Homogenization of Non-symmetric Convolution Type Operators

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.