2.1. Basic notation

Here, we collect the basic notation we are going to use throughout the manuscript. Let $Y\subset\mathbb{R}^N$ be a periodicity cell, namely

\begin{equation*} Y = \left\{\, \sum_{i=1}^N \lambda_i v_i \,:\, 0 \lt \lambda_i \lt 1\right\}, \end{equation*}

where $v_1,\dots,v_N$ is a basis of $\mathbb{R}^N$. Without loss of generality, up to a translation, we can even assume that $Y$ has its barycentre at the origin. This assumption is just to simplify the writing of the main result.

The space $H^1_{{\rm per}}(Y)$ is the subset of $H^1(Y)$ of functions with periodic boundary conditions. More precisely, $u\in H^1(Y)$ if and only if the function $\widetilde{u}: \mathbb{R}^N\to\mathbb{R}$ defined as $\widetilde{u}(y):= u(\widetilde{y})$ belongs to $H^1_{\mathrm{loc}}(\mathbb{R}^N)$, where

\begin{equation*} y=\sum_{i=1}^N \lambda_i v_i,\quad\quad\quad \widetilde{y}:=\sum_{i=1}^N \{\lambda_i\} v_i, \end{equation*}

and $\{\lambda_i\}:= \lambda_i - \lfloor \lambda_i \rfloor$.

Given a function $f$, the notation $f^\varepsilon$ stands for $f^\varepsilon(x):= f(x/\varepsilon)$. Moreover, we denote by $\partial_i$ the $i^{th}$ partial derivative operator with respect to the variable $x$, and by $\partial_{y_i}$ the $i^{th}$ partial derivative operator with respect to the variable $y$. In particular, we have that

\begin{equation*} \partial_i f^\varepsilon(x) = \frac{1}{\varepsilon} \partial_{y_i}f^\varepsilon(x). \end{equation*}

Finally, the symbol $\langle\cdot\rangle_Y$ denotes the average over $Y$, i.e.,

\begin{equation*} \langle f\rangle_Y:= {1\over |Y|}\int_Y f(y)\,\mathrm{d} y, \end{equation*}

with $|Y|$ being the $N$-dimensional Lebesgue measure of $Y$.

2.2. Main result

Let $\Omega\subset\mathbb{R}^N$ be an open, bounded set, and let $f\in L^2(\Omega)$. Let $A:\mathbb{R}^N\to\mathbb{R}^{N\times N}$ be a matrix-valued function in $L^\infty$ such that

1. (H1) $A$ is $Y$-periodic;

2. (H2) $A$ is symmetric, i.e., $a_{ij}(y)=a_{ji}(y)$;

3. (H3) there exist two positive constants $\alpha, \beta$ such that

   \begin{equation*} \notag \alpha|\xi|^2\leq A(y)\xi\cdot\xi\leq \beta |\xi|^2, \end{equation*}

   for all $\xi\in\mathbb{R}^N$.

For $\varepsilon \gt 0$, let $F_{\varepsilon}: L^2(\Omega)\to \mathbb{R}\cup \{+\infty\}$ be the functional defined as

\begin{equation*} \notag F_{\varepsilon}(u):= \int_\Omega A^{\varepsilon}\left(x\right) \nabla u(x)\cdot\nabla u(x) \,\mathrm{d}x - \int_\Omega f(x) u(x)\,\mathrm{d}x, \end{equation*}

if $u\in H^1_0(\Omega)$, and as $F_{\varepsilon}(u):=+\infty$ otherwise in $L^2(\Omega)$.

Under Assumptions (H1)–(H3), we know (see, for instance, [Reference Marcellini28], [Reference Müller27, Theorem 1.3], or [Reference Maso29, Corollary 24.5]) that $\{F_{\varepsilon}\}_\varepsilon$ $\Gamma$-converges with respect to the weak topology of $H^1(\Omega)$, or equivalently the strong topology of $L^2(\Omega)$, to the effective functional $F^0_{\rm hom}: L^2(\Omega)\to\mathbb{R}\cup\{+\infty\}$ given by

\begin{equation*} \notag F^0_{\rm hom}(u):= \int_{\Omega} A_{{\rm hom}}\nabla u(x)\cdot\nabla u(x) \,\mathrm{d}x - \int_\Omega f(x) u(x)\,\mathrm{d}x, \end{equation*}

if $u\in H^1_0(\Omega)$, and by $F^0_{\rm hom}(u):=+\infty$ otherwise in $L^2(\Omega)$. Here, the effective matrix $A_{{\rm hom}}$ is a constant matrix given by the cell-formula

(2.1)\begin{equation} A_{{\rm hom}}\xi\cdot\xi := \inf\Big\{\int_{Y} A(y) (\xi + \nabla \varphi(y))\cdot(\xi+ \nabla \varphi(y)) \,\mathrm{d} y : \varphi \in H^1_{{\rm per}}(Y)\Big\}. \end{equation}

We refer to this $\Gamma$-convergence result as the zeroth-order term in the expansion by $\Gamma$-convergence of $F_\varepsilon$. For our analysis, we need to recall the following. Using the fact that the functional in (2.1) is quadratic, for each $\xi\in\mathbb{R}^N$, the minimization problem defining $A_{{\rm hom}}\xi\cdot\xi$ has a unique solution (up to an additive constant), denoted by $\psi_\xi$.

Definition 2.1. Let $e_1,\dots,e_N$ be the standard orthonormal basis of $\mathbb{R}^N$. For each $i=1,\dots,N$, let $\psi_i\in H^1_{{\rm per}}(Y)$ be the unique solution to

\begin{equation*} \inf\left\{\int_{Y} A(y) (e_i + \nabla \varphi(y))\cdot(e_i+ \nabla \varphi(y)) \,\mathrm{d} y : \varphi \in H^1_{{\rm per}}(Y),\,\, \int_Y \varphi(y)\,\mathrm{d} y =0 \right\}. \end{equation*}

The function $\psi_i$ is called the first-order corrector for $A$ associated with the vector $e_i$.

Remark 2.2. It turns out that the map $\xi\mapsto \psi_\xi$ is linear (see [Reference Maso29, Example 25.5]). Namely,

\begin{equation*} \psi_\xi = \sum_{i=1}^N \psi_i \xi_i, \end{equation*}

where $\xi=(\xi_1,\dots,\xi_N)$. Therefore, the knowledge of the first-order correctors for $A$ is sufficient to obtain $A_{{\rm hom}}$.

The goal of this paper is to develop further the expansion by $\Gamma$-convergence of $F_\varepsilon$ (see [Reference Anzellotti and Baldo5] for further details). As explained in the Introduction, there are two issues with boundary effects in obtaining such a result. Thus, in order to carry out our analysis, we need to assume the following

1. (H4) It holds that

   \begin{equation*} f = -\mathrm{div}(A_{{\rm hom}}\nabla g), \end{equation*}

   for some $g\in C^\infty_c(Y)$.

Assumption (H4) is in order to ensure that the solution $u_0^{\min}$ has compact support in $\Omega$. This allows us to avoid using boundary layers, which have two main issues: they do not have a variational definition, and their contribution to the energy is not clearly quantifiable in terms of the parameter $\varepsilon$, which prevents us from getting an order of the energy.

