Asymptotic analysis for non-local problems in composites with different imperfect contact conditions

Abstract

We consider a composite material made up of a hosting medium containing an ε-periodic array of perfect thermal conductors. Comparing with the previous contributions in the literature, in the present paper, the inclusions are completely disconnected and form two families with dissimilar physical behaviour. More specifically, the imperfect contact between the hosting medium and the inclusions obeys two different laws, according to the two different types of inclusions. The contact conditions involve the small parameter ε and two positive constants $D_1,D_2$. We investigate the homogenization limit $ϵ\to 0$ and the limits for $D_1,D_2$ going to 0 or $+\mathrm{\infty }$, taken in any order, with the aim to find out the cases in which the two limits commute.

Keywords:

• Homogenization
• time-dependent unfolding
• total flux boundary conditions
• imperfect contact conditions
• parabolic problems
• exchange of asymptotic limits

AMS-MSC:

• 35B27
• 35Q79
• 35K20

Acknowledgments

The first author is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The second author is member of the Gruppo Nazionale per la Fisica Matematica (GNFM) of the Istituto Nazionale di Alta Matematica (INdAM). The last author wishes to thank Dipartimento di Scienze di Base e Applicate per l'Ingegneria for the warm hospitality and Università ‘La Sapienza’ of Rome for the financial support. We are grateful to the editor and to the anonymous referees for their useful remarks which improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).