On homogenization problems with oscillating Dirichlet conditions in space-time domains

We prove the homogenization of fully nonlinear parabolic equations with periodic oscillating Dirichlet boundary conditions on certain general prescribed space-time domains. It was proved in (Feldman, J. Math. Pures Appl. 101 (2014), no. 5, 599-622; Feldman and Kim, Ann. Sci. ec. Norm. Super 50 (2017), no. 4, 1017-1064) that for elliptic equations, the homogenized boundary data exist at boundary points with irrational normal directions, and it is generically discontinuous elsewhere. However, for parabolic problems, on a flat moving part of the boundary, we prove the existence of continuous homogenized boundary data g over bar $\bar{g}$. We also show that, unlike the elliptic case, g over bