Ingredients from Regularity Theory
Our estimates crucially rely on three basic regularity estimates for elliptic PDEs, the first two being the Caccioppoli inequality and the hole-filling estimate for nonlinear elliptic equations (and systems) with monotone nonlinearity and the last one being a weighted Meyers estimate for linear elliptic equations (and systems). We first state the Caccioppoli inequality and the hole-filling estimate in the nonlinear setting. The (standard) proofs are provided in “Appendix A”.
Lemma 43
(Caccioppoli inequality and hole-filling estimate for monotone systems). Let \(A(x,\xi )\) be a monotone operator subject to the assumptions (A1)–(A2). Let \(0<T\leqq \infty \) and let u be a solution to the system of PDEs
$$\begin{aligned} -\nabla \cdot (A(x,\nabla u)) + \frac{1}{T} u=\nabla \cdot g + \frac{1}{T}f \end{aligned}$$
for some \(f\in L^2({\mathbb {R}^d};{\mathbb {R}^{m}})\) and some \(g\in L^2({\mathbb {R}^d};{\mathbb {R}^{m\times d}})\). Then there exist constants \(C>0\) and \(\delta >0\) depending only on d, m, \(\lambda \), and \(\Lambda \) with the following property: For any \(R,r>0\) with \(R\geqq r\) we have the Caccioppoli inequality
$$\begin{aligned}&\fint _{B_{R/2}(x_0)} |\nabla u|^2 + \frac{1}{T} |u|^2 \,\mathrm{{d}}x\nonumber \\&\quad \leqq \frac{C}{R^2} \fint _{B_R(x_0)} |u-b|^2 \,\mathrm{{d}}x + \frac{C}{T} |b|^2 + C \fint _{B_R(x_0)} |g|^2 + \frac{1}{T}|f|^2 \,\mathrm{{d}}x \end{aligned}$$
(54)
for any \(b\in {\mathbb {R}^{m}}\) and the hole-filling estimate
$$\begin{aligned}&\int _{B_r(x_0)} |\nabla u|^2 + \frac{1}{T} |u|^2 \,\mathrm{{d}}x \nonumber \\&\quad \leqq C \bigg (\frac{r}{R}\bigg )^{\delta } \bigg (\int _{B_R(x_0)} |\nabla u|^2 +\frac{1}{T} |u|^2 \,\mathrm{{d}}x\bigg ) \nonumber \\&\qquad + C \int _{B_R(x_0)} \bigg (\frac{r}{r+|x-x_0|}\bigg )^\delta \Big (|g|^2+\frac{1}{T}|f|^2\Big ) \,\mathrm{{d}}x. \end{aligned}$$
(55)
We next state a weighted Meyers-type estimate for linear uniformly elliptic equations and systems. Its proof (which is provided in “Appendix C”) relies on the usual Meyers estimate, along with a duality argument and a hole-filling estimate for the adjoint operator. The details are provided in [13] for the case \(T=\infty \); however, the proof applies verbatim to the case \(T>0\), as the only ingredients are the Meyers estimate for the PDE and the hole-filling estimate for the adjoint PDE.
Lemma 44
(Weighted Meyers estimate for linear elliptic systems). Let \(a:{\mathbb {R}^d}\rightarrow {\mathbb {R}^{m\times d}}\otimes {\mathbb {R}^{m\times d}}\) be a uniformly elliptic and bounded coefficient field with ellipticity and boundedness constants \(\lambda \) and \(\Lambda \). Let \(r>0\) be arbitrary. Let \(v \in H^1({\mathbb {R}^d};{\mathbb {R}^{m}})\) and \(g \in L^2({\mathbb {R}^d};{\mathbb {R}^{m\times d}})\), \(f\in L^2({\mathbb {R}^d};{\mathbb {R}^{m}})\) be functions related through
$$\begin{aligned} -\nabla \cdot (a \nabla v) + \frac{1}{T} v = \nabla \cdot g + \frac{1}{\sqrt{T}} f. \end{aligned}$$
There exists a Meyers exponent \(\bar{p}>2\) and a constant \(c>0\), which both only depend on d, m, \(\lambda \), and \(\Lambda \), such that for all \(2\leqq p<\bar{p}\) and all \(0<\alpha _0<c\) we have
$$\begin{aligned}&\left( \int _{\mathbb {R}^d}\Big (|\nabla v|^{p}+\Big |\frac{1}{\sqrt{T}}v\Big |^p \Big ) \bigg (1+\frac{|x|}{r}\bigg )^{\alpha _0} \,\mathrm{{d}}x \right) ^{\frac{1}{p}} \nonumber \\&\quad \leqq C \left( \int _{\mathbb {R}^d}(|g|^{p} + |f|^p) \bigg (1+\frac{|x|}{r}\bigg )^{\alpha _0} \,\mathrm{{d}}x\right) ^{\frac{1}{p}}, \end{aligned}$$
(56)
where the constant C depends only on d, m, \(\lambda \), \(\Lambda \), p, and \(\alpha _0\).
The localization ansatz for the correctors relies crucially on the following elementary deterministic energy estimate with exponential localization. As the proof is short and elementary, we directly provide it here.
Lemma 45
(Exponential localization). Suppose that \(A:{\mathbb {R}^d}\times {\mathbb {R}^{m\times d}}\rightarrow {\mathbb {R}^{m\times d}}\) is a monotone operator satisfying (A1) and (A2). Let \(T>0\) and \(L\geqq \sqrt{T}\). Consider \(u\in H^1_{{\mathrm {loc}}}({\mathbb {R}^d};{\mathbb {R}^{m}})\) and \(f\in L^2_{{\mathrm {loc}}}({\mathbb {R}^d};{\mathbb {R}^{m}})\), \(F\in L^2_{{\mathrm {loc}}}({\mathbb {R}^d};{\mathbb {R}^{m\times d}})\) related by
$$\begin{aligned} -\nabla \cdot (A(x,\nabla u))+\frac{1}{T} u=\nabla \cdot F+\frac{1}{T} f \end{aligned}$$
in a distributional sense in \({\mathbb {R}^d}\). Suppose that u, f, and F have at most polynomial growth in the sense that
$$\begin{aligned} \exists k\in \mathbb {N}\,:\qquad \limsup \limits _{R\rightarrow \infty }R^{-k}\left( \fint _{B_R}(|u|+|\nabla u|+|f|+|F|)^2\right) ^\frac{1}{2}=0. \end{aligned}$$
Then for \(0<\gamma \leqq c(d,m,\lambda ,\Lambda )\) we have
$$\begin{aligned}&\int _{\mathbb {R}^d}\Big (|\nabla u|^2+\frac{1}{T}|u|^2\Big )\exp (-\gamma |x|/L)\,\mathrm{{d}}x \\&\quad \leqq C(d,m,\lambda ,\Lambda ) \int _{\mathbb {R}^d}\Big (|F|^2+\frac{1}{T}|f|^2\Big )\exp (-\gamma |x|/L)\,\mathrm{{d}}x. \end{aligned}$$
Proof
Set \(\eta (x){:}{=}\exp (-\gamma |x|/L)\). We test the equation with \(u\eta \) (which can be justified by approximation thanks to the polynomial growth assumption). By an integration by parts, the ellipticity and Lipschitz continuity of A, and using \(|\nabla \eta |\leqq \tfrac{\gamma }{L} \eta \leqq \tfrac{\gamma }{\sqrt{T}}\eta \), we get
$$\begin{aligned}&\lambda \int _{\mathbb {R}^d}|\nabla u|^2\eta +\frac{1}{T} |u|^2\eta \,\text {d}x\\&\quad \leqq \gamma \int _{\mathbb {R}^d}\Lambda |\nabla u|\frac{1}{\sqrt{T}}|u|\eta \,\text {d}x+\int _{\mathbb {R}^d}|F|\Big (|\nabla u|+\gamma \frac{1}{\sqrt{T}}|u|\Big )\eta \,\text {d}x+\frac{1}{T}\int _{\mathbb {R}^d}|f||u|\eta \,\text {d}x \end{aligned}$$
The claim now follows for \(\gamma \leqq c\) by absorbing the terms with u and \(\nabla u\) on the right-hand side into the left-hand side with help of Young’s inequality. \(\quad \square \)
Remark 46
We frequently apply the exponential localization in the following form: Suppose that \(A:{\mathbb {R}^d}\times {\mathbb {R}^{m\times d}}\rightarrow {\mathbb {R}^{m\times d}}\) is a monotone operator satisfying (A1) and (A2). Let \(T>0\) and \(L\geqq \sqrt{T}\). Consider \(u_1,u_2\in H^1_{{\mathrm {loc}}}({\mathbb {R}^d};{\mathbb {R}^{m}})\) and \(f\in L^2_{{\mathrm {loc}}}({\mathbb {R}^d};{\mathbb {R}^{m}})\), \(F\in L^2_{{\mathrm {loc}}}({\mathbb {R}^d};{\mathbb {R}^{m\times d}})\), all with at most polynomial growth and related by
$$\begin{aligned} -\nabla \cdot (A(x,\nabla u_1)-A(x,\nabla u_2))+\frac{1}{T}(u_1-u_2)=\nabla \cdot F+\frac{1}{T}f \end{aligned}$$
in a distributional sense in \({\mathbb {R}}^d\). Then, for \(0<\gamma \leqq c(d,m,\lambda ,\Lambda )\), we have
$$\begin{aligned}&\int _{\mathbb {R}^d}\Big (|\nabla u_1-\nabla u_2|^2+\frac{1}{T}|u_1-u_2|^2\Big )\exp (-\gamma |x|/L)\,\text {d}x\\&\quad \leqq C(d,m,\lambda ,\Lambda ) \int _{\mathbb {R}^d}\Big (|F|^2+\frac{1}{T}|f|^2\Big )\exp (-\gamma |x|/L)\,\text {d}x. \end{aligned}$$
Indeed, with \(a(x){:}{=}\int _0^1 \partial _\xi A(x,\xi +(1-s)\nabla u_1(x)+s\nabla u_2(x))\,ds\) and \(\delta u{:}{=}u_1-u_2\), we have \(-\nabla \cdot (a(x)\nabla \delta u) + \frac{1}{T}\delta u=\nabla \cdot F+\frac{1}{T} f\). Since A is a monotone operator satisfying (A1) and (A2), the derivative \(\partial _\xi A(x,\xi )\) is a uniformly elliptic matrix field; hence, a(x) is a uniformly elliptic coefficient field and the claimed estimate follows from the linear version of Lemma 45.
The Convergence Rate of the Solutions
We first provide the proof of the error estimate for \(||u_\varepsilon -u_{\mathsf {hom}}||_{L^2}\). It is based on a two-scale expansion with a piecewise constant approximation for the slope of the limiting solution \(u_{\mathsf {hom}}\), whose approximation properties are stated in Lemma 34 and Proposition 36.
Proof of Lemma 34
Step 1. Construction of the partition of unity. The construction of the partition of unity in the case \({\mathcal {O}}={\mathbb {R}}^d\) is elementary, see Remark 35. We thus only discuss the case of a bounded domain. In the following \(\bar{C}>1\) denotes a constant that may vary from line to line, but that can be chosen only depending on \({\mathcal {O}}\) and the dimension d. We fix the length scale \(\delta \) with \(0<\delta \leqq \frac{1}{\bar{C}}\). Thanks to the assumptions on \({\mathcal {O}}\) (\(C^1\)-boundary or Lipschitz boundary & convexity), for \(\bar{C}>0\) large enough, we have that
$$\begin{aligned} B_{6\delta }(x)\cap {\mathcal {O}}\text { is connected for all }x\in {\mathcal {O}}, \end{aligned}$$
(57)
and we may cover \({\mathcal {O}}\) (enlarged by a layer of thickness \(\delta /2\)) by balls of radius \(B_{\delta }\), whose volume interesected with \({\mathcal {O}}\) is comparable with \(\delta ^d\); more precisely,
$$\begin{aligned} {\mathcal {O}}+B_{\delta /2}(0)\subset \bigcup _{x\in K''}B_{\delta }(k),\qquad K''{:}{=}\{\,x\in {\mathcal {O}}\,:\,B_{\delta /\bar{C}}(x)\subseteq {\mathcal {O}}\,\}. \end{aligned}$$
Since the set on the left is compact, we can find a finite subset \(K'\subset K''\) such that the above inclusion holds with \(K''\) replaced by \(K'\). Moreover, by the Vitali covering lemma we can find a finite subset \(K\subseteq K'\) such that the balls \(\{B_{\delta /3}(k)\}_{k\in K}\) are disjoint and \({\mathcal {O}}+B_{\delta /2}\subseteq \bigcup _{k\in K}B_\delta (k)\). With this covering at hand, we may iteratively define (measurable) functions \(\chi _k:{\mathbb {R}}^d\rightarrow \{0,1\}\) such that
$$\begin{aligned} 1_{B_{\delta /3}(k)}\leqq \chi _k\leqq 1_{B_{\delta }(k)}\qquad \text {and}\qquad \sum _{k\in K}\chi _k=1\text { on }{\mathcal {O}}+B_{\delta /2}(0). \end{aligned}$$
(58)
Let \(\eta _{\delta /2}\) denote the standard mollifier with support in \(B_{\delta /2}(0)\). For \(k\in K\) set \(\eta _k{:}{=}\chi _k*\eta _{\delta /2}\). By construction \(\{\eta _k\}_{k\in K}\) is a smooth partition of unity for \({\mathcal {O}}\) satisfying (45a) and (45b) (the latter is a consequence of (58) and the fact that the balls \(B_{\delta /\bar{C}}(k)\), \(k\in K\), are disjoint and contained in \({\mathcal {O}}\)) as well as (45c).