We are now in position to write the asymptotic expansion through $\Gamma$-convergence we will study.

Definition 2.3. Let $\{\varepsilon_n\}_n$ be an infinitesimal sequence. For $n\in\mathbb{N}\setminus\{0\}$, we define the functional $F_n^1: L^2(\Omega)\to \mathbb{R}\cup\{+\infty\}$ as

(2.2)\begin{equation} F_n^1 (u):= {F_{n}(u) - \min_{H^1_0(\Omega)}F^0_{{\rm hom}} \over \varepsilon_n}. \end{equation}

Remark 2.4. Note that, using standard estimates (see the proof of Proposition 5.1), it is possible to prove that, for each $n\in\mathbb N\setminus\{0\}$, the minimization problem

\begin{equation*} \min_{u\in H^1_0(\Omega)} F_n(u) \end{equation*}

admits a unique solution $u^{\rm min}_n\in H^1(\Omega)$. In a similar way, it is possible to prove that the minimization problem

\begin{equation*} \min_{u\in H^1_0(\Omega)} F^0_{\rm hom}(u) \end{equation*}

admits a unique minimizer $u^{\rm min}_0\in H^1(\Omega)$. In particular, we have that

\begin{equation*} F_{n}^1 (u) = {F_{n}(u) - F^0_{{\rm hom}}(u^{\rm min}_0) \over \varepsilon_n}. \end{equation*}

Moreover, the $\Gamma$-convergence of $F_{n}$ to $F^0_{\rm hom}$ together with the compactness of $\{u^{\rm min}_n\}_{n}$ yields that $u^{\rm min}_n\to u^{\rm min}_0$ in $L^2(\Omega)$ as $n\to\infty$.

Our goal is to study the $\Gamma$-limit of the family of functionals $\{F_n^1\}_{n}$. To this end, we introduce the candidate limiting functional.

Definition 2.6. Define the functional $F^1_{{\rm hom}}: L^2(\Omega)\to \mathbb{R}\cup\{+\infty\}$ as

(2.3)\begin{align} F^1_{{\rm hom}}(u)&:= \sum_{i,j=1}^N \int_\Omega \nabla(\partial_i u^{\rm min}_0\partial_j u^{\rm min}_0)(x)\,\mathrm{d}x \cdot\left\langle y \left[ a_{ij} + 2A e_j \cdot \nabla \psi_i + A \nabla \psi_i \cdot \nabla \psi_j \right]\right\rangle_Y \nonumber \\ &\quad+2 \sum_{i=1}^N \int_\Omega \langle \psi_i A\rangle_Y \nabla u^{\rm min}_0(x)\cdot \partial_i\nabla u^{\rm min}_0(x)\,\mathrm{d}x \nonumber \\ &\quad+ 2 \sum_{j=1}^N\sum_{i=1}^N \int_{\Omega} \partial_j u^{\rm min}_0(x) \langle \psi_i A\nabla_y\psi_j \rangle_Y \cdot \partial_i \nabla u^{\rm min}_0(x) \,\mathrm{d}x, \end{align}

if $u=u^{\min}_0$, and $+\infty$ else in $L^2(\Omega)$. Here, $\psi_i$ are the first-order correctors defined in Definition 2.1.

We now state the main result of the present paper.

We show that the analogous first-order $\Gamma$-expansion is trivial in the case of functionals defined on $L^p$. We are able to prove this statement in a more general setting than that considered above. Fix $p\in(1,\infty)$ and $M\geq1$. Let $V:\Omega\times\mathbb{R}^M\to\mathbb{R}$ be a Carathéodory function such that

1. (A1) For each $z\in\mathbb{R}^M$, the function $x\mapsto V(x,z)$ is $Y$-periodic;

2. (A2) There exists $0 \lt c_1 \lt c_2 \lt +\infty$ such that

   \begin{equation*} c_1(|z|^p-1) \leq V(x,z)\leq c_2(|z|^p+1), \end{equation*}

   for all $x\in\Omega$ and $z\in\mathbb{R}^M$.

Note that in this case, the source term would only be a part of the function $V$. That is why there is no analogue of assumption (H4).

Definition 2.10. For $n\in\mathbb{N}\setminus\{0\}$, define $G_n: L^p(\Omega;\mathbb{R}^M)\to\mathbb{R}\cup\{+\infty\}$ as

\begin{equation*} G_n(u) := \int_\Omega V\left( \frac{x}{\varepsilon_n}, u(x) \right)\,\mathrm{d}x. \end{equation*}

Remark 2.11. Note that there is no loss of generality in assuming the function $V$ to be convex in the second variable. Indeed, the relaxation of $G_n$ with respect to the weak- $L^p(\Omega)$ topology is given by

\begin{equation*} \overline{G_n}(u) = \int_\Omega V^c\left( \frac{x}{\varepsilon_n}, u(x) \right)\,\mathrm{d}x \end{equation*}

where $V^c$ denotes the convex envelope of the function $V$ in the second variable.

We now introduce the homogenized functional.

Definition 2.12. For $z\in\mathbb{R}^M$, let

\begin{equation*} V_{\rm hom}(z):= \inf\left\{\frac{1}{|Y|} \int_Y V^c(y,z + \varphi(y))\,\mathrm{d} y \,: \, \varphi\in L^p(Y;\mathbb{R}^M),\, \int_Y \varphi(y)\,\mathrm{d} y = 0 \right\}. \end{equation*}

Define the functional $G_{\rm hom}: L^p(\Omega;\mathbb{R}^M)\to\mathbb{R}\cup\{+\infty\}$ as

\begin{equation*} G_{\rm hom} := \int_\Omega V_{\rm hom} (u(x))\,\mathrm{d}x. \end{equation*}

Using the same arguments as in the proof of [Reference Cristoferi, Fonseca and Ganedi17, Theorem 3.3], we get the following.

Next result justifies our claim that the $\Gamma$-expansion is trivial in such a case.

Theorem 2.14. Assume that Assumptions (A1)-(A2), as well as (H4) hold and that, for all $y\in Y$, the function $u\mapsto V(y,u)$ is strictly convex. Let $m\in\mathbb{R}^M$. Then,

\begin{align*} &\min\left\{G_n(v) : v\in L^p(\Omega;\mathbb{R}^M),\, \int_\Omega u(x)\,\mathrm{d}x =m \right\} \\ & = \min\left\{G_{{\rm hom}}(v) : v\in L^p(\Omega;\mathbb{R}^M),\, \int_\Omega u(x)\,\mathrm{d}x =m \right\}, \end{align*}

for all $n\in\mathbb{N}\setminus\{0\}$.

Remark 2.15. The assumption of strict convexity is needed only to simplify the strategy of the proof, since we need to invert the relation

\begin{equation*} \partial_u V(x,v)=c. \end{equation*}

Strict convexity gives us a unique inverse, while with convexity alone, we would have to get a slightly more involved argument. Since this result is to show that for these functionals there is no need for a first order $\Gamma$-expansion, we decided to use this extra technical assumption.

