I will discuss the homogenization of periodic oscillating Dirichlet
boundary problems in general domains for second order uniformly elliptic
equations. These problems are connected with the study of boundary layers
in fluid mechanics and with the study of higher order asymptotic expansions
in interior homogenization theory. The talk will be aimed at a general
audience. I will explain some recent progress about the continuity
properties of the homogenized problem which displays a sharp contrast
between the case of linear and nonlinear interior equations. This is based
on joint work with Inwon Kim.