Limiting distribution of elliptic homogenization error with periodic diffusion and random potential. (English) Zbl 1335.35313

This paper deals with the limiting distribution of the elliptic homogenization error with periodic diffusion and random potential. Let \(D\subset \mathbb{R}^{d}\) be an open bounded \(C^{1,1}\) domain and \(u^{\varepsilon}\),\(u\) be the solutions to the problem
\[ \begin{cases} -\dfrac{\partial}{\partial x_{i}}\left(a_{ij}\left(\dfrac{x}{\varepsilon}\right)\dfrac{\partial u^{\varepsilon}}{\partial x_{j}}(x,\omega)\right)+q(\dfrac{x}{\varepsilon},\omega)u^{\varepsilon}(x,\omega)=f(x), & x\in D, \\ u^{\varepsilon}(x)=0, & x\in\partial D,\end{cases} \]
and to the deterministic homogenized problem
\[ \begin{cases} -\bar a_{ij}\dfrac{\partial^2 u}{\partial x_{i}\partial x_{j}}(x)+\bar q u(x)=f(x), & x\in D,\\ u(x)=0, & x\in\partial D,\end{cases} \] respectively. Here, \[ \bar a_{ij}=\int_{[0,1]^{d}}a_{ik}(y)\left(\delta_{kj}+\dfrac{\partial\chi^{k}}{\partial x_{j}}(y)\right)dy, \]
\(\chi^{k}\) is the unique solution of the corrector equation, \(\bar q=E(q(0,\omega))\). Suppose that the diffusion coefficients \(A=(a_{ij}):\mathbb{R}^{d}\to \mathbb{R}^{d\times d}\) are smooth, periodic and satisfy the condition of uniform ellipticity, the random potential \(q(x,\omega)\) satisfies the conditions of stationarity, ergodicity and has short-range correlations, \(f\in L^2(D)\) and \(2\leq d\leq7\). Then, there exists \(C>0\), such that \(E\| u^{\varepsilon}-u\|_{L^2}\leq C\varepsilon\| f\|_{L^2}\). Moreover,
\[ E\| u^{\varepsilon}-Eu^{\varepsilon}\|_{L^2}\leq C\varepsilon^{2\wedge (d/2)}\| f\|_{L^2} \quad\text{if } d\neq 4 \] and
\[ E\| u^{\varepsilon}-Eu^{\varepsilon}\|_{L^2}\leq C\varepsilon^{2}|\log\varepsilon|^{1/2}\| f\|_{L^2}\quad\text{if }d=4. \]
Furthermore, for any \(\phi\in L^2(D)\), \(E|(u^{\varepsilon}-Eu^{\varepsilon},\phi)_{L^2}|\leq C\varepsilon^{d/2}\|\phi\|_{L^2}\| f\|_{L^2}\).
Let us denote by \(q(x,\omega)=\bar q+\nu(x,\omega)\), \(R(x)=E(\nu(x+y,\omega)\nu(y,\omega))\), \(\sigma^2=\int_{R^{d}}R(x)dx\) and let \(G(x,y)\) be the Green’s function of the homogenized problem. Let \(W(y)\) denote the standard \(d\)-parameter Wiener process. Then, for \(d=2,3\), as \(\varepsilon\to0\), \((u^{\varepsilon}-Eu^{\varepsilon})/\varepsilon^{d/2}\) converges in distribution to
\[ \sigma\int_{D}G(x,y)u(y)dW(y)\quad\text{ in }L^2(D). \]
For \(d=4,5\), as \(\varepsilon\to0\), the above holds as convergence in law in \(H^{-1}(D)\).


35R60 PDEs with randomness, stochastic partial differential equations
60F25 \(L^p\)-limit theorems
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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