Abstract
In this paper, we study high order correctors in stochastic homogenization. We consider elliptic equations in divergence form on \(\mathbb {Z}^d\), with the random coefficients constructed from i.i.d. random variables. We prove moment bounds on the high order correctors and their gradients under dimensional constraints. It implies the existence of stationary correctors and stationary gradients in high dimensions. As an application, we prove a two-scale expansion of the solutions to the random PDE, which identifies the first and higher order random fluctuations in a strong sense.
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Acknowledgements
We would like to thank the anonymous referee for several helpful suggestions. The author’s research is partially funded by Grant DMS-1613301 from the US National Science Foundation.
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Appendices
Appendix A: The existence of stationary correctors: removing the massive term
The goal in this section is to show that for the corrector equation (2.8) defined when \(d\geqslant 2n-1\)
the uniform estimates \(\Vert \psi _n^\lambda \Vert _p\lesssim 1\) and \(\Vert \nabla \psi _n^\lambda \Vert _p\lesssim 1\) implies the existence of a stationary corrector and a stationary gradient, respectively.
In Sect. 2.3.1, we proved that when \(d\geqslant 2n-1\), there exists a stationary random field \(\Psi \) such that \(\Psi (0)\in L^2(\Omega )\) and
The rest is standard, and we present it for the convenience of the reader.
We first introduce some notations. Let \(\omega =(\omega _e)_{e\in \mathbb {B}}\in \Omega \) denote the sample point, and for \(x\in \mathbb {Z}^d\), we define the shift operator \(\tau _x\) on \(\Omega \) by \((\tau _x \omega )_e=\omega _{x+e}\), where \(x+e:=(x+\underline{e},x+\bar{e})\) is the edge obtained by shifting e by x. Since \(\{\omega _e\}_{e\in \mathbb {B}}\) are i.i.d., \(\{\tau _x\}_{x\in \mathbb {Z}^d}\) is a group of measure-preserving transformations. We can define the operator
for any measurable function f on \(\Omega \), and the generators of \(T_x\), denoted by \(\{D_i\}_{i=1}^d\), are defined by \(D_if:=T_{e_i}f-f\). The adjoint \(D_i^*\) is defined by \(D_i^* f:=T_{-e_i}f-f\). We denote the gradient on \(\Omega \) by \(D=(D_1,\ldots ,D_d)\) and the divergence \(D^*F:=\sum _{i=1}^d D_i^*F_i\) for \(F:\Omega \rightarrow \mathbb {R}^d\).
Lemma A.1
Let \(\phi _\lambda \) solve \((\lambda +\nabla ^* a\nabla )\phi _\lambda =\nabla ^*\Psi \) with \(\Psi \) a stationary random field on \(\mathbb {Z}^d\) such that \(\Psi (0)\in L^2(\Omega )\). For the equation
-
(i)
there exists a random field \(\phi \) solving (A.2) such that \(\nabla \phi \) is stationary and \(\nabla \phi (0)\in L^2(\Omega )\).
-
(ii)
if \(\langle |\phi _\lambda |^2\rangle \lesssim 1\), then there exists a stationary random field \(\phi \) solving (A.2) such that \(\phi (0)\in L^2(\Omega )\).
Proof
Part (i) comes from [31, Theorem 3]. For Part (ii), let \(\tilde{a}(\omega )=(\omega _{e_1},\ldots ,\omega _{e_d})\) and \(\tilde{\Psi }\in L^2(\Omega )\) so that \(\Psi (x)=\tilde{\Psi }(\tau _x\omega )\), we lift the equation to the probability space
so it is clear that \(\phi _\lambda (x)=\tilde{\phi }_\lambda (\tau _x\omega )\). Since \(\langle |\tilde{\phi }_\lambda |^2\rangle \lesssim 1\), we can extract a subsequence \(\tilde{\phi }_\lambda \rightarrow \tilde{\phi }\) weakly in \(L^2\), which implies \(D\tilde{\phi }_\lambda \rightarrow D\tilde{\phi }\) weakly in \(L^2(\Omega )\).
For any \(G\in L^2(\Omega )\), we have
and sending \(\lambda \rightarrow 0\) leads to
so \(D^*\tilde{a}D\tilde{\phi }=D^*\tilde{\Psi }\). Now we define \(\phi (x)=\tilde{\phi }(\tau _x\omega )\), and it is clear that
The proof is complete. \(\square \)
Appendix B: Finite difference approximation
The following are some classical results and we present it for the convenience of the reader. We recall the classical Green’s function estimates:
for some \(c>0\), with \(\mathcal {G}^\alpha \) the Green’s function of \(\alpha +\nabla ^*\nabla \) on \(\mathbb {Z}^d\). Let \(\mathcal {G}_\varepsilon ^\alpha (x,y)\) be the Green’s function of \(\alpha +\nabla _\varepsilon ^*\nabla _\varepsilon \) on \(\varepsilon \mathbb {Z}^d\), it is clear that
Lemma B.1
Let \(u_\varepsilon \) solve
on \(\varepsilon \mathbb {Z}^d\), with \(f_\varepsilon \) satisfying (i) there exists \(\lambda >0\) s.t. \(|f_\varepsilon (x)|\lesssim e^{-\lambda |x|}\), (ii) all discrete derivatives of \(f_\varepsilon \) satisfies (i), then for \(v_\varepsilon =u_\varepsilon \) or any derivative of \(u_\varepsilon \), we have
for constant \(c>0\) independent of \(\varepsilon >0,x\in \varepsilon \mathbb {Z}^d\).
