Homogenization

Definition

Let $Y = [0, 1)^n$ be the unit cell and let $a = (a_{ij}) \colon \R^n \to \R^{n \times n}$ be measurable and $Y$-periodic. Assume $a$ is uniformly elliptic: for constants $0 < \alpha \le \beta$, $a(y)\xi \cdot \xi \ge \alpha \abs{\xi}^2$ and $\abs{a(y)\xi} \le \beta \abs{\xi}$ for all $y, \xi \in \R^n$. Let $\Omega \subseteq \R^n$ be a bounded open set and $f \in L^2(\Omega)$. For $\varepsilon > 0$, let $u_\varepsilon \in H^1_0(\Omega)$ be the weak solution of the elliptic problem $$-\operatorname{div}\big(a(x/\varepsilon) \nabla u_\varepsilon\big) = f \text{ in } \Omega, \qquad u_\varepsilon = 0 \text{ on } \partial\Omega.$$ For $j = 1, \dots, n$, the cell problem asks for a $Y$-periodic $\chi_j \in H^1_{\mathrm{loc}}(\R^n)$ with mean $0$ over $Y$ and $$-\operatorname{div}_y\big(a(y)(e_j + \nabla_y \chi_j)\big) = 0 \text{ in } \R^n,$$ where $e_j$ is the $j$-th unit vector. The homogenized matrix is the constant matrix $a^0$ with entries $$a^0_{ij} = \int_Y e_i \cdot a(y)\big(e_j + \nabla_y \chi_j(y)\big) \, dy,$$ and the homogenized problem is $-\operatorname{div}(a^0 \nabla u_0) = f$ in $\Omega$, $u_0 = 0$ on $\partial\Omega$. Homogenization is the passage from the problems for $u_\varepsilon$ to the homogenized problem as $\varepsilon \to 0$.