## 2 Principles of Homogenization
Let us decompose the temperature distribution onto slow and highly oscillated parts:  
\[T(\mathbf{x})=u_{0}(\mathbf{x})+\varepsilon u(\mathbf{x},\mathbf{y}). \tag{5}\]  
Then, the differential operator becomes  
\[\frac{d}{dx_{i}}=\frac{\partial}{\partial x_{i}}+\varepsilon^{-1}\frac{ \partial}{\partial y_{i}}, \tag{6}\]  
and the operator  
\[A_{\varepsilon}\equiv\
abla_{\mathbf{x}}\cdot\left(\lambda(\varepsilon^{-1} \mathbf{x})\
abla_{\mathbf{x}}\right)=\varepsilon^{-2}A_{1}+\varepsilon^{-1} A_{2}+A_{3}, \tag{7}\]where the main term  
\[A_{1}=\
abla_{\mathbf{y}}\cdot\big{(}\boldsymbol{\lambda}(\mathbf{y})\
abla_{ \mathbf{y}}\big{)}. \tag{8}\]  
The main task of homogenization theory is to study the limit boundary value problem for \(Q^{\prime}\) when \(\varepsilon\to 0\). The solution of the boundary value problem (3) converges to a solution to the averaged boundary value problem. The results of convergence, corresponding to \(T(\mathbf{x})\) in a Sobolev space and to the flux (1) in the Lebesgue space \(L^{2}(Q^{\prime})\), reflect the clear physical fact that the macroscopic behavior of the sample \(Q^{\prime}\) is described by the effective constant tensor \(\boldsymbol{\lambda}_{e}\). Homogenization theory deals with the limit case \(\varepsilon\to 0\). The effect of the finite small parameter \(\varepsilon\) is discussed in [39]. It is convenient to present the results of homogenization theory by rescaling the sample size \(Q^{\prime}\) and the small periodicity cell \(Q\). Instead of the limit \(\varepsilon\to 0\), we will consider the equivalent limit when the area of \(Q\) is fixed and normalized to unity, and the sample boundary \(\partial Q^{\prime}\) tends to infinity. Therefore, the boundary \(\partial Q^{\prime}\to\infty\) is considered. The next result of homogenization theory asserts that the tensor \(\boldsymbol{\lambda}_{e}\) can be found from the problem for a doubly periodic composite represented by the small cell \(Q\):  
\[\
abla\cdot(\boldsymbol{\lambda}(\mathbf{x})\
abla u(\mathbf{x}))=0,\quad \mathbf{x}\in Q, \tag{9}\]  
with the jump conditions per cell:  
\[\mathbf{x}\mapsto\mathbf{x}+\sum_{j=1,2}m_{j}\omega_{j},\]0  
where \(\zeta_{j}\) are given constants. Sometimes, the constants \(\zeta_{j}\) are introduced into the differential equation Equation (9) by the substitution \(u_{j}(\mathbf{x})=u(\mathbf{x})+\xi_{j}x_{j}\) (\(j=1,2\)). Here, \(\mathbf{x}=(x_{1},x_{2})\). The problems (9) and (10) become  
\[\mathbf{x}\mapsto\mathbf{x}+\sum_{j=1,2}m_{j}\omega_{j},\]1  
\[\mathbf{x}\mapsto\mathbf{x}+\sum_{j=1,2}m_{j}\omega_{j},\]2  
A dispersed composite can be described by the cell \(Q=D\cup D^{+}\cup\partial D^{+}\), where \(D^{+}=\cup_{k=1}^{N}D_{k}\) denotes the union of inclusions, \(\partial D^{+}\) the union of their boundaries, and \(D\) the host domain. The parameter \(f\) denotes the concentration of inclusions. It is introduced as the ratio of areas (volumes) \(f=\frac{|D^{+}|}{|Q|}\). A piece-wise constant scalar function  
\[\lambda(\mathbf{x})= \begin{cases}\lambda_{k}&\mathbf{x}\in D_{k}\quad(k=1,2,\ldots,N)\\ \lambda=1&\mathbf{x}\in D\end{cases}  \tag{13}\]  
describes a dispersed composite, where the conductivity of \(D\) is normalized to unity and the conductivity of \(D_{k}\) is \(\lambda_{k}\). The number of inclusions per periodicity cell \(Q\) is always equal to \(N\). Let us introduce the dimensionless contrast parameters:  
\[\varrho_{k}=\frac{\lambda_{k}-1}{\lambda_{k}+1}. \tag{14}\]  
The parameter \(\lambda_{k}\geq 0\) represents the steady heat conduction and may be complex when referring to the dielectric permittivity. The designation \(\varrho\) is used for two-phase composites when \(\lambda_{k}=\lambda_{1}\) for all \(k\). For a successful homogenization, one should respect the following homogenization principles:* In the framework of homogenization theory of composites [7; 8; 9], \(Q\) must be a fundamental domain. In particular, a parallelepiped in \(\mathbb{R}^{3}\) is the most suitable to represent the fundamental domain.