3.1. Recall of homogenization

We recall the foundations of the homogenization theory for elliptic equations. Even if this theory is classical by now, we revisit it since we will use a slightly different definition of the correctors than that in [Reference Allaire and Amar2, Reference Bensoussan, Lions and Papanicolaou8].

Let $\Omega$ be a bounded and open subset of $\mathbb{R}^N$. Let $A$ be the matrix-valued function satisfying Assumptions (H1)-(H3). For a given $f\in L^2(\Omega)$, we consider the following equation

(3.1)\begin{equation} \left\{ \begin{aligned} -{\rm div}(A^\varepsilon(x)\nabla u_\varepsilon(x)) &= f(x) && \mbox{in }\Omega,\\ u_\varepsilon(x)&=0 && \mbox{on } \partial\Omega. \end{aligned} \right. \end{equation}

It is well-known that this problem admits a unique solution in $H^1_0(\Omega)$ (see, e.g., [Reference Bensoussan, Lions and Papanicolaou8]). To carry out a homogenization procedure, the solution $u_\varepsilon$ is assumed to have the following two-scale expansion

\begin{equation*} u_\varepsilon(x) = u_0\left(x, {x\over \varepsilon}\right) + \varepsilon u_1\left(x, {x\over \varepsilon}\right) + \varepsilon^2u_2\left(x, {x\over \varepsilon}\right) + \dots, \end{equation*}

where each $u_i$ is $Y$-periodic with respect to the fast variable $y={x\over\varepsilon}$. The variables $x$ and $y$ are treated as independent. Plugging such an expansion into (3.1) and identifying powers of $\varepsilon$, we get a cascade of equations. Here, we only care about the second-order expansion. Therefore, setting $\mathcal{A}_\varepsilon \varphi(x):= -{\rm div}(A^\varepsilon(x)\nabla \varphi(x))$, we deduce that

\begin{equation*} \mathcal{A}_\varepsilon= \varepsilon^{-2}\mathcal{A}_0+\varepsilon^{-1}\mathcal{A}_1+ \mathcal{A}_2, \end{equation*}

where

\begin{align*} \mathcal{A}_0 &:= -\sum_{i=1}^N{\partial_ {y_i}}\left(\sum_{j=1}^N a_{ij}(y){\partial_{y_j}}\right),\notag\\ \mathcal{A}_1 &:= -\sum_{i=1}^N{\partial_{y_i}}\left(\sum_{j=1}^N a_{ij}(y){\partial_j}\right) - \sum_{i=1}^N{\partial_i}\left(\sum_{j=1}^N a_{ij}(y){\partial_{y_j}}\right),\notag\\ \mathcal{A}_2 &:= -\sum_{i=1}^N{\partial_i}\left(\sum_{j=1}^N a_{ij}(y){\partial_j}\right)\notag. \end{align*}

Using (3.1), matching powers of $\varepsilon$ up to second order gives us the following equations

(3.2)\begin{equation} \qquad\qquad\qquad\,\,\,\,\,\,\mathcal{A}_0u_0=0, \end{equation}

(3.3)\begin{equation} \qquad\,\,\,\,\,\,\,\,\,\mathcal{A}_0u_1 + \mathcal{A}_1u_0=0, \end{equation}

(3.4)\begin{equation} \mathcal{A}_0u_2 + \mathcal{A}_1u_1 + \mathcal{A}_2u_0=f, \end{equation}

(3.5)\begin{equation} \mathcal{A}_0u_3 + \mathcal{A}_1u_3 + \mathcal{A}_2u_1=0. \end{equation}

From (3.2), it follows that $u_0(x,y)\equiv u_0(x)$. The solution $u_1$ to (3.3) is given by

(3.6)\begin{equation} u_1\left(x, {x\over \varepsilon}\right)= \sum_{j=1}^N \psi_j^{\varepsilon}\left(x\right) {\partial_j}u_0(x) + \widetilde{u}_1(x), \end{equation}

where, for $j=1,\dots, N$, $\psi_j$ is the unique solution in $H^1_{\rm per}(Y)$ to the problem

(3.7)\begin{equation} \begin{cases} \mathcal{A}_0 \psi_j(y)= \sum_{i=1}^N\partial_{y_i} a_{ij}(y) & \mbox{in } Y,\\ \int_Y \psi_j(y)\,\mathrm{d} y=0, \\ y\mapsto \psi_j(y) &Y\mbox{-periodic}. \end{cases} \end{equation}

Namely, for $j=1,\dots, N$, the function $\psi_j$ are the first-order corrector defined in Definition 2.1. The solution to equation (3.4) is given by

(3.8)\begin{equation} u_2\left(x, {x\over \varepsilon} \right) = \sum_{i,j=1}^N\chi^{\varepsilon}_{ij}\left(x\right) {\partial^2_{ij}} u_0(x) +\sum_{j=1}^N \psi_j^{\varepsilon}\left(x\right) {\partial_j}\widetilde{u}_1(x)+ \widetilde{u}_2(x), \end{equation}

where for all $i,j=1,\dots, n$, the second-order corrector $\chi_{ij}\in H^1_{{\rm per}}(Y)$ satisfies the following auxiliary problem

(3.9)\begin{equation} \begin{cases} \mathcal{A}_0 \chi_{ij}(y) = b_{ij}(y)-\int_Y b_{ij}(y)\,\mathrm{d} y &\mbox{in } Y,\\ \int_Y \chi_{ij}(y)\,\mathrm{d} y=0,\\ y\mapsto \chi_{ij}(y)&Y\mbox{-periodic}, \end{cases} \end{equation}

with

(3.10)\begin{equation} b_{ij}(y):= a_{ij}(y) +\sum_{k=1}^N a_{ik}(y){\partial_{y_k} \psi_j}(y) + \sum_{k=1}^N{\partial_{y_k}}(a_{ki}\psi_j)(y). \end{equation}

It is worth recalling that the function $\widetilde{u}_1$ in (3.6) can be taken identically equal to zero if one is only interested in the first-order expansion of $u_\varepsilon$. Otherwise, $\widetilde{u}_1$ is determined by the compatibility condition of equation (3.5), namely,

(3.11)\begin{equation} {\rm div}(A_{{\rm hom}}\nabla \widetilde{u}_1(x))= -\sum_{i,j,k=1}^N c_{ijk}{\partial^3_{ijk} u_0^{\min}}(x), \end{equation}

where $A_{{\rm hom}}$ is the homogenized matrix defined by (2.1) and for $i,j,k=1,\dots,N$, the constant $c_{ijk}$ is defined as

(3.12)\begin{equation} c_{ijk}:= \left\langle \sum_{l=1}^N a_{kl}{\partial_{y_l} \chi_{ij}} + a_{ij}\psi_k \right\rangle_Y. \end{equation}

Since in the present paper, we are interested in expanding $u_\varepsilon$ up to the second-order, the function $\widetilde{u}_2$ in (3.8) can be taken identically equal to zero.

3.2. Periodic functions

We collect some useful properties of the space $L^2_{{\rm per}}(Y; \mathbb{R}^N)$ of the periodic functions (see [Reference Jikov, Kozlov and Oleinik25, Chapter 1] for further details).

