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

Abstract

A method is developed for analysing asymptotic behaviour of terms involving an arbitrary integer order powers of 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 formulæ 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 allows of its generalisation and application to the corresponding non-stationary equation, as well as to some other small amplitude homogenisation problems.

Keywords:

• Small amplitude homogenisation
• H-measures
• non-quadratic terms

AMS Subject Classifications:

• 35B27
• 35J15
• 35S05

Acknowledgements

The author acknowledges N. Antonić and M. Vrdoljak for interesting discussions on the subject, as well as the anonymous referees for the careful reading and useful remarks that have improved the final version of the paper.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported in part by the Croatian Science Foundation [grant number 9780 WeConMApp].