Step 2. Estimates. The arguments for (47a) and (47b) are standard. We prove (47c). We first note that for all \(k,{\tilde{k}}\in K\) with \(|k-{\tilde{k}}|\leqq 4\delta \) we have \({\text {supp}}\eta _k\cup {\text {supp}}\eta _{{\tilde{k}}}\subset B_{6\delta }({\tilde{k}})\). Moreover, by (57) the set \(B_{6\delta }({\tilde{k}})\cap {\mathcal {O}}\) is connected. Denote by \(v\in H^1({\mathcal {O}}\cap B_{6\delta }({\tilde{k}}))\) the unique mean-free weak solution to the Neumann problem
$$\begin{aligned} -\triangle v= & {} \frac{\eta _k}{\int _{\mathcal {O}}\eta _k\,\text {d}x}-\frac{\eta _{{\tilde{k}}}}{\int _{\mathcal {O}}\eta _{{\tilde{k}}}\,\text {d}x}\qquad \text {in }B_{6\delta }({\tilde{k}})\cap {\mathcal {O}},\\ \partial _\nu v= & {} 0\qquad \text {on }\partial (B_{6\delta }({\tilde{k}})\cap {\mathcal {O}}). \end{aligned}$$
The standard a priori estimate, Poincaré’s inequality and (45b) yield
$$\begin{aligned} \int _{B_{6\delta }({\tilde{k}})\cap {\mathcal {O}}}|\nabla v|^2 \,\text {d}x\leqq C\delta ^{2-d}, \end{aligned}$$
for a constant C only depending on \(\bar{C}\). This implies (47c), since
$$\begin{aligned} \xi _k-\xi _{{\tilde{k}}}=\int _{{\mathcal {O}}}\bigg (\frac{\eta _k}{\int _{\mathcal {O}}\eta _k}-\frac{\eta _{{\tilde{k}}}}{\int _{\mathcal {O}}\eta _{{\tilde{k}}}}\bigg )g\,\text {d}x=\int _{{\mathcal {O}}\cap B_{6\delta }({\tilde{k}})}\nabla g\cdot \nabla v\,\text {d}x, \end{aligned}$$
as can be seen by an integration by parts. \(\quad \square \)
Proof of Proposition 36
To shorten the notation, we implicitly assume that \(k,\tilde{k}\in K\) and we shall use the shorthand notation
$$\begin{aligned} k\sim \tilde{k}\qquad :\Leftrightarrow \qquad k,\tilde{k}\text { are nearby sites, that is }|k-\tilde{k}|\leqq 4\delta . \end{aligned}$$
Note that \({\text {supp}}\eta _k\cap {\text {supp}}\eta _{\tilde{k}}\ne \emptyset \) implies \(k\sim \tilde{k}\). We shall also use the notation \(A\lesssim B\) if \(A\leqq CB\) for a constant C that only depends on \(d,m,\lambda ,\Lambda \), \(\bar{C}\) and \({\mathcal {O}}\). First, we note that in the sense of distribution in \({\mathcal {O}}\), we have
$$\begin{aligned}&\nabla \cdot (A_\varepsilon (x,\nabla \hat{u}_\varepsilon )) \\&\quad =\nabla \cdot \Big (A_\varepsilon \Big (x,\nabla \bar{u}+\sum _{k} \eta _k \nabla \phi _{k}\Big )\Big ) \\&\qquad +\nabla \cdot \Big (A_\varepsilon \Big (x,\nabla \bar{u}+ \sum _{k} \eta _k \nabla \phi _{k} + \sum _{k\in K} \phi _{k} \nabla \eta _k \Big ) \\&\qquad \qquad \qquad -A_\varepsilon \Big (x,\nabla \bar{u}+\sum _{k} \eta _k \nabla \phi _{k}\Big )\Big ). \end{aligned}$$
Adding and subtracting intermediate terms and using the fact that the \(\eta _k\) form a partition of unity (that is \(\sum _{k}\eta _k=1\)), we get
$$\begin{aligned}&\nabla \cdot (A_\varepsilon (x,\nabla \hat{u}_\varepsilon )) \\&\quad = \nabla \cdot \big (A_{\mathsf {hom}}\big (\nabla \bar{u}\big )\big ) \\&\qquad +\nabla \cdot \bigg (\sum _{k} \eta _k \big (A_{\mathsf {hom}}(\xi _k)-A_{\mathsf {hom}}(\nabla \bar{u})\big )\bigg ) \\&\qquad +\nabla \cdot \bigg (\sum _{k} \eta _k \big (A_\varepsilon \big (x,\xi _k+\nabla \phi _{k}\big )-A_{\mathsf {hom}}\big (\xi _k\big )\big )\bigg ) \\&\qquad +\nabla \cdot \bigg (\sum _{k} \eta _k \big (A_\varepsilon \big (x,\nabla \bar{u}+\nabla \phi _{k}\big )-A_\varepsilon \big (x,\xi _k+\nabla \phi _{k}\big )\big )\bigg ) \\&\qquad +\nabla \cdot \bigg (A_\varepsilon \Big (x,\nabla \bar{u}+\sum _{k} \eta _k \nabla \phi _{k}\Big )-\sum _{k} \eta _k A_\varepsilon \big (x,\nabla \bar{u}+\nabla \phi _{k}\big )\bigg ) \\&\qquad +\nabla \cdot \bigg (A_\varepsilon \Big (x,\nabla \bar{u}+ \sum _{k} \eta _k \nabla \phi _{k} + \sum _{k} \phi _{k} \nabla \eta _k \bigg ) \\&\qquad \qquad \qquad -A_\varepsilon \Big (x,\nabla \bar{u}+\sum _{k} \eta _k \nabla \phi _{k}\Big )\Big ). \end{aligned}$$
Using the equation for the flux corrector \(\nabla \cdot \sigma _{k} = A_\varepsilon \big (x,\xi _k+\nabla \phi _{k}\big )-A_{\mathsf {hom}}\big (\xi _k\big )\) (see (9)) and the skew-symmetry of \(\sigma _{k}\) (which implies \(\nabla \cdot (\eta \nabla \cdot \sigma _{k})=-\nabla \cdot (\sigma _{k}\nabla \eta )\)), we obtain
$$\begin{aligned} \nabla \cdot (A_\varepsilon (x,\nabla \hat{u}_\varepsilon ))\,=\, \nabla \cdot \big (A_{\mathsf {hom}}\big (\nabla \bar{u}\big )\big ) +\nabla \cdot R, \end{aligned}$$
with a residuum \(R{:}{=}I+II+III\) where
$$\begin{aligned} I{:}{=}&\sum _{k} \eta _k \big (A_{\mathsf {hom}}(\xi _k)-A_{\mathsf {hom}}(\nabla \bar{u})\big ) \\&+\sum _{k} \eta _k \big (A_\varepsilon \big (x,\nabla \bar{u}+\nabla \phi _{k}\big )-A_\varepsilon \big (x,\xi _k+\nabla \phi _{k}\big )\big ) \end{aligned}$$
and
$$\begin{aligned} II{:}{=}&-\sum _{k} \sigma _{k} \nabla \eta _k \\&+A_\varepsilon \Big (x,\nabla \bar{u}+ \sum _{k} \eta _k \nabla \phi _{k} + \sum _{k} \phi _{k} \nabla \eta _k \Big ) -A_\varepsilon \Big (x,\nabla \bar{u}+\sum _{k} \eta _k \nabla \phi _{k}\Big ) \end{aligned}$$
as well as
$$\begin{aligned} III{:}{=}A_\varepsilon \Big (x,\nabla \bar{u}+ \sum _{k} \eta _k \nabla \phi _{k}\Big )-\sum _{k} \eta _k A_\varepsilon \big (x,\nabla \bar{u}+\nabla \phi _{k}\big ). \end{aligned}$$
It is our goal to show that R can be estimated for all \(\ell \) as
$$\begin{aligned}&\int _{{\mathcal {O}}}\eta _\ell |R|^2 \,\text {d}x\lesssim \int _{{\mathcal {O}}}\eta _\ell (|I|^2+|II|^2+|III|^2) \,\text {d}x \nonumber \\&\quad \lesssim \delta ^2 \int _{B_{2\delta }(\ell )\cap {\mathcal {O}}} |\nabla ^2 \bar{u}|^2\,\text {d}x +\frac{1}{\delta ^2}\sum _{k\,:\, k\sim \ell }\int _{B_{6\delta }(\ell )}|\phi _{\ell }-\phi _{k}|^2 + |\sigma _{\ell }-\sigma _{k}|^2 \,\text {d}x. \end{aligned}$$
(59)
For the argument we first argue that \(\int \eta _\ell |I|^2\) may be bounded by the first term on the right-hand side of (59). Indeed, by Lipschitz continuity of \(A_\varepsilon \) and \(A_{{\mathsf {hom}}}\) in \(\xi \) (see (A2) and Theorem 11a, respectively), we have the pointwise bound \(|I|\lesssim \sum _{k}\eta _k|\nabla \bar{u}-\xi _k|\), and thus (with help of (47b)),
$$\begin{aligned} \int _{{\mathcal {O}}}\eta _\ell |I|^2\,\text {d}x\lesssim & {} \sum _{k\,:\,k\sim \ell }\int _{B_{2\delta }(k)\cap {\mathcal {O}}}|\nabla \bar{u}-\xi _k|^2\,\text {d}x\\\lesssim & {} \delta ^2\sum _{k\,:\,k\sim \ell }\int _{B_{2\delta }(k)\cap {\mathcal {O}}}|\nabla ^2\bar{u}|^2\,\text {d}x\lesssim \delta ^2\int _{B_{6\delta }(\ell )\cap {\mathcal {O}}}|\nabla ^2\bar{u}|^2\,\text {d}x. \end{aligned}$$
Next, we show that \(\int \eta _\ell |II|^2\,\text {d}x\) may be bounded by the second term on the right-hand side of (59). By the Lipschitz-continuity of \(A_\varepsilon \) in \(\xi \) and the fact that \(\sum _k\nabla \eta _k=0\) (as the \(\eta _k\) form a partition of unity), we have \(|II|\lesssim \big |\,\sum _{k}(\sigma _{\ell }-\sigma _{k})\nabla \eta _k\,\big |+\big |\sum _{k}(\phi _{\ell }-\phi _{k})\nabla \eta _k\big |\). Since \(|\nabla \eta _k|\leqq \bar{C}\delta ^{-1}\), we deduce that
$$\begin{aligned} \int _{{\mathcal {O}}}\eta _\ell |II|^2\,\text {d}x\lesssim & {} \delta ^{-2}\sum _{k:k\sim \ell } \int _{B_{2\delta }(\ell )}|\phi _{\ell }-\phi _{k}|^2+|\sigma _{\ell }-\sigma _{k}|^2\,\text {d}x. \end{aligned}$$
Finally, we estimate the third term III. Since \(\sum _{\tilde{k}}\eta _{\tilde{k}}=1\) we have
$$\begin{aligned} III=\sum _{\tilde{k}}\eta _{\tilde{k}}\Big (A_\varepsilon \big (x,\nabla \bar{u}+ \sum _{k} \eta _k \nabla \phi _{k}\big )-A_\varepsilon \big (x,\nabla \bar{u}+\nabla \phi _{{\tilde{k}}}\big )\Big ). \end{aligned}$$
Hence, the Lipschitz-continuity of \(A_\varepsilon \) in \(\xi \) yields \( |III|\lesssim \sum _{\tilde{k},k}\eta _{\tilde{k}}\eta _k |\nabla \phi _{k}-\nabla \phi _{{\tilde{k}}}|\). By the support property of \(\eta _k\) and \(\eta _{\tilde{k}}\) we get
$$\begin{aligned} \int _{{\mathcal {O}}}\eta _\ell |III|^2\,\text {d}x\leqq C\sum _{k\,:\,k\sim \ell }\int _{B_{2\delta }(\ell )}|\nabla \phi _{\ell }-\nabla \phi _{k}|^2\,\text {d}x. \end{aligned}$$
(60)
Thus, we need to bound the difference of the two corrector gradients \(\nabla \phi _{\ell }\) and \(\nabla \phi _{k}\) for nearby grid points k and \(\ell \). To this aim, note that the corrector equation (4) implies
$$\begin{aligned} -\nabla \cdot (A_\varepsilon (x,\xi _\ell +\nabla \phi _{\ell })-A_\varepsilon (x,\xi _{k}+\nabla \phi _{{k}}))&=0. \end{aligned}$$
An energy estimate based on assumptions (A1) and (A2) thus yields
$$\begin{aligned} \fint _{B_{2\delta }(\ell )} |\nabla \phi _{\ell }-\nabla \phi _{k}|^2 \,\text {d}x \lesssim |\xi _\ell -\xi _{k}|^2 + \frac{1}{\delta ^2} \fint _{B_{6\delta }(\ell )} |\phi _{\ell }-\phi _{k}|^2 \,\text {d}x. \end{aligned}$$
In conjunction with (47b) and (60), we see that \(\int \eta _\ell |III|^2 \,\text {d}x\) may be bounded by the right-hand side of (59). \(\quad \square \)
Proof of Theorem 2, Theorem 4, and Theorem 7
We use the notation \(A\lesssim B\) if \(A\leqq CB\) for a constant C that only depends on \(d,m,\lambda ,\Lambda \), \(\bar{C}\) and \({\mathcal {O}}\).
Step 1: Proof of Theorem 7– the case of a bounded domain \({\mathcal {O}}\). For the proof we appeal to the two-scale expansion introduced in Proposition 36. Let \(\delta \) denote a discretization scale that satisfies \(\varepsilon \leqq \delta \leqq \frac{1}{C}\) and that we fix later. According to Proposition 36 we denote by \(\{\eta _k\}_{k\in K}\) the partition of unity of Lemma 34 and by \(\hat{u}_\varepsilon \) the two-scale expansion defined in (48) associated with \(\bar{u}{:}{=}u_{{\mathsf {hom}}}\). Moreover, we define the correctors \(\phi _k,\sigma _k\) as in Proposition 36. Note that by the proposition and the identity \(\nabla \cdot (A(\omega _\varepsilon ,\nabla u_\varepsilon ))=\nabla \cdot (A_{{\mathsf {hom}}}(\nabla \bar{u}))\) in \({\mathcal {O}}\), we have
$$\begin{aligned} \nabla \cdot (A(\omega _\varepsilon ,\nabla u_\varepsilon )-A(\omega _\varepsilon ,\nabla \hat{u}_\varepsilon )) = \nabla \cdot R\qquad \text {in a distributional sense in } {\mathcal {O}}, \end{aligned}$$
(61)
with a residuum \(R\in L^2({\mathcal {O}};{\mathbb {R}^{m\times d}})\).
Step 1.1: We claim that
$$\begin{aligned} \int _{\mathcal {O}}|R|^2 \,\text {d}x \,\leqq \,\mathcal {C}^2\Vert \nabla u_{{\mathsf {hom}}}\Vert _{H^1({\mathcal {O}})}^2\,\Big (\delta ^2+\Big (\frac{\varepsilon }{\delta }\Big )^2\mu _d\Big (\frac{\delta }{\varepsilon }\Big )\Big ). \end{aligned}$$
(62)
Here and below we denote by \(\mathcal {C}\) a random constant that might change from line to line, but that satisfies the stretched exponential moment bound
$$\begin{aligned} \mathbb {E}\Big [\exp \Big (\frac{\mathcal {C}^{\bar{\nu }}}{C}\Big )\Big ]\leqq 2, \end{aligned}$$
(63)
with \(\bar{\nu }\) only depending on \(d,m,\lambda ,\Lambda ,\rho \) and with \(C>0\) depending only on \(d,m,\lambda ,\Lambda ,\) \(\rho ,\) and \({\mathcal {O}}\). Below, we shall tacitly use the calculus rules for random constants with stretched exponential moments, see Lemma 47.
The starting point for the argument is the local residuum estimate (49), which we post-process by appealing to the triangle inequality:
$$\begin{aligned} \int _{\mathcal {O}}|R|^2 \,=\,&\sum _{\ell \in K}\int _{{\mathcal {O}}}\eta _\ell |R|^2\nonumber \\ \lesssim \,&\delta ^2 \int _{\mathcal {O}}|\nabla ^2u_{{\mathsf {hom}}}|^2\,\text {d}x\,+\delta ^{-2}\sum _{\ell \in K}\sum _{k\in K\atop |k-\ell |\leqq 4\delta } \int _{B_{6\delta }(\ell )}(|\phi _{k}|^2 + |\sigma _{k}|^2) \,\text {d}x. \end{aligned}$$
(64)
As we shall show in Step 3, from Corollary 21 we may obtain the corrector estimate
$$\begin{aligned} \sum _{\ell \in K}\sum _{k\in K\atop |k-\ell |\leqq 4\delta }\int _{B_{6\delta }(\ell )}|\phi _{k}|^2 + |\sigma _{k}|^2 \,\text {d}x\leqq & {} \mathcal {C}^2\,\int _{{\mathcal {O}}}|\nabla u_{{\mathsf {hom}}}|^2\,\varepsilon ^2\mu _d\Big (\frac{\delta }{\varepsilon }\Big ), \end{aligned}$$
(65)
where
$$\begin{aligned} \mu _d(s){:}{=} {\left\{ \begin{array}{ll} s&{}\text {for }d=1,\\ |\log (s)|&{}\text {for }d=2,\\ 1&{}\text {for }d\geqq 3. \end{array}\right. } \end{aligned}$$
Now, (62) follows by combining the previous two estimates.