The space $L^2_{\rm sol}(Y)$ of solenoidal periodic functions is defined as

\begin{equation*} L^2_{\rm sol}(Y):=\{f\in L^2_{{\rm per}}(Y; \mathbb{R}^N)\; : \; \hbox{div}f=0\hbox{in } \mathbb{R}^N \}, \end{equation*}

where the equality $\hbox{div}f=0$ is to be intended in distributional sense, i.e.,

\begin{equation*} \int_{\mathbb{R}^N} f(x)\cdot \nabla \varphi(x)\; dx=0, \;\;\;\hbox{for any } \varphi \in C^\infty_c(\mathbb{R}^N). \end{equation*}

The space $L^2_{\rm sol}(Y)$ turns out to be a closed subspace of $L^2(Y; \mathbb{R}^N)$. Setting

\begin{equation*} \notag \mathcal V^2_{\rm pot}(Y):= \{\nabla u\;: u\in H^1_{{\rm per}}(Y; \mathbb{R}^N)\}, \end{equation*}

we immediately deduce the following orthogonal representation

\begin{equation*} \notag L^2_{{\rm per}}(Y; \mathbb{R}^N) = L^2_{\rm sol}(Y) \oplus \mathcal V^2_{\rm pot}(Y). \end{equation*}

The next proposition provides a representation of solenoidal functions, which will be a key tool in Proposition 4.8.

Proposition 3.2. Let $f\in L^2_{\rm sol}(Y)$. Then, $f=(f_1, \dots, f_N)$ can be represented in the form

\begin{equation*} \notag f_j(x)= \langle f_j\rangle_Y +\sum_{i=1}^N {\partial_i} \alpha_{ij}(x),\;\;\;\hbox{for all } j=1,\dots, N, \end{equation*}

where $\alpha_{ij}\in H^1(Y)$ is such that $\alpha_{ij}=-\alpha_{ji}$ and $\langle \alpha_{ij}\rangle_Y=0$, for all $i,j=1, \dots, N$.

Proof. Without loss of generality, we can assume that $Y=[0, 2\pi)^N$. Using the Fourier series of $f$, we have that

\begin{equation*} \notag f=\langle f \rangle_Y + \sum_{\substack{k\in\mathbb{Z}^N\\k\neq 0}} f^k e^{\mathrm{i}k\cdot x}, \end{equation*}

where $f^k$ is the Fourier coefficient of $f$. We claim that $f^k\cdot k=0$, for each $k\in\mathbb{Z}^N$. Indeed, for fixed $k\in \mathbb{Z}^N\setminus\{0\}$, we can decompose $f^k$ as $f^k = C_1^k k + C_2^k v$, for some constants $C_1^k, C_2^k$ and $v\in ({\rm span}(k))^{\perp}$. Therefore,

\begin{equation*} \notag f=\langle f\rangle_Y + \sum_{\substack{k\in\mathbb{Z}^N\\k\neq 0}}C_1^k ke^{\mathrm{i}k\cdot x} + \sum_{\substack{k\in\mathbb{Z}^N\\k\neq 0}}C_2^k v e^{\mathrm{i}k\cdot x}. \end{equation*}

Noticing that $\mathrm{i} ke^{\mathrm{i}k\cdot x}=\nabla(e^{\mathrm{i}k\cdot x}) \in \mathcal V^2_{\rm pot}(Y)$ and since by assumption, we know that $f\in L^2_{\rm sol}(Y)$, we obtain that $C_1^k=0$, for any $k\in\mathbb{Z}^N\setminus\{0\}$. This leads us to conclude that $f^k\cdot k=0$, and that $C_2^k v = f^k$.

Therefore,

\begin{equation*} f^k = f^k - \frac{f^k\cdot k}{|k|^2}k, \end{equation*}

which writes, component-wise, as

\begin{equation*} f_j^k = \sum_{i=1}^N\ g_{ij}^k k_i, \;\;\;\hbox{with }\,\,\,\,\, g_{ij}:={f_j^k k_i- f_i^k k_j \over |k|^2}. \end{equation*}

Therefore, for each $j=1,\dots, N$, we get that

\begin{align*} f_j &= \langle f_j \rangle_Y + \sum_{\substack{k\in\mathbb{Z}^N\\k\neq 0}} f^k_j e^{\mathrm{i}k\cdot x} = \langle f_j \rangle_Y + \sum_{\substack{k\in\mathbb{Z}^N\\k\neq 0}} \sum_{i=1}^N\ g_{ij}^k k_i e^{\mathrm{i}k\cdot x}\\& = \langle f_j \rangle_Y + \sum_{i=1}^N \sum_{\substack{k\in\mathbb{Z}^N\\k\neq 0}} \ g_{ij}^k k_i e^{\mathrm{i}k\cdot x}. \end{align*}

Thus, by defining

\begin{equation*} \notag \alpha_{ij}(x) := -\mathrm{i} \sum_{\substack{k\in\mathbb{Z}^N\\k\neq 0}} g_{ij}^k e^{\mathrm{i}k\cdot x}, \end{equation*}

we get the desired result.

4.1. First-order Riemann–Lebesgue lemma

We prove a quantitative version of the Riemann–Lebesgue lemma. Since we will use this latter result several times, we recall it here for the reader’s convenience.

Lemma 4.1. (Riemann–Lebesgue lemma)

Let $p\in[1,\infty]$, and let $\Omega\subset\mathbb{R}^N$ be an open bounded set. Let $\{\varepsilon_n\}_{n}$ be an infinitesimal sequence. Let $g\in L^p(\mathbb{R}^N)$ be an $Y$-periodic function. Then,

\begin{equation*} \lim_{n\to\infty}\int_\Omega g\left(\frac{x}{\varepsilon_n}\right)\varphi(x)\, dx = \langle g \rangle_Y \int_\Omega \varphi(x)\, dx, \end{equation*}

for all $\varphi\in L^{p'}(\Omega)$, with $\tfrac{1}{p}+\tfrac{1}{p'}=1$ if $p \lt \infty$, and $p':= 1$ if $p=\infty$.

We will present two versions of the refined result, which will require more stringent assumptions on the test functions $\varphi$. We start by assuming our test function $\varphi$ to belong to $C^2(\Omega)\cap L^1(\Omega)$. This will require us to also impose geometric requirement on $\Omega$ and on the infinitesimal sequence.

Proposition 4.2. (First-order Riemann–Lebesgue lemma)

Let $p\in[1,\infty]$, and let $g\in L^p_{\mathrm{loc}}(\mathbb{R}^N)$ be an $Y$-periodic function. Let $\Omega\subset\mathbb{R}^N$ be an open bounded set such that, up to a set of Lebesgue measure zero, can be written as

(4.1)\begin{equation} \Omega = \bigcup_{i=1}^k (x_i + Y),\quad x_i\in \mathbb{Z}^N,\quad\quad (x_i+Y)\cap(x_j+Y)=\emptyset\, \text{if }\, i\neq j. \end{equation}

Then,

\begin{align*} &\lim_{n\to\infty} n\left[ \int_\Omega g(nx)\varphi(x)\, dx - \langle g \rangle_Y \int_\Omega \varphi(x)\, dx \right] = \int_\Omega \nabla \varphi(x) \,\mathrm{d}x \cdot \left[ \langle y g \rangle_Y - \langle y \rangle_Y\, \langle g\rangle_Y \right], \end{align*}

for all $\varphi\in C^2(\Omega)\cap L^1(\Omega)$.

