Homogenization of Non-symmetric Convolution Type Operators

The paper studies homogenization problem for a bounded in L₂(Rᵈ) convolution type operator A_ε , ε >0 , of the form $${aligned} ({A}_ε u) ({x}) = ε ⁻ᵈ⁻² ∫ R^d a(({x}-{y})/ε ) μ ({x}/ε , {y}/ε ) ( u({x}) - u({y}) ) \,d{y}. {aligned}$$ It is assumed that $$a({x})$$ is a non-negative function from $$L_1(R^d)$$ , and $$μ ({x},{y})$$ is a periodic in $${x}$$ and $${y}$$ function such that $$0< μ _- μ ({x},{y}) μ _+< ∞ $$ . No symmetry assumption on $$a(· )$$ and $$μ (· )$$ is imposed, so the operator $${A}_ε $$ need not be self-adjoint. Under the assumption that the moments $$M_k = ∫ R^d |{x}|^k a({x})\,d{x}$$ , $$k=1,2,3$$ , are finite we obtain, for small $$ε >0$$ , sharp in order approximation of the resolvent $$({A}_ε + I)⁻¹$$ in the operator norm in $$L_2(R^d)$$ , the discrepancy being of order $$O(ε )$$ . The approximation is given by an operator of the form $$({A}^0 + ε ⁻¹ {α },∇ + I)⁻¹$$ multiplied on the right by a periodic function $$q_0({x}/ε )$$ ; here $${A}^0 = - divg^0 ∇ $$ is the effective operator, and $${α }$$ is a constant vector.