up to extraction of a subsequence as $\epsilon \downarrow 0$,

$$
\begin{align}
q_E(\dot{\epsilon}) &\to E \quad \text{weakly in } L^2(U), \nonumber \\
\tilde{q}_E(\dot{\epsilon}) &\to \bar{B}E \quad \text{weakly in } L^\alpha(U), \nonumber \\
\tilde{\Sigma}_E(\dot{\epsilon}) &\to -\bar{b} : E \quad \text{weakly in } L^\alpha(U), \nonumber \\
p_\epsilon &\to D(u_0) \quad \text{weakly in } L^2(U), \nonumber \\
\tilde{p}_\epsilon &\to \tilde{p}_0 \quad \text{weakly in } L^\alpha(U), \nonumber \\
\tilde{S}_\epsilon &\to \tilde{S}_0 \quad \text{weakly in } L^\alpha(U), \tag{4.26}
\end{align}
$$

for some $u_0 \in H_0^1(U)^d$, $\tilde{p}_0 \in L^\alpha(U)_{\text{sym}}^{d \times d}$, and $\tilde{S}_0 \in L^\alpha(U)$. In the case $\alpha > \frac{2d}{d+2}$ (hence $\alpha' < \frac{2d}{d-2}$), further appealing to the compact Rellich embedding and to the sublinearity of $\psi_E^\kappa$, cf. (2.9), we further deduce almost surely, up to extraction of a subsequence,

$$
\begin{align}
\epsilon \psi_E (\dot{\epsilon}) &\to 0 \quad \text{strongly in } L^{\alpha'} (U), \tag{4.27} \\
u_\epsilon &\to u_0 \quad \text{strongly in } L^{\alpha'} (U). \nonumber
\end{align}
$$

If the inclusions $\{I_n\}_n$ were uniformly separated as assumed in [7], then we could choose $\alpha = 2$, cf. (4.22), so that a standard div-curl argument in form of e.g. [20, Lemma 12.12] would allow us to use (4.26) and pass to the limit on both sides of identity (4.25) (along the subsequence), to the effect that

$$
\bar{B}E : D(u_0) = E : \tilde{p}_0 \quad \text{in } U. \tag{4.28}
$$

In the present situation, with $\alpha < 2$, we need to repeat the proof of the div-curl lemma and show that this identity (4.28) still holds. Once this is proven, the conclusion (4.23) easily follows: passing to the weak limit in (4.12) (along the subsequence) yields

$$
- \operatorname{div}(2\tilde{p}_0 - \tilde{S}_0 \operatorname{Id}) = (1 - \lambda)f \quad \text{in } U,
$$

and thus, inserting (4.28) in form $\tilde{p}_0 = \bar{B} D(u_0)$, we deduce that $(u_0, \tilde{S}_0 - f_U \tilde{S}_0)$ coincides with the unique solution $(\tilde{u}, \tilde{S})$ of the homogenized equation (2.14). With this characterization of the limit, the conclusion (4.23) now follows from (4.26).

It remains to prove (4.28), and we split the proof into two further substeps. We start with the case $\frac{2d}{d+2} < \alpha < 2$, and next we discuss the critical case $\alpha = \frac{2d}{d+2}$.

Substep 1.1. Proof of (4.28) in the case $\frac{2d}{d+2} < \alpha < 2$.

We shall pass to the limit on both sides of (4.25) and we start with the analysis of the left-hand side. Given a test function $h \in C_c^1(U)$ supported in $U^\kappa$ for some fixed $\kappa > 2\epsilon$, integrating by parts, using property (4.11) of the extension $(\tilde{q}_E, \tilde{\Sigma}_E)$, and using the constraint $\text{div}(u_\epsilon) = 0$, we find

$$
\begin{align*}
\int_U h \tilde{q}_E (\dot{\epsilon}) : p_\epsilon &= \int_U h \tilde{q}_E (\dot{\epsilon}) : D(u_\epsilon) \\
&= \int_U D(h u_\epsilon) : \tilde{q}_E (\dot{\epsilon}) - \int_U (\nabla h \otimes u_\epsilon) : \tilde{q}_E (\dot{\epsilon})
\end{align*}
$$
up to extraction of a subsequence as $\epsilon \downarrow 0$,

$$
\begin{align}
q_E(\dot{\epsilon}) &\to E \quad \text{weakly in } L^2(U), \nonumber \\
\tilde{q}_E(\dot{\epsilon}) &\to \bar{B}E \quad \text{weakly in } L^\alpha(U), \nonumber \\
\tilde{\Sigma}_E(\dot{\epsilon}) &\to -\bar{b} : E \quad \text{weakly in } L^\alpha(U), \nonumber \\
p_\epsilon &\to D(u_0) \quad \text{weakly in } L^2(U), \nonumber \\
\tilde{p}_\epsilon &\to \tilde{p}_0 \quad \text{weakly in } L^\alpha(U), \nonumber \\
\tilde{S}_\epsilon &\to \tilde{S}_0 \quad \text{weakly in } L^\alpha(U), \tag{4.26}
\end{align}
$$

for some $u_0 \in H_0^1(U)^d$, $\tilde{p}_0 \in L^\alpha(U)_{\text{sym}}^{d \times d}$, and $\tilde{S}_0 \in L^\alpha(U)$. In the case $\alpha > \frac{2d}{d+2}$ (hence $\alpha' < \frac{2d}{d-2}$), further appealing to the compact Rellich embedding and to the sublinearity of $\psi_E^\kappa$, cf. (2.9), we further deduce almost surely, up to extraction of a subsequence,

$$
\begin{equation}
\begin{aligned}
\epsilon \psi_E (\dot{\epsilon}) &\to 0 && \text{strongly in } L^{\alpha'} (U), \\
u_\epsilon &\to u_0 && \text{strongly in } L^{\alpha'} (U).
\end{aligned}
\tag{4.27}
\end{equation}
$$

If the inclusions $\{I_n\}_n$ were uniformly separated as assumed in [7], then we could choose $\alpha = 2$, cf. (4.22), so that a standard div-curl argument in form of e.g. [20, Lemma 12.12] would allow us to use (4.26) and pass to the limit on both sides of identity (4.25) (along the subsequence), to the effect that

$$
\bar{B}E : D(u_0) = E : \tilde{p}_0 \quad \text{in } U. \tag{4.28}
$$

In the present situation, with $\alpha < 2$, we need to repeat the proof of the div-curl lemma and show that this identity (4.28) still holds. Once this is proven, the conclusion (4.23) easily follows: passing to the weak limit in (4.12) (along the subsequence) yields

$$
- \operatorname{div}(2\tilde{p}_0 - \tilde{S}_0 \operatorname{Id}) = (1 - \lambda)f \quad \text{in } U,
$$

and thus, inserting (4.28) in form $\tilde{p}_0 = \bar{B} D(u_0)$, we deduce that $(u_0, \tilde{S}_0 - f_U \tilde{S}_0)$ coincides with the unique solution $(\tilde{u}, \tilde{S})$ of the homogenized equation (2.14). With this characterization of the limit, the conclusion (4.23) now follows from (4.26).

It remains to prove (4.28), and we split the proof into two further substeps. We start with the case $\frac{2d}{d+2} < \alpha < 2$, and next we discuss the critical case $\alpha = \frac{2d}{d+2}$.

Substep 1.1. Proof of (4.28) in the case $\frac{2d}{d+2} < \alpha < 2$.

We shall pass to the limit on both sides of (4.25) and we start with the analysis of the left-hand side. Given a test function $h \in C_c^1(U)$ supported in $U^\kappa$ for some fixed $\kappa > 2\epsilon$, integrating by parts, using property (4.11) of the extension $(\tilde{q}_E, \tilde{\Sigma}_E)$, and using the constraint $\text{div}(u_\epsilon) = 0$, we find

$$
\begin{align*}
\int_U h \tilde{q}_E (\dot{\epsilon}) : p_\epsilon &= \int_U h \tilde{q}_E (\dot{\epsilon}) : D(u_\epsilon) \\
&= \int_U D(h u_\epsilon) : \tilde{q}_E (\dot{\epsilon}) - \int_U (\nabla h \otimes u_\epsilon) : \tilde{q}_E (\dot{\epsilon})
\end{align*}
$$
$$
\begin{align}
&= \frac{1}{2} \int_U \tilde{\Sigma}_E(\dot{\epsilon}) \operatorname{div}(h u_\epsilon) - \int_U (\nabla h \otimes u_\epsilon) : \tilde{q}_E(\dot{\epsilon}) \nonumber \\
&= -\frac{1}{2} \int_U (\nabla h \otimes u_\epsilon) : (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id})(\dot{\epsilon}). \tag{4.29}
\end{align}
$$

Note that the relation $\operatorname{div}(u_\epsilon) = 0$ entails $\int_U \nabla h \cdot u_\epsilon = 0$, which allows us to add any constant to the pressure $\tilde{\Sigma}_E$, for instance replacing it by $\tilde{\Sigma}_E - \mathbb{E}[\tilde{\Sigma}_E]$. In view of (4.26) and (4.27), we may now pass to the limit in the above, to the effect that

$$
\lim_{\epsilon \downarrow 0} \int_U h \tilde{q}_E(\dot{\epsilon}) : p_\epsilon = - \int_U (\nabla h \otimes u_0) : \bar{B} E = \int_U h \bar{B} E : D(u_0). \quad (4.30)
$$

