Homogenization for non-self-adjoint periodic elliptic operators on an infinite cylinder

2015/08/20 by Senik, Nikita N.
#35B27 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1508.04963

Abstract

We consider the problem of homogenization for non-self-adjoint second-order elliptic differential operators~A^ε of divergence form on L₂(ℝ^d₁×\mathbbT^d₂), where d₁ is positive and~d₂ is non-negative. The~coefficients of the operator~A^ε are periodic in the first variable with period~ε and smooth in a certain sense in the second. We show that, as ε gets small, (A^ε-μ)^-1 and~D_x₂(A^ε-μ)^-1 converge in the operator norm to, respectively, (A⁰-μ)^-1 and~D_x₂(A⁰-μ)^-1, where A⁰ is an operator whose coefficients depend only on~x₂. We also obtain an approximation for D_x₁(A^ε-μ)^-1 and find the next term in the approximation for~(A^ε-μ)^-1. Estimates for the rates of convergence and the rates of approximation are provided and are sharp with respect to the order.

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