Proof. Let $\{z_i\}_{i}$ be an enumeration of $\mathbb{Z}^N$. For each $n\in\mathbb{N}\setminus\{0\}$, let

\begin{equation*} I_n:= \left\{i\in\mathbb{N} : \frac{1}{n} (z_i + Y)\subset \Omega \right\}. \end{equation*}

Note that, by (4.1), we get that (up to a set of Lebesgue measure zero)

\begin{equation*} \Omega = \bigcup_{i\in I_n} Y^n_i, \quad \quad \quad Y^n_i:= \frac{1}{n}(z_i + Y), \end{equation*}

where the sets $Y_i^n$ are pairwise disjoint. Thus,

\begin{align*} &\int_\Omega g(nx)\varphi(x)\, dx - \frac{1}{|Y|}\int_Y g(y)\, dy \, \int_\Omega \varphi(x)\, dx \\ &\quad =\sum_{i\in I_n}\left[ \int_{Y^n_i} g(nx)\varphi(x)\, dx - \frac{1}{|Y|}\int_Y g(y)\, dy \, \int_{Y^n_i} \varphi(x)\, dx \right]. \end{align*}

Then, using the change of variable $x = z_i + y/n$ in every set $Y^n_i$, and recalling that, by periodicity of $g$, we have that $g(z_i + y) = g(y)$ for all $z_i\in\mathbb{Z}^N$, we obtain

\begin{align*} \int_\Omega & g(nx)\varphi(x)\, dx - \frac{1}{|Y|}\int_Y g(y)\, dy \, \int_\Omega \varphi(x)\, dx \\ &\quad =\sum_{i\in I_n} \frac{1}{n^N }\left[ \int_Y g(y)\varphi\left(\frac{z_i + y}{n}\right)\, dy - \frac{1}{|Y|}\int_Y \varphi\left(\frac{z_i + y}{n}\right)\, dy \, \int_Y g(y)\, dy \right] \\ &\quad =\sum_{i\in I_n} \frac{1}{n^N} \left[ \int_Y g(y) \left[\varphi\left(\frac{z_i + y}{n}\right) - \varphi\left(\frac{z_i}{n}\right) \right]\,\right. \\ & \qquad \left.dy - \frac{1}{|Y|}\int_Y \left[\varphi\left(\frac{z_i + y}{n}\right) - \varphi\left(\frac{z_i}{n}\right) \right]\, dy \, \int_Y g(y)\, dy \right]. \end{align*}

We now use a second-order Taylor expansion for $\varphi$ to write

\begin{equation*} \varphi\left(\frac{z_i + y}{n}\right) - \varphi\left(\frac{z_i}{n}\right) = \frac{1}{n} \nabla \varphi\left(\frac{z_i}{n}\right)\cdot y + \frac{1}{2n^2} H\varphi\left(\frac{z_i}{n}\right)[y,y] + o\left(\frac{1}{n^2}\right), \end{equation*}

where $H\varphi(x)$ denotes the Hessian matrix of $\varphi$ at $x$. Thus, we get that

\begin{align*} &\int_\Omega g(nx)\varphi(x)\, dx - \frac{1}{|Y|}\int_Y g(y)\, dy \, \int_Y \varphi(x)\, dx \\ & = \frac{1}{n} \sum_{i\in I_n} \frac{1}{n^N} \nabla \varphi\left(\frac{z_i}{n}\right) \cdot \left[ \int_Y g(y)y\, dy - \frac{1}{|Y|}\int_Y y\, dy\, \int_Y g(y)\, dy \right] +O\left(\frac{1}{n^2}\right) \\ & = \frac{1}{n} \sum_{i\in I_n} \frac{|Y|}{n^N} \nabla \varphi\left(\frac{z_i}{n}\right) \cdot \left[ \langle y g \rangle_Y - \langle y \rangle_Y\, \langle g\rangle_Y \right] +O\left(\frac{1}{n^2}\right). \end{align*}

Now, since by assumption $\varphi\in C^0(\Omega)\cap L^1(\Omega)$, we have that

\begin{equation*} \lim_{n\to\infty} \sum_{i\in I_n}\frac{|Y|}{n^N} \nabla \varphi\left(\frac{z_i}{n}\right) = \int_\Omega |\nabla \varphi(x)| dx. \end{equation*}

This concludes the proof of the result.

Remark 4.3. Note that the above result is invariant under translation of the periodicity grid and of the function $g$. In particular, there is no loss of generality in assuming that $\langle y\rangle_Y=0$, namely that the barycentre of the periodicity cell is at the origin. Therefore, assuming without loss of generality that $\langle y \rangle_Y=0$, the above result yields

\begin{align*} &\lim_{n\to\infty} n\left[ \int_\Omega g(nx)\varphi(x)\, dx - \langle g \rangle_Y \int_\Omega \varphi(x)\, dx \right] = \int_\Omega \nabla \varphi(x) \,\mathrm{d}x \cdot \langle y g \rangle_Y, \end{align*}

for all $\varphi\in C^2(\Omega)\cap L^1(\Omega)$. Note that $\langle y g \rangle_Y$ is the barycentre of the periodicity cell $Y$ with respect to the density $g$.

Remark 4.4. In particular, the above result implies that

\begin{equation*} \lim_{n\to\infty} n\left[ \int_\Omega g(nx)\varphi(x)\, dx - \langle g\rangle_Y \, \int_\Omega \varphi(x)\, dx \right] =0, \end{equation*}

for all $Y$-periodic function $\varphi\in C^2(\mathbb{R}^N)$.

We now present the version of the first-order Riemann–Lebesgue lemma that we will use in the proof of our main result (Theorem 2.7). Thanks to Assumption (H4), we will only need to consider functions $\varphi\in C^2(\Omega)\cap C_c(\Omega)$. This assumption allows us to consider a general infinitesimal sequence $(\varepsilon_n)_n$, and a general open set $\Omega\subset\mathbb{R}^N$.

Proposition 4.6. (First-order Riemann–Lebesgue lemma - test functions with compact support)

Let $p\in[1,\infty]$, and let $g\in L^p_{\mathrm{loc}}(\mathbb{R}^N)$ be an $Y$-periodic function. Let $\Omega\subset\mathbb{R}^N$ be an open set, and let $\{\varepsilon_n\}_n$ be an infinitesimal sequence. Then,

\begin{align*} &\lim_{n\to\infty} \frac{1}{\varepsilon_n}\left[ \int_\Omega g\left(\frac{x}{\varepsilon_n}\right)\varphi(x)\, dx - \langle g \rangle_Y \int_\Omega \varphi(x)\, dx \right] \\&\quad = \int_\Omega \nabla \varphi(x) \,\mathrm{d}x \cdot \left[ \langle y g \rangle_Y - \langle y \rangle_Y\, \langle g\rangle_Y \right], \end{align*}

for all $\varphi\in C^2(\Omega)\cap C_c(\Omega)$.

