## Mathematical definition on perforated domains

To proceed what we have introduced in “The first glance at homogenization“, this present note shall focus on the mathematical point of view of special domains in which the homogenization is conducted. Roughly speaking, we consider here the perforated domains.

Let $\Omega$ be a bounded open set in $\mathbb{R}^{n}$ with a smooth boundary $\partial\Omega$. Let $Y$ be the representative cell, i.e.

$Y=(0,\ell_{1})\times...\times(0,\ell_{n}).$

Let $T$ be an open subset of $Y$ with a smooth boundary $\partial T$ in such a way that $\overline{T}\subset \overline{Y}$. Setting $Y^{*} = Y\backslash T$ and denoting $\tau(\varepsilon\overline{T})$ by the set of all translated images of the form $\varepsilon(k\ell + \overline{T})$ for $k\in\mathbb{Z}^{n}$ and $k\ell = (k_{1}\ell_{1},...,k_{n}\ell_{n})$, we define the set, denoted by $T_{\varepsilon}$, of periodically distributed holes as follows:

$T_{\varepsilon} = \Omega \cap \tau(\varepsilon\overline{T}).$

Thus, the domain $\Omega_{\varepsilon}$, called perforated, is defined by

$\Omega_{\varepsilon} = \Omega \backslash \overline{T}_{\varepsilon}.$

Notice that its boundary is composed of two parts. The first part is denoted by $\partial_{\text{int}}\Omega_{\varepsilon}$ describing the union of the boundaries of the holes strictly contained in $\Omega$. Mathematically, it is given by

$\partial_{\text{int}}\Omega_{\varepsilon} = \bigcup \left\{\partial(\varepsilon(k\ell+\overline{T})) | \varepsilon(k\ell + \overline{T}\subset \Omega)\right\}.$

Governed by this way, the second one is certainly its exterior boundary, denoted by $\partial_{\text{ext}}\Omega_{\varepsilon}$, and given by

$\partial_{\text{ext}}\Omega_{\varepsilon} = \partial\Omega_{\varepsilon}\backslash \partial_{\text{int}}\Omega_{\varepsilon}.$

In order to reach our analysis in further notes, it suffices to make one of the following assumptions on the hole $T$:

1. $T$ is a smooth open set with a $C^2$ boundary;
2. $Y^{*}$ is locally on one side of $\partial T$, and $\partial T$ is a union of a finite number of segments if $n=2$ or plane faces if $n>2$.

In addition, the perforated domain $\Omega_{\varepsilon}$ is supposed to be connected. One important thing should be noticed here is that the second boundary $\partial_{\text{ext}}\Omega_{\varepsilon}$ is not necessarily smooth. In general, it may have angles, and the perforated one may not be locally on one side of this boundary.

Finally, we stop this brief note by showing some depicting figures closely related to practical applications. The first picture is usually seen in many building construction area, the perforated metal sheet while the second one describes the lower part of a wireless router, made of soft perforated plastic.

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