---
tag: J3FQ13SP85
kind: definition
name: Homogenization
msc: [35B27, 74Q05, 35J15]
defines: 5RSREPYN0F
uses: [B4TTNHSTRX, SJZX81A783, M4CEZ5K9HP]
sources:
  - url: https://en.wikipedia.org/wiki/Asymptotic_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|SJZX81A783]] of the [[elliptic|B4TTNHSTRX]] 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 $\varepsilon \to 0$.

## Remarks

The basic theorem of periodic homogenization states that $a^0$ is again uniformly elliptic and that $u_\varepsilon \to u_0$ weakly in the [[Sobolev space|M4CEZ5K9HP]] $H^1_0(\Omega)$ and strongly in $L^2(\Omega)$.
The formula for $a^0$ comes from the two-scale expansion $u_\varepsilon(x) \approx u_0(x) + \varepsilon \sum_j \chi_j(x/\varepsilon) \partial_j u_0(x)$.

Example: in one dimension, let $a$ be a $1$-periodic function with $\alpha \le a \le \beta$.
The cell problem gives $a(y)(1 + \chi') = c$ for a constant $c$, and the mean-zero condition $\int_0^1 \chi' = 0$ gives $c = \big(\int_0^1 a^{-1} \, dy\big)^{-1}$.
So $a^0 = \big(\int_0^1 a^{-1} \, dy\big)^{-1}$ is the harmonic mean of $a$.
For $a$ equal to $1$ on half the period and $3$ on the other half, $a^0 = 3/2$, while the arithmetic mean is $2$.

Non-example: the naive guess $a^0 = \int_Y a \, dy$, the arithmetic mean, is wrong in general, as the example shows; the weak limit of $a(x/\varepsilon) \nabla u_\varepsilon$ is not the product of the weak limits.

## Explanations

### L1 human

Homogenization replaces a finely striped material by one uniform material that behaves the same at large scale.
The effective coefficients come from solving one small problem on a single period of the stripes.