Proof. Let $\varphi\in C^2(\Omega)\cap C_c(\Omega)$. Let $\{z_i\}_{i}$ be an enumeration of $\mathbb{Z}^N$. Let $\bar{n}\in\mathbb{N}$ be such that, for all $n\geq\bar{n}$, there exists $I_n\subset\mathbb{N}$ for which

(4.2)\begin{equation} \mathrm{supp}(\varphi) \subset \bigcup_{i\in I_n} \varepsilon_n(z_i + Y) \subset \Omega. \end{equation}

where $\mathrm{supp}(\varphi)$ denotes the support of $\varphi$. Then, by using the same argument as that employed in the proof of Proposition 4.2, we obtain

\begin{align*} &\int_\Omega g\left(\frac{x}{\varepsilon_n}\right)\varphi(x)\, dx - \frac{1}{|Y|}\int_Y g(y)\, dy \, \int_Y \varphi(x)\, dx \\ &\quad = \varepsilon_n\sum_{i\in I_n} \varepsilon_n^N \nabla \varphi\left(\frac{z_i}{n}\right) \cdot \left[ \int_Y g(y)y\, dy - \frac{1}{|Y|}\int_Y y\, dy\, \int_Y g(y)\, dy \right] +O\left(\varepsilon_n^2\right) \\ &\quad = \varepsilon_n \sum_{i\in I_n} |Y|\varepsilon_n^N \nabla \varphi\left(\frac{z_i}{n}\right) \cdot \left[ \langle y g \rangle_Y - \langle y \rangle_Y\, \langle g\rangle_Y \right] +O\left(\varepsilon_n^2\right). \end{align*}

Using (4.2), we get that

\begin{equation*} \lim_{n\to\infty}\sum_{i\in I_n} |Y|\varepsilon_n^N \nabla \varphi\left(\frac{z_i}{n}\right) = \int_\Omega |\nabla\varphi(x)|dx. \end{equation*}

This gives the desired result.

Finally, we present the version of the first-order Riemann–Lebesgue lemma in the nonlinear case.

Proposition 4.7. Let $V:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ be a function such that

1. (i) For all $p\in\mathbb{R}$, the function $x\mapsto V(x,p)$ is $Y$-periodic;

2. (ii) For all $x\in\mathbb{R}^N$, the function $p\mapsto V(x,p)$ is of class $C^2$;

3. (iii) The function

   \begin{equation*} t\mapsto \int_Y \partial_p V(y,t) \, dy,\quad\quad\quad \end{equation*}

   is Riemann integrable, where $\partial_p V$ denotes the partial derivative of $V$ with respect to the second variable.

Let $\{\varepsilon_n\}_n$ be an infinitesimal sequence. Then, the following holds

\begin{align*} &\lim_{n\to\infty} \frac{1}{\varepsilon_n}\left[ \int_\Omega V\left(\frac{x}{\varepsilon_n},\varphi(x)\right)\, dx - \frac{1}{|Y|}\int_\Omega \int_Y V(t, \varphi(x) \, dt \, dx \right] \\ &\quad= \int_\Omega \nabla \varphi(x) \cdot \left[ \frac{1}{|Y|} \int_Y \partial_p V(y,\varphi(x)) y \, dy - \frac{1}{|Y|^2} \int_Y\int_Y \partial_p V(t,\varphi(x)) y \, dt \, dy \right] dx, \end{align*}

for all $\varphi\in C^2(\Omega)\cap L^1(\Omega)$, provided either one of the following assumptions is in force:

1. (a) For each $n\in\mathbb{N}\setminus\{0\}$, $\varepsilon_n=1/n$, and assumption (4.1) holds;

2. (b) The function $\varphi$ has compact support in $\Omega$.

The argument to prove the nonlinear version of the first-order Riemann–Lebesgue lemma follows a similar strategy to the one used in the proof of the two results above, and therefore it will be omitted.

4.2. Estimates

In this section, we prove the fundamental estimate that allows us to consider the ‘surrogate’ sequence given by the expansion with first and second-order correctors in place of the minimizer $u^{min}_\varepsilon$. Note that while $u_\varepsilon$ only approximates the formal expansion $u_0+\varepsilon u_1$ to an order of $O(\sqrt{\varepsilon})$ in the $H^1$ norm, we now show that the higher-order expansion $u_{\varepsilon}^{(2)}$ satisfies the underlying PDE to a much higher order, $O(\varepsilon^2)$, in the $H^{-1}$ norm. This refined approximation in the dual space is the key to overcoming the $H^1$-based limitations in the variational analysis. Recall that $u_0^{\min}\in C^\infty_c(\Omega)$ (see Remark 2.5).

Proposition 4.8. Assume that Assumption (H4) holds. Let $u^{(2)}_\varepsilon$ be the function defined as

(4.3)\begin{equation} u^{(2)}_\varepsilon(x, y) := u_0^{\min}(x)+ \varepsilon u_1\left(x, y\right)+ \varepsilon^2 u_2\left(x, y\right), \end{equation}

with $u_1$ and $u_2$ being defined as in (3.6) and (3.8), respectively. Then, it holds that

\begin{equation*} \notag \|{\rm div}(A^\varepsilon \nabla u^{(2)}_\varepsilon)-{\rm div}(A^{{\rm hom}} \nabla u^{\rm min}_0) \|_{H^{-1}(\Omega)}\leq C \varepsilon^2, \end{equation*}

for some $C \lt +\infty$.

Proof. Recall that $\widetilde{u}_2\equiv 0$ since we are only interested in the asymptotic expansion up to the second-order. Note that

\begin{equation*} \notag {\rm div}(A^\varepsilon(x) \nabla u^{(2)}(x)_\varepsilon-A^{{\rm hom}} \nabla u^{\rm min}_0(x)) \!=\! \sum_{i=1}^N \partial_i\left( (A^\varepsilon(x)\nabla u^{(2)}_\varepsilon(x)\!-\! A^{{\rm hom}}\nabla u^{\rm min}_0(x))_i\right), \end{equation*}

where, for $i=1,\dots, N$, the $i$-th component $(A^\varepsilon(x)\nabla u^{(2)}_\varepsilon(x)- A^{{\rm hom}}\nabla u^{\rm min}_0(x))_i$ is given by