Step 1.2: To derive a bound on the difference \(\nabla u_\varepsilon -\nabla \hat{u}_\varepsilon \) from the above estimate on the residuum, we would like to test the equation with \(u_\varepsilon -\hat{u}_\varepsilon \). Thus, we need to modify \(\hat{u}_\varepsilon \) close to the boundary to obtain an admissible test function. Let \(\tau \geqq \varepsilon \). Set \((\partial {\mathcal {O}})_\tau {:}{=}\{x\in {\mathcal {O}}\,:\,{\text {dist}}(x,\partial {\mathcal {O}})<\tau \}\). We denote by \(\psi \) a cut-off function with \(\psi \equiv 1\) in \(\{x\in {\mathcal {O}}:{\text {dist}}(x,\partial {\mathcal {O}})>\tau \}\), \(\psi =0\) on \(\partial {\mathcal {O}}\), and \(|\nabla \psi |\lesssim \tau ^{-1}\). We claim that
$$\begin{aligned}&\int _{\mathcal {O}}|\nabla u_\varepsilon -\nabla \hat{u}_\varepsilon - \nabla ((1-\psi )(\bar{u}-\hat{u}_\varepsilon ))|^2 \,\text {d}x \nonumber \\&\quad \leqq \mathcal {C}^2\,\Vert \nabla u_{{\mathsf {hom}}}\Vert _{H^1({\mathcal {O}})}^2\,\Big (\tau +\delta ^2+\Big (\frac{\varepsilon }{\delta }\Big )^2\mu _d\Big (\frac{\delta }{\varepsilon }\Big ) + \frac{\varepsilon ^2}{\tau }\mu _d\Big (\frac{\delta }{\varepsilon }\Big )\Big ). \end{aligned}$$
(66)
Note that the terms with the factors \(\tau \) and \(\frac{\varepsilon ^2}{\tau }\mu _d(\frac{\delta }{\varepsilon })\) are due to the cut-off close to the boundary. For the argument, we test the equation (61) with the difference \((u_\varepsilon -\hat{u}_\varepsilon +(1-\psi )(\hat{u}_\varepsilon -\bar{u}))\in H^1_0({\mathcal {O}})\). By the monotonicity property (A1) and the Lipschitz continuity (A2) we get
$$\begin{aligned} \int _{\mathcal {O}}\lambda |\nabla u_\varepsilon -\nabla \hat{u}_\varepsilon |^2 \,\text {d}x \leqq&-\int _{\mathcal {O}}R \cdot \nabla (u_\varepsilon -\hat{u}_\varepsilon ) \,\text {d}x \\&+\Lambda \int _{{\mathcal {O}}} |\nabla u_\varepsilon -\nabla \hat{u}_\varepsilon | |\nabla ((1-\psi )(\hat{u}_\varepsilon -\bar{u}))| \,\text {d}x \\&-\int _{{\mathcal {O}}} R \cdot \nabla ((1-\psi )(\hat{u}_\varepsilon -\bar{u})) \,\text {d}x. \end{aligned}$$
This entails the estimate
$$\begin{aligned} \int _{\mathcal {O}}|\nabla u_\varepsilon -\nabla \hat{u}_\varepsilon - \nabla ((1-\psi )(\bar{u}-\hat{u}_\varepsilon ))|^2 \,\text {d}x \lesssim&\int _{\mathcal {O}}|R|^2 \,\text {d}x + \int _{(\partial {\mathcal {O}})_\tau } |\nabla \hat{u}_\varepsilon -\nabla \bar{u}|^2 \,\text {d}x \\&+ \tau ^{-2} \int _{(\partial {\mathcal {O}})_\tau } |\hat{u}_\varepsilon -\bar{u}|^2 \,\text {d}x. \end{aligned}$$
By the definition of \(\hat{u}_\varepsilon \) and the properties of the partition of unity (cf. (45a),(45b)), this implies
$$\begin{aligned}&\int _{\mathcal {O}}|\nabla u_\varepsilon -\nabla \hat{u}_\varepsilon - \nabla ((1-\psi )(\bar{u}-\hat{u}_\varepsilon ))|^2 \,\text {d}x \\&\quad \lesssim \int _{\mathcal {O}}|R|^2 \,\text {d}x + \sum _{k\in K} \int _{(\partial {\mathcal {O}})_\tau }\eta _k(|\nabla \phi _{k}|^2 + \tau ^{-2}|\phi _{k}|^2) \,\text {d}x. \end{aligned}$$
We combine this estimate with the following estimate, the proof of which is postponed to Step 3:
$$\begin{aligned} \int _{(\partial {\mathcal {O}})_\tau }\sum _{k\in K}\eta _k\big (|\nabla \phi _k|^2+\tau ^{-2}|\phi _k|^2) \,\text {d}x \leqq \Big (C \tau +\mathcal {C}^2\,\frac{\varepsilon ^2}{\tau }\mu _d\Big (\frac{\delta }{\varepsilon }\Big ) \Big ) \,\Vert \nabla u_{{\mathsf {hom}}}\Vert _{H^1({\mathcal {O}})}^2. \end{aligned}$$
(67)
Thus, (66) follows from the previous two estimates and (62).
Step 1.3: From the definition of the two-scale expansion \(\hat{u}_\varepsilon \) and the Poincaré inequality we obtain
$$\begin{aligned}&\Vert u_\varepsilon -u_{{\mathsf {hom}}}\Vert _{L^2({\mathcal {O}})}=\bigg |\bigg |u_\varepsilon -\hat{u}_\varepsilon +(1-\psi )\sum _{k\in K}\eta _k\phi _k\bigg |\bigg |_{L^2({\mathcal {O}})}+\bigg |\bigg |\sum _{k\in K}\eta _k\phi _k\bigg |\bigg |_{L^2({\mathcal {O}})}\\&\quad \lesssim \Vert \nabla u_\varepsilon -\nabla \hat{u}_\varepsilon - \nabla ((1-\psi )(\bar{u}-\hat{u}_\varepsilon ))\Vert _{L^2({\mathcal {O}})}+\left( \sum _{k\in K}\int _{B_{2\delta }(k)}|\phi _k|^2\,\text {d}x\right) ^\frac{1}{2} . \end{aligned}$$
Thus, with (66) and (65) we obtain
$$\begin{aligned} \Vert u_\varepsilon -u_{{\mathsf {hom}}}\Vert _{L^2({\mathcal {O}})}^2 \leqq \mathcal {C}^2\,\Vert \nabla u_{{\mathsf {hom}}}\Vert _{H^1({\mathcal {O}})}^2\,\Big (\tau +\delta ^2+\Big (\frac{\varepsilon }{\delta }\Big )^2\mu _d\Big (\frac{\delta }{\varepsilon }\Big ) + \frac{\varepsilon ^2}{\tau }\mu _d\Big (\frac{\delta }{\varepsilon }\Big )\Big ). \end{aligned}$$
By setting \(\tau {:}{=}\delta ^2\) and
$$\begin{aligned} \delta {:}{=} {\left\{ \begin{array}{ll} \varepsilon ^{1/3}&{}\text {for }d=1,\\ \varepsilon ^{1/2} |\log \varepsilon |^{1/4}&{}\text {for }d=2,\\ \varepsilon ^{1/2}&{}\text {for }d\geqq 3, \end{array}\right. } \end{aligned}$$
this completes the proof of Theorem 7.
Step 2: Proof of Theorem 2and Theorem 4– the cases with \({\mathcal {O}}={\mathbb {R}^d}\). As before we appeal to the two-scale expansion \(\hat{u}_\varepsilon \) of Proposition 36 and thus recall the definitions of \(\{\eta _k\}_{k\in K}\), \(\phi _k,\sigma _k\) and \(\hat{u}_\varepsilon \) from Step 1. To improve the scaling of the error, we will crucially use the improved estimate on the difference of correctors \(\phi _{\xi _k}-\phi _{\xi _{\ell }}\) and \(\sigma _{\xi _k}-\sigma _{\xi _{\ell }}\) provided by Corollary 21; however, as these estimates grow with a factor of \((1+|\xi _k|^C+|\xi _\ell |^C)\), the \(L^\infty \)-norm for the gradient \(\nabla u_{\mathsf {hom}}\) appears on the right-hand side of the estimate. As in Step 1 we deduce from Proposition 36 that
$$\begin{aligned} \nabla \cdot (A(\omega _\varepsilon ,\nabla u_\varepsilon )-A(\omega _\varepsilon ,\nabla \hat{u}_\varepsilon )) = \nabla \cdot R,\qquad \text {for }d\geqq 3, \end{aligned}$$
(68)
and
$$\begin{aligned} \nabla \cdot (A(\omega _\varepsilon ,\nabla u_\varepsilon )-A(\omega _\varepsilon ,\nabla \hat{u}_\varepsilon )) - (u_\varepsilon -\hat{u}_\varepsilon ) = \nabla \cdot R +\sum _{k\in K} \eta _k \phi _{k}~\text {for }d=1,2. \end{aligned}$$
(69)
Step 2.1: We claim that
$$\begin{aligned} \int _{\mathbb {R}^d}|R|^2 \,\leqq \,\mathcal {C}^2\widehat{C}(\nabla u_{{\mathsf {hom}}})^2\,\Big (\delta ^2+\varepsilon ^2\mu _d\Big (\frac{\delta }{\varepsilon }\Big )\Big ) \end{aligned}$$
(70)
where
$$\begin{aligned} \widehat{C}(\nabla u_{{\mathsf {hom}}})^2{:}{=}\Vert \nabla u_{{\mathsf {hom}}}\Vert _{H^1({\mathbb {R}^d})}^2+(1+\sup _{{\mathbb {R}^d}}|\nabla u_{\mathsf {hom}}|)^{2C}\int _{{\mathbb {R}^d}}|\nabla ^2u_{{\mathsf {hom}}}|^2\,\text {d}x. \end{aligned}$$
Here and below we denote by \(\mathcal {C}\) a random constant that might change from line to line, but that satisfies the stretched exponential moment bound
$$\begin{aligned} \mathbb {E}\Big [\exp \Big (\frac{\mathcal {C}^{\bar{\nu }}}{C}\Big )\Big ]\leqq 2, \end{aligned}$$
with \(\bar{\nu }\) only depending on \(d,m,\lambda ,\Lambda ,\rho ,\nu \) and with \(C>0\) only depending on \(d,m,\lambda ,\Lambda ,\rho ,\nu \) as well as \({\mathcal {O}}\).
Indeed, the local residuum estimate (49) yields
$$\begin{aligned} \int _{{\mathbb {R}}^d} |R|^2 \,\text {d}x \,\lesssim \,&\delta ^2 \int _{{\mathbb {R}^d}}|\nabla ^2u_{{\mathsf {hom}}}|^2\,\text {d}x\nonumber \\&+\delta ^{-2}\sum _{\ell \in K}\sum _{k\in K\atop |k-\ell |\leqq 4\delta } \int _{B_{6\delta }(\ell )}(|\phi _{k}-\phi _\ell |^2 + |\sigma _{k}-\sigma _\ell |^2) \,\text {d}x. \end{aligned}$$
(71)
We combine it with the improved corrector estimate
$$\begin{aligned}&\delta ^{-2}\sum _{\ell \in K}\sum _{k\in K\atop |k-\ell |\leqq 4\delta }\int _{B_{6\delta }(\ell )}(|\phi _{k}-\phi _\ell |^2 + |\sigma _{k}-\sigma _\ell |^2) \,\text {d}x\nonumber \\&\quad \leqq \mathcal {C}^2\,(1+\sup _{{\mathbb {R}^d}}|\nabla u_{\mathsf {hom}}|)^{2C}\Big (\int _{{\mathbb {R}^d}}|\nabla ^2u_{{\mathsf {hom}}}|^2\,\text {d}x\Big )\,\varepsilon ^2\mu _d\Big (\frac{\delta }{\varepsilon }\Big ). \end{aligned}$$
(72)
The proof of the above estimate is postponed to Step 3. It exploits the regularity assumption (R). The combination of the previous two estimates yields (70).
Step 2.2: We fix \(\delta \) as follows:
$$\begin{aligned} \delta {:}{=} {\left\{ \begin{array}{ll} \varepsilon |\log \varepsilon |^{\frac{1}{2}}&{}d=2,\\ \varepsilon &{}d\ne 2. \end{array}\right. } \end{aligned}$$
We first note that with this choice, we have for \(d\geqq 3\)
$$\begin{aligned} \Vert u_\varepsilon -u_{\mathsf {hom}}\Vert _{L^{2d/(d-2)}({\mathbb {R}^d})}&\leqq \bigg |\bigg |\sum _{k\in K}\eta _k\phi _k\bigg |\bigg |_{L^{2d/(d-2)}({\mathbb {R}^d})}+\,\mathcal {C}\,\widehat{C}(\nabla u_{{\mathsf {hom}}})\,\varepsilon , \end{aligned}$$
(73)
while for \(d=1,2\) we have
$$\begin{aligned} \Vert u_\varepsilon -u_{\mathsf {hom}}\Vert _{L^{2}({\mathbb {R}^d})} \leqq \,&\,\bigg |\bigg |\sum _{k\in K}\eta _k\phi _k\bigg |\bigg |_{L^{2}({\mathbb {R}^d})}\\\nonumber&\qquad +\,\mathcal {C}\,\widehat{C}(\nabla u_{{\mathsf {hom}}})\,\left\{ \begin{aligned}&\varepsilon&\text {for }d=1,\\&\varepsilon |\log \varepsilon |^\frac{1}{2}&\text {for }d=2. \end{aligned}\right. \end{aligned}$$
(74)
Indeed, for \(d=3\) the estimate follows from the energy energy estimate for the PDE (68) combined with (70) and the Sobolev embedding. For \(d=1,2\) the estimate can be directly obtained from the energy estimate for the PDE (69) in combination with (70). As we shall prove in Step 3, we have
$$\begin{aligned} \bigg \Vert \sum _{k\in K} \eta _k\phi _{\xi _k}\bigg \Vert _{L^p({\mathbb {R}^d})}\leqq \mathcal {C} \,\widehat{C}(\nabla u_{{\mathsf {hom}}})\, \left\{ \begin{aligned}&\varepsilon ^\frac{1}{2}&\text {for }d=1\text { and }p=2,\\&\varepsilon |\log \varepsilon |^\frac{1}{2}&\text {for }d=2\text { and }p=2,\\&\varepsilon&\text {for }d\geqq 3\text { and }p=\frac{2d}{d-2}. \end{aligned}\right. \end{aligned}$$
(75)
Combining the previous three estimates, we obtain the statements of Theorem 2 and Theorem 4.
Step 3: Proofs of the corrector estimates (65), (67), (72), and (75).
Step 3.1 - Proof of (65). We first claim that there exists a positive constant \(C\lesssim 1\) and \(\bar{\nu }>0\) only depending on \(d,m,\lambda ,\Lambda \) and \(\rho \) such that, for \(k,\ell \in K\) with \(|k-\ell |\leqq 4\delta \), we have
$$\begin{aligned} \int _{B_{6\delta }(\ell )}|\phi _k|^2+|\sigma _k|^2 \,\text {d}x \leqq \mathcal {C}_k^2\,\Big (\int _{B_{2\delta }(k)\cap {\mathcal {O}}}|\nabla u_{{\mathsf {hom}}}|^2\,\text {d}x\Big )\,\varepsilon ^2\mu _d\Big (\frac{\delta }{\varepsilon }\Big ), \end{aligned}$$
(76)
with a random constant \(\mathcal {C}_k\) satisfying
$$\begin{aligned} \mathbb {E}\Big [\exp \Big (\frac{\mathcal {C}_k^{\bar{\nu }}}{C}\Big )\Big ]\leqq 2. \end{aligned}$$
(77)
For the argument, we first recall that \(\phi _k(\omega _\varepsilon ,x)=\phi _{\xi _k}(\omega _\varepsilon ,x)-\fint _{B_\varepsilon (k)}\phi _{\xi _k}(\omega _\varepsilon ,y)\,dy\). Thus, by translation of \(\omega _\varepsilon \), we may assume without loss of generality that \(k=0\) and \(|\ell |\leqq 4\delta \). The claim then follows from Corollary 21 (applied with \(r=10\delta \) and \(x_0=\ell \in B_{4\delta }(0)\)) and the estimate of \(|\xi _k|\) via (47a).
Summation of (76) thus yields (65) with a random constant \(\mathcal {C}\) given by the expression
$$\begin{aligned} \mathcal {C}=\left( \int _{\mathcal {O}}|\nabla u_{{\mathsf {hom}}}|^2\,\text {d}x\right) ^{-\frac{1}{2}}\left( \sum _{\ell ,k\in K\atop |\ell -k|\leqq 4\delta }\mathcal {C}_k^2\Big (\int _{B_{2\delta }(k)\cap {\mathcal {O}}}|\nabla u_{{\mathsf {hom}}}|^2\,\text {d}x\Big )\,\right) ^\frac{1}{2}. \end{aligned}$$
Since \(\sum _{\ell ,k\in K\atop |\ell -k|\leqq 4\delta }\int _{B_{2\delta }(k)\cap {\mathcal {O}}}|\nabla u_{{\mathsf {hom}}}|^2\,\text {d}x\lesssim \int _{{\mathcal {O}}}|\nabla u_{{\mathsf {hom}}}|^2\,\text {d}x\), we deduce with help of the calculus rules for random variables with stretched exponential moments (see Lemma 47) that \(\mathcal {C}\) satisfies the claimed moments bounds.