Proof
It suffices to consider the case \(v_\varepsilon =u_\varepsilon \). For derivatives, e.g., \(v_\varepsilon =\nabla _{\varepsilon ,i} u_\varepsilon \), using the fact that \(\nabla _{\varepsilon ,i}\) commutes with \(\nabla _{\varepsilon ,j}\) and \(\nabla _{\varepsilon ,j}^*\), we have
so we only need to apply the result when \(v_\varepsilon =u_\varepsilon \).
For \(u_\varepsilon (x)\), we use the Green’s function representation
and by (B.1), we have
for some constant \(\rho >0\). Thus
For the first term on the r.h.s. of the above expression, we only need to decompose the summation into \(\sum _{|y|<|x|/2}\) and \(\sum _{|y|\geqslant |x|/2}\) to complete the proof. \(\square \)
Lemma B.2
Let \(u_\varepsilon \) solve
on \(\varepsilon \mathbb {Z}^d\), with \(f_\varepsilon \) satisfying (i) there exists \(\lambda >0\) s.t. \(|f_\varepsilon (x)|\lesssim e^{-\lambda |x|}\), (ii) there exists a continuous function \(\bar{f}:\mathbb {R}^d\rightarrow \mathbb {R}\) such that \(\Vert f_\varepsilon -\bar{f}\Vert _{2,\varepsilon }\rightarrow 0\) as \(\varepsilon \rightarrow 0\), then for \(\bar{u}\) solving
on \(\mathbb {R}^d\), we have \(\Vert u_\varepsilon -\bar{u}\Vert _{2,\varepsilon }\rightarrow 0\) as \(\varepsilon \rightarrow 0\).
Proof
We first note that \(|\bar{f}(x)|\lesssim e^{-\lambda |x|}\) for \(x\in \mathbb {R}^d\). The rest of the proof is decomposed into three steps.
Step 1. We write
and define
on \(\varepsilon \mathbb {Z}^d\). The goal is to show \(\Vert u_\varepsilon -\bar{u}_\varepsilon \Vert _{2,\varepsilon }\rightarrow 0\) as \(\varepsilon \rightarrow 0\). By (B.1), we deduce
thus
It is clear by assumption that \(I_2\rightarrow 0\), and
as \(\varepsilon \rightarrow 0\).
Step 2. Define
on \(\varepsilon \mathbb {Z}^d\), with \(\mathscr {G}^\alpha \) the continuous Green’s function of \(\alpha -\Delta \) in \(\mathbb {R}^d\). The goal is to show \(\Vert \bar{u}_\varepsilon -\tilde{u}_\varepsilon \Vert _{2,\varepsilon }\rightarrow 0\) as \(\varepsilon \rightarrow 0\). We first have
where we used the scaling property \(\mathscr {G}^{\varepsilon ^2\alpha }(x/\varepsilon ,y/\varepsilon )=\varepsilon ^{d-2}\mathscr {G}^\alpha (x,y)\). By [11, Lemma 3.1],
for some \(c>0\), so
for some \(c>0\), which implies \(\Vert \bar{u}_\varepsilon -\tilde{u}_\varepsilon \Vert _{2,\varepsilon }\rightarrow 0\).
Step 3. We first extend \(\tilde{u}_\varepsilon (x)\) from \(\varepsilon \mathbb {Z}^d\) to \(\mathbb {R}^d\) such that \(\tilde{u}_\varepsilon (x)=\tilde{u}_\varepsilon ([x]_\varepsilon )\) with \([x]_\varepsilon \) the \(\varepsilon \)-integer part of x. Then we consider
It is clear that \(\int _{\mathbb {R}^d} |\bar{u}([x]_\varepsilon )-\bar{u}(x)|^2dx\rightarrow 0\) as \(\varepsilon \rightarrow 0\), so we only need to show
Since \(\bar{u}(x)=\int _{\mathbb {R}^d} \mathscr {G}^\alpha (x,y)\bar{f}(y)dy\), we have \(\tilde{u}_\varepsilon (x)\rightarrow \bar{u}(x)\) for \(x\in \mathbb {R}^d\). Now by Lemma B.1, \(|\tilde{u}_\varepsilon (x)|+|\bar{u}(x)|\lesssim e^{-c|x|}\) for some \(c>0\), so by dominated convergence theorem, the proof is complete. \(\square \)
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Gu, Y. High order correctors and two-scale expansions in stochastic homogenization. Probab. Theory Relat. Fields 169, 1221–1259 (2017). https://doi.org/10.1007/s00440-016-0750-0
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DOI: https://doi.org/10.1007/s00440-016-0750-0
Keywords
- Quantitative homogenization
- High order corrector
- Two-scale expansion
- Random fluctuation
Mathematics Subject Classification
- 35B27
- 35K05
- 34E05
- 35R60