(4.4)\begin{equation} (A^\varepsilon(x)\nabla u^{(2)}_\varepsilon(x)- A^{{\rm hom}}\nabla u^{\rm min}_0(x))_i= P_{i, \varepsilon}(x)+ \varepsilon Q_{i, \varepsilon}(x)+ \varepsilon^2R_{i,\varepsilon}(x), \end{equation}

with

(4.5)\begin{align} P_{i, \varepsilon}(x) &:= \sum_{j=1}^N a^\varepsilon_{ij}(x) \partial_j u^{\rm min}_0(x) \!+\! \sum_{j,k=1}^N a_{ik}^\varepsilon(x)\partial_{y_k}\psi^\varepsilon_j(x)\partial_j u^{\rm min}_0 (x)\!-\! \sum_{j=1}^N a_{ij}^{{\rm hom}}\partial_j u^{\rm min}_0(x),\nonumber\\ Q_{i, \varepsilon}(x)&:= \sum_{j,k=1}^N \psi^\varepsilon_j(x) a^\varepsilon_{ik}(x)\partial^2_{kj} u^{\rm min}_0(x) \!+\! \sum_{k,j,l=1}^N a_{il}^\varepsilon(x) \partial_{y_l}\chi^{\varepsilon}_{kj}(x)\partial^2_{kj}u^{\rm min}_0(x)\nonumber\\ &\quad + \sum_{j=1}^N a_{ij}^\varepsilon(x)\partial_j\widetilde{u}_1(x) + \sum_{j,k=1}^N a_{ik}^\varepsilon(x)\partial_{y_k}\psi^\varepsilon_j(x)\partial_j\widetilde{u}_1(x),\nonumber\\ R_{i, \varepsilon}(x)&:= \sum_{k,j,l=1}^N \chi_{kj}^\varepsilon(x) a_{il}^\varepsilon(x)\partial^3_{lkj}u^{\rm min}_0(x) + \sum_{j,k=1}^N \psi_j^\varepsilon(x) a^{\varepsilon}_{ik}(x)\partial^2_{kj}\widetilde{u}_1(x) . \end{align}

We now estimate the above terms, starting from $P_{i, \varepsilon}(x)$. Note that $P_{i, \varepsilon}(x)$ rewrites as

\begin{equation*} \notag P_{i, \varepsilon}(x) = \sum_{j=1}^N g_i^{j, \varepsilon}(x) \partial_j u^{\rm min}_0(x), \;\;\; \mbox{for all } i=1,\dots, N, \end{equation*}

where $g_i^j$ is defined as

\begin{align*} \notag g_i^j(y) := a_{ij}(y)+ \sum_{k=1}^Na_{ik}(y)\partial_{y_k}\psi_{j}(y) -a_{ij}^{{\rm hom}}, \;\;\; \hbox{for all } i, j=1,\cdots, N. \end{align*}

For fixed $j=1,\dots, N$, set $G^j:=(g_i^1,\dots, g_i^N)$. We have that $G^j\in L^2_{\rm sol}(Y)$. Indeed, thanks to problem (3.7) satisfied by $\psi_j$, it follows that

\begin{equation*} \notag \mbox{div } G^j(y)= \sum_{i=1}^N \partial_{y_i} g_{i}^j(y)=0. \end{equation*}

Therefore, by applying Proposition 3.2, the components of $G^j$ are represented by

(4.6)\begin{equation} g_i^j(y) = \sum_{k=1}^N \partial_{y_k}\alpha^j_{ik}(y). \end{equation}

Note that $\langle g_i^j\rangle_Y=0$, since

(4.7)\begin{equation} a_{ij}^{{\rm hom}}= \left\langle a_{ij}(y)+ \sum_{k=1}^Na_{ik}(y)\partial_{y_k}\psi_{j}(y) \right\rangle_Y. \end{equation}

Using the representation (4.6) as well as the Leibniz rule, we get that

\begin{align*} P_{i, \varepsilon}(x) &= \sum_{j,k=1}^N \partial_{y_k} \alpha_{ik}^{j, \varepsilon}(x)\partial_ju^{\rm min}_0(x) \notag\\ &\quad= \varepsilon\left(\sum_{j,k=1}^N \partial_{k}(\alpha_{ik}^{j,\varepsilon}\partial_ju^{\rm min}_0)(x) - \sum_{j,k=1}^N \alpha_{ik}^{j, \varepsilon}(x)\partial^2_{kj}u^{\rm min}_0(x) \right).\notag \end{align*}

Hence, (4.4) turns into

(4.8)\begin{align} &(A^\varepsilon(x)\nabla u^{(2)}_\varepsilon(x)- A^{{\rm hom}}\nabla u^{\rm min}_0(x))_i \nonumber \\&\quad = \varepsilon\left( \sum_{j,k=1}^N \partial_{k}(\alpha_{ik}^{j,\varepsilon}\partial_ju^{\rm min}_0)(x) + \widetilde{Q}_{i,\varepsilon}(x)\right)+ \varepsilon^2R_{i,\varepsilon}(x), \end{align}

where

\begin{equation*} \notag \widetilde{Q}_{i,\varepsilon}(x) := Q_{i,\varepsilon}(x)- \sum_{j,k=1}^N \alpha_{ik}^{j, \varepsilon}(x)\partial^2_{kj}u^{\rm min}_0(x), \end{equation*}

with $Q_{i,\varepsilon}$ being defined as in (4.5). We now estimate $\widetilde{Q}_{i,\varepsilon}(x)$. As for $P_{i,\varepsilon}$, $\widetilde{Q}_{i,\varepsilon}(x)$ can be rewritten as

(4.9)\begin{equation} \widetilde{Q}_{i,\varepsilon}(x)= \sum_{j,k=1}^N h_i^{jk, \varepsilon}(x)\partial_{kj}^2u^{\rm min}_0(x) +\sum_{j=1}^N t_i^{j,\varepsilon}(x)\partial_j\widetilde{u}_1(x), \;\;\; \hbox{for all } i=1,\dots, N, \end{equation}

where for any fixed $j,k=1,\dots, N$, the functions $h^{kj}_i$ and $t^j_i$ are defined as

(4.10)\begin{equation} h_i^{kj} (y):= \psi_j(y) a_{ik}(y) + \sum_{l=1}^N a_{il}(y) \partial_{y_l}\chi_{kj}(y) -\alpha_{ik}^j(y), \end{equation}

(4.11)\begin{equation} t_i^j(y):= a_{ij}(y) +\sum_{k=1}^N a_{ik}(y) \partial_{y_k}\psi_j(y). \end{equation}

We claim that $H^{kj}:= (h_1^{kj}, \dots, h_N^{kj})$ as well as $T^j:= (t_1^j, \dots, t_N^j)$ belong to $L^2_{\rm sol}(Y)$. Indeed, bearing in mind that $\chi_{kj}^\varepsilon$ is the solution to (3.9) together with the fact that

\begin{equation*}\sum_{i=1}^N\partial_{y_i}(-\alpha_{ik}^j)=\sum_{i=1}^N\partial_{y_i}(\alpha_{ki}^j) = g_k^j,\end{equation*}

we deduce that for fixed $j,k=1,\dots, N$,

\begin{align*} {\rm div}(H^{kj}) &= \sum_{i=1}^N \partial_{y_i} h_i^{kj}(y)\\&= \sum_{i=1}^N\partial_{y_i}(\psi_j^\varepsilon a_{ik}^\varepsilon)(y) +\sum_{i=1}^N\partial_{y_i}\left(\sum_{l=1}^N a_{il}\partial_l\chi_{kj}\right)(y) +\sum_{i=1}^N \partial_{y_i}(-\alpha_{ik}^j)(y) \notag\\ &=\sum_{i=1}^N \partial_{y_i}(\psi_j(y) a_{ik}(y)) -b_{kj}(y)+\int_Y b_{kj}dy + g_k^j(y) =0, \notag \end{align*}

where we have used the equality $a_{kj}^{{\rm hom}}=\langle b_{kj}\rangle_Y$ (cf. (3.10) and (4.7)). Likewise, since $\psi_j$ is the solution to (3.7), we immediately conclude that

\begin{equation*} \notag {\rm div}(T^j) = \sum_{i=1}^N \partial_{y_i} t_i^j (y)=\sum_{i=1}^N\partial_{y_i}\left(a_{ij}(y) +\sum_{k=1}^N a_{ik}(y)\partial_{y_k}\psi_j(y)\right) =0. \end{equation*}