Step 3.2 - Proof of (67). We first claim that there exists a positive constant \(C\lesssim 1\) and \(\bar{\nu }>0\) only depending on \(d,m,\lambda ,\Lambda \) and \(\rho \) such that for \(k\in K\) we have
$$\begin{aligned} \int _{(B_{2\delta }(k)\cap \partial {\mathcal {O}})_{\tau }}|\nabla \phi _k|^2+\tau ^{-2}|\phi _k|^2\,\text {d}x\,\lesssim \, \frac{\tau }{\delta }\bigg (1+\mathcal {C}_{k}^2\,\Big (\frac{\varepsilon }{\tau }\Big )^2\mu _d\Big (\frac{\delta }{\varepsilon }\Big ) \bigg )\int _{{\mathcal {O}}\cap B_{3\delta }(k)}|\nabla u_{{\mathsf {hom}}}|^2 \,\text {d}x \end{aligned}$$
(78)
with a random constant \(\mathcal {C}_k\) satisfying (77). For the argument, we first note that \(\fint _{B_{2\tau }(y)}|\nabla \phi _k|^2\,\text {d}x\lesssim |\xi _k|^2+\tau ^{-2}\fint _{B_{4\tau }(y)}|\phi _k|^2\,\text {d}x\) thanks to the Caccioppoli inequality. The remaining argument is similar to the one in Step 3.1, covering the set \((B_{2\delta }(k)\cap \partial {\mathcal {O}})_\tau \) with balls of the form \(B_{4\tau }(y)\).
Thanks to the properties of the partition of unity (45b), we deduce from (78),
$$\begin{aligned}&\sum _{k\in K}\int _{(\partial {\mathcal {O}})_\tau }\eta _k(|\nabla \phi _k|^2+\tau ^{-2}|\phi _k|^2)\,\text {d}x\\&\quad \lesssim \frac{\tau }{\delta } \sum _{k\in K\atop {\text {dist}}(k,\partial \Omega )\leqq 3\delta }\bigg (1+\mathcal {C}_{k}^2\,\Big (\frac{\varepsilon }{\tau }\Big )^2\mu _d\Big (\frac{\delta }{\varepsilon }\Big )\bigg )\Big (\int _{{\mathcal {O}}\cap B_{2\delta }(k)}|\nabla u_{{\mathsf {hom}}}|^2\,\text {d}x\Big )\\&\quad \lesssim \frac{\tau }{\delta } \bigg (1+\mathcal {C}^2 \,\Big (\frac{\varepsilon }{\tau }\Big )^2\mu _d\Big (\frac{\delta }{\varepsilon }\Big )\bigg )\int _{(\partial {\mathcal {O}})_{5\delta }}|\nabla u_{{\mathsf {hom}}}|^2\,\text {d}x, \end{aligned}$$
with a random constant given by
$$\begin{aligned} \mathcal {C}=\left( \int _{(\partial {\mathcal {O}})_{5\delta }}|\nabla u_{{\mathsf {hom}}}|^2\,\text {d}x\right) ^{-\frac{1}{2}}\left( \sum _{k\in K\atop {\text {dist}}(k,\partial \Omega )\leqq 3\delta }\mathcal {C}_k^2\int _{{\mathcal {O}}\cap B_{3\delta }(k)}|\nabla u_{{\mathsf {hom}}}|^2\,\text {d}x\right) ^\frac{1}{2}. \end{aligned}$$
As in Step 3.1 we deduce that \(\mathcal {C}\) satisfies the claimed moments bounds. To conclude (67), it remains to prove the trace-type estimate
$$\begin{aligned} \int _{(\partial {\mathcal {O}})_r}|\nabla u_{{\mathsf {hom}}}|^2\,\text {d}x\lesssim r \Vert \nabla u_{{\mathsf {hom}}}\Vert _{H^1({\mathcal {O}})}^2 \end{aligned}$$
(79)
for \(r=5\delta \). In fact, the estimate holds for any \(v\in H^1({\mathcal {O}})\) (instead of \(\nabla u_{{\mathsf {hom}}}\)). To see this it suffices to consider a smooth \(v:{\mathbb {R}^d}_+\rightarrow {\mathbb {R}}\) with \({\mathbb {R}^d}_+{:}{=}{\mathbb {R}}^{d-1}\times (0,\infty )\). Then
$$\begin{aligned} \int _{{\mathbb {R}}^{d-1}\times (0,r)}|v|^2\,\text {d}x=&\int _{{\mathbb {R}}^{d-1}}\int _0^r |v(x',s)|^2\,{\text {ds}}\,\text {d}x'\\ \leqq&2r \int _{{\mathbb {R}}^{d-1}}|v(x',0)|^2\,\text {d}x'+2\int _{{\mathbb {R}}^{d-1}}\int _0^r \bigg |\int _0^s\partial _d v(x',t)\,\text {dt} \bigg |^2\,\text {ds}\,\text {d}x'\\ \leqq&2r \int _{{\mathbb {R}}^{d-1}}|v(x',0)|^2\,\text {d}x'+2r^2\int _{{\mathbb {R}}^{d-1}}\int _0^r|\partial _d v(x',t)|^2\,\text {dt}\,\text {d}x'\\ \lesssim&r \Vert v\Vert _{H^1({\mathcal {O}})}^2, \end{aligned}$$
where the last line holds thanks to the trace estimate. The case of a general Lipschitz or \(C^1\)-domain can be reduced to \(\mathbb {R}^d_+\) by appealing to a partition of unity and a local straightening of the boundary.
Step 3.3 - Proof of (72). With the regularity assumption (R) at hand, Corollary 21 in combination with (47c) yields
$$\begin{aligned}&\delta ^{-2}\sum _{k\in K\atop |k-\ell |\leqq 4\delta }\int _{B_{6\delta }(\ell )}(|\phi _{k}-\phi _\ell |^2 + |\sigma _{k}-\sigma _\ell |^2) \,\text {d}x\\&\quad \leqq \,\, \mathcal {C}_{\ell }^2 (1+\sup _{{\mathbb {R}^d}}|\nabla u_{\mathsf {hom}}|)^{2C}\int _{B_{6\delta }(\ell )\cap {\mathcal {O}}}|\nabla ^2u_{{\mathsf {hom}}}|^2\,\text {d}x~ \varepsilon ^2\mu _d\Big (\frac{\delta }{\varepsilon }\Big ) \end{aligned}$$
where \(\mathcal {C}_\ell \) denotes a random constant satisfying \(\mathbb {E}[\exp (\tfrac{\mathcal {C}_\ell ^{\bar{\nu }}}{C})]\leqq 2\) with \(C, \bar{\nu }\) only depend on \(d,m,\lambda ,\Lambda ,\rho \) and \(\nu \). A summation in \(\ell \in K\) and the calculus rules for random variable with stretched exponential moments (see Lemma 47) yield (72).
Step 3.4 - Proof of (75). The estimate follows from Corollary 21 and (47a) by an argument similar to the one in Step 3.1. \(\quad \square \)
Lemma 47
Let J denote a countable index set. For \(j\in J\) let \(\mathcal {C}_j\) be a non-negative random variable with stretched exponential moments of the form
$$\begin{aligned} \mathbb {E}\bigg [\exp \bigg (\frac{\mathcal {C}_j^{\nu _j}}{C_j}\bigg )\bigg ]\leqq 2, \end{aligned}$$
with positive constants \(C_j\) and exponents \(0< \nu _j\leqq 2\). Suppose that \((a_j)_{j\in J}\) is a summable sequence of non-negative numbers. Then the random variable
$$\begin{aligned} \mathcal {C}{:}{=}\frac{\sum _{j\in J}a_j\mathcal {C}_j}{\sum _{j\in J}a_j}\qquad \text {satisfies}\qquad \mathbb {E}\bigg [\exp \bigg (\frac{\mathcal {C}^{\hat{\nu }}}{\hat{C}}\bigg )\bigg ]\leqq 2, \end{aligned}$$
where \(\hat{\nu }{:}{=}\inf _{j\in J}\nu _j\) and \(\hat{C}{:}{=}C(\nu _1,\ldots ,\nu _J)\sup _{j\in J}C_j^{\hat{\nu }/\nu _j}\).
Proof
We first note that there exist universal constants \(0<c'\leqq C'<\infty \) such that for any non-negative random variable \(\mathcal {Z}\) we have the chain of implications
$$\begin{aligned} \mathbb {E}\bigg [\exp (C'\mathcal {Z})\bigg ]\leqq 2 \quad \quad \Rightarrow \quad \quad \forall p\geqq 1\,:\, \mathbb {E}\big [\mathcal {Z}^p\big ]^\frac{1}{p}\leqq p \quad \quad \Rightarrow \quad \quad \mathbb {E}\bigg [\exp (c'\mathcal {Z})\bigg ]\leqq 2. \end{aligned}$$
We now estimate
$$\begin{aligned} \mathbb {E}\bigg [\Big (\frac{\mathcal {C}^\nu }{\hat{C}}\Big )^{p}\bigg ]^\frac{1}{\hat{\nu } p}= & {} \frac{1}{\hat{C}^{1/\hat{\nu }}}\mathbb {E}\bigg [\Big (\Big (\frac{\sum _{j\in J}a_j\mathcal {C}_j}{\sum _{j\in J}a_j}\Big )^{\hat{\nu }}\Big )^{p}\bigg ]^\frac{1}{\hat{\nu }p} \leqq \frac{1}{\hat{C}^{1/\hat{\nu }}}\frac{\sum _{j\in J}a_j\mathbb {E} \Big [\big (\mathcal {C}_j^{\hat{\nu }}\big )^p \Big ]^{\frac{1}{\hat{\nu }p}}}{\sum _{j\in J}a_j}\\\leqq & {} \frac{1}{\hat{C}^{1/\hat{\nu }}} \frac{\sum _{j\in J}a_j\mathbb {E} \Big [\big (\mathcal {C}_j^{\nu _j}\big )^p \Big ]^{\frac{1}{\nu _j p}}}{\sum _{j\in J}a_j} \leqq \frac{1}{\hat{C}^{1/\hat{\nu }}} \frac{\sum _{j\in J} a_j (p C_j)^{\frac{1}{\nu _j}} }{\sum _{j\in J}a_j}, \\\leqq & {} \big (c' p\big )^{\frac{1}{\hat{\nu }}}, \end{aligned}$$
and thus the claimed estimate follows. \(\quad \square \)
Estimates on the Random Fluctuations of the RVE Approximation for the Effective Material Law
We next establish the estimates on fluctuations of the representative volume approximation for the effective material law stated in Theorem 14a.
Proof of Theorem 14a
To ease the notation we drop the indices \(\varepsilon \) and L and simply write \(\omega \) instead of \(\omega _{\varepsilon ,L}\). Consider the random variable
$$\begin{aligned} F(\omega ){:}{=}A^{{{\text {RVE}}},L}(\omega ,\xi )\cdot \Xi =\fint _{[0,L]^d}A(\omega (x),\xi +\nabla \phi _\xi (\omega ,x)) \cdot \Xi \,\text {d}x. \end{aligned}$$
Let \(\delta \omega \) denote a periodic infinitesimal perturbation in the sense of Definition 16b. Then for all L-periodic parameter fields \(\omega \) we have
$$\begin{aligned} \delta F(\omega ){:}{=}&\lim \limits _{t\rightarrow 0}\frac{F(\omega +t\delta \omega )-F(\omega )}{t}\nonumber \\ =&\fint _{[0,L]^d}\partial _\omega A(\omega (x),\xi +\nabla \phi _\xi (\omega ,x))\delta \omega (x)\cdot \Xi \nonumber \\&\qquad \qquad +a_\xi (x)\nabla \delta \phi _\xi (\omega ,x) \cdot \Xi \,\text {d}x, \end{aligned}$$
(80)
where
$$\begin{aligned} a_\xi (x)=\partial _\xi A(\omega (x),\xi +\nabla \phi _\xi (\omega ,x)), \end{aligned}$$
and where \(\delta \phi _\xi =\delta \phi _\xi (\omega ,\cdot )\) is the unique (L-periodic) solution with mean zero to
$$\begin{aligned} -\nabla \cdot (a_\xi \nabla \delta \phi _\xi )=\nabla \cdot (\partial _\omega A(\omega (x),\xi +\nabla \phi _\xi (\omega ,x))\delta \omega (x)). \end{aligned}$$
(81)
Introducing the unique L-periodic solution h with vanishing mean to the PDE
$$\begin{aligned} -\nabla \cdot (a_\xi ^{*}\nabla h)=\nabla \cdot (a_\xi ^*(x)\Xi ), \end{aligned}$$
(82)
we deduce by testing (82) with \(\delta \phi _{\xi }\) and testing (81) with h
$$\begin{aligned} \delta F(\omega ) =&\fint _{[0,L]^d}\partial _\omega A(\omega (x),\xi +\nabla \phi _\xi (\omega ,x))\delta \omega (x)\cdot \Xi \,\text {d}x \\&+\fint _{[0,L]^d}\partial _\omega A(\omega (x),\xi +\nabla \phi _\xi (\omega ,x))\delta \omega (x) \cdot \nabla h\,\text {d}x. \end{aligned}$$
This establishes by (A3)
$$\begin{aligned} \bigg |\frac{\partial F(\omega )}{\partial \omega }\bigg | \leqq C L^{-d} |\xi +\nabla \phi _\xi | (|\Xi |+|\nabla h|) \end{aligned}$$
which yields by the q-th moment version of the spectral gap inequality in Lemma 23
$$\begin{aligned}&\mathbb {E}_L\Big [\big |F-\mathbb {E}_L[F]\big |^{2q}\Big ]^{1/2q}\\&\quad \leqq C q \varepsilon ^{\frac{d}{2}}L^{-d} \mathbb {E}_L\bigg [\bigg (\int _{[0,L]^d} \bigg (\fint _{B_\varepsilon (x)} |\xi +\nabla \phi _\xi | (|\Xi |+|\nabla h|)\,\mathrm{{d}}\tilde{x}\bigg )^2 \,d x \bigg )^q\bigg ]^{1/2q}. \end{aligned}$$
By Hölder’s inequality, we infer for any \(p>2\)
$$\begin{aligned}&\mathbb {E}_L\Big [\big |F-\mathbb {E}_L[F]\big |^{2q}\Big ]^{1/2q} \\&\quad \leqq C q \Big (\frac{\varepsilon }{L}\Big )^{\frac{d}{2}} \mathbb {E}_L\bigg [\bigg (\fint _{[0,L]^d} \bigg |\fint _{B_\varepsilon (x)} |\xi +\nabla \phi _\xi |^2 \,\text {d}\tilde{x}\bigg |^{p/(p-2)} \,\text {d}x\bigg )^{q(p-2)/p} \\&\qquad \qquad \qquad \qquad \qquad \times \bigg (\fint _{[0,L]^d} (|\Xi |+|\nabla h|)^p \,\text {d}x\bigg )^{2q/p}\bigg ]^{1/2q}. \end{aligned}$$
Bounding the last integral by \(C|\Xi |\) by Meyers estimate for (82) (see Lemma 44) and using the estimate (99), we obtain
$$\begin{aligned}&\mathbb {E}_L\Big [\big |F-\mathbb {E}_L[F]\big |^{2q}\Big ]^{1/2q} \\&\quad \leqq C q \Big (\frac{\varepsilon }{L}\Big )^{\frac{d}{2}} |\xi | |\Xi | \, \mathbb {E}_L \bigg [\bigg (\fint _{[0,L]^d} \big |r_{*,L,\xi }(x)\big |^{(d-\delta )p/(p-2)} \,\text {d}x\bigg )^{q(p-2)/p}\bigg ]^{1/2q}. \end{aligned}$$
Using stationarity of \(r_{*,L,\xi }\) and the moment bound of Lemma 26 (which we prove explicitly for the probability distribution \(\mathbb {P}\), but which may be established for \(\mathbb {P}_L\) analogously; furthermore, while the estimates of Lemma 26 are stated for finite \(T<\infty \), they are uniform in \(T\geqq \varepsilon ^2\) and therefore also hold in the limit \(T\rightarrow \infty \)), we deduce for q large enough
$$\begin{aligned} \mathbb {E}_L\Big [\big |F-\mathbb {E}_L[F]\big |^{2q}\Big ]^{1/2q} \leqq C q^C\Big (\frac{\varepsilon }{L}\Big )^{\frac{d}{2}} |\xi | |\Xi |. \end{aligned}$$
This is the assertion of Theorem 14a. \(\quad \square \)
Estimates for the Error Introduced by Localization
We next establish the estimates from Lemma 40, Corollary 41, and Lemma 42 for the error introduced in the correctors by the exponential localization on scale \(\sqrt{T}\) via the massive term. We then prove Proposition 39, which estimates the systematic error in the approximation for the effective coefficient \(\mathbb {E}[A^{{{\text {RVE}}},\eta ,T}]\) introduced by the finite localization parameter \(T<\infty \).