We turn to the analysis of the right-hand side of (4.25). Integrating by parts, using prop-
erty (4.12) of the extension $(\tilde{p}_{\epsilon}, \tilde{S}_{\epsilon})$, and using the constraint $\operatorname{div}(\psi_{\epsilon}) = 0$, we find

$$
\begin{align}
\int_U h q_E(\dot{\epsilon}) : \tilde{p}_\epsilon &= E : \int_U h \tilde{p}_\epsilon + \int_U h D(\psi_E)(\dot{\epsilon}) : \tilde{p}_\epsilon \nonumber \\
&= E : \int_U h \tilde{p}_\epsilon + \int_U D(h \epsilon \psi_E(\dot{\epsilon})) : \tilde{p}_\epsilon - \int_U (\nabla h \otimes \epsilon \psi_E(\dot{\epsilon})) : \tilde{p}_\epsilon \nonumber \\
&= E : \int_U h \tilde{p}_\epsilon + \frac{1}{2} \int_{U \setminus I_\epsilon(U)} h \epsilon \psi_E(\dot{\epsilon}) \cdot f \nonumber \\
&\quad - \frac{1}{2} \int_U (\nabla h \otimes \epsilon \psi_E(\dot{\epsilon})) : (2 \tilde{p}_\epsilon - \tilde{S}_\epsilon \operatorname{Id}). \tag{4.31}
\end{align}
$$

In view of (4.26) and (4.27), we may now pass to the limit in the above, to the effect that

$$
\lim_{\epsilon \downarrow 0} \int_U h q_E(\dot{\epsilon}) : \tilde{p}_\epsilon = E : \int_U h \tilde{p}_0. \quad (4.32)
$$

Combining this with (4.25) and (4.30) and choosing an arbitrary test function $h \in C_c^\infty(U)$, this proves claim (4.28).

Substep 1.2. Proof of (4.28) in the critical case $\alpha = \frac{2d}{d+2}$.

It suffices to prove that (4.30) and (4.32) still hold in this case. Due to the failure of the compact Rellich embedding (4.27), we can no longer pass to the limit directly in (4.29) and (4.31), so a finer analysis is needed. We appeal again to a two-scale argument as inspired by [21, Lemma 1.15].

We start with the proof of (4.30). Given $\eta > 0$, we choose a partition $\{Q_i\}_i$ of $U$ into measurable subsets with $|Q_i| \simeq \eta^d$. In these terms, we can decompose (4.29) as

$$
\begin{align}
\int_U h \tilde{q}_E(\dot{\epsilon}) : p_\epsilon ={}& -\frac{1}{2} \sum_i \left( \int_{Q_i} u_\epsilon \right) \cdot \int_{Q_i} (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id})(\dot{\epsilon}) \nabla h \nonumber \\
& -\frac{1}{2} \sum_i \int_{Q_i} \nabla h \otimes \left( u_\epsilon - \int_{Q_i} u_\epsilon \right) : (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id})(\dot{\epsilon}). \tag{4.33}
\end{align}
$$
$$
\begin{align}
&= \frac{1}{2} \int_U \tilde{\Sigma}_E(\dot{\epsilon}) \operatorname{div}(h u_\epsilon) - \int_U (\nabla h \otimes u_\epsilon) : \tilde{q}_E(\dot{\epsilon}) \nonumber \\
&= -\frac{1}{2} \int_U (\nabla h \otimes u_\epsilon) : (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id})(\dot{\epsilon}). \tag{4.29}
\end{align}
$$

Note that the relation $\operatorname{div}(u_\epsilon) = 0$ entails $\int_U \nabla h \cdot u_\epsilon = 0$, which allows us to add any constant to the pressure $\tilde{\Sigma}_E$, for instance replacing it by $\tilde{\Sigma}_E - \mathbb{E}[\tilde{\Sigma}_E]$. In view of (4.26) and (4.27), we may now pass to the limit in the above, to the effect that

$$
\lim_{\epsilon \downarrow 0} \int_U h \tilde{q}_E(\dot{\epsilon}) : p_\epsilon = - \int_U (\nabla h \otimes u_0) : \bar{B} E = \int_U h \bar{B} E : D(u_0). \quad (4.30)
$$

We turn to the analysis of the right-hand side of (4.25). Integrating by parts, using prop-
erty (4.12) of the extension $(\tilde{p}_{\epsilon}, \tilde{S}_{\epsilon})$, and using the constraint $\operatorname{div}(\psi_{\epsilon}) = 0$, we find

$$
\begin{align}
\int_U h q_E(\dot{\epsilon}) : \tilde{p}_\epsilon &= E : \int_U h \tilde{p}_\epsilon + \int_U h D(\psi_E)(\dot{\epsilon}) : \tilde{p}_\epsilon \nonumber \\
&= E : \int_U h \tilde{p}_\epsilon + \int_U D(h \epsilon \psi_E(\dot{\epsilon})) : \tilde{p}_\epsilon - \int_U (\nabla h \otimes \epsilon \psi_E(\dot{\epsilon})) : \tilde{p}_\epsilon \nonumber \\
&= E : \int_U h \tilde{p}_\epsilon + \frac{1}{2} \int_{U \setminus I_\epsilon(U)} h \epsilon \psi_E(\dot{\epsilon}) \cdot f \nonumber \\
&\quad - \frac{1}{2} \int_U (\nabla h \otimes \epsilon \psi_E(\dot{\epsilon})) : (2 \tilde{p}_\epsilon - \tilde{S}_\epsilon \operatorname{Id}). \tag{4.31}
\end{align}
$$

In view of (4.26) and (4.27), we may now pass to the limit in the above, to the effect that

$$
\lim_{\epsilon \downarrow 0} \int_U h q_E(\dot{\epsilon}) : \tilde{p}_\epsilon = E : \int_U h \tilde{p}_0. \quad (4.32)
$$

Combining this with (4.25) and (4.30) and choosing an arbitrary test function $h \in C_c^\infty(U)$, this proves claim (4.28).

Substep 1.2. Proof of (4.28) in the critical case $\alpha = \frac{2d}{d+2}$.

It suffices to prove that (4.30) and (4.32) still hold in this case. Due to the failure of the compact Rellich embedding (4.27), we can no longer pass to the limit directly in (4.29) and (4.31), so a finer analysis is needed. We appeal again to a two-scale argument as inspired by [21, Lemma 1.15].

We start with the proof of (4.30). Given $\eta > 0$, we choose a partition $\{Q_i\}_i$ of $U$ into measurable subsets with $|Q_i| \simeq \eta^d$. In these terms, we can decompose (4.29) as

$$
\begin{align}
\int_U h \tilde{q}_E(\dot{\epsilon}) : p_\epsilon ={}& -\frac{1}{2} \sum_i \left( \int_{Q_i} u_\epsilon \right) \cdot \int_{Q_i} (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id})(\dot{\epsilon}) \nabla h \nonumber \\
& -\frac{1}{2} \sum_i \int_{Q_i} \nabla h \otimes \left( u_\epsilon - \int_{Q_i} u_\epsilon \right) : (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id})(\dot{\epsilon}). \tag{4.33}
\end{align}
$$
On the one hand, using the compact Rellich embedding in the form of the almost sure strong convergence $u_\epsilon \to u_0$ in $L^1(U)$, and using Lemma 4.3, we find

$$
\begin{align*}
\lim_{\epsilon \downarrow 0} \frac{1}{2} \sum_i \left( \int_{Q_i} u_\epsilon \right) \cdot \int_{Q_i} \nabla h \cdot (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id})\left(\frac{\cdot}{\epsilon}\right) \\
&= \frac{1}{2} \sum_i \left( \int_{Q_i} \nabla h \right) \otimes \left( \int_{Q_i} u_0 \right) : (2\bar{B}E + (\bar{b} : E) \operatorname{Id}),
\end{align*}
$$

hence, letting the mesh $\eta$ of the partition $\{Q_i\}_i$ tend to 0, using that the constraint $\operatorname{div}(u_\epsilon) = 0$ entails $\int_U \nabla h \cdot u_0 = 0$, and integrating by parts,

$$
\begin{align}
& \lim_{\eta \downarrow 0} \lim_{\epsilon \downarrow 0} \frac{1}{2} \sum_i \left( \int_{Q_i} u_\epsilon \right) \cdot \int_{Q_i} \nabla h \cdot (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id})\left(\frac{\cdot}{\epsilon}\right) \nonumber \\
&= \frac{1}{2} \left( \int_U \nabla h \otimes u_0 \right) : (2\bar{B}E + (\bar{b} : E) \operatorname{Id}) = - \int_U h \bar{B}E : \operatorname{D}(u_0). \tag{4.34}
\end{align}
$$

On the other hand, using Hölder's inequality and the Poincaré–Sobolev embedding, the
second right-hand side term in (4.33) can be estimated as

$$
\begin{align*}
\left| \sum_i \int_{Q_i} \nabla h \otimes \left( u_\epsilon - \int_{Q_i} u_\epsilon \right) : (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id}) \left( \frac{\cdot}{\epsilon} \right) \right|
& \le \|\nabla h\|_{L^\infty(U)} \sum_i \left\| u_\epsilon - \int_{Q_i} u_\epsilon \right\|_{L^{\frac{2d}{d-2}}(Q_i)} \| (\tilde{q}_E, \tilde{\Sigma}_E) \left( \frac{\cdot}{\epsilon} \right) \|_{L^{\frac{2d}{d+2}}(Q_i)} \\
& \lesssim \|\nabla h\|_{L^\infty(U)} \sum_i \|\nabla u_\epsilon\|_{L^2(Q_i)} \| (\tilde{q}_E, \tilde{\Sigma}_E) \left( \frac{\cdot}{\epsilon} \right) \|_{L^{\frac{2d}{d+2}}(Q_i)} \\
& \le \|\nabla h\|_{L^\infty(U)} \|\nabla u_\epsilon\|_{L^2(U)} \left( \sum_i \| (\tilde{q}_E, \tilde{\Sigma}_E) \left( \frac{\cdot}{\epsilon} \right) \|_{L^{\frac{2d}{d+2}}(Q_i)} \right)^{\frac{1}{2}},
\end{align*}
$$

hence, passing to the limit $\epsilon \downarrow 0$, using the boundedness of $\nabla u_\epsilon$ in $L^2(U)$, and using the
stationarity and the boundedness of $(\tilde{q}_E, \tilde{\Sigma}_E)$ in $L^{\frac{2d}{d+2}}(\Omega)$, cf. Corollary 4.1 (i),

$$
\limsup_{\epsilon \downarrow 0} \left| \sum_i \int_{Q_i} \nabla h \otimes \left( u_\epsilon - \int_{Q_i} u_\epsilon \right) : (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id}) \left( \frac{\cdot}{\epsilon} \right) \right| \\
\lesssim_f \|\nabla h\|_{L^\infty(U)} \left( \sum_i |Q_i|^{\frac{d+2}{d}} \right)^{\frac{1}{2}} \lesssim \eta \|\nabla h\|_{L^\infty(U)}.
$$

Now letting the mesh $\eta$ of the partition $\{Q_i\}_i$ tend to 0, and combining this with (4.33)
and (4.34), we deduce (4.30).

