Homogenization

Homogenization refers to the phenomenon (and the corresponding method of analysis) where solutions of a class of equations with highly oscillatory coefficients behave approximately like the solutions of a regular (say translation invariant) equation (effective equation) the approximation becoming more accurate as the oscillation of the coefficients is higher.

Typically, one needs some assumption in the way the oscillations organize across space, the most common in increasing order of generality include periodic coefficients, quasi-periodic coefficients, and stationary ergodic coefficients.

An illustrative elementary example is given by the family of linear elliptic equations:

\[(\epsilon)\;\;\;\; \left \{ \begin{array}{rl} a_{ij} \left (\frac{x}{\epsilon} \right ) u_{ij}^{(\epsilon)} = f \left (\frac{x}{\epsilon} \right ) & x\in \Omega \\ u^{(\epsilon)} = g(x) & x \in \partial \Omega \end{array}\right. \]

Where the functions $a_{ij},f$ are assumed to be $\mathbb{Z}^n$-periodic, $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain and $g: \partial \Omega \to \mathbb{R}$ is continuous. Of course the matrix $a_{ij}$ is assumed to be uniformly periodic.

In this setting, homogenization refers to the fact that as $\epsilon \to 0$, the unique solution $u^{(\epsilon)}$ of the problem above converges uniformly to a function $\bar u: \Omega\to\R$ which is the unique solution of

\[ \left \{ \begin{array}{rl} \bar a_{ij} \bar u_{ij} = \bar f & x\in \Omega \\ \bar u = g(x) & x \in \partial \Omega \end{array} \right. \]