Proof of Lemma 40
We will use the exponential weight \(\eta (x){:}{=}\exp (-\gamma |x|/\sqrt{T})\) with \(0<\gamma \ll 1\). Note that
$$\begin{aligned} |\nabla \eta |\leqq \frac{\gamma }{\sqrt{T}}\eta . \end{aligned}$$
(83)
By the localized corrector equation (11a) we have
$$\begin{aligned} -\nabla \cdot (A(\omega _\varepsilon ,\xi +\nabla \phi _\xi ^{2T})-A(\omega _\varepsilon ,\xi +\nabla \phi _\xi ^T)) + \frac{1}{2T}(\phi _\xi ^{2T}-\phi _\xi ^T) =\frac{1}{2T}\phi _\xi ^{T}. \end{aligned}$$
Testing with \((\phi _\xi ^{2T}-\phi _\xi ^{T})\eta \) and using the monotonicity and Lipschitz continuity of A (see (A1)-(A2)) as well as (83), we get
$$\begin{aligned}&\int _{\mathbb {R}^d}\Big ( \lambda \big |\nabla \phi _\xi ^{2T}-\nabla \phi _\xi ^T\big |^2 + \frac{1}{2T}\big |\phi _\xi ^{2T}-\phi _\xi ^T\big |^2\Big )\eta \,\text {d}x \\&\quad \leqq \frac{1}{2T} \int _{\mathbb {R}^d}\phi _\xi ^{T} (\phi _\xi ^{2T}-\phi _\xi ^T)\eta \,\text {d}x+\gamma \int _{\mathbb {R}^d}\Lambda |\nabla \phi _\xi ^{2T}-\nabla \phi _\xi ^T| \frac{1}{\sqrt{T}} |\phi _\xi ^{2T}-\phi _\xi ^T|\eta \,\text {d}x. \end{aligned}$$
Choosing \(\gamma \leqq c(d,m,\lambda ,\Lambda )\), we may absorb the second term on the RHS into the LHS to obtain
$$\begin{aligned} \int _{\mathbb {R}^d}\Big (\big |\nabla \phi _\xi ^{2T}-\nabla \phi _\xi ^T\big |^2 + \frac{1}{T}\big |\phi _\xi ^{2T}-\phi _\xi ^T\big |^2\Big )\eta \,\text {d}x \leqq \frac{C}{T} \int _{\mathbb {R}^d}\phi _\xi ^{T} (\phi _\xi ^{2T}-\phi _\xi ^T)\eta \,\text {d}x. \end{aligned}$$
(84)
In the following, we treat dimensions \(d\leqq 2\) and \(d\geqq 3\) separately. In the case \(d\leqq 2\), by Young’s inequality and absorption the previous inequality yields
$$\begin{aligned} \int _{\mathbb {R}^d}\Big (\big |\nabla \phi _\xi ^{2T}-\nabla \phi _\xi ^T\big |^2 + \frac{1}{T}\big |\phi _\xi ^{2T}-\phi _\xi ^T\big |^2\Big )\eta \,\text {d}x \leqq \frac{C}{T}\int _{\mathbb {R}^d}|\phi _\xi ^T|^2\eta \,\text {d}x. \end{aligned}$$
Note that by Proposition 19 (in connection with a dyadic decomposition of \({\mathbb {R}}^d\) for \(d=1,2\) into the ball \(B_{\sqrt{T}}(0)\) and the annuli \(\{2^i\sqrt{T}\leqq |x|<2^{i+1}\sqrt{T}\}\), \(i=0,1,2,\ldots \)) we obtain
$$\begin{aligned} \frac{1}{T}\int _{{\mathbb {R}^d}}|\phi _\xi ^T|^2\eta \,\text {d}x \leqq {\left\{ \begin{array}{ll} \mathcal {C} |\xi |^2 \varepsilon &{}\text {for }d=1, \\ \mathcal {C} |\xi |^2 \varepsilon ^2 \Big |\log \frac{\sqrt{T}}{\varepsilon }\Big | &{}\text {for }d=2. \end{array}\right. } \end{aligned}$$
Since \(\eta \geqq \exp (-1)\) on \(B_{\sqrt{T}}\), the claimed estimate follows for \(d\leqq 2\).
In the case \(d\geqq 3\) we can use in (84) the representation \(\phi ^{T}_{\xi }=\nabla \cdot (\theta ^{T}_{\xi }-b)\) for any \(b\in {\mathbb {R}^{m\times d}}\) (see (18b)). We obtain by an integration by parts and by the Cauchy-Schwarz inequality the estimate
$$\begin{aligned}&\frac{1}{T} \int _{\mathbb {R}^d}\phi _\xi ^{T} (\phi _\xi ^{2T}-\phi _\xi ^T)\eta \,\text {d}x\\&\quad \leqq \frac{1}{T}\int _{\mathbb {R}^d}|\theta ^{T}_{\xi }-b|\Big (|\nabla \phi _{\xi }^{2T}-\nabla \phi _{\xi }^{T}|+\frac{\gamma }{\sqrt{T}}|\phi _{\xi }^{2T}-\phi _{\xi }^{T}|\Big )\eta \,\text {d}x \end{aligned}$$
With Young’s inequality we may absorb the second factor into the LHS of (84). We thus obtain
$$\begin{aligned} \int _{\mathbb {R}^d}\Big (\big |\nabla \phi _\xi ^{2T}-\nabla \phi _\xi ^T\big |^2 + \frac{1}{T}\big |\phi _\xi ^{2T}-\phi _\xi ^T\big |^2\Big )\eta \,\text {d}x\leqq \frac{C}{T^2} \int _{\mathbb {R}^d}|\theta ^{T}_{\xi }-b|^2\eta \,\text {d}x. \end{aligned}$$
By appealing to Proposition 19 (in connection with a dyadic decomposition of \({\mathbb {R}^d}\)), and the fact that \(\eta \geqq \exp (-1)\) on \(B_{\sqrt{T}}\), the claimed estimate follows for \(d\geqq 3\). \(\quad \square \)
Proof of Corollary 41
Set \(u{:}{=}\phi ^{2T}_\xi -\phi ^T_\xi \) and note that with \(\hat{a}(\xi ){:}{=}\int _0^1 \partial _\xi A(\omega _\varepsilon ,\xi +(1-s)\nabla \phi _\xi ^T+s\nabla \phi _\xi ^{2T})\,ds\) we have by (11a)
$$\begin{aligned} -\nabla \cdot (\hat{a}\nabla u)+\frac{1}{T} u=\frac{1}{2T}\phi _\xi ^{2T}. \end{aligned}$$
Applying the Meyers estimate of Lemma 54 to this PDE – upon rewriting the right-hand side using (18b) in case \(d\geqq 3\) – we obtain, for \(0\leqq p-2\ll 1\), that
$$\begin{aligned}&\bigg (\fint _{B_{\sqrt{T}}(x_0)}|\nabla (\phi ^{2T}_\xi -\phi ^T_\xi )|^p \,\text {d}x\bigg )^{1/p} \\&\quad \leqq C(d,m,\lambda ,\Lambda ,p) \bigg (\fint _{B_{2\sqrt{T}}(x_0)}|\nabla u|^2 + \Big |\frac{1}{\sqrt{T}}u\Big |^2 \,\text {d}x\bigg )^{1/2} \\&\qquad + {\left\{ \begin{array}{ll} C(d,m,\lambda ,\Lambda ,p) \bigg (\fint _{B_{2\sqrt{T}}(x_0)} \Big |\frac{1}{\sqrt{T}}\phi _\xi ^{2T}\Big |^p \,\text {d}x\bigg )^{1/p} &{}\text {in case }d\leqq 2, \\ C(d,m,\lambda ,\Lambda ,p) \bigg (\fint _{B_{2\sqrt{T}}(x_0)} \Big |\frac{1}{T}(\theta _\xi ^{2T}-b)\Big |^p \,\text {d}x\bigg )^{1/p} &{}\text {in case }d\geqq 3. \end{array}\right. } \end{aligned}$$
This implies, by Proposition 19 and Lemma 40, that
$$\begin{aligned}&\bigg (\fint _{B_{\sqrt{T}}(x_0)}|\nabla (\phi ^{2T}_\xi -\phi ^T_\xi )|^p \,\text {d}x\bigg )^{1/p} \\&\quad \leqq \mathcal {C}|\xi | \bigg (\frac{\varepsilon }{\sqrt{T}}\bigg )^{\frac{d\wedge 4}{2}} {\left\{ \begin{array}{ll} \big |\log (\sqrt{T}/\varepsilon ) \big | &{}\text {for }d\in \{2,4\},\\ 1&{}\text {for }d=3,\ d=1,\text { and }d\geqq 5. \end{array}\right. } \end{aligned}$$
Taking the p-th stochastic moment and using stationarity, we conclude. \(\quad \square \)
Proof of Lemma 42
Again, we use the weight \(\eta (x){:}{=}\exp (-\gamma |x|/\sqrt{T})\) for \(0<\gamma \ll 1\). We subtract the equations for \(\phi ^{2T}_{\xi ,\Xi }\) and \(\phi ^{T}_{\xi ,\Xi }\) (see (17a)) to obtain
$$\begin{aligned} -\nabla \cdot \Big (a_\xi ^{2T}(\Xi +\nabla \phi ^{2T}_{\xi ,\Xi })-a_\xi ^T(\Xi +\nabla \phi ^{T}_{\xi ,\Xi })\Big )+\frac{1}{T}(\phi ^{2T}_{\xi ,\Xi }-\phi ^{T}_{\xi ,\Xi })=\frac{1}{2T}\phi ^{2T}_{\xi ,\Xi }. \end{aligned}$$
By adding and subtracting \(a_\xi ^T(\Xi +\phi ^{2T}_{\xi ,\Xi })\) and by additionally appealing to the representation \(\nabla \cdot (\theta ^{2T}_{\xi ,\Xi }-b)=\phi ^{2T}_{\xi ,\Xi }\) for any \(b\in {\mathbb {R}^{m\times d}}\) (see (19b)) in the case \(d\geqq 3\), we get
$$\begin{aligned}&-\nabla \cdot \big (a_\xi ^{T}(\nabla \phi ^{2T}_{\xi ,\Xi }-\nabla \phi ^{T}_{\xi ,\Xi })\big )+\frac{1}{T}(\phi ^{2T}_{\xi ,\Xi }-\phi ^{T}_{\xi ,\Xi })\nonumber \\&\quad =\,\nabla \cdot \Big ((a^{2T}_\xi -a_\xi ^T)(\Xi +\nabla \phi ^{2T}_{\xi ,\Xi })\Big )+ {\left\{ \begin{array}{ll} \frac{1}{2T}\phi ^{2T}_{\xi ,\Xi }&{}\text {for }d\leqq 2,\\ \nabla \cdot (\frac{1}{2T}(\theta ^{2T}_{\xi ,\Xi }-b))&{}\text {for }d\geqq 3. \end{array}\right. } \end{aligned}$$
(85)
Testing the equation with \((\phi ^{2T}_{\xi ,\Xi }-\phi ^{T}_{\xi ,\Xi })\eta \) (with \(0<\gamma \ll 1\)) yields the exponentially localized energy estimate
$$\begin{aligned}&\int _{{\mathbb {R}^d}}\Big (\lambda |\nabla \phi ^{2T}_{\xi ,\Xi }-\nabla \phi ^{T}_{\xi ,\Xi }|^2+\frac{1}{T}|\phi ^{2T}_{\xi ,\Xi }-\phi ^{T}_{\xi ,\Xi }|^2\Big )\eta \,\text {d}x\\&\quad \leqq C\int _{\mathbb {R}^d}|a^{2T}_\xi -a_\xi ^T|^2|\Xi +\nabla \phi ^{2T}_{\xi ,\Xi }|^2\eta \,\text {d}x +C{\left\{ \begin{array}{ll} \frac{1}{T}\int _{\mathbb {R}^d}|\phi ^{2T}_{\xi ,\Xi }|^2\eta \,\text {d}x&{}\text {for }d\leqq 2,\\ \frac{1}{T^2}\int _{\mathbb {R}^d}|\theta ^{2T}_{\xi ,\Xi }-b|^2\eta \,\text {d}x&{}\text {for }d\geqq 3. \end{array}\right. } \end{aligned}$$
By taking the expectation and exploiting stationarity of the LHS, we get
$$\begin{aligned}&\mathbb {E}\Big [|\nabla \phi ^{2T}_{\xi ,\Xi }-\nabla \phi ^{T}_{\xi ,\Xi }|^2+\frac{1}{T}|\phi ^{2T}_{\xi ,\Xi }-\phi ^{T}_{\xi ,\Xi }|^2\Big ]\\&\quad \leqq C\mathbb {E}\left[ |a^{2T}_\xi -a_\xi ^T|^2|\Xi +\nabla \phi ^{2T}_{\xi ,\Xi }|^2\right] +C\left\{ \begin{aligned}\displaystyle&\frac{1}{T}\mathbb {E}\left[ |\phi ^{2T}_{\xi ,\Xi }|^2\right]&\text {for }d\leqq 2,\\ \displaystyle&\frac{1}{T^2}\mathbb {E}\left[ \int _{\mathbb {R}^d}|\theta ^{2T}_{\xi ,\Xi }-b|^2\frac{\eta }{\sqrt{T}^d}\,\text {d}x\right]&\text {for }d\geqq 3. \end{aligned}\right. \end{aligned}$$
The second term on the RHS can be estimated with help of Proposition 20.
We estimate the first term on the RHS: Since \(\partial _\xi A\) is Lipschitz by assumption (R) and since \(a_\xi ^T(x)=A(\omega _\varepsilon (x),\xi +\nabla \phi _\xi ^T)\), we have \(|a_\xi ^{2T}-a_\xi ^T|\leqq C |\nabla \phi _\xi ^{2T}-\nabla \phi _\xi ^T|\). Hence, with Hölder’s inequality with exponents \(0<p-1 \leqq c(d,m,\lambda ,\Lambda )\) and \(\frac{p}{p-1}\), the bound on \(\Xi +\nabla \phi _{\xi ,\Xi }^T\) from (42) and Lemma 30, and Corollary 41, we get
$$\begin{aligned}&\mathbb {E}\left[ |a^{2T}_\xi -a_\xi ^T|^2|\Xi +\nabla \phi ^{2T}_{\xi ,\Xi }|^2\right] \\&\quad \leqq C \mathbb {E}\left[ |\nabla \phi _\xi ^{2T}-\nabla \phi _\xi ^{T}|^{2p}\right] ^\frac{1}{p}\mathbb {E}\left[ |\Xi +\nabla \phi ^{2T}_{\xi ,\Xi }|^{2\frac{p}{p-1}}\right] ^\frac{p-1}{p}\\&\quad \leqq C |\Xi |^2(1+|\xi |)^{C} |\xi |^2\bigg (\frac{\varepsilon }{\sqrt{T}}\bigg )^{d\wedge 4} \times {\left\{ \begin{array}{ll} \big |\log (\sqrt{T}/\varepsilon )\big |^2 &{}\text {for }d\in \{2,4\},\\ 1&{}\text {for }d=3,\ d=1,\text { and }d\geqq 5. \end{array}\right. } \end{aligned}$$
\(\square \)
Proof of Proposition 39
Step 1: Proof of (a). This is a direct consequence of stationarity of \(\mathbb {P}\) (see assumption (P1)) and stationarity of the random field \(A(\omega _\varepsilon ,\xi +\nabla \phi ^T_\xi ) \cdot \Xi -\frac{1}{T}\phi _\xi ^T\phi _{\xi ,\Xi }^{*,T}\), the latter of which is a consequence of the former.