We turn to the proof of (4.32). Given $\eta > 0$, we consider as above a partition $\{Q_i\}_i$
of $U$ into measurable subsets with $|Q_i| \simeq \eta^d$. The starting point is the Poincaré–Sobolev
embedding in the form

$$
\| \epsilon \psi_E (\dot{\epsilon}) \|_{L^{\frac{2d}{d-2}}(Q_i)} \lesssim \| \nabla \psi_E (\dot{\epsilon}) \|_{L^2(Q_i)} + |Q_i|^{\frac{d-2}{2d}} \int_{Q_i} |\epsilon \psi_E (\dot{\epsilon})|.
$$
On the one hand, using the compact Rellich embedding in the form of the almost sure strong convergence $u_\epsilon \to u_0$ in $L^1(U)$, and using Lemma 4.3, we find

$$ 
\begin{aligned}
& \lim_{\epsilon \downarrow 0} \frac{1}{2} \sum_i \left( \int_{Q_i} u_\epsilon \right) \cdot \int_{Q_i} \nabla h \cdot (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id})\left(\frac{\cdot}{\epsilon}\right) \\
& = \frac{1}{2} \sum_i \left( \int_{Q_i} \nabla h \right) \otimes \left( \int_{Q_i} u_0 \right) : (2\bar{B}E + (\bar{b} : E) \operatorname{Id}),
\end{aligned}
$$

hence, letting the mesh $\eta$ of the partition $\{Q_i\}_i$ tend to 0, using that the constraint $\operatorname{div}(u_\epsilon) = 0$ entails $\int_U \nabla h \cdot u_0 = 0$, and integrating by parts,

$$ 
\begin{aligned}
& \lim_{\eta \downarrow 0} \lim_{\epsilon \downarrow 0} \frac{1}{2} \sum_i \left( \int_{Q_i} u_\epsilon \right) \cdot \int_{Q_i} \nabla h \cdot (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id})\left(\frac{\cdot}{\epsilon}\right) \\
& = \frac{1}{2} \left( \int_U \nabla h \otimes u_0 \right) : (2\bar{B}E + (\bar{b} : E) \operatorname{Id}) = - \int_U h \bar{B}E : \operatorname{D}(u_0). \quad (4.34)
\end{aligned}
$$

On the other hand, using Hölder's inequality and the Poincaré–Sobolev embedding, the second right-hand side term in (4.33) can be estimated as

$$ 
\begin{aligned}
& \left| \sum_i \int_{Q_i} \nabla h \otimes \left( u_\epsilon - \int_{Q_i} u_\epsilon \right) : (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id})\left(\frac{\cdot}{\epsilon}\right) \right| \\
& \le \|\nabla h\|_{L^\infty(U)} \sum_i \left\| u_\epsilon - \int_{Q_i} u_\epsilon \right\|_{L^{\frac{2d}{d-2}}(Q_i)} \| (\tilde{q}_E, \tilde{\Sigma}_E)\left(\frac{\cdot}{\epsilon}\right) \|_{L^{\frac{2d}{d+2}}(Q_i)} \\
& \lesssim \|\nabla h\|_{L^\infty(U)} \sum_i \|\nabla u_\epsilon\|_{L^2(Q_i)} \| (\tilde{q}_E, \tilde{\Sigma}_E)\left(\frac{\cdot}{\epsilon}\right) \|_{L^{\frac{2d}{d+2}}(Q_i)} \\
& \le \|\nabla h\|_{L^\infty(U)} \|\nabla u_\epsilon\|_{L^2(U)} \left( \sum_i \| (\tilde{q}_E, \tilde{\Sigma}_E)\left(\frac{\cdot}{\epsilon}\right) \|_{L^{\frac{2d}{d+2}}(Q_i)}^2 \right)^{\frac{1}{2}},
\end{aligned}
$$

hence, passing to the limit $\epsilon \downarrow 0$, using the boundedness of $\nabla u_\epsilon$ in $L^2(U)$, and using the stationarity and the boundedness of $(\tilde{q}_E, \tilde{\Sigma}_E)$ in $L^{\frac{2d}{d+2}}(\Omega)$, cf. Corollary 4.1 (i),

$$ 
\begin{aligned}
& \limsup_{\epsilon \downarrow 0} \left| \sum_i \int_{Q_i} \nabla h \otimes \left( u_\epsilon - \int_{Q_i} u_\epsilon \right) : (2\tilde{q}_E - \tilde{\Sigma}_E \operatorname{Id})\left(\frac{\cdot}{\epsilon}\right) \right| \\
& \lesssim_f \|\nabla h\|_{L^\infty(U)} \left( \sum_i |Q_i|^{\frac{d+2}{d}} \right)^{\frac{1}{2}} \lesssim \eta \|\nabla h\|_{L^\infty(U)}.
\end{aligned}
$$

Now letting the mesh $\eta$ of the partition $\{Q_i\}_i$ tend to 0, and combining this with (4.33) and (4.34), we deduce (4.30).

We turn to the proof of (4.32). Given $\eta > 0$, we consider as above a partition $\{Q_i\}_i$ of $U$ into measurable subsets with $|Q_i| \simeq \eta^d$. The starting point is the Poincaré–Sobolev embedding in the form

$$ \|\epsilon \psi_E(\dot{\epsilon})\|_{L^{\frac{2d}{d-2}}(Q_i)} \lesssim \|\nabla \psi_E(\dot{\epsilon})\|_{L^2(Q_i)} + |Q_i|^{\frac{d-2}{2d}} \int_{Q_i} |\epsilon \psi_E(\dot{\epsilon})|. $$
By the stationarity and the boundedness of $\nabla\psi_E$ in $L^2(\Omega)$, and by the sublinearity of $\psi_E$ in $L^1$, cf. (2.9), we deduce almost surely,

$$ \limsup_{\epsilon \downarrow 0} \|\epsilon \psi_E(\dot{\epsilon})\|_{L^{\frac{2d}{d-2}}(Q_i)} \lesssim |Q_i|^{\frac{1}{2}} \|\nabla \psi_E\|_{L^2(\Omega)} \lesssim |Q_i|^{\frac{1}{2}}. $$

Summing over $i$, this yields

$$ \limsup_{\epsilon \downarrow 0} \|\epsilon \psi_E(\dot{\epsilon})\|_{L^{\frac{2d}{d-2}}(U)} \lesssim \left( \sum_i |Q_i|^{\frac{d}{d-2}} \right)^{\frac{d-2}{2d}} \lesssim \eta, $$

and thus, letting the mesh $\eta$ of the partition $\{Q_i\}_i$ tend to 0,

$$ \lim_{\epsilon \downarrow 0} \|\epsilon \psi_E(\dot{\epsilon})\|_{L^{\frac{2d}{d-2}}(U)} = 0, \quad (4.35) $$

which proves that $\psi_E$ is in fact still sublinear in $L^{\alpha'} = L^{\frac{2d}{d-2}}$. This allows us to pass to the limit in (4.31), and claim (4.32) follows.

### Step 2. Convergence of the pressure.

While it is already shown in Step 1, cf. (4.23), that almost surely $\tilde{S}_\epsilon - f_U \tilde{S}_\epsilon \to \bar{S}$ weakly in $L^\alpha(U)$, we turn to the weak convergence of the restricted pressure $S_\epsilon 1_{U\setminus I_\epsilon}(U) = \tilde{S}_\epsilon 1_{U\setminus I_\epsilon}(U)$, and we establish at the same time the corrector result for the pressure, cf. (2.15). For this purpose, we start by examining the two-scale expansion errors

$$ \begin{aligned} w_\epsilon &:= u_\epsilon - \bar{u} - \sum_{E \in \mathcal{E}} \epsilon \psi_E(\dot{\epsilon}) \nabla_E \bar{u}, \\ Q_\epsilon &:= S_\epsilon 1_{U \setminus I_\epsilon}(U) - \bar{S} - \bar{b} : D(\bar{u}) - \sum_{E \in \mathcal{E}} (\Sigma_E 1_{\mathbb{R}^d \setminus I})(\dot{\epsilon}) \nabla_E \bar{u}. \end{aligned} $$

Without loss of generality, we may assume that $f \in W^{1,\infty}(U)^d$ and $\bar{u} \in W_0^{3,\infty}(U)^d$, while the general case easily follows by an approximation argument as in [7, Step 8.4 of the proof of Proposition 2.1].