Therefore, applying Proposition 3.2, it follows that the components of $H^{kj}$ and $T^j$ are represented by

\begin{align*}\notag h_i^{kj}(y)= \langle h_i^{kj}\rangle_Y+ \sum_{l=1}^N \partial_{y_l} \beta_{il}^{kj}(y)\;\;\;\hbox{and}\;\;\; t_i^j(y)= \langle t_i^{j}\rangle_Y+\sum_{k=1}^N \partial_{y_k}\gamma_{ik}^j(y), \end{align*}

for all $i=1,\dots, N$. This implies that

\begin{align*} \widetilde{Q}_{i,\varepsilon}(x) &= \sum_{j,k=1}^N \langle h_i^{kj}\rangle_Y\partial^2_{kj}u_0^{\min}(x)+ \sum_{j,k,l=1}^N \partial_{y_l} \beta_{il}^{kj,\varepsilon}(x)\partial^2_{kj}u_0^{\min}(x) \notag\\ &\quad+ \sum_{j=1}^N\langle t_i^{j}\rangle_Y\partial_j\widetilde{u}_1(x)+\sum_{j,k=1}^N \sum_{k=1}^N \partial_{y_k}\gamma_{ik}^{j, \varepsilon}(x) \partial_j\widetilde{u}_1(x)\notag\\ & =\sum_{j,k=1}^N \langle h_i^{kj}\rangle_Y\partial^2_{kj}u_0^{\min}(x)+ \sum_{j=1}^N\langle t_i^{j}\rangle_Y\partial_j\widetilde{u}_1(x) \notag\\ &\quad +\varepsilon\biggl( \sum_{j,k,l=1}^N \partial_l(\beta_{il}^{kj, \varepsilon}\partial_{kj}^2u^{\rm min}_0)(x)- \sum_{j,k,l=1}^N \beta_{il}^{kj,\varepsilon}(x)\partial_{lkj}^3u^{\rm min}_0(x)\notag\\ &\quad+ \sum_{j,k=1}^N\partial_i(\gamma_{ik}^{j,\varepsilon}\partial_j\widetilde{u}_1)(x) -\sum_{j,k=1}^N\gamma_{ik}^{j,\varepsilon}(x)\partial^2_{kj}\widetilde{u}_1)(x) \biggr)\notag \end{align*}

From (4.4) together with (4.8), we conclude that

\begin{align*} &(A^\varepsilon(x)\nabla u^{(2)}_\varepsilon(x)- A^{{\rm hom}}\nabla u^{\rm min}_0(x))_i \\& = \varepsilon\left( \sum_{j,k=1}^N \partial_{k}(\alpha_{ik}^{j,\varepsilon}\partial_ju^{\rm min}_0)(x)+\sum_{j,k=1}^N \langle h_i^{kj}\rangle_Y\partial^2_{kj}u_0^{\min}(x)+ \sum_{j=1}^N\langle t_i^{j}\rangle_Y\partial_j\widetilde{u}_1(x)\right)\notag\\ &\quad + \varepsilon^2\left( \sum_{j,k,l=1}^N \partial_l(\beta_{il}^{kj, \varepsilon}\partial_{kj}^2u^{\rm min}_0)(x)+ \sum_{j,k=1}^N\partial_i(\gamma_{ik}^{j,\varepsilon}\partial_j\widetilde{u}_1)(x) + \widetilde{R}_{i,\varepsilon}(x) \right),\notag \end{align*}

where $\widetilde{R}_{i,\varepsilon}(x)$ is defined as

\begin{align*} \widetilde{R}_{i,\varepsilon}(x)&:= R_{i,\varepsilon}(x) - \sum_{j,k,l=1}^N \beta_{il}^{kj,\varepsilon}(x)\partial_{lkj}^3u^{\rm min}_0(x)-\sum_{j,k=1}^N\gamma_{ik}^{j,\varepsilon}(x)\partial^2_{kj}\widetilde{u}_1(x). \notag \end{align*}

Noticing that $\langle h_i^{kj}\rangle_Y= c_{ijk}$ (cf. (3.12) and (4.10)) and $\langle t_i^{j}\rangle_Y= a_{ij}^{{\rm hom}}$ (cf. (4.7) and (4.11)) as well as bearing in mind problem (3.11), we obtain that

\begin{align*} \notag \sum_{i=1}^N \partial_i\left(\sum_{j,k=1}^N \langle h_i^{kj}\rangle_Y\partial^2_{kj}u_0^{\min}(x)+ \sum_{j=1}^N\langle t_i^{j}\rangle_Y\partial_j\widetilde{u}_1(x) \right)=0. \end{align*}

Moreover, since $\alpha_{ik}^j=-\alpha_{ki}^j$, it follows that

\begin{equation*} \notag \sum_{i=1}^N\partial_i\left(\sum_{j,k=1}^N \partial_{k}(\alpha_{ik}^{j,\varepsilon}\partial_ju^{\rm min}_0)(x) \right)=0. \end{equation*}

Likewise, since $\beta_{il}^{kj} = - \beta_{li}^{kj}$ and $\gamma_{ik}^{j}=-\gamma_{ki}^{j}$,

\begin{equation*} \notag \sum_{i=1}^N\partial_i\left( \sum_{j,k,l=1}^N \partial_l(\beta_{il}^{kj, \varepsilon}\partial_{kj}^2u^{\rm min}_0)(x)\right)= \sum_{i=1}^N\partial_i\left(\sum_{j,k=1}^N\partial_i(\gamma_{ik}^{j,\varepsilon}\partial_j\widetilde{u}_1)(x)\right)=0. \end{equation*}

Therefore,

\begin{equation*} \notag {\rm div}(A^\varepsilon(x) \nabla u^{(2)}(x)_\varepsilon-A^{{\rm hom}} \nabla u^{\rm min}_0(x)) = \varepsilon^2 {\rm div}(\widetilde{R}_\varepsilon(x)), \end{equation*}

with $\widetilde{R}_\varepsilon=(\widetilde{R}_{1,\varepsilon}, \dots, \widetilde{R}_{N,\varepsilon}).$ Now,

\begin{equation*} \notag \|{\rm div}(A^\varepsilon \nabla u^{(2)}_\varepsilon-A^{{\rm hom}} \nabla u^{\rm min}_0) \|_{H^{-1}(\Omega)} = \|{\rm div}\widetilde{R}_\varepsilon\|_{H^{-1}(\Omega)} \leq \|\widetilde{R}_\varepsilon\|_{L^{2}(\Omega)} \leq C \varepsilon^2, \end{equation*}

which concludes the proof.