  In this note we study periodic homogenization of Dirichlet problem for
divergence type elliptic systems when both the coefficients and the boundary
data are oscillating. One of the key difficulties here is the determination of
the fixed boundary data corresponding to the limiting (homogenized) problem.
This issue has been addressed in recent papers by D. G\'{e}rard-Varet and N.
Masmoudi, and by C. Prange, however, not much is known about the regularity of
this fixed data. The main objective of this note is to initiate a study of this
problem, and to prove several regularity results in this connection.
