On the nature of the temperature distribution in a perforated body with given values on the external boundary under conditions of heat transfer by Newton's law on the boundary of the cavities

For $\varepsilon \in (0,1)$ let $\Omega _\varepsilon =\Omega \cap \varepsilon \omega$, where $\Omega \subset \mathbb R^d$ is a bounded domain $\varepsilon \omega$ is the set obtained by an $\varepsilon ^{-1}$-fold contraction from an unbounded domain $\omega$ with a $1$-periodic structure, the set $\mathbb R^d \setminus \omega$ being dispersible. Then $\partial \Omega _\varepsilon =\Gamma _\varepsilon \cup S_\varepsilon$, where $\Gamma _\varepsilon$ is the external boundary of $\Omega _\varepsilon$ and $S_\varepsilon$ is the boundary of the cavities lying in $\Omega _\varepsilon$. We study the effect of the exponentially damping (as $\varepsilon \to 0$) influence of a non-zero temperature regime established on $\Gamma _\varepsilon$ on the temperature distribution inside an isotropic body occupying $\Omega _\varepsilon$ under the condition that the heat exchange on $S_\varepsilon$ with the medium filling the cavities of the body follows Newton's law with coefficient of proportionality $a_\varepsilon (x)=a(x/\varepsilon )$, where $a(y)$ is a $1$-periodic function defined on $\partial \omega~$ such that $\int _S a(y)\,ds>0$, if $S=\partial \omega \cap \bigl \{x\in \mathbb R^d:|x_i|<1/2,\ i=\overline {1,d}\bigr \}$.