Consider a test function $g \in C_c^\infty(U)^d$ with $D(g)|_{\epsilon I} = 0$. Inserting the above definition of $(w_\epsilon, Q_\epsilon)$ and reorganizing the terms, we compute

$$ \begin{aligned} \int_U D(g) : (2 D(w_\epsilon) - Q_\epsilon \mathrm{Id}) &= \int_U D(g) : (2 p_\epsilon - S_\epsilon \mathrm{Id}) - \int_U D(g) : (2 \bar{B} D(\bar{u}) - \bar{S} \mathrm{Id}) \\ &\quad - \sum_{E \in \mathcal{E}} \int_U D(g) : (2 q_E - \Sigma_E 1_{\mathbb{R}^d \setminus I} \mathrm{Id})(\dot{\epsilon}) \nabla_E \bar{u} \\ &\quad + \sum_{E \in \mathcal{E}} \int_U D(g) : (2 \bar{B} E + (\bar{b} : E) \mathrm{Id}) \nabla_E \bar{u} \\ &\quad - 2 \sum_{E \in \mathcal{E}} \int_U D(g) : (\nabla \nabla_E \bar{u} \otimes \epsilon \psi_E(\dot{\epsilon})). \end{aligned} $$
By the stationarity and the boundedness of $\nabla\psi_E$ in $L^2(\Omega)$, and by the sublinearity of $\psi_E$ in $L^1$, cf. (2.9), we deduce almost surely,

$$ \limsup_{\epsilon \downarrow 0} \|\epsilon \psi_E(\dot{\epsilon})\|_{L^{\frac{2d}{d-2}}(Q_i)} \lesssim |Q_i|^{\frac{1}{2}} \|\nabla \psi_E\|_{L^2(\Omega)} \lesssim |Q_i|^{\frac{1}{2}}. $$

Summing over $i$, this yields

$$ \limsup_{\epsilon \downarrow 0} \|\epsilon \psi_E(\dot{\epsilon})\|_{L^{\frac{2d}{d-2}}(U)} \lesssim \left( \sum_i |Q_i|^{\frac{d}{d-2}} \right)^{\frac{d-2}{2d}} \lesssim \eta, $$

and thus, letting the mesh $\eta$ of the partition $\{Q_i\}_i$ tend to 0,

$$ \lim_{\epsilon \downarrow 0} \|\epsilon \psi_E(\dot{\epsilon})\|_{L^{\frac{2d}{d-2}}(U)} = 0, \quad (4.35) $$

which proves that $\psi_E$ is in fact still sublinear in $L^{\alpha'} = L^{\frac{2d}{d-2}}$. This allows us to pass to the limit in (4.31), and claim (4.32) follows.

**Step 2.** Convergence of the pressure.

While it is already shown in Step 1, cf. (4.23), that almost surely $\tilde{S}_\epsilon - f_U \tilde{S}_\epsilon \to \bar{S}$ weakly in $L^\alpha(U)$, we turn to the weak convergence of the restricted pressure $S_\epsilon \mathbb{1}_{U\setminus I_\epsilon}(U) = \tilde{S}_\epsilon \mathbb{1}_{U\setminus I_\epsilon}(U)$, and we establish at the same time the corrector result for the pressure, cf. (2.15). For this purpose, we start by examining the two-scale expansion errors

$$ \begin{aligned} w_\epsilon &:= u_\epsilon - \bar{u} - \sum_{E \in \mathcal{E}} \epsilon \psi_E(\dot{\epsilon}) \nabla_E \bar{u}, \\ Q_\epsilon &:= S_\epsilon \mathbb{1}_{U \setminus I_\epsilon}(U) - \bar{S} - \bar{b} : D(\bar{u}) - \sum_{E \in \mathcal{E}} (\Sigma_E \mathbb{1}_{\mathbb{R}^d \setminus I})(\dot{\epsilon}) \nabla_E \bar{u}. \end{aligned} $$

Without loss of generality, we may assume that $f \in W^{1,\infty}(U)^d$ and $\bar{u} \in W_0^{3,\infty}(U)^d$, while the general case easily follows by an approximation argument as in [7, Step 8.4 of the proof of Proposition 2.1].

Consider a test function $g \in C_c^\infty(U)^d$ with $D(g)|_{\epsilon I} = 0$. Inserting the above definition of $(w_\epsilon, Q_\epsilon)$ and reorganizing the terms, we compute

$$ \begin{aligned} \int_U D(g) : (2 D(w_\epsilon) - Q_\epsilon \mathrm{Id}) &= \int_U D(g) : (2 p_\epsilon - S_\epsilon \mathrm{Id}) - \int_U D(g) : (2 \bar{B} D(\bar{u}) - \bar{S} \mathrm{Id}) \\ &\quad - \sum_{E \in \mathcal{E}} \int_U D(g) : (2 q_E - \Sigma_E \mathbb{1}_{\mathbb{R}^d \setminus I} \mathrm{Id})(\dot{\epsilon}) \nabla_E \bar{u} \\ &\quad + \sum_{E \in \mathcal{E}} \int_U D(g) : (2 \bar{B} E + (\bar{b} : E) \mathrm{Id}) \nabla_E \bar{u} \\ &\quad - 2 \sum_{E \in \mathcal{E}} \int_U D(g) : (\nabla \nabla_E \bar{u} \otimes \epsilon \psi_E(\dot{\epsilon})). \end{aligned} $$
Since $D(g)$ vanishes in $\varepsilon I$, recalling that $(q_E, \Sigma_E)(\dot{\varepsilon})$ and $(p_E, S_E)$ coincide with $(\tilde{q}_E, \tilde{\Sigma}_E)(\dot{\varepsilon})$ and $(\tilde{p}_E, \tilde{S}_E)$ in $U \setminus \varepsilon I \subset U \setminus I_E(U)$, and appealing to (4.11) and (4.12), and to the homogenized equation (2.14), we easily find

$$ \int_U D(g) : (2D(w_\varepsilon) - Q_\varepsilon \mathrm{Id}) = \mathcal{F}_\varepsilon(g), \quad (4.36) $$

in terms of

$$ \begin{aligned} \mathcal{F}_\varepsilon(g) := & - \int_U g \cdot (\mathbb{1}_I(\dot{\varepsilon}) - \lambda) f - 2 \sum_{E \in \varepsilon} \int_U D(g) : (\nabla \nabla_E \bar{u} \otimes \varepsilon \psi_E(\dot{\varepsilon})) \\ & + \sum_{E \in \varepsilon} \int_U (\nabla \nabla_E \bar{u} \otimes g) : ((2\tilde{q}_E - \tilde{\Sigma}_E \mathrm{Id})(\dot{\varepsilon}) - (2\bar{B}E + (\bar{b}:E)\mathrm{Id})). \end{aligned} \quad (4.37) $$

We now appeal to Lemma 3.2 in the following form: there exists $z_\varepsilon \in H_0^1(U)^d$ with $D(z_\varepsilon)|_{\varepsilon I} = 0$, such that

$$ \mathrm{div}(z_\varepsilon) = \left( T_\varepsilon |T_\varepsilon|^{\alpha-2} - \int_{U \setminus \varepsilon I} T_\varepsilon |T_\varepsilon|^{\alpha-2} \right) \mathbb{1}_{U \setminus \varepsilon I}, \quad T_\varepsilon := Q_\varepsilon - \int_{U \setminus \varepsilon I} Q_\varepsilon, $$

and

$$ \begin{aligned} \|\nabla z_\varepsilon\|_{L^2(U)} & \lesssim_{U, \alpha, r} \Lambda_\varepsilon(U; r, \frac{2\alpha}{2-\alpha}) \|T_\varepsilon|T_\varepsilon|^{\alpha-2}\|_{L^{\alpha'}(U \setminus \varepsilon I)} \\ & \lesssim \Lambda_\varepsilon(U; r, \frac{2\alpha}{2-\alpha}) \left\| Q_\varepsilon - \int_{U \setminus \varepsilon I} Q_\varepsilon \right\|_{L^\alpha(U \setminus \varepsilon I)}^{\alpha-1}, \end{aligned} \quad (4.38) $$

where we have set

$$ \Lambda_{\varepsilon}(U; r, p) := \left( |U| + \varepsilon^d \sum_{n:\varepsilon I_n \cap U \neq \emptyset} \mu_r(\rho_{n;U,\varepsilon})^p \right)^{\frac{1}{p}}. $$

Testing (4.36) with $g = z_\varepsilon$, and using the properties of $z_\varepsilon$, we find

$$ \left\| Q_{\varepsilon} - \int_{U \setminus \varepsilon I} Q_{\varepsilon} \right\|_{L^{\alpha}(U \setminus \varepsilon I)}^{\alpha} = - \mathcal{F}_{\varepsilon}(z_{\varepsilon}) + 2 \int_{U} D(z_{\varepsilon}) : D(w_{\varepsilon}). \quad (4.39) $$

