Homogenization for non-self-adjoint locally periodic elliptic operators

2017/03/06 by Nikita N. Senik, Senik, Nikita N.
Computer Science · Engineering · #35B27 (Primary) #35J15 #35J47 (Secondary) #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1703.02023

openalex publication_date 2017/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the homogenization problem for matrix strongly elliptic operators on L₂(\mathbb R^d)ⁿ of the form \mathcal A^ε=-divA(x,x/ε)∇. The function A is Lipschitz in the first variable and periodic in the second. We do not require that A^*=A, so \mathcal A^ε need not be self-adjoint. In this paper, we provide, for small ε, two terms in the uniform approximation for (\mathcal A^ε-μ)^-1 and a first term in the uniform approximation for ∇(\mathcal A^ε-μ)^-1. Primary attention is paid to proving sharp-order bounds on the errors of the approximations.

Related

• Homogenization for locally periodic elliptic problems on a domain
• Homogenization for non-self-adjoint periodic elliptic operators on an infinite cylinder
• Homogenization of high order elliptic operators with periodic coefficients
• Homogenization estimates for high order elliptic operators
• Improved homogenization estimates for high order elliptic systems