Homogenization of non-symmetric convolution type operators

Abstract

The paper studies homogenization problem for a bounded in $L_2({\mathbb{R}}^d)$ convolution type operator ${\mathbb A}_\eps$, $\eps >0$, of the form $$ ({\mathbb A}_\eps u) (\x) = \eps^{-d-2} \int_{\R^d} a((\x-\y)/\eps) μ(\x/\eps, \y/\eps) \left( u(\x) - u(\y) \right)\,d\y. $$ It is assumed that $a(\x)$ is a non-negative function from $L_1({\mathbb{R}}^d)$, and $μ(\x,\y)$ is a periodic in $\x$ and $\y$ function such that $0< μ_- \leqslant μ(\x,\y) \leqslant μ_+< \infty$. No symmetry assumption on $a(\cdot )$ and $\hat{I}¼(\cdot )$ is imposed, so the operator ${\mathbb A}_\eps$ need not be self-adjoint. Under the assumption that the moments $M_k = \int_{\R^d} |\x|^k a(\x)\,d\x$, $k=1,2,3$, are finite we obtain, for small $\eps>0$, sharp in order approximation of the resolvent $({\mathbb A}_\eps + I)^{-1}$ in the operator norm in $L_2({\mathbb{R}}^d)$, the discrepancy being of order $O(\eps)$. The approximation is given by an operator of the form $({\mathbb A}^0 + \eps^{-1} \langle \boldsymbolα,\nabla \rangle + I)^{-1}$ multiplied on the right by a periodic function $q_0(\x/\eps)$; here ${\mathbb{A}}^0=-\mathrm{div}{g}^0\nabla$ is the effective operator, and $\hat{\mathbf{I}}\pm$ is a constant vector.