Homogenization

Homogenization refers to the phenomenon-- and the corresponding methods of analysis-- where solutions of a class of equations with highly oscillatory coefficients behave approximately like the solutions of a regular, effective equation which is usually translation invariant. The approximation becomes more accurate as the oscillation of the coefficients is higher. At its core, the topic of homogenization should be viewed as a complexity reduction problem in the sense that the original equations typically contain too much detail to be efficiently solvable, whereas if one is willing to make the (hopefully small) error in passing to the translation invariant effective equation, then computing solutions and describing their properties is typically much easier. Especially in many nonlinear problems there can be large gaps between the theorems of existence for an effective equation and the results which provide refined information about the effective equation which can be practically implemented.

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:

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

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 elliptic.

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

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

In the elliptic nonlocal setting, the high frequency oscillations are often modeled in in the equation by considering a linear operator of the form

\[ L^\epsilon(u,x):= \int_{\mathbb{R}^n}(u(x+y)+u(x-y)-2u(x))\left[\epsilon^{-n-\alpha}K(\frac{x}{\epsilon},\frac{y}{\epsilon})\right]dy, \] where $K$ is a fixed uniformly elliptic kernel. Further simplifications can be made for the sake of presentation if $K(x,\lambda y)=\lambda^{-n-\alpha}K(x,y)$, in which case the rescaled operator is

\[ L^\epsilon(u,x):= \int_{\mathbb{R}^n}(u(x+y)+u(x-y)-2u(x))K(\frac{x}{\epsilon},y)dy \]

(The reason for this choice of scaling can be seen to be the natural one with its relationship to the behavior of a rescaled jump process in an inhomogeneous environment, $X^\epsilon_t:=\epsilon X_{t\epsilon^{-\alpha}}$, which in the 2nd order case is the familiar diffusive scaling of time and space.)

As hinted at above, the main goals of homogenization are: to identify the effective equation, prove the convergence of the $u^\epsilon$, and hopefully provide more refined information about the effective equation and possibly rates at which the $u^\epsilon$ converge.

In the non-divergence elliptic theory, the identification of the effective operator, say $\bar L$, is heuristically somewhat simple and technically quite complicated. At least from one point of view, the main ideas can all be linked back to a simple heuristic ansatz for the $u^\epsilon$ combined with a need to make an equation independent of $\epsilon$ in order to capture a limit. The ansatz gives that $u^\epsilon$ should be (locally!) expanded as a "smooth" term, $\bar u$, plus a small high frequency perturbation, $\epsilon^\alpha v(x/\epsilon)$:

\[ u^\epsilon(x) = \bar u(x) + \epsilon^\alpha v(x/\epsilon) + o(\epsilon^\alpha). \]

Plugging this into the equation (with the simplified kernels) and ignoring higher order terms eventually gives after a change of variables

\[ \int_{\mathbb{R}^n}(\bar u(x+y)+\bar u(x-y)-2\bar u(x))K(\frac{x}{\epsilon},y)dy + \int_{\mathbb{R}^n}(v(\frac{x}{\epsilon}+y)+v(\frac{x}{\epsilon}-y)-2v(\frac{x}{\epsilon}))K(\frac{x}{\epsilon},y)dy = f(\frac{x}{\epsilon}). \] The unknown in this equation is the function, $v$, and heuristically it should be chosen so that the whole line can be made independent of $\epsilon$. In the simplest case of periodic coefficients, this has become known as solving the corrector equation, and homogenization follows from it. It states that there exists a unique choice of $\lambda$ such that there exists a bounded, periodic solution, $v(z)$, of the equation (posed in the $z$ variable, with $\bar u$ and $x$ fixed parameters!)

\[ \int_{\mathbb{R}^n}(\bar u(x+y)+\bar u(x-y)-2\bar u(x))K(z,y)dy + \int_{\mathbb{R}^n}(v(z+y)+v(z-y)-2v(z))K(z,y)dy = \lambda. \]