Averaging of symmetric diffusion in random medium
The accuracy of convergence is studied of the solutions \(u_{\epsilon}\) of the elliptic equations depending on a random parameter \(L_{\epsilon}u_{\epsilon}=f\), \(x\in Q\subseteq {\mathbb{R}}^ d\), \(u_{\epsilon}=g\), \(x\in \partial Q\), \(\epsilon >0\), to the solution of the averaging problem \(\bar LU=f\), \(x\in Q\), \(U=g\), \(x\in \partial Q\) where \(L_{\epsilon}u=(1/2)D_ i(a_{ij}(y,\omega)D_ ju)\), \(D_ i=\partial /\partial x_ i\), \(y=x/\epsilon\), \(\bar Lu=(1/2)\bar a_{ij}D_ iD_ ju\), \(\bar a{}_{ij}\) constants, \(d\geq 3\), and \(L_{\epsilon}\) is approaching \(\bar L\) using the corresponding resolvent operators. The diffusion coefficients are supposed to be homogeneous, \(a_{ij}(y+z,\omega)=a_{ij}(y,\omega)\), \(z\in {\mathbb{Z}}^ d\), strictly elliptic, \(C_ 1| \xi |^ 2\leq a_{ij}(y,\omega)\xi_ i\xi_ j\leq C_ 2| \xi |^ 2\), \(\xi \in {\mathbb{R}}^ d\), and to satisfy an ergodic property with respect to the random parameter \(\omega\in \Omega\). The main theorem states the following estimate \[ E| u_{\epsilon}-U|^ 2_{\infty}\leq \tilde C\epsilon^{\gamma},\quad \tilde C,\gamma >0 \] where \(| v|_{\infty}=\sup \{| v(x)|\), \(x\in Q\}\).
scientific article; zbMATH DE number 3994669