.e. $\overline{u}_{0}$ has compact support in $\varOmega$). Then, the solution $u_{0}$ to equation (3.24) satisfying the homogeneous boundary condition on $\partial\varOmega\times[0,T]$ and the initial condition $u(x,0)=\overline{u}_{0}(x)$ in $\varOmega$ belongs to $\mathcal{C}^{\infty}(\overline{\varOmega}\times[0,T])$.

Proof.

The result can be obtained applying [20, Theorem 12 in Section 5]. ∎

Remark 3.4*.*

Actually, the asserted $\mathcal{C}^{\infty}$-regularity of the homogenized solution $u_{0}$ is far from being optimal in order to obtain the error estimate proved in Section 4. Indeed, to this purpose, it is enough to have that u_{0}\in\mathcal{C}^{0}\big{(}[0,T];\mathcal{C}^{3}(\overline{\varOmega})\big{)} and this is guaranteed if, for instance $\overline{u}_{0}\in\mathcal{C}^{4}(\overline{\varOmega})$ and satisfies the compatibility conditions

[TABLE]

where L_{hom}=-\operatorname{div}\big{(}(\lambda_{0}I+A^{hom})\nabla\big{)}, with $\lambda_{0}$ and $A^{hom}$ defined in (3.25). However, we prefer the simpler assumptions of Proposition 3.3, since we are not interested in stating which are the minimal conditions to be satisfied by the initial data in order to obtain the optimal regularity of the homogenized solution.

For further use (taking into account the system satisfied by $u_{2}$ and (3.24)), we introduce the factorization of the function $u_{2}$ in terms of the homogenized solution $u_{0}$; i.e.,

[TABLE]

where the functions $\widetilde{\chi}_{ij}:Y\to\mathbb{R}$ satisfy

[TABLE]

The functions $\widetilde{\chi}_{ij}$ are also required to be periodic in $Y$, with zero integral average on $Y$. In order to obtain (3.30)–(3.32) we have taken into account (3.12), which gives

[TABLE]

with $\phi(x,y,t)=u_{1}(x,y,t)=-\chi(y)\cdot\nabla_{x}u_{0}(x,t)$ and the usual summation convention for repeated indexes. By [12], problem (3.30)–(3.32) admits a unique solution $\widetilde{\chi}_{ij}\in\mathcal{C}^{\infty}_{\#}(Y)$, for $i,j=1,\dots,N$, since it is easy to check that

[TABLE]

4. Error estimate

In this section we prove that the limit $u$ of the sequence $\{u_{\varepsilon}\}$ of the solutions of problem (2.6)–(2.10) coincides with the solution $u_{0}$ of equation (3.24). In order to achieve this result, we will state an error estimate for the sequence $\{u_{\varepsilon}\}$, which gives the rate of convergence of such a sequence to the homogenized function $u_{0}$, in a suitable norm, thus obtaining a stronger convergence result with respect to the one obtained in our previous paper [13]. However, this result needs extra-regularity assumptions on the initial data $\overline{u}_{0}(x)$ (see Proposition 3.3 and Remark 3.4), which assure more regularity of the homogenized solution $u_{0}$.

Theorem 4.1**.**

Assume that $\overline{u}_{0}\in\mathcal{C}^{\infty}_{c}(\varOmega)$. Let $u_{0}$ be the smooth solution of (3.24), satisfying the initial condition $u_{0}(x,0)=\overline{u}_{0}(x)$ in $\varOmega$ and the boundary condition $u_{0}(x,t)=0$ on $\partial\varOmega\times(0,T)$; moreover, let $u_{1}$ be the function defined in (3.13). Then

[TABLE]

for a proper constant $\gamma>0$, independent of $\varepsilon$.

Proof.

Let us define the rest function

[TABLE]

Separately in $\varOmega_{\textup{int}}^{\varepsilon}$ and in $\varOmega_{\textup{out}}^{\varepsilon}$, we get

[TABLE]

Moreover,

[TABLE]

and

[TABLE]

where we have taken into account the problems satisfied by $u_{1}$ and $u_{2}$ ($u_{1}$ and $u_{2}$ are defined in Subsection 3.1) and the fact that $\operatorname{div}^{B}_{x}\nabla^{B}_{y}u_{0}=0$ and $\Delta^{\!\!B}_{y}u_{0}=0$.

Let us now introduce the corrected rest function

[TABLE]

where $\phi_{\varepsilon}$ is a cut-off function equal to $1$ in a neighbourhood of $\partial\varOmega$, and such that

[TABLE]

Clearly, $\phi_{\varepsilon}\equiv 0$ on $\varGamma^{\varepsilon}$ (since $\operatorname{dist}(\varGamma^{\varepsilon},\partial\varOmega)\geq\gamma_{0}\varepsilon$, by the assumptions made in Subsection 2.2), so that $r_{\varepsilon}=\tilde{r}_{\varepsilon}$ on $\varGamma^{\varepsilon}$. We may assume $0\leq\phi_{\varepsilon}\leq 1$, $\lvert\nabla\phi_{\varepsilon}\rvert\leq\gamma/\varepsilon$. The function $\tilde{r}_{\varepsilon}$ satisfies $[\tilde{r}_{\varepsilon}]=0$ on $\varGamma^{\varepsilon}$ and

[TABLE]

and on $\varGamma^{\varepsilon}$

[TABLE]

Note that the correction $u_{1}\phi_{\varepsilon}$ has been introduced precisely in order to guarantee (4.5). Multiply (4.3) by $\tilde{r}_{\varepsilon}$ and integrate by parts; by virtue of (4.5), we get

[TABLE]

This implies

[TABLE]

Next, compute

[TABLE]

Note that the last integral in (4.8) can be bounded in the following way

[TABLE]

where $\delta>0$ will be chosen in the following. We exploit here the estimate

[TABLE]

which is a consequence of the regularity of the cell functions $\chi$ and $\widetilde{\chi}$ (recall (3.13)–(3.16) and (3.29)–(3.32)) and of the homogenized function $u_{0}$. Similarly, for $\delta^{\prime}=\min(\lambda_{\textup{int}},\lambda_{\textup{out}})/2$,

[TABLE]

where, again due to the stated regularity of $\chi$ and $u_{0}$, we used

[TABLE]

Moreover, for $\delta^{\prime\prime}$ which will be chosen later, we obtain

[TABLE]

Here, we use

[TABLE]

which is again a consequence of the regularity of $\widetilde{\chi}$ and $u_{0}$.

Combining the previous estimates, we have

[TABLE]

where $\delta^{\prime\prime\prime}$ will be chosen later. Finally, using Poincaré’s inequality, Gronwall’s lemma and absorbing the gradient term in (4.12) into the left hand side (which is possible choosing $\delta,\delta^{\prime},\delta^{\prime\prime},\delta^{\prime\prime\prime}$ sufficiently small), we get

[TABLE]

On recalling the definition of $\tilde{r}_{\varepsilon}$, and invoking again Poincaré?s inequality, we obtain

[TABLE]

Moreover, taking into account that $r_{\varepsilon}=\tilde{r}_{\varepsilon}-u_{1}\phi_{\varepsilon}$ and using (4.13), it follows that

[TABLE]

where we recall the estimate for $\nabla(u_{1}\phi_{\varepsilon})$ done in (4.10). Hence, by (4.14) and (4.15), we obtain (4.1). Finally, (4.2) can be obtained making use of (4.14) and taking into account that

[TABLE]

This concludes the proof. ∎