Noting that definition (4.37) of $\mathcal{F}_\varepsilon$ yields

$$ \begin{aligned} |\mathcal{F}_\varepsilon(g)| & \lesssim \|g\|_{H^1(U)} (\|f\|_{W^{1,\infty}(U)} + \|\nabla \bar{u}\|_{W^{2,\infty}(U)}) \\ & \times \sup_{E \in \varepsilon} (\|\varepsilon \psi_E(\dot{\varepsilon})\|_{L^2(U)} + \|\mathbb{1}_I(\dot{\varepsilon}) - \lambda\|_{H^{-1}(U)} \\ & \qquad + \|\tilde{q}_E(\dot{\varepsilon}) - \bar{B}E\|_{H^{-1}(U)} + \|\tilde{\Sigma}_E(\dot{\varepsilon}) + \bar{b}:E\|_{H^{-1}(U)}), \end{aligned} $$
Since $D(g)$ vanishes in $\varepsilon I$, recalling that $(q_E, \Sigma_E)(\dot{\varepsilon})$ and $(p_E, S_E)$ coincide with $(\tilde{q}_E, \tilde{\Sigma}_E)(\dot{\varepsilon})$ and $(\tilde{p}_E, \tilde{S}_E)$ in $U \setminus \varepsilon I \subset U \setminus I_E(U)$, and appealing to (4.11) and (4.12), and to the homogenized equation (2.14), we easily find

$$ \int_U D(g) : (2D(w_\varepsilon) - Q_\varepsilon \mathrm{Id}) = \mathcal{F}_\varepsilon(g), \quad (4.36) $$

in terms of

$$ \begin{aligned} \mathcal{F}_{\varepsilon}(g) := & - \int_U g \cdot (\mathbb{1}_I(\dot{\varepsilon}) - \lambda) f - 2 \sum_{E \in \varepsilon} \int_U D(g) : (\nabla \nabla_E \bar{u} \otimes \varepsilon \psi_E(\dot{\varepsilon})) \\ & + \sum_{E \in \varepsilon} \int_U (\nabla \nabla_E \bar{u} \otimes g) : ((2\tilde{q}_E - \tilde{\Sigma}_E \mathrm{Id})(\dot{\varepsilon}) - (2\bar{B}E + (\bar{b}:E)\mathrm{Id})). \end{aligned} \quad (4.37) $$

We now appeal to Lemma 3.2 in the following form: there exists $z_\varepsilon \in H_0^1(U)^d$ with $D(z_\varepsilon)|_{\varepsilon I} = 0$, such that

$$ \mathrm{div}(z_\varepsilon) = \left( T_\varepsilon |T_\varepsilon|^{\alpha-2} - \int_{U \setminus \varepsilon I} T_\varepsilon |T_\varepsilon|^{\alpha-2} \right) \mathbb{1}_{U \setminus \varepsilon I}, \quad T_\varepsilon := Q_\varepsilon - \int_{U \setminus \varepsilon I} Q_\varepsilon, $$

and

$$ \begin{aligned} \|\nabla z_\varepsilon\|_{L^2(U)} & \lesssim_{U, \alpha, r} \Lambda_\varepsilon(U; r, \frac{2\alpha}{2-\alpha}) \|T_\varepsilon|T_\varepsilon|^{\alpha-2}\|_{L^{\alpha'}(U \setminus \varepsilon I)} \\ & \lesssim \Lambda_\varepsilon(U; r, \frac{2\alpha}{2-\alpha}) \left\| Q_\varepsilon - \int_{U \setminus \varepsilon I} Q_\varepsilon \right\|_{L^\alpha(U \setminus \varepsilon I)}^{\alpha-1}, \end{aligned} \quad (4.38) $$

where we have set

$$ \Lambda_{\varepsilon}(U; r, p) := \left( |U| + \varepsilon^d \sum_{n:\varepsilon I_n \cap U \neq \emptyset} \mu_r(\rho_{n;U,\varepsilon})^p \right)^{\frac{1}{p}}. $$

Testing (4.36) with $g = z_\varepsilon$, and using the properties of $z_\varepsilon$, we find

$$ \left\| Q_{\varepsilon} - \int_{U \setminus \varepsilon I} Q_{\varepsilon} \right\|_{L^{\alpha}(U \setminus \varepsilon I)}^{\alpha} = - \mathcal{F}_{\varepsilon}(z_{\varepsilon}) + 2 \int_{U} D(z_{\varepsilon}) : D(w_{\varepsilon}). \quad (4.39) $$

Noting that definition (4.37) of $\mathcal{F}_\varepsilon$ yields

$$ \begin{aligned} |\mathcal{F}_\varepsilon(g)| & \lesssim \|g\|_{H^1(U)} (\|f\|_{W^{1,\infty}(U)} + \|\nabla \bar{u}\|_{W^{2,\infty}(U)}) \\ & \times \sup_{E \in \varepsilon} (\|\varepsilon \psi_E(\dot{\varepsilon})\|_{L^2(U)} + \|\mathbb{1}_I(\dot{\varepsilon}) - \lambda\|_{H^{-1}(U)} \\ & \qquad + \|\tilde{q}_E(\dot{\varepsilon}) - \bar{B}E\|_{H^{-1}(U)} + \|\tilde{\Sigma}_E(\dot{\varepsilon}) + \bar{b}:E\|_{H^{-1}(U)}), \end{aligned} $$
inserting this into (4.39), and using (4.38), we deduce

$$
\begin{align*}
& \left\| Q_{\varepsilon} - \int_{U \setminus \varepsilon I} Q_{\varepsilon} \right\|_{L^{\alpha}(U \setminus \varepsilon I)} \\
& \lesssim_{U, \alpha, r} \Lambda_{\varepsilon}(U; r, \frac{2\alpha}{2-\alpha}) \|w_{\varepsilon}\|_{H^1(U)} \\
& \qquad + \Lambda_{\varepsilon}(U; r, \frac{2\alpha}{2-\alpha}) (\|f\|_{W^{1,\infty}(U)} + \|\nabla \bar{u}\|_{W^{2,\infty}(U)}) \\
& \qquad \times \sup_{E \in \varepsilon} (\|\varepsilon \psi_E(\dot{\varepsilon})\|_{L^2(U)} + \|1_I(\dot{\varepsilon}) - \lambda\|_{H^{-1}(U)}) \\
& \qquad + \|\tilde{q}_E(\dot{\varepsilon}) - \bar{B}E\|_{H^{-1}(U)} + \|\tilde{\Sigma}_E(\dot{\varepsilon}) + \tilde{b} : E\|_{H^{-1}(U)}).
\end{align*}
$$

Noting that the moment condition (4.22) entails $\lim_{\varepsilon \downarrow 0} \Lambda_{\varepsilon}(U; r, \frac{2\alpha}{2-\alpha}) < \infty$, and using (2.9), (4.24), and Lemma 4.3, together with the ergodic theorem in the form of the almost sure weak convergence $1_I(\dot{\varepsilon}) \to \lambda$ in $L_{\text{loc}}^2(\mathbb{R}^d)$, the above right-hand side tends to 0 almost surely as $\varepsilon \downarrow 0$. This concludes the proof of (2.15). ■

**5. Further technical tools**

This last section is devoted to the proofs of Corollaries 5 and 6, which are further technical tools for the analysis of particle suspensions without uniform separation.

*Proof of Corollary 5.* Note that the Stokes equation (2.22) entails $\operatorname{div}(\sigma(u, S)) = 0$ in $\mathbb{R}^d \setminus I$. For all $n$, in terms of the cutoff function $w_n \in H_0^1(I_n^+)$ with $w_n|_{I_n} = 1$ that we constructed in Lemma 3.3, an integration by parts then yields

$$
\int_{\partial I_n} g \cdot \sigma(u, S) v = - \int_{I_n^+ \setminus I_n} \operatorname{div}(w_n \sigma(u, S) g) = - \int_{I_n^+ \setminus I_n} \mathcal{D}(w_n g) : \sigma(u, S). \quad (5.1)
$$

In order to reformulate the right-hand side, we appeal to the extension result of Theorem 4.
More precisely, given $\beta \in (1, \infty)$ and $\alpha, r$ as in (2.17), since the Stokes equation (2.22)
ensures that the flux $p = \mathcal{D}(u) \in L_{\text{loc}}^2(\mathbb{R}^d)^{d \times d}$ satisfies $\operatorname{tr}(p) = 0$ and

$$
\int_{\mathbb{R}^d} \mathcal{D}(g) : p = 0 \quad \forall g \in C_c^1(\mathbb{R}^d)^d : \operatorname{div}(g) = 0, \mathcal{D}(g)|_{I} = 0,
$$

Theorem 4 provides an extension $\tilde{p} \in L^\alpha(I_n^+)_{\text{sym}}^{d \times d}$ with $\operatorname{tr}(\tilde{p}) = 0$, and an associated pressure field $\tilde{S} \in L_{\text{loc}}^\alpha(\mathbb{R}^d)$, such that

$$
(\tilde{p}, \tilde{S})|_{\mathbb{R}^d \setminus I} = (p, S)|_{\mathbb{R}^d \setminus I}, \quad \text{and} \quad \operatorname{div}(2\tilde{p} - \tilde{S} \operatorname{Id}) = 0 \text{ in } \mathbb{R}^d,
$$

and such that the following estimate holds, for all $n$:

$$
\| (\tilde{p}, \tilde{S}) \|_{L^{\alpha}(I_n^{+})} \lesssim_{\alpha, \beta, r} \mu_r(\rho_n) \| \mathcal{D}(u) \|_{L^{\beta}(I_n^{+} \setminus I_n)}.
$$
inserting this into (4.39), and using (4.38), we deduce

$$
\begin{align*}
& \left\| Q_{\varepsilon} - \int_{U \setminus \varepsilon I} Q_{\varepsilon} \right\|_{L^{\alpha}(U \setminus \varepsilon I)} \\
& \lesssim_{U, \alpha, r} \Lambda_{\varepsilon}(U; r, \frac{2\alpha}{2-\alpha}) \|w_{\varepsilon}\|_{H^1(U)} \\
& \qquad + \Lambda_{\varepsilon}(U; r, \frac{2\alpha}{2-\alpha}) (\|f\|_{W^{1,\infty}(U)} + \|\nabla \bar{u}\|_{W^{2,\infty}(U)}) \\
& \qquad \times \sup_{E \in \varepsilon} (\|\varepsilon \psi_E(\dot{\varepsilon})\|_{L^2(U)} + \|1_I(\dot{\varepsilon}) - \lambda\|_{H^{-1}(U)}) \\
& \qquad + \|\tilde{q}_E(\dot{\varepsilon}) - \bar{B}E\|_{H^{-1}(U)} + \|\tilde{\Sigma}_E(\dot{\varepsilon}) + \tilde{b}\|_E \|H^{-1}(U)\).
\end{align*}
$$

