Exploring limit behaviour of non-quadratic terms via H-measures. Application to small amplitude homogenisation

TL;DR

This paper introduces a method using H-measures to analyze the asymptotic behavior of non-quadratic terms in homogenization problems, providing explicit formulas and extending to non-stationary cases.

Contribution

It develops a novel approach with H-measures for higher-order analysis in small amplitude homogenization, including explicit Fourier-based formulas under periodic conditions.

Findings

Derived explicit formulas for higher-order correction terms.

Extended the method to non-stationary diffusion equations.

Applicable to general non-periodic homogenization problems.

Abstract

A method is developed for analysing asymptotic behaviour of terms involving an arbitrary integer order powers of L p functions by means of H-measures. It is applied to the small amplitude homogenisation problem for a stationary diffusion equation, in which coefficients are assumed to be analytic perturbations of a constant, enabling formul{\ae} for higher order correction terms in a general, non-periodic setting. Explicit expressions in terms of Fourier coefficients are obtained under periodicity assumption. The method enables its generalisation and application to the corresponding non-stationary equation, as well as to some other small amplitude homogenisation problems.