Step 2: Proof of (b). First note that it suffices to prove for any \(T\geqq 2\varepsilon ^2\) the estimate
$$\begin{aligned}&\left| \mathbb {E}\left[ A^{{{\text {RVE}}},\eta ,2T}(\xi ) \cdot \Xi \right] -\mathbb {E}\left[ A^{{{\text {RVE}}},\eta ,T}(\xi ) \cdot \Xi \right] \right| \nonumber \\&\quad \leqq C (1+|\xi |)^{2C}|\xi ||\Xi | \bigg (\frac{\varepsilon }{\sqrt{T}}\bigg )^{d\wedge 4} {\left\{ \begin{array}{ll} \big |\log (\sqrt{T}/\varepsilon )\big |^2&{}\text {for }d=2\text { and }d=4,\\ 1&{}\text {for }d=3,\ d=1,\text { and }d\geqq 5. \end{array}\right. } \end{aligned}$$
(86)
Indeed, the claimed estimate then follows by rewriting the systematic localization error as a telescopic sum,
$$\begin{aligned}&A_{{\mathsf {hom}}}(\xi ) \cdot \Xi \, - \,\mathbb {E}\left[ A^{{{\text {RVE}}},\eta ,T}(\xi ) \cdot \Xi \right] \\&\quad =\sum _{i=0}^\infty \bigg (\mathbb {E}\left[ A^{{{\text {RVE}}},\eta ,2^{i+1}T}(\xi ) \cdot \Xi \right] -\mathbb {E}\left[ A^{{{\text {RVE}}},\eta ,2^iT}(\xi ) \cdot \Xi \right] \bigg ), \end{aligned}$$
which holds, since
$$\begin{aligned} \lim _{T\rightarrow \infty } \mathbb {E}\left[ A^{{{\text {RVE}}},\eta ,T}(\xi ) \cdot \Xi \right] = A_{{\mathsf {hom}}}(\xi ) \cdot \Xi \qquad \mathbb {P}\text {-almost surely}. \end{aligned}$$
We present the argument for (86). In view of (a), we may assume without loss of generality that the weight \(\eta \) satisfies
$$\begin{aligned} {\text {supp}}\eta \subset B_{\sqrt{T}},\qquad \int _{\mathbb {R}^d}\eta \,\text {d}x=1,\qquad |\eta |+\sqrt{T}|\nabla \eta | \leqq C(d) \sqrt{T}^{-d}. \end{aligned}$$
Let \(\phi ^{*,T}_{\xi ,\Xi }\) denote the localized, linearized, adjoint corrector (that is the T-localized homogenization corrector associated with the linear elliptic PDE with coefficient field \((a_\xi ^{T})^*\)). The localized corrector equation (11a) yields
$$\begin{aligned}&-\int _{\mathbb {R}^d}\eta \Big (A(\omega _\varepsilon ,\xi +\nabla \phi ^{2T}_\xi )-A(\omega _\varepsilon ,\xi +\nabla \phi ^{T}_\xi )\Big ) \cdot \nabla \phi ^{*,T}_{\xi ,\Xi }\,\text {d}x\nonumber \\&\quad =\int _{\mathbb {R}^d}\Big (A(\omega _\varepsilon ,\xi +\nabla \phi ^{2T}_\xi )-A(\omega _\varepsilon ,\xi +\nabla \phi ^{T}_\xi )\Big ) \cdot (\phi ^{*,T}_{\xi ,\Xi }\nabla \eta )\nonumber \\&\qquad \quad \qquad +\eta \Big (\frac{1}{2T}\phi ^{2T}_\xi -\frac{1}{T}\phi ^T_\xi \Big )\phi ^{*,T}_{\xi ,\Xi }\,\text {d}x. \end{aligned}$$
(87)
Combined with the definition of the localized RVE approximation in Definition 37, we get
$$\begin{aligned}&\Big (A^{{{\text {RVE}}},\eta ,2T}(\xi )- A^{{{\text {RVE}}},\eta ,T}(\xi )\Big ) \cdot \Xi \\&\quad =\int _{\mathbb {R}^d}\eta \Bigg (\Big (A(\omega _\varepsilon ,\xi +\nabla \phi ^{2T}_\xi )-A(\omega _\varepsilon ,\xi +\nabla \phi ^{T}_\xi )\Big ) \cdot \Xi -\frac{1}{2T}\phi _\xi ^{2T}\phi _{\xi ,\Xi }^{*,2T}\\&\qquad \qquad \qquad +\frac{1}{T}\phi _\xi ^{T}\phi _{\xi ,\Xi }^{*,T}\Bigg )\,\text {d}x\\&\quad =\int _{\mathbb {R}^d}\eta \Bigg (\Big (A(\omega _\varepsilon ,\xi +\nabla \phi ^{2T}_\xi )-A(\omega _\varepsilon ,\xi +\nabla \phi ^{T}_\xi )\Big ) \cdot (\Xi +\nabla \phi ^{*,T}_{\xi ,\Xi })\\&\qquad \qquad \qquad -\frac{1}{2T}\phi _\xi ^{2T}\phi _{\xi ,\Xi }^{*,2T} +\frac{1}{T}\phi _\xi ^{T}\phi _{\xi ,\Xi }^{*,T}\Bigg )\,\text {d}x\\&\qquad +\int _{\mathbb {R}^d}\Big (A(\omega _\varepsilon ,\xi +\nabla \phi ^{2T}_\xi )-A(\omega _\varepsilon ,\xi +\nabla \phi ^{T}_\xi )\Big ) \cdot (\phi ^{*,T}_{\xi ,\Xi }\nabla \eta )\\&\qquad \qquad +\eta \Big (\frac{1}{2T}\phi ^{2T}_\xi -\frac{1}{T}\phi ^T_\xi \Big )\phi ^{*,T}_{\xi ,\Xi }\,\text {d}x\\&\quad =\int _{\mathbb {R}^d}\eta \Bigg (\Big (A(\omega _\varepsilon ,\xi +\nabla \phi ^{2T}_\xi )-A(\omega _\varepsilon ,\xi +\nabla \phi ^{T}_\xi )\Big ) \cdot (\Xi +\nabla \phi ^{*,T}_{\xi ,\Xi })\\&\qquad \qquad \qquad \quad -\frac{1}{2T}\phi _\xi ^{2T}(\phi _{\xi ,\Xi }^{*,2T}-\phi _{\xi ,\Xi }^{*,T})\Bigg )\,\text {d}x\\&\qquad +\int _{\mathbb {R}^d}\Big (A(\omega _\varepsilon ,\xi +\nabla \phi ^{2T}_\xi )-A(\omega _\varepsilon ,\xi +\nabla \phi ^{T}_\xi )\Big ) \cdot (\phi ^{*,T}_{\xi ,\Xi }\nabla \eta )\,\text {d}x. \end{aligned}$$
We subtract the linearized corrector equation (17a) in its form for the adjoint coefficient field and the corresponding corrector
$$\begin{aligned} \int _{\mathbb {R}^d}a_\xi ^{T,*}\big (\Xi +\nabla \phi _{\xi ,\Xi }^{*,T}\big ) \cdot \nabla \big (\eta (\phi ^{2T}_\xi -\phi ^T_\xi )\big )+\frac{1}{T}\int _{\mathbb {R}^d}\eta \phi _{\xi ,\Xi }^{*,T}(\phi _\xi ^{2T}-\phi _\xi ^T)\,\text {d}x=0, \end{aligned}$$
where \(a_\xi ^{T,*}{:}{=}(\partial _\xi A(\xi +\nabla \phi _\xi ^T))^*\). We get
$$\begin{aligned}&\Big (A^{{{\text {RVE}}},\eta ,2T}(\xi )- A^{{{\text {RVE}}},\eta ,T}(\xi )\Big ) \cdot \Xi \nonumber \\&\quad =\int _{\mathbb {R}^d}\eta \Big (A(\omega _\varepsilon ,\xi +\nabla \phi ^{2T}_\xi )-A(\omega _\varepsilon ,\xi +\nabla \phi ^{T}_\xi )-a_\xi ^{T}(\nabla \phi _\xi ^{2T}-\nabla \phi _\xi ^T)\Big ) \nonumber \\&\qquad \qquad \cdot (\Xi +\nabla \phi ^{*,T}_{\xi ,\Xi })\,\text {d}x\nonumber \\&\qquad +\int _{\mathbb {R}^d}\Big (A(\omega _\varepsilon ,\xi +\nabla \phi ^{2T}_\xi )-A(\omega _\varepsilon ,\xi +\nabla \phi ^{T}_\xi )\Big )\cdot \phi ^{*,T}_{\xi ,\Xi }\nabla \eta \nonumber \\&\qquad \qquad -a_\xi ^T(\Xi +\nabla \phi _{\xi ,\Xi }^{*,T}) \cdot (\phi _\xi ^{2T}-\phi _\xi ^T)\nabla \eta \,\text {d}x \nonumber \\&\qquad -\int _{\mathbb {R}^d}\eta \Big (\frac{1}{2T}\phi _\xi ^{2T}(\phi _{\xi ,\Xi }^{*,2T}-\phi _{\xi ,\Xi }^{*,T})+\frac{1}{T}\phi _{\xi ,\Xi }^{*,T}(\phi _\xi ^{2T}-\phi _\xi ^T)\Big )\,\text {d}x. \end{aligned}$$
(88)
We take the expectation of this identity and note that the expectation of the second integral on the right-hand side vanishes: Indeed, since it is of the form \(\mathbb {E}\big [\int _{\mathbb {R}^d}B\nabla \eta \big ]\) where B is a stationary random field and \(\eta \) is compactly supported, we have \(\mathbb {E}\big [\int _{\mathbb {R}^d}B\nabla \eta \big ]=\mathbb {E}\big [B\big ]\int _{\mathbb {R}^d}\nabla \eta =0\). Moreover, for the first term on the RHS of (88) we appeal to the uniform bound on \(\partial _\xi ^2 A\) from assumption (R) in form of (recall that \(a_\xi ^T=\partial _\xi A(\omega _\varepsilon ,\xi +\nabla \phi _\xi ^T)\))
$$\begin{aligned} \big |A(\omega _\varepsilon ,\xi +\nabla \phi _\xi ^{2T})-A(\omega _\varepsilon ,\xi +\nabla \phi _\xi ^T)-a_\xi ^T(\nabla \phi _\xi ^{2T}-\nabla \phi _\xi ^{T})\big | \leqq C \big |\nabla \phi _\xi ^{2T}-\nabla \phi _\xi ^T\big |^2. \end{aligned}$$
We thus get
$$\begin{aligned}&\Big |\mathbb {E}\Big [\Big (A^{{{\text {RVE}}},\eta ,2T}(\xi )- A^{{{\text {RVE}}},\eta ,T}(\xi )\Big ) \cdot \Xi \Big ]\Big |\nonumber \\&\quad \leqq C \mathbb {E}\Big [|\nabla \phi _\xi ^{2T}-\nabla \phi _\xi ^{T}|^2|\Xi +\nabla \phi ^{*,T}_{\xi ,\Xi }|\Big ]\nonumber \\&\qquad +\ \Bigg |\mathbb {E}\Bigg [\int _{\mathbb {R}^d}\eta \Big (\frac{1}{2T}\phi _\xi ^{2T}(\phi _{\xi ,\Xi }^{*,2T}-\phi _{\xi ,\Xi }^{*,T})+\frac{1}{T}\phi _{\xi ,\Xi }^{*,T}(\phi _\xi ^{2T}-\phi _\xi ^T)\Big )\,\text {d}x\Bigg ]\Bigg |\nonumber \\&\quad {=}{:}I_1+I_2. \end{aligned}$$
(89)
To estimate \(I_1\) we first apply Hölder’s inequality with exponents p and \(\frac{p}{p-1}\) with \(0<p-1 \leqq c(d,m,\lambda ,\Lambda )\) to obtain
$$\begin{aligned} I_1 \leqq C \mathbb {E}\big [\big |\nabla \phi _\xi ^{2T} - \nabla \phi _\xi ^{T}\big |^{2p}\big ]^\frac{1}{p} \mathbb {E}\big [\big |\Xi +\nabla \phi _{\xi ,\Xi }^{T}\big |^{\frac{p}{p-1}}\big ]^\frac{p-1}{p}. \end{aligned}$$
We then appeal to Corollary 41 and the moment bound on the linearized corrector from (42) as well as Lemma 30 to deduce
$$\begin{aligned} I_1 \leqq C |\xi |^2(1+|\xi |)^C|\Xi | \bigg (\frac{\varepsilon }{\sqrt{T}}\bigg )^{d\wedge 4} {\left\{ \begin{array}{ll} \big |\log (\sqrt{T}/\varepsilon )\big |^2 &{}\text {for }d\in \{2,4\},\\ 1&{}\text {for }d=3,\ d=1,\text { and }d\geqq 5.\\ \end{array}\right. } \end{aligned}$$
Regarding \(I_2\) we distinguish the cases \(d\leqq 2\) and \(d\geqq 3\). In the case \(d= 2\), we apply Lemma 42, Lemma 40, as well as Proposition 19 and Proposition 20 to obtain
$$\begin{aligned} I_2&\leqq C\mathbb {E}\left[ \frac{1}{T}|\phi _\xi ^{2T}|^2\right] ^\frac{1}{2}\mathbb {E}\left[ \frac{1}{T}|\phi _{\xi ,\Xi }^{*,2T}-\phi _{\xi ,\Xi }^{*,T}|^2\right] ^\frac{1}{2}+ \mathbb {E}\left[ \frac{1}{T}|\phi _{\xi ,\Xi }^{*,T}|^2\right] ^\frac{1}{2}\mathbb {E}\left[ \frac{1}{T}|\phi _\xi ^{2T}-\phi _\xi ^T|^2\right] ^\frac{1}{2}\\&\leqq C(1+|\xi |)^{C}|\xi ||\Xi | \varepsilon ^2 \sqrt{T}^{-2} \big |\log (\sqrt{T}/\varepsilon )\big |^2, \end{aligned}$$
and in case \(d=1\) we proceed similarly.
In the case \(d\geqq 3\) we appeal to the representation of \(\smash {\phi _{\xi }^{2T}}\) and \(\smash {\phi _{\xi ,\Xi }^{*,T}}\) as \(\smash {\nabla \cdot \theta _{\xi }^{2T}}\) and \(\smash {\nabla \cdot \theta _{\xi ,\Xi }^{*,2T}}\) by (18b) and (19b), respectively. To shorten the notation, in the following we assume without loss of generality that \(\smash {\fint _{B_{\sqrt{T}}}\theta _\xi ^{2T}=\fint _{B_{\sqrt{T}}}\theta _{\xi ,\Xi }^{*,T}=0}\). An integration by parts thus yields
$$\begin{aligned} I_2= & {} \Bigg |\mathbb {E}\Bigg [\int _{\mathbb {R}^d}\eta \Big (\frac{1}{2T}\theta _\xi ^{2T}\cdot (\nabla \phi _{\xi ,\Xi }^{*,2T}-\nabla \phi _{\xi ,\Xi }^{*,T})+\frac{1}{T}\theta _{\xi ,\Xi }^{*,T}\cdot (\nabla \phi _\xi ^{2T}-\nabla \phi _\xi ^T)\Big )\,\text {d}x\\&\qquad +\int _{\mathbb {R}^d}\nabla \eta \cdot \Big (\frac{1}{2T}\theta _\xi ^{2T}(\phi _{\xi ,\Xi }^{*,2T}-\phi _{\xi ,\Xi }^{*,T})+\frac{1}{T}\theta _{\xi ,\Xi }^{*,T}(\phi _\xi ^{2T}-\phi _\xi ^T)\Big )\,\text {d}x\Bigg ]\Bigg |. \end{aligned}$$
With the properties of \(\eta \) (in particular, \(|\nabla \eta |\lesssim \sqrt{T}^{-d-1}\) and \({\text {supp}}\eta \subset B_{\sqrt{T}}\)), by the Cauchy-Schwarz inequality, and by stationarity of the localized correctors, we get
$$\begin{aligned} I_2&\leqq \frac{C}{T} \mathbb {E}\Bigg [\fint _{B_{\sqrt{T}}}|\theta _\xi ^{2T}|^2\,\text {d}x\Bigg ]^\frac{1}{2} \mathbb {E}\bigg [|\nabla \phi _{\xi ,\Xi }^{*,2T}-\nabla \phi _{\xi ,\Xi }^{*,T}|^2 +\frac{1}{T}|\phi _{\xi ,\Xi }^{*,2T}-\phi _{\xi ,\Xi }^{*,T}|^2 \bigg ]^\frac{1}{2} \\&\quad +\frac{C}{T} \mathbb {E}\Bigg [\fint _{B_{\sqrt{T}}}|\theta _{\xi ,\Xi }^{*,2T}|^2\,\text {d}x\Bigg ]^\frac{1}{2}\mathbb {E}\Bigg [|\nabla \phi _{\xi }^{2T}-\nabla \phi _{\xi }^{T}|^2+\frac{1}{T}|\phi _{\xi }^{2T}-\phi _{\xi }^{T}|^2\Bigg ]^\frac{1}{2}. \end{aligned}$$
By appealing to Proposition 19 and Lemma 42 for the first term and to Proposition 20 and Lemma 40 for the second term, we obtain
$$\begin{aligned} I_2\leqq C (1+|\xi |)^{C}|\xi ||\Xi |\bigg (\frac{\varepsilon }{\sqrt{T}}\bigg )^{d\wedge 4} {\left\{ \begin{array}{ll} 1 &{}\text {for }d=3 \text { and }d\geqq 5,\\ \big |\log (\sqrt{T}/\varepsilon )\big |^2 &{}\text {for }d=4. \end{array}\right. } \end{aligned}$$
Plugging in the estimates on \(I_1\) and \(I_2\) into (89), this establishes the estimate on the localization error for the representative volume element method for \(\mathbb {P}\).
Step 3: Proof of (c). For the periodized probability distribution \(\mathbb {P}_L\), one may proceed analogously to (b), deriving an error estimate on \(\mathbb {E}_L[A^{{{\text {RVE}}},\eta ,T}]-\mathbb {E}_L[A^{{{\text {RVE}}},L}]\). \(\quad \square \)
Coupling Error for RVEs
Proof of Lemma 38
To shorten the presentation we set \(\widehat{\omega }{:}{=}\pi _L\widetilde{\omega }\) and similarly mark quantities that are associated with \(\widehat{\omega }\), that is,
$$\begin{aligned} \widehat{\phi }_\xi ^T(x)=\phi _\xi ^T(\widehat{\omega },x),\qquad \widehat{\phi }_{\xi ,\Xi }^T(x){:}{=}\phi _{\xi ,\Xi }^T(\widehat{\omega },x), \qquad \widehat{a}_\xi ^T(x){:}{=}A(\widehat{\omega },\xi +\nabla \widehat{\phi }_\xi ^T). \end{aligned}$$
Moreover, we shall use the following notation for the differences:
$$\begin{aligned} \hat{\delta }\phi _\xi ^T{:}{=}\phi _\xi ^T-\widehat{\phi }_\xi ^T,\qquad \hat{\delta }\phi _{\xi ,\Xi }^T{:}{=}\phi _{\xi ,\Xi }^T-\widehat{\phi }_{\xi ,\Xi }^T \end{aligned}$$
(we use the symbol \(\hat{\delta }\) to distinguish the quantity from the sensitivities considered in Sects. 5 and 6). In the proof we make use of the exponential test-functions
$$\begin{aligned} \eta (x){:}{=}\exp (-\gamma |x|/\sqrt{T}), \end{aligned}$$
where \(0<\gamma \ll 1\) is chosen such that the exponential localization estimate Lemma 45 applies.
Step 1. Estimate for \(\hat{\delta }\phi ^T_\xi \). We claim that
$$\begin{aligned} \sup _{x_0\in B_{L/8}}\int _{{\mathbb {R}^d}}\Big (|\nabla \hat{\delta }\phi ^T_\xi |^2+\frac{1}{T}|\hat{\delta }\phi ^T_\xi |^2\Big )\eta (x-x_0)\,\text {d}x\leqq C \sqrt{T}^d\exp (-\tfrac{\gamma }{16}L/\sqrt{T})|\xi |^2. \end{aligned}$$
(90)
Indeed, by subtracting the equations for \(\phi _\xi ^T\) and \(\widehat{\phi }_\xi ^T\), we find that
$$\begin{aligned} -\nabla \cdot (a(x)\nabla \hat{\delta }\phi _\xi ^T)+\frac{1}{T}\hat{\delta }\phi _\xi ^T=\nabla \cdot F, \end{aligned}$$
(91)
where
$$\begin{aligned} a(x)&{:}{=}\int _0^1 \partial _\xi A(\tilde{\omega },\xi +(1-s)\nabla \phi _\xi ^T+s\nabla \widehat{\phi }_\xi ^T)\,\text {ds},\\ F&{:}{=}A(\tilde{\omega },\xi +\nabla \widehat{\phi }^\xi _T)-A(\widehat{\omega },\xi +\nabla \widehat{\phi }^\xi _T). \end{aligned}$$
By the exponentially localized energy estimate of Lemma 45, we have
$$\begin{aligned} \int _{\mathbb {R}^d}\Big (|\nabla \hat{\delta }\phi ^T_\xi |^2+\frac{1}{T}|\hat{\delta }\phi ^T_\xi |^2\Big )\eta (x-x_0)\,\text {d}x\leqq C \int _{\mathbb {R}^d}|F|^2\eta (x-x_0)\,\text {d}x. \end{aligned}$$
Since \(|x_0|\leqq \frac{L}{8}\), for all x with \(|x|\geqq \frac{L}{4}\) the estimate \(|x-x_0|\geqq \frac{L}{16} +\frac{1}{2}|x-x_0|\) applies. This yields, for such x, that
$$\begin{aligned} \eta (x-x_0)\leqq \exp (-\tfrac{\gamma }{16}L/\sqrt{T})\exp (-\tfrac{\gamma }{2} |x-x_0|/\sqrt{T}) \end{aligned}$$
(92)
As \(\hat{\omega }(x)=\tilde{\omega }(x)\) holds for \(|x|\leqq \frac{L}{4}\), we see that F vanishes on \(B_{L/4}\). We thus obtain
$$\begin{aligned} \int _{\mathbb {R}^d}|F|^2\eta (x-x_0)\,\text {d}x&\leqq C \exp (-\tfrac{\gamma }{16}L/\sqrt{T})\int _{\mathbb {R}^d}|\xi +\nabla \widehat{\phi }_T|^2\exp (-\tfrac{\gamma }{2}|x-x_0|/\sqrt{T})\,\text {d}x\\&\leqq C \exp (-\tfrac{\gamma }{16}L/\sqrt{T})\sqrt{T}^d|\xi |^2, \end{aligned}$$
where for the last estimate we appealed to the localized energy estimate for \(\widehat{\phi }_\xi ^T\), see Lemma 45. We conclude that (90) holds. For further reference, we note that we may similarly derive that
$$\begin{aligned} \sup _{x_0\in B_{L/8}}\int _{{\mathbb {R}^d}}\Big (|\nabla \hat{\delta }\phi ^T_\xi |^p+\Big |\frac{1}{\sqrt{T}}\hat{\delta }\phi ^T_\xi \Big |^p\Big )\eta (x-x_0)\,\text {d}x\leqq C \sqrt{T}^d\exp (-\tfrac{\gamma }{16}L/\sqrt{T})|\xi |^p \end{aligned}$$
(93)
for some \(p=p(d,m,\lambda ,\Lambda )>2\) by applying the Meyers estimate of Lemma 54 to (91) with the dyadic decomposition \({\mathbb {R}^d}=B_{\sqrt{T}}\cup \bigcup _{k=1}^\infty (B_{2^k\sqrt{T}}\setminus B_{2^{k-1}\sqrt{T}})\), the estimate (90), and the bound
$$\begin{aligned} \int _{\mathbb {R}^d}|F|^p\eta (x-x_0)\,\text {d}x&\leqq C \exp (-\tfrac{\gamma }{16}L/\sqrt{T})\int _{\mathbb {R}^d}|\xi +\nabla \widehat{\phi }_T|^p\exp (-\tfrac{\gamma }{2}|x-x_0|/\sqrt{T})\,\text {d}x\\&\leqq C \exp (-\tfrac{\gamma }{16}L/\sqrt{T})\sqrt{T}^d|\xi |^p. \end{aligned}$$
Note that in the last step of of the last inequality we have again used the Meyers estimate of Lemma 54 together with the localized energy estimate of Lemma 45 and a dyadic decomposition.
Step 2. Estimate for \(\hat{\delta }\phi _{\xi ,\Xi }^T\). We claim that there exists \(q=q(d,m,\lambda ,\Lambda )\) such that for all \(x_0\in B_{\frac{L}{8}}\) we have
$$\begin{aligned}&\int _{{\mathbb {R}^d}}\Big (|\nabla \hat{\delta }\phi ^T_{\xi ,\Xi }|^2+\frac{1}{T}|\hat{\delta }\phi ^T_{\xi ,\Xi }|^2\Big )\eta (x-x_0)\,\text {d}x\nonumber \\&\quad \leqq C \sqrt{T}^d\exp (-\tfrac{\gamma }{32}L/\sqrt{T})(1+|\xi |^2)\Vert \Xi +\nabla \phi _{\xi ,\Xi }^T\Vert _{q,T,x_0}^2, \end{aligned}$$
(94)
where
$$\begin{aligned} \Vert \Xi +\nabla \phi _{\xi ,\Xi }^T\Vert _{q,T,x_0}{:}{=}\left( \sqrt{T}^{-d}\int _{\mathbb {R}^d}|\Xi +\nabla \phi _{\xi ,\Xi }^T|^{q}\exp (-\tfrac{\gamma }{2}|x-x_0|/\sqrt{T})\,\text {d}x\right) ^\frac{1}{q}. \end{aligned}$$
Indeed, by subtracting the equations (17a) for \(\phi _{\xi ,\Xi }^T\) and \(\widehat{\phi }_{\xi ,\Xi }^T\), we get
$$\begin{aligned} -\nabla \cdot (\widehat{a}_\xi ^T\nabla \hat{\delta }\phi _{\xi ,\Xi }^T)+\frac{1}{T} \hat{\delta }\phi _{\xi ,\Xi }^T =\nabla \cdot \big ((a_\xi ^T-\widehat{a}_\xi ^T)(\Xi +\nabla \phi _{\xi ,\Xi }^T)\big ). \end{aligned}$$
Note that
$$\begin{aligned} a_\xi ^T-\widehat{a}_\xi ^T&=\partial _\xi A(\tilde{\omega },\xi +\nabla \phi _\xi ^T)-\partial _\xi A(\widehat{\omega },\xi +\nabla \widehat{\phi }_\xi ^T)\\&=\big (\partial _\xi A(\tilde{\omega },\xi +\nabla \phi _\xi ^T)-\partial _\xi A(\widehat{\omega },\xi +\nabla \phi _\xi ^T)\big ) \\&\quad +\big (\partial _\xi A(\widehat{\omega },\xi +\nabla \phi _\xi ^T)-\partial _\xi A(\widehat{\omega },\xi +\nabla \widehat{\phi }_\xi ^T)\big )\\&{=}{:}F_1+F_2. \end{aligned}$$
By exponential localization in form of Lemma 45, the Lipschitz continuity of \(\partial _\xi A\) (see (R)), the fact that \(F_1\) vanishes on \(B_{\frac{L}{4}}\) by \(\tilde{\omega }=\widehat{\omega }\) on \(B_{\frac{L}{4}}\), and the uniform bound on \(\partial _\xi A\) from (A2), we get
$$\begin{aligned}&\int _{\mathbb {R}^d}\Big (|\nabla \hat{\delta }\phi ^T_{\xi ,\Xi }|^2+\frac{1}{T}|\hat{\delta }\phi ^T_{\xi ,\Xi }|^2\Big )\eta (x-x_0)\,\text {d}x\nonumber \\&\quad \leqq C \int _{\mathbb {R}^d}|F_1|^2|\Xi +\nabla \phi _{\xi ,\Xi }^T|^2\eta (x-x_0)\,\text {d}x+C\int _{\mathbb {R}^d}|F_2|^2|\Xi +\nabla \phi _{\xi ,\Xi }^T|^2\eta (x-x_0)\,\text {d}x\nonumber \\&\quad \leqq C \int _{\{|x|>\frac{L}{4}\}} |\Xi +\nabla \phi _{\xi ,\Xi }^T|^2\eta (x-x_0)\,\text {d}x\nonumber \\&\qquad +C\int _{\mathbb {R}^d}|\nabla \hat{\delta }\phi _{\xi }^T|^2|\Xi +\nabla \phi _{\xi ,\Xi }^T|^2\eta (x-x_0)\,\text {d}x. \end{aligned}$$
(95)
We estimate the first term on the RHS by Hölder’s inequality as
$$\begin{aligned}&\int _{\{|x|>\frac{L}{4}\}}|\Xi +\nabla \phi _{\xi ,\Xi }^T|^2\eta (x-x_0)\,\text {d}x\\&\quad \leqq C(d,m,\lambda ,\Lambda ,q) \sqrt{T}^d\exp \big (-\tfrac{\gamma }{16}L/\sqrt{T}\big ) \Vert \Xi +\nabla \phi _{\xi ,\Xi }^T\Vert ^2_{q,T,x_0}. \end{aligned}$$
Next, we estimate the second term on the RHS in (95). Using Hölder’s inequality with exponents p/2 and \(\frac{p}{p-2}\) (with \(0<p-2\ll 1\)), setting \(q{:}{=}\frac{2p}{p-2}\), and recalling (93) from Step 1, we get
$$\begin{aligned}&\int _{\mathbb {R}^d}|\nabla \hat{\delta }\phi _{\xi }^T|^2|\Xi +\nabla \phi _{\xi ,\Xi }^T|^2\eta (x-x_0)\,\text {d}x \\&\quad \leqq C \sqrt{T}^d\left( \sqrt{T}^{-d}\int _{{\mathbb {R}}^d}|\nabla \hat{\delta }\phi _\xi ^T|^{p}\eta (x-x_0)\,\text {d}x\right) ^\frac{2}{p} \Vert \Xi +\nabla \phi _{\xi ,\Xi }^T\Vert ^2_{q,T,x_0}\\&\quad \leqq C \sqrt{T}^d\exp (-\tfrac{\gamma }{32}L/\sqrt{T})|\xi |^2\Vert \Xi +\nabla \phi _{\xi ,\Xi }^T\Vert ^2_{q,T,x_0}. \end{aligned}$$
This completes the argument for (94).