Noting that the moment condition (4.22) entails $\lim_{\varepsilon \downarrow 0} \Lambda_{\varepsilon}(U; r, \frac{2\alpha}{2-\alpha}) < \infty$, and using (2.9), (4.24), and Lemma 4.3, together with the ergodic theorem in the form of the almost sure weak convergence $1_I(\dot{\varepsilon}) \to \lambda$ in $L_{\text{loc}}^2(\mathbb{R}^d)$, the above right-hand side tends to 0 almost surely as $\varepsilon \downarrow 0$. This concludes the proof of (2.15). ■

**5. Further technical tools**

This last section is devoted to the proofs of Corollaries 5 and 6, which are further technical tools for the analysis of particle suspensions without uniform separation.

*Proof of Corollary 5.* Note that the Stokes equation (2.22) entails $\operatorname{div}(\sigma(u, S)) = 0$ in $\mathbb{R}^d \setminus I$. For all $n$, in terms of the cutoff function $w_n \in H_0^1(I_n^+)$ with $w_n|_{I_n} = 1$ that we constructed in Lemma 3.3, an integration by parts then yields

$$
\int_{\partial I_n} g \cdot \sigma(u, S) v = - \int_{I_n^+ \setminus I_n} \operatorname{div}(w_n \sigma(u, S) g) = - \int_{I_n^+ \setminus I_n} \mathcal{D}(w_n g) : \sigma(u, S). \quad (5.1)
$$

In order to reformulate the right-hand side, we appeal to the extension result of Theorem 4.
More precisely, given $\beta \in (1, \infty)$ and $\alpha, r$ as in (2.17), since the Stokes equation (2.22)
ensures that the flux $p = \mathcal{D}(u) \in L_{\text{loc}}^2(\mathbb{R}^d)^{d \times d}$ satisfies $\operatorname{tr}(p) = 0$ and

$$
\int_{\mathbb{R}^d} \mathcal{D}(g) : p = 0 \quad \forall g \in C_c^1(\mathbb{R}^d)^d : \operatorname{div}(g) = 0, \mathcal{D}(g)|_{I} = 0,
$$

Theorem 4 provides an extension $\tilde{p} \in L^\alpha(I_n^+)_{\text{sym}}^{d \times d}$ with $\operatorname{tr}(\tilde{p}) = 0$, and an associated pressure field $\tilde{S} \in L_{\text{loc}}^\alpha(\mathbb{R}^d)$, such that

$$
(\tilde{p}, \tilde{S})|_{\mathbb{R}^d \setminus I} = (p, S)|_{\mathbb{R}^d \setminus I}, \quad \text{and} \quad \operatorname{div}(2\tilde{p} - \tilde{S} \operatorname{Id}) = 0 \text{ in } \mathbb{R}^d,
$$

and such that the following estimate holds, for all $n$:

$$
\| (\tilde{p}, \tilde{S}) \|_{L^{\alpha}(I_n^{+})} \lesssim_{\alpha, \beta, r} \mu_r(\rho_n) \| \mathcal{D}(u) \|_{L^{\beta}(I_n^{+} \setminus I_n)}.
$$
Writing $\sigma(u, S) = 2p - S$ Id in (5.1), and using these extensions, we find

$$ \int_{\partial I_n} g \cdot \sigma(u, S) \nu = - \int_{I_n^+ \setminus I_n} D(w_n g) : (2p - S) \mathrm{Id} = \int_{I_n} D(w_n g) : (2\tilde{p} - \tilde{S}) \mathrm{Id}, $$

and we may then estimate

$$ \begin{aligned} \left| \int_{\partial I_n} g \cdot \sigma(u, S) \nu \right| & \lesssim \|w_n\|_{W^{1,\alpha'}(I_n^+)} \|g\|_{W^{1,\infty}(I_n^+)} \|(\tilde{p}, \tilde{S})\|_{L^\alpha(I_n)} \\ & \lesssim_{\alpha, \beta, r} \mu_r(\rho_n) \|w_n\|_{W^{1,\alpha'}(I_n^+)} \|g\|_{W^{1,\infty}(I_n^+)} \|D(u)\|_{L^\beta(I_n^+ \setminus I_n)}. \end{aligned} $$

Combining this with the bound on norms of $w_n$ in Lemma 3.3, choosing $\beta = 2$, and optimizing the choice of $\alpha, r$, the conclusion follows. ■

*Proof of Corollary 6.* For $R \ge 5$, choose $\zeta_R \in C_c^\infty(B_{2R-4}; \mathbb{R}^+)$ with $\zeta_R|_{B_R} = 1$ and with $|\nabla\zeta_R| \lesssim R^{-1}$. For any $V \in \mathbb{R}^d$ and $c \in \mathbb{R}$, testing the Stokes equation (2.22) with $\zeta_R(u - V)$, and replacing the pressure $S$ by $S - c$, we find

$$ \int_{\mathbb{R}^d} \zeta_R |\nabla u|^2 = - \int_{\mathbb{R}^d} ((u-V) \otimes \nabla \zeta_R) : (\nabla u - (S-c) \mathrm{Id} \mathbb{1}_{\mathbb{R}^d \setminus I}) \\ - \sum_{n: I_n^+ \subset B_{2R}} \int_{\partial I_n} \zeta_R(u-V) \cdot \sigma(u, S-c) \nu. $$

Since $D(u) = 0$ in $I_n$, we may write $u = V_n + \Theta_n(x - x_n)$ in $I_n$ for some $V_n \in \mathbb{R}^d$ and $\Theta_n \in \mathbb{M}^{\text{skew}}$. The boundary conditions for $u$ then allow us to add any constant to the test function $\zeta_R$ in the last right-hand-side term, and we obtain

$$ \int_{\mathbb{R}^d} \zeta_R |\nabla u|^2 = - \int_{\mathbb{R}^d} ((u-V) \otimes \nabla \zeta_R) : (\nabla u - (S-c) \mathrm{Id} \mathbb{1}_{\mathbb{R}^d \setminus I}) \\ - \sum_{n: I_n^+ \subset B_{2R}} \int_{\partial I_n} \left( \zeta_R - \int_{I_n} \zeta_R \right) (V_n - V + \Theta_n(x-x_n)) \cdot \sigma(u, S-c) \nu. $$

Hence, using the properties of $\zeta_R$, Hölder's inequality, and appealing to the trace estimate of Corollary 5 to bound the last right-hand-side term, we deduce for all $s \ge 1$,

$$ \begin{aligned} \|\nabla u\|^2_{L^2(B_R)} \lesssim R^{-1} \|u-V\|_{L^s(B_{2R})} (\|\nabla u\|_{L^{s'}(B_{2R})} + \|S-c\|_{L^{s'}(B_{2R}\setminus I)}) \\ + R^{-1} \left( \sum_{n: I_n^+ \subset B_{2R}} \mu'(\rho_n)^2 (|V_n-V|^2 + |\Theta_n|^2) \right)^{\frac{1}{2}} \|\nabla u\|_{L^2(B_{2R})}, \end{aligned} $$

where we have set for abbreviation,

$$ \mu'(\rho_n) := \begin{cases} \rho_n^{\frac{1}{4d}(d+1)(d+2)-\frac{5}{2}} & : d \le 6, \\ 1 & : d > 6. \end{cases} $$
Writing $\sigma(u, S) = 2p - S$ Id in (5.1), and using these extensions, we find

$$ \int_{\partial I_n} g \cdot \sigma(u, S) v = - \int_{I_n^+ \setminus I_n} D(w_n g) : (2p - S) \mathrm{Id} = \int_{I_n} D(w_n g) : (2\tilde{p} - \tilde{S}) \mathrm{Id}, $$

and we may then estimate

$$ \begin{aligned} \left| \int_{\partial I_n} g \cdot \sigma(u, S) v \right| & \lesssim \|w_n\|_{W^{1,\alpha'}(I_n^+)} \|g\|_{W^{1,\infty}(I_n^+)} \|(\tilde{p}, \tilde{S})\|_{L^\alpha(I_n)} \\ & \lesssim_{\alpha, \beta, r} \mu_r(\rho_n) \|w_n\|_{W^{1,\alpha'}(I_n^+)} \|g\|_{W^{1,\infty}(I_n^+)} \|D(u)\|_{L^\beta(I_n^+ \setminus I_n)}. \end{aligned} $$

Combining this with the bound on norms of $w_n$ in Lemma 3.3, choosing $\beta = 2$, and optimizing the choice of $\alpha, r$, the conclusion follows. ■

