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 \({\mathbb {A}}_\varepsilon \), \(\varepsilon >0\), of the form

$$\begin{aligned} ({\mathbb {A}}_\varepsilon u) ({\textbf{x}}) = \varepsilon ^{-d-2} \int _{\mathbb {R}^d} a(({\textbf{x}}-{\textbf{y}})/\varepsilon ) \mu ({\textbf{x}}/\varepsilon , {\textbf{y}}/\varepsilon ) \left( u({\textbf{x}}) - u({\textbf{y}}) \right) \,d{\textbf{y}}. \end{aligned}$$

It is assumed that \(a({\textbf{x}})\) is a non-negative function from \(L_1(\mathbb {R}^d)\), and \(\mu ({\textbf{x}},{\textbf{y}})\) is a periodic in \({\textbf{x}}\) and \({\textbf{y}}\) function such that \(0< \mu _- \leqslant \mu ({\textbf{x}},{\textbf{y}}) \leqslant \mu _+< \infty \). No symmetry assumption on \(a(\cdot )\) and \(\mu (\cdot )\) is imposed, so the operator \({\mathbb {A}}_\varepsilon \) need not be self-adjoint. Under the assumption that the moments \(M_k = \int _{\mathbb {R}^d} |{\textbf{x}}|^k a({\textbf{x}})\,d{\textbf{x}}\), \(k=1,2,3\), are finite we obtain, for small \(\varepsilon >0\), sharp in order approximation of the resolvent \(({\mathbb {A}}_\varepsilon + I)^{-1}\) in the operator norm in \(L_2(\mathbb {R}^d)\), the discrepancy being of order \(O(\varepsilon )\). The approximation is given by an operator of the form \(({\mathbb {A}}^0 + \varepsilon ^{-1} \langle \varvec{\alpha },\nabla \rangle + I)^{-1}\) multiplied on the right by a periodic function \(q_0({\textbf{x}}/\varepsilon )\); here \({\mathbb {A}}^0 = - \operatorname {div}g^0 \nabla \) is the effective operator, and \(\varvec{\alpha }\) is a constant vector.