  This note is a summary of the recent paper [9]. Here, we study the
homogenization of elliptic systems with Dirichlet boundary condition, when both
the coefficients and the boundary datum are oscillating.
  In particular, in the paper [9], we showed that, the solutions converge in L2
with a power rate, and we identified the homogenized limit system and the
homogenized boundary data. Due to a boundary layer phenomenon, this homogenized
system depends in a non trivial way on the boundary. The analysis in [9]
answers a longstanding open problem, raised for instance in [4]
