On the homogenization of a Signorini-type problem in a domain with inclusions

TL;DR

This paper studies how Signorini-type interface conditions affect the asymptotic behavior of heat exchange problems in periodic media with inclusions as the period shrinks, revealing different limit behaviors based on a parameter.

Contribution

It provides a detailed analysis of the asymptotic limits of Signorini-type problems in periodic domains with inclusions, depending on the parameter b3, which was not previously characterized.

Findings

Different limit problems are derived for various b3 values.

The Signorini condition influences the effective behavior of the medium.

As b4 tends to zero, the interface conditions lead to distinct homogenized models.

Abstract

In this paper we investigate the effect of a Signorini-type interface condition on the asymptotic behaviour, as $\varepsilon$ tends to zero, of problems posed in $\varepsilon$-periodic domains with inclusions. The Signorini-type condition is expressed in terms of two complementary equalities involving the jump of the solution on the interface and its conormal derivative via a parameter $\gamma$. Our problem models the heat exchange in a medium hosting an $\varepsilon$-periodic array of thermal conductors in presence of impurities distributed on some regions of the interface. Different limit problems are obtained according to different values of $\gamma$.