In Dimension “Zero”

We will really start studying homogenization theory in Chap. 2 . For now, we only propose an informal introduction to the problem. How can we expect the solutions of (1) to behave in the limit ε → 0 ? What information can we obtain “for free”? What additional price should we pay to answer to more difficult questions? We will see that, specifically, two ingredients show up: weak convergence of sequences of functions, and the (related) notion of mean value of a function. The material we collect in the present chapter will be used throughout this textbook. It will prove useful in (at least) two ways: both in getting an intuition about more elaborate problems and in building mathematical proofs.