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

Jing, Wenjia

Anal. PDE 9, No. 1, 193-228 (2016).

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 \(H^{-1}(D)\).

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

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

Keywords:

elliptic equation in divergence form; highly oscillatory coefficients; elliptic homogenization error; periodic diffusion; random potential; limiting distribution; Green’s function