Step 3. Conclusion. Set
$$\begin{aligned} \zeta (\widetilde{\omega },x){:}{=}\bigg (A(\widetilde{\omega }(x),\xi +\nabla \phi ^T_\xi (\widetilde{\omega },x)) \cdot \Xi -\frac{1}{T}\phi _\xi ^T(\widetilde{\omega },x)\phi _{\xi ,\Xi }^{*,T}(\widetilde{\omega },x)\bigg ). \end{aligned}$$
Since \(\eta _L\) is supported in \(B_{\frac{L}{8}}\) and \(\eta _L \leqq L^{-d}\), we have
$$\begin{aligned} I_1{:}{=}\Big |\big (A^{{{\text {RVE}}},\eta _L,T}(\widetilde{\omega },\xi )- A^{{{\text {RVE}}},\eta _L,T}(\widehat{\omega },\xi )\big ) \cdot \Xi \Big | \leqq C L^{-d}\int _{B_{\frac{L}{8}}}|\zeta (\widetilde{\omega },x)-\zeta (\widehat{\omega },x)|\,\text {d}x. \end{aligned}$$
We cover \(B_{\frac{L}{8}}\) by balls of radius \(\sqrt{T}\leqq L\) and centers in \(B_{\frac{L}{8}}\); more precisely, there exists a set \(X_{L,T}\subset B_{\frac{L}{8}}\) with \(\# X_{L,T}\leqq C(d) \big (L/{\sqrt{T}}\big )^d\) and \(\cup _{x_0\in X_{L,T}}B_{\sqrt{T}}(x_0)\supset B_{\frac{L}{8}}\). Thus,
$$\begin{aligned} I_1&\leqq C L^{-d}\sum _{x_0\in X_{L,T}}\int _{B_{\sqrt{T}}(x_0)\cap B_{L/8}}|\zeta (\widetilde{\omega },x)-\zeta (\widehat{\omega },x)|\,\text {d}x \\&\leqq C \sqrt{T}^{-d}\Big (\frac{1}{\# X_{L,T}}\sum _{x_0\in X_{L,T}}\int _{B_{L/8}(x_0)\cap B_{L/8}}|\zeta (\widetilde{\omega },x)-\zeta (\widehat{\omega },x)|\eta (x-x_0)\,\text {d}x\Big ). \end{aligned}$$
By the definition of \(\zeta \), the estimates in (90) and (94) from Step 1 and Step 2, and the deterministic exponentially localized bounds on \(\frac{1}{\sqrt{T}}\phi _\xi ^T\) and \(\frac{1}{\sqrt{T}}\phi _{\xi ,\Xi }^{*,T}\) (which are a consequence of Lemma 45), and the Lipschitz continuity of A with respect to the second variable (see (A2)), we conclude for \(x_0\in X_{L,T}\subset B_{\frac{L}{8}}\) that
$$\begin{aligned}&\int _{B_{L/8}(x_0)\cap B_{L/8}} |\zeta (\widetilde{\omega },x)-\zeta (\widehat{\omega },x)|\eta (x-x_0)\,\text {d}x\\&\quad \leqq C \int _{{\mathbb {R}^d}}\big (\big |A(\tilde{\omega },\xi +\nabla \phi _\xi ^T)-A(\tilde{\omega },\xi +\nabla \widehat{\phi }_\xi ^T)\big ||\Xi | \\&\qquad +\frac{1}{T}|\phi _\xi ^T||\hat{\delta }\phi _{\xi ,\Xi }^{*,T}|+\frac{1}{T}|\hat{\delta }\phi _\xi ^T||\widehat{\phi }_{\xi ,\Xi }^{*,T}|\big )\eta (x-x_0)\,\text {d}x \\&\quad \leqq C \left( \int _{\mathbb {R}^d}|\Xi |^2\eta (x-x_0)\,\text {d}x\right) ^\frac{1}{2}\left( \int _{{\mathbb {R}^d}}|\nabla \hat{\delta }\phi _\xi ^T|^2\eta (x-x_0) \,\text {d}x\right) ^\frac{1}{2} \\&\qquad +C \left( \int _{{\mathbb {R}^d}}\frac{1}{T}|\phi _\xi ^T|^2\eta (x-x_0) \,\text {d}x\right) ^\frac{1}{2}\left( \int _{{\mathbb {R}^d}}\frac{1}{T}|\hat{\delta }\phi _{\xi ,\Xi }^{*,T}|^2\eta (x-x_0) \,\text {d}x\right) ^\frac{1}{2}\\&\qquad +C \left( \int _{{\mathbb {R}^d}}\frac{1}{T}|\hat{\delta }\phi _\xi ^T|^2\eta (x-x_0) \,\text {d}x\right) ^\frac{1}{2}\left( \int _{{\mathbb {R}^d}}\frac{1}{T}|\widehat{\phi }_{\xi ,\Xi }^{*,T}|^2\eta (x-x_0) \,\text {d}x\right) ^\frac{1}{2} \\&\quad \leqq C \sqrt{T}^d\Big (\exp (-\tfrac{\gamma }{32}L/\sqrt{T})\Big )^\frac{1}{2}\Big (|\xi |^2 |\Xi |^2+(1+|\xi |^2)|\xi |^2\Vert \Xi +\nabla \phi _{\xi ,\Xi }^T\Vert _{q,T,x_0}^2 \Big )^\frac{1}{2}. \end{aligned}$$
In total, we have shown the desired deterministic estimate
$$\begin{aligned}&|A^{{{\text {RVE}}},\eta _L,T}(\widetilde{\omega },\xi )-A^{{{\text {RVE}}},\eta _L,T}(\pi _L\widetilde{\omega },\xi )| \\&\quad \leqq C \exp \Big (-\frac{\gamma }{64}\cdot \frac{L}{\sqrt{T}}\Big ) \Big (|\xi ||\Xi |+(1+|\xi |)|\xi |\Vert \Xi +\nabla \phi _{\xi ,\Xi }^{T}\Vert _{q,L,T}\Big ). \end{aligned}$$
\(\square \)
Estimate on the Systematic Error of the RVE Method
We now estimate the systematic error of the RVE approximation for the effective material law. We begin with the case in the presence of the regularity condition (R).
Proof of Theorem 14b – the case with (R)
By rescaling we may assume without loss of generality that \(\varepsilon =1\). In the following \(\eta :{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) denotes a non-negative weight supported in \(B_{\frac{L}{8}}\) with \(|\eta |\leqq C(d) L^{-d}\) and \(\int _{{\mathbb {R}^d}}\eta \,\text {d}x=1\). Moreover, we consider a localization parameter T according to
$$\begin{aligned} \sqrt{T}=\frac{\gamma }{64}\frac{L}{\log ((L/\varepsilon )^{d\wedge 4})} \end{aligned}$$
(96)
for the \(\gamma =\gamma (d,m,\lambda ,\Lambda )\) from Lemma 38. Our starting point is the error decomposition (53), which yields for any \(\Xi \in {\mathbb {R}^{m\times d}}\)
$$\begin{aligned}&\mathbb {E}_L\left[ A^{{{\text {RVE}}},L}(\xi )\right] \cdot \Xi -A_{{\text {hom}}}(\xi )\cdot \Xi \nonumber \\&\quad = \mathbb {E}_L\left[ A^{{{\text {RVE}}},L}(\xi )\cdot \Xi \,\right] -\mathbb {E}_L\left[ A^{{{\text {RVE}}},\eta ,T}(\xi )\cdot \Xi \,\right] \nonumber \\&\qquad +\mathbb {E}_L\left[ A^{{{\text {RVE}}},\eta , T}(\xi )\cdot \Xi \,\right] -\mathbb {E}_L\left[ A^{{{\text {RVE}}},\eta , T}(\pi _L\omega _{\varepsilon ,L},\xi )\cdot \Xi \,\right] \nonumber \\&\qquad +\mathbb {E}\left[ A^{{{\text {RVE}}},\eta , T}(\pi _L\omega _\varepsilon ,\xi )\cdot \Xi \,\right] -\mathbb {E}\left[ A^{{{\text {RVE}}},\eta , T}(\xi )\cdot \Xi \,\right] \nonumber \\&\qquad +\mathbb {E}\left[ A^{{{\text {RVE}}},\eta ,T}(\xi )\cdot \Xi \,\right] -A_{{\text {hom}}}(\xi )\cdot \Xi \nonumber \\&\quad {=}{:} I_1+I_2+I_3+I_4. \end{aligned}$$
(97)
Note that in the above decomposition we already used the equality
$$\begin{aligned} \mathbb {E}_L\left[ A^{{{\text {RVE}}},\eta , T}(\pi _L\omega _{\varepsilon ,L},\xi )\right] =\mathbb {E}\left[ A^{{{\text {RVE}}},\eta , T}(\pi _L\omega _\varepsilon ,\xi )\right] , \end{aligned}$$
which is valid since \(\mathbb {P}_L\) is assumed to be a L-periodic approximation of \(\mathbb {P}\) in the sense of Definition 13 (recall also (51)). The terms \(I_2\) and \(I_3\) are coupling errors that can be estimated deterministically with help of Lemma 38. Combined with the choice of T in (96) and the bound on high moments of \(\nabla \phi _{\xi ,\Xi }^T\) obtained by combining (42) and Lemma 30, we arrive at
$$\begin{aligned} |I_2|+|I_3|\leqq C (1+|\xi |^C) |\xi ||\Xi | \bigg (\frac{L}{\varepsilon }\bigg )^{-(d\wedge 4)}. \end{aligned}$$
The terms \(I_1\) and \(I_4\) are systematic localization errors that can be estimated with help of Proposition 39b,c. We obtain using again (96)
$$\begin{aligned} |I_1|+|I_4|\leqq C (1+|\xi |)^{C}|\xi ||\Xi | \bigg (\frac{L}{\varepsilon }\bigg )^{-(d\wedge 4)}|\log (L/\varepsilon )|^{\alpha _d}. \end{aligned}$$
Having estimated all terms in (97), this establishes the first estimate in Theorem 14b upon taking the supremum with respect to \(\Xi \), \(|\Xi |\leqq 1\). \(\quad \square \)
We next establish the suboptimal estimate for the systematic error of the RVE method in the case without the small-scale regularity condition (R).
Proof of Theorem 14b – the case without (R)
As in the case with small scale regularity (R), we denote by \(\eta :{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) a non-negative weight supported in \(B_{\frac{L}{8}}\) with \(|\eta |\leqq C(d) L^{-d}\) and \(\int _{{\mathbb {R}^d}}\eta =1\). Moreover, we consider a localization parameter \(\sqrt{T}\leqq L\), whose relative scaling with respect to L will be specified below in Step 3. For any parameter field \(\widetilde{\omega }\) we consider the localized RVE-approximation
$$\begin{aligned} A^{{{\text {RVE}}},\eta ,T}(\widetilde{\omega },\xi ){:}{=}\int _{\mathbb {R}^d}\eta A(\widetilde{\omega },\xi +\nabla \phi ^T_\xi )\,\text {d}x. \end{aligned}$$
Note that it has a simpler form compared to the quantity introduced in Definition 37. In particular, the above expression does not invoke the linearized corrector (for which we cannot derive suitable estimates without the small scale regularity condition (R)). As in the case with small scale regularity, the starting point is estimate (97), that is the decomposition of the systematic error
$$\begin{aligned} \mathbb {E}_L[A^{{{\text {RVE}}},L}(\xi )]-A_{\mathsf {hom}}(\xi ) = I_1+I_2+I_3+I_4 \end{aligned}$$
into the two coupling errors
$$\begin{aligned} I_2&{:}{=}\mathbb {E}_L\left[ A^{{{\text {RVE}}},\eta , T}(\xi )\right] -\mathbb {E}_L\left[ A^{{{\text {RVE}}},\eta , T}(\pi _L\omega _{\varepsilon ,L},\xi )\right] ,\\ I_3&{:}{=}\mathbb {E}\left[ A^{{{\text {RVE}}},\eta , T}(\pi _L\omega _\varepsilon ,\xi )\right] -\mathbb {E}\left[ A^{{{\text {RVE}}},\eta , T}(\xi )\right] , \end{aligned}$$
and the two systematic localization errors
$$\begin{aligned} I_1&{:}{=} \mathbb {E}_L\left[ A^{{{\text {RVE}}},L}(\xi )\right] -\mathbb {E}_L\left[ A^{{{\text {RVE}}},\eta ,T}(\xi )\right] ,\\ I_4&{:}{=}\mathbb {E}\left[ A^{{{\text {RVE}}},\eta ,T}(\xi )\right] -A_{{\text {hom}}}(\xi ). \end{aligned}$$
Step 1. Estimate of the coupling errors. We claim that
$$\begin{aligned} |I_2|+|I_3|\leqq C \exp \Big (-\frac{\gamma }{32}L/\sqrt{T}\Big )|\xi |. \end{aligned}$$
Indeed, this can be seen by an argument similar to the proof of Lemma 38. In fact the argument is significantly simpler thanks to the absence of the linearized corrector in the definition of the localized RVE-approximation. We only discuss the argument for \(I_3\), since the one for \(I_2\) is analogous. We first note that, thanks to the assumptions on \(\eta \) and the Lipschitz continuity of A (see (A2)) we have
$$\begin{aligned} \Big |A^{{{\text {RVE}}},\eta , T}(\omega _\varepsilon ,\xi )-A^{{{\text {RVE}}},\eta , T}(\pi _L\omega _\varepsilon ,\xi )\Big |\leqq C \fint _{B_{\frac{L}{8}}}|\nabla \hat{\delta }\phi _\xi ^T| \,\text {d}x, \end{aligned}$$
(98)
where \(\hat{\delta }\phi _\xi ^T\) is defined by (91). As in Sect. 4.6 we consider a minimal cover of \(B_{\frac{L}{8}}\) by balls of radius \(\sqrt{T}\); more precisely, let \(X_{L,T}\subset B_{\frac{L}{8}}\) denote a finite set of points such that \(\# X_{L,T}\leqq C(d) (L/\sqrt{T})^d\) and \(\cup _{x_0\in X_{L,T}}B_{\sqrt{T}}(x_0)\supset B_{\frac{L}{8}}\). Then
$$\begin{aligned}{}[\text {RHS of }(98)]&\leqq C \sqrt{T}^{-d}\frac{1}{\# X_{L,T}}\sum _{x_0\in X_{L,T}}\int _{B_{\frac{L}{8}}(x_0)}|\nabla \hat{\delta }\phi _\xi ^T|\exp (-\gamma |x-x_0|/\sqrt{T})\,\text {d}x~~~~~~~~\\&\leqq C \sqrt{T}^{-\frac{d}{2}}\frac{1}{\# X_{L,T}}\sum _{x_0\in X_{L,T}}\left( \int _{{\mathbb {R}}^d}|\nabla \hat{\delta }\phi _\xi ^T|^2\exp (-\gamma |x-x_0|/\sqrt{T})\,\text {d}x\right) ^\frac{1}{2} \end{aligned}$$
where \(0<\gamma \ll 1\) is chosen such that the exponential localization estimate of Lemma 45 applies. Combining the estimate with (90) thus yields
$$\begin{aligned}&\Big |A^{{{\text {RVE}}},\eta , T}(\omega _\varepsilon ,\xi )-A^{{{\text {RVE}}},\eta , T}(\pi _L\omega _\varepsilon ,\xi )\Big |\\&\quad \leqq C {\sqrt{T}}^{-d/2} \sup _{x_0\in B_{\frac{L}{8}}}\left( \int _{{\mathbb {R}}^d}|\nabla \hat{\delta }\phi _\xi ^T|^2\exp (-\gamma |x-x_0|/\sqrt{T}) \,\text {d}x\right) ^\frac{1}{2}\\&\quad \leqq C \exp \big (-\tfrac{\gamma }{32}L/\sqrt{T} \big )|\xi |. \end{aligned}$$
Step 2. Estimate of the systematic localization errors. We claim that
$$\begin{aligned} |I_1|+|I_4|\leqq C |\xi | \bigg (\frac{\varepsilon }{\sqrt{T}}\bigg )^{\frac{d\wedge 4}{2}} {\left\{ \begin{array}{ll} \big |\log \sqrt{T}\big |^\frac{1}{2} &{}\text {for }d\in \{2,4\},\\ 1&{}\text {for }d=3\text { and }d\geqq 5.\\ \end{array}\right. } \end{aligned}$$
We only discuss \(I_4\), since the argument for \(I_1\) is similar. The Lipschitz continuity of A (see (A2)) and the localization error estimate for the corrector in form of Lemma 40 yield
$$\begin{aligned}&\Big |\mathbb {E}\left[ A^{{{\text {RVE}}},\eta ,2T}(\xi )-A^{{{\text {RVE}}},\eta ,T}(\xi )\right] \Big | \\&\quad \leqq C \mathbb {E}\left[ |\nabla \phi _\xi ^{2T}-\nabla \phi _\xi ^{T}|^2\right] ^\frac{1}{2} \\&\quad \leqq C|\xi | \bigg (\frac{\varepsilon }{\sqrt{T}}\bigg )^{\frac{d\wedge 4}{2}}\times {\left\{ \begin{array}{ll} \big |\log (\varepsilon /\sqrt{T})\big |^\frac{1}{2} &{}\text {for }d\in \{2,4\},\\ 1&{}\text {for }d=3\text { and }d\geqq 5.\\ \end{array}\right. } \end{aligned}$$
The claimed estimate now follows by a telescopic sum argument similar to Step 2 of the proof of Proposition 39.
Step 3. Conclusion. The combination of the previous steps yields
$$\begin{aligned}&\frac{|I_1|+|I_2|+|I_3|+|I_4|}{|\xi |} \\&\quad \leqq C \exp \Big (-\frac{\gamma }{32} \frac{L}{\sqrt{T}}\Big )+\bigg (\frac{\varepsilon }{\sqrt{T}}\bigg )^{\frac{d\wedge 4}{2}}\times {\left\{ \begin{array}{ll} \big |\log (\sqrt{T}/\varepsilon )\big |^\frac{1}{2} &{}\text {for }d\in \{2,4\},\\ 1&{}\text {otherwise}. \end{array}\right. } \end{aligned}$$
With \(\sqrt{T}=\frac{\gamma }{32}L\big (\log {((L/\varepsilon )^{\frac{d\wedge 4}{2}}})\big )^{-1}\), the RHS turns into
$$\begin{aligned} C\bigg (\frac{\varepsilon }{L}\bigg )^{\frac{d\wedge 4}{2}}+C\bigg (\frac{\varepsilon }{L}\bigg )^{\frac{d\wedge 4}{2}}(\log (L/\varepsilon ))^{\frac{d\wedge 4}{2}} \times {\left\{ \begin{array}{ll} \big |\log L\big |^\frac{1}{2} &{}\text {for }d\in \{2,4\},\\ 1&{}\text {otherwise}.\\ \end{array}\right. } \end{aligned}$$
This establishes the result. \(\quad \square \)