*Proof of Corollary 6.* For $R \ge 5$, choose $\zeta_R \in C_c^\infty(B_{2R-4}; \mathbb{R}^+)$ with $\zeta_R|_{B_R} = 1$ and with $|\nabla\zeta_R| \lesssim R^{-1}$. For any $V \in \mathbb{R}^d$ and $c \in \mathbb{R}$, testing the Stokes equation (2.22) with $\zeta_R(u - V)$, and replacing the pressure $S$ by $S - c$, we find

$$ \int_{\mathbb{R}^d} \zeta_R |\nabla u|^2 = - \int_{\mathbb{R}^d} ((u - V) \otimes \nabla \zeta_R) : (\nabla u - (S - c) \mathrm{Id} \mathbb{1}_{\mathbb{R}^d \setminus I}) \\ - \sum_{n: I_n^+ \subset B_{2R}} \int_{\partial I_n} \zeta_R (u - V) \cdot \sigma(u, S - c) v. $$

Since $D(u) = 0$ in $I_n$, we may write $u = V_n + \Theta_n(x - x_n)$ in $I_n$ for some $V_n \in \mathbb{R}^d$ and $\Theta_n \in \mathbb{M}^{\text{skew}}$. The boundary conditions for $u$ then allow us to add any constant to the test function $\zeta_R$ in the last right-hand-side term, and we obtain

$$ \int_{\mathbb{R}^d} \zeta_R |\nabla u|^2 = - \int_{\mathbb{R}^d} ((u - V) \otimes \nabla \zeta_R) : (\nabla u - (S - c) \mathrm{Id} \mathbb{1}_{\mathbb{R}^d \setminus I}) \\ - \sum_{n: I_n^+ \subset B_{2R}} \int_{\partial I_n} \left( \zeta_R - \int_{I_n} \zeta_R \right) (V_n - V + \Theta_n(x - x_n)) \cdot \sigma(u, S - c) v. $$

Hence, using the properties of $\zeta_R$, Hölder's inequality, and appealing to the trace estimate of Corollary 5 to bound the last right-hand-side term, we deduce for all $s \ge 1$,

$$ \begin{aligned} \|\nabla u\|^2_{L^2(B_R)} \lesssim R^{-1} \|u - V\|_{L^s(B_{2R})} (\|\nabla u\|_{L^{s'}(B_{2R})} + \|S - c\|_{L^{s'}(B_{2R} \setminus I)}) \\ + R^{-1} \left( \sum_{n: I_n^+ \subset B_{2R}} \mu'(\rho_n)^2 (|V_n - V|^2 + |\Theta_n|^2) \right)^{\frac{1}{2}} \|\nabla u\|_{L^2(B_{2R})}, \end{aligned} $$

where we have set for abbreviation,

$$ \mu'(\rho_n) := \begin{cases} \rho_n^{\frac{1}{4d}(d+1)(d+2)-\frac{5}{2}} & : d \le 6, \\ 1 & : d > 6. \end{cases} $$
Choosing $c = \int_{B_{2R} \setminus I} S$ and appealing to a pressure estimate as in (4.7) (with $\alpha = s'$), this becomes, for all $2 \le r \ne \frac{2d}{d-2}$ and $2 \vee \frac{2dr}{d(r-2)+2r} \le s < \infty$, with $s > d$ if $r = 2$,

$$ \|\nabla u\|_{L^2(B_R)}^2 \lesssim R^{-1} \left( |B_R| + \sum_{n: I_n^+ \subset B_{2R}} \mu_r(\rho_n)^{\frac{2s}{s-2}} \right)^{\frac{s-2}{2s}} \|u-V\|_{L^s(B_{2R})} \|\nabla u\|_{L^2(B_{2R})} \\ + R^{-1} \left( \sum_{n: I_n^+ \subset B_{2R}} \mu'(\rho_n)^2 (|V_n-V|^2 + |\Theta_n|^2) \right)^{\frac{1}{2}} \|\nabla u\|_{L^2(B_{2R})}. $$

Noting that

$$ |V_n - V|^2 \lesssim \int_{I_n} |u - V|^2, \quad |\Theta_n|^2 \lesssim \int_{I_n} |\nabla u|^2, $$

Hölder's inequality yields

$$ \sum_{n: I_n^+ \subset B_{2R}} \mu'(\rho_n)^2 (|V_n - V|^2 + |\Theta_n|^2) \\ \lesssim \left( \sum_{n: I_n^+ \subset B_{2R}} \mu'(\rho_n)^{\frac{2s}{s-2}} \right)^{\frac{s-2}{s}} \|u - V\|_{L^s(B_{2R})}^2 + \left( \sup_{n: I_n^+ \subset B_{2R}} \mu'(\rho_n)^2 \right) \|\nabla u\|_{L^2(B_{2R})}^2 \\ \lesssim \left( \sum_{n: I_n^+ \subset B_{2R}} \mu'(\rho_n)^{\frac{2s}{s-2}} \right)^{\frac{s-2}{s}} (\|u - V\|_{L^s(B_{2R})}^2 + \|\nabla u\|_{L^2(B_{2R})}^2). $$

Inserting this into the above, choosing $V := \int_{B_{2R}} u$, and optimizing in $r$, the conclusion follows. ■

**Acknowledgments.** We thank David Gérard-Varet for pointing out a mistake in a previous version of this work and for explaining some related computations in [16]. We also thank Antoine Gloria for motivating discussions on the topic and we thank Roxane Verdikt for drawing the figures.

**Funding.** We acknowledge financial support from the CNRS-Momentum program and from F.R.S.-FNRS.

## References

[1] G. K. Batchelor and J. T. Green, The determination of the bulk stress in suspension of spherical particles to order $c^2$. *J. Fluid Mech.* **56** (1972), no. 3, 401–427 Zbl 0246.76108

[2] G. K. Batchelor and J. T. Green, The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. *J. Fluid Mech.* **56** (1972), no. 2, 375–400 Zbl 0247.76088

[3] P. Bella, B. Fehrman, and F. Otto, A Liouville theorem for elliptic systems with degenerate ergodic coefficients. *Ann. Appl. Probab.* **28** (2018), no. 3, 1379–1422 Zbl 1400.35129 MR 3809467
Choosing $c = \int_{B_{2R} \setminus I} S$ and appealing to a pressure estimate as in (4.7) (with $\alpha = s'$), this becomes, for all $2 \le r \ne \frac{2d}{d-2}$ and $2 \vee \frac{2dr}{d(r-2)+2r} \le s < \infty$, with $s > d$ if $r = 2$,

$$ \|\nabla u\|^2_{L^2(B_R)} \lesssim R^{-1} \left( |B_R| + \sum_{n: I_n^+ \subset B_{2R}} \mu_r(\rho_n)^{\frac{2s}{s-2}} \right)^{\frac{s-2}{2s}} \|u-V\|_{L^s(B_{2R})} \|\nabla u\|_{L^2(B_{2R})} \\ + R^{-1} \left( \sum_{n: I_n^+ \subset B_{2R}} \mu'(\rho_n)^2 (|V_n-V|^2 + |\Theta_n|^2) \right)^{\frac{1}{2}} \|\nabla u\|_{L^2(B_{2R})}. $$

Noting that

$$ |V_n - V|^2 \lesssim \int_{I_n} |u - V|^2, \quad |\Theta_n|^2 \lesssim \int_{I_n} |\nabla u|^2, $$

Hölder's inequality yields

$$ \sum_{n: I_n^+ \subset B_{2R}} \mu'(\rho_n)^2 (|V_n - V|^2 + |\Theta_n|^2) \\ \lesssim \left( \sum_{n: I_n^+ \subset B_{2R}} \mu'(\rho_n)^{\frac{2s}{s-2}} \right)^{\frac{s-2}{s}} \|u-V\|_{L^s(B_{2R})}^2 + \left( \sup_{n: I_n^+ \subset B_{2R}} \mu'(\rho_n)^2 \right) \|\nabla u\|_{L^2(B_{2R})}^2 \\ \lesssim \left( \sum_{n: I_n^+ \subset B_{2R}} \mu'(\rho_n)^{\frac{2s}{s-2}} \right)^{\frac{s-2}{s}} (\|u-V\|_{L^s(B_{2R})}^2 + \|\nabla u\|_{L^2(B_{2R})}^2). $$

Inserting this into the above, choosing $V := \int_{B_{2R}} u$, and optimizing in $r$, the conclusion follows. ■

**Acknowledgments.** We thank David Gérard-Varet for pointing out a mistake in a previous version of this work and for explaining some related computations in [16]. We also thank Antoine Gloria for motivating discussions on the topic and we thank Roxane Verdikt for drawing the figures.

