Work

An Operator-Asymptotic Approach to Periodic Homogenization for Equations of Linearized Elasticity

Year: 2025

Type: article

Abstract: We present an operator-asymptotic approach to the problem of homogenization of periodic composite media in the setting of three-dimensional linearized elasticity. This is based on a uniform approximation with respect to the inverse wavelength | χ | for the solution to the resolvent problem when written as a superposition of elementary plane waves with wave vector (“quasimomentum”) χ . We develop an asymptotic procedure in powers of | χ | , combined with a new uniform version of the classical Korn inequality. As a consequence, we obtain L 2 → L 2 , L 2 → H 1 , and higher-order L 2 → L 2 norm-resolvent estimates in R 3 . The L 2 → H 1 and higher-order L 2 → L 2 correctors emerge naturally from the asymptotic procedure, and the former is shown to coincide with the classical formulae.

Source: Asymptotic Analysis

DOI: 10.1177/09217134241308360

Authors: Yi-Sheng Lim, Josip Žubrinić

Institutions: University of Bath, University of Zagreb, Faculty of Electrical Engineering and Computing in Zagreb

Language: English

Topic: Advanced Mathematical Modeling in Engineering

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: Hrvatska Zaklada za Znanost, Engineering and Physical Sciences Research Council, Centre for Doctoral Training in Statistical Applied Mathematics, University of Bath

Awards: Homogenization, dimension reduction and structural optimization in continuum mechanics, G3205148109, Homogenization and Optimization in Mechanics of Solids, EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath