On Homogenization for Non-Self-Adjoint Periodic Elliptic Operators on an Infinite Cylinder

Abstract

We consider an operator A^ε on L₂(\({\mathbb{R}^{{d_1}}} \times {T^{{d_2}}}\)) (d₁ is positive, while d₂ can be zero) given by A^ε = −div A(ε⁻¹x₁,x₂)∇, where A is periodic in the first variable and smooth in a sense in the second. We present approximations for (A^ε − μ)⁻¹ and ∇(A^ε − μ)⁻¹ (with appropriate μ) in the operator norm when ε is small. We also provide estimates for the rates of approximation that are sharp with respect to the order.