**Funding.** We acknowledge financial support from the CNRS-Momentum program and from F.R.S.-FNRS.

## References

[1] G. K. Batchelor and J. T. Green, The determination of the bulk stress in suspension of spherical particles to order $c^2$. *J. Fluid Mech.* **56** (1972), no. 3, 401–427 Zbl 0246.76108

[2] G. K. Batchelor and J. T. Green, The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. *J. Fluid Mech.* **56** (1972), no. 2, 375–400 Zbl 0247.76088

[3] P. Bella, B. Fehrman, and F. Otto, A Liouville theorem for elliptic systems with degenerate ergodic coefficients. *Ann. Appl. Probab.* **28** (2018), no. 3, 1379–1422 Zbl 1400.35129 MR 3809467
[4] P. Bella and M. Schäffner, Local boundedness and Harnack inequality for solutions of linear nonuniformly elliptic equations. *Comm. Pure Appl. Math.* **74** (2021), no. 3, 453–477
Zbl **1469.35073** MR **4201290**

[5] A. Chiarini and J.-D. Deuschel, Invariance principle for symmetric diffusions in a degenerate and unbounded stationary and ergodic random medium. *Ann. Inst. Henri Poincaré Probab. Stat.* **52** (2016), no. 4, 1535–1563 Zbl **1355.60037** MR **3573286**

[6] M. Duerinckx and A. Gloria, On Einstein's effective viscosity formula. 2020, arXiv:2008.03837

[7] M. Duerinckx and A. Gloria, Corrector equations in fluid mechanics: effective viscosity of colloidal suspensions. *Arch. Ration. Mech. Anal.* **239** (2021), no. 2, 1025–1060
Zbl **1456.76134** MR **4201621**

[8] M. Duerinckx and A. Gloria, Quantitative homogenization theory for random suspensions in steady Stokes flow. 2021, arXiv:2103.06414

[9] M. Duerinckx and A. Gloria, Continuum percolation in stochastic homogenization and the effective viscosity problem. 2022, arXiv:2108.09654

[10] M. Duerinckx and A. Gloria, Sedimentation of random suspensions and the effect of hyper-uniformity. *Ann. PDE* **8** (2022), no. 1, Paper No. 2 MR **4366081**

[11] F. Flegel, M. Heida, and M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps. *Ann. Inst. Henri Poincaré Probab. Stat.* **55** (2019), no. 3, 1226–1257 Zbl **1442.60071** MR **4010934**

[12] G. A. Francfort, Homogenisation of a class of fourth order equations with application to incompressible elasticity. *Proc. Roy. Soc. Edinburgh Sect. A* **120** (1992), no. 1-2, 25–46
Zbl **0770.35006** MR **1149982**

[13] G. P. Galdi, *An introduction to the mathematical theory of the Navier-Stokes equations*. 2nd edn., Springer Monogr. Math., Springer, New York, 2011 Zbl **1245.35002**
MR **2808162**

[14] D. Gérard-Varet, Derivation of the Batchelor-Green formula for random suspensions. *J. Math. Pures Appl.* **(9)** **152** (2021), 211–250 Zbl **1475.35275** MR **4280836**

[15] D. Gérard-Varet and A. Girodroux-Lavigne, Homogenization of stiff inclusions through network approximation. 2021, arXiv:2106.06299

[16] D. Gérard-Varet and M. Hillairet, Computation of the drag force on a sphere close to a wall: the roughness issue. *ESAIM Math. Model. Numer. Anal.* **46** (2012), no. 5, 1201–1224
Zbl **1267.76020** MR **2916378**

[17] D. Gérard-Varet and M. Hillairet, Analysis of the viscosity of dilute suspensions beyond Einstein's formula. *Arch. Ration. Mech. Anal.* **238** (2020), no. 3, 1349–1411 Zbl **1454.76101**
MR **4160802**

[18] D. Gérard-Varet and R. M. Höfer, Mild assumptions for the derivation of Einstein's effective viscosity formula. *Comm. Partial Differential Equations* **46** (2021), no. 4, 611–629
Zbl **1469.76131** MR **4260456**

[19] D. Gérard-Varet and A. Mecherbet, On the correction to Einstein's formula for the effective viscosity. 2020, arXiv:2004.05601

[20] V. V. Jikov, S. M. Kozlov, and O. A. Olešnik, *Homogenization of differential operators and integral functionals*. Springer, Berlin, 1994 Zbl **0838.35001** MR **1329546**

[21] S. Neukamm, M. Schäffner, and A. Schlömerkemper, Stochastic homogenization of non-convex discrete energies with degenerate growth. *SIAM J. Math. Anal.* **49** (2017), no. 3, 1761–1809 Zbl **1362.74028** MR **3650427**
[4] P. Bella and M. Schäffner, Local boundedness and Harnack inequality for solutions of linear nonuniformly elliptic equations. *Comm. Pure Appl. Math.* **74** (2021), no. 3, 453–477
Zbl **1469.35073** MR **4201290**

[5] A. Chiarini and J.-D. Deuschel, Invariance principle for symmetric diffusions in a degenerate and unbounded stationary and ergodic random medium. *Ann. Inst. Henri Poincaré Probab. Stat.* **52** (2016), no. 4, 1535–1563 Zbl **1355.60037** MR **3573286**

[6] M. Duerinckx and A. Gloria, On Einstein's effective viscosity formula. 2020, arXiv:2008.03837

[7] M. Duerinckx and A. Gloria, Corrector equations in fluid mechanics: effective viscosity of colloidal suspensions. *Arch. Ration. Mech. Anal.* **239** (2021), no. 2, 1025–1060
Zbl **1456.76134** MR **4201621**

[8] M. Duerinckx and A. Gloria, Quantitative homogenization theory for random suspensions in steady Stokes flow. 2021, arXiv:2103.06414

[9] M. Duerinckx and A. Gloria, Continuum percolation in stochastic homogenization and the effective viscosity problem. 2022, arXiv:2108.09654

[10] M. Duerinckx and A. Gloria, Sedimentation of random suspensions and the effect of hyper-uniformity. *Ann. PDE* **8** (2022), no. 1, Paper No. 2 MR **4366081**

[11] F. Flegel, M. Heida, and M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps. *Ann. Inst. Henri Poincaré Probab. Stat.* **55** (2019), no. 3, 1226–1257 Zbl **1442.60071** MR **4010934**

[12] G. A. Francfort, Homogenisation of a class of fourth order equations with application to incompressible elasticity. *Proc. Roy. Soc. Edinburgh Sect. A* **120** (1992), no. 1-2, 25–46
Zbl **0770.35006** MR **1149982**

[13] G. P. Galdi, *An introduction to the mathematical theory of the Navier-Stokes equations*. 2nd edn., Springer Monogr. Math., Springer, New York, 2011 Zbl **1245.35002**
MR **2808162**

[14] D. Gérard-Varet, Derivation of the Batchelor-Green formula for random suspensions. *J. Math. Pures Appl.* **(9)** **152** (2021), 211–250 Zbl **1475.35275** MR **4280836**

[15] D. Gérard-Varet and A. Girodroux-Lavigne, Homogenization of stiff inclusions through network approximation. 2021, arXiv:2106.06299

[16] D. Gérard-Varet and M. Hillairet, Computation of the drag force on a sphere close to a wall: the roughness issue. *ESAIM Math. Model. Numer. Anal.* **46** (2012), no. 5, 1201–1224
Zbl **1267.76020** MR **2916378**

[17] D. Gérard-Varet and M. Hillairet, Analysis of the viscosity of dilute suspensions beyond Einstein's formula. *Arch. Ration. Mech. Anal.* **238** (2020), no. 3, 1349–1411 Zbl **1454.76101**
MR **4160802**

[18] D. Gérard-Varet and R. M. Höfer, Mild assumptions for the derivation of Einstein's effective viscosity formula. *Comm. Partial Differential Equations* **46** (2021), no. 4, 611–629
Zbl **1469.76131** MR **4260456**

[19] D. Gérard-Varet and A. Mecherbet, On the correction to Einstein's formula for the effective viscosity. 2020, arXiv:2004.05601

[20] V. V. Jikov, S. M. Kozlov, and O. A. Olešnik, *Homogenization of differential operators and integral functionals*. Springer, Berlin, 1994 Zbl **0838.35001** MR **1329546**

[21] S. Neukamm, M. Schäffner, and A. Schlömerkemper, Stochastic homogenization of non-convex discrete energies with degenerate growth. *SIAM J. Math. Anal.* **49** (2017), no. 3, 1761–1809 Zbl **1362.74028** MR **3650427**
[22] B. Niethammer and R. Schubert, A local version of Einstein's formula for the effective viscosity of suspensions. *SIAM J. Math. Anal.* **52** (2020), no. 3, 2561–2591 Zbl **1473.35454** MR **4102716**

[23] B. Opic and A. Kufner, *Hardy-type inequalities*. Pitman Res. Not. Math. Ser. 219, Longman Scientific & Technical, Harlow, 1990 Zbl **0698.26007** MR **1069756**

[24] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory. *Izv. Akad. Nauk SSSR Ser. Mat.* **50** (1986), no. 4, 675–710, 877 Zbl **0599.49031** MR **864171**

[25] V. V. Zhikov, Some problems of extension of functions arising in connection with the homogenization theory. *Differ. Equ.* **26** (1990), no. 1, 34–44 Zbl **0701.35034** MR **1050358**

Received 22 September 2020; revised 1 September 2021; accepted 10 September 2021.

Mitia Duerinckx

Département de Mathématique, Université Libre de Bruxelles, Boulevard du Triomphe, 1050 Bruxelles, Belgium; and Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay, 307 rue Michel Magat, 91400 Orsay Cedex, France; mitia.duerinckx@ulb.be
[22] B. Niethammer and R. Schubert, A local version of Einstein's formula for the effective viscosity of suspensions. *SIAM J. Math. Anal.* **52** (2020), no. 3, 2561–2591 Zbl **1473.35454** MR **4102716**

[23] B. Opic and A. Kufner, *Hardy-type inequalities*. Pitman Res. Not. Math. Ser. 219, Longman Scientific & Technical, Harlow, 1990 Zbl **0698.26007** MR **1069756**

[24] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory. *Izv. Akad. Nauk SSSR Ser. Mat.* **50** (1986), no. 4, 675–710, 877 Zbl **0599.49031** MR **864171**

[25] V. V. Zhikov, Some problems of extension of functions arising in connection with the homogenization theory. *Differ. Equ.* **26** (1990), no. 1, 34–44 Zbl **0701.35034** MR **1050358**

Received 22 September 2020; revised 1 September 2021; accepted 10 September 2021.

Mitia Duerinckx

Département de Mathématique, Université Libre de Bruxelles, Boulevard du Triomphe, 1050 Bruxelles, Belgium; and Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay, 307 rue Michel Magat, 91400 Orsay Cedex, France; mitia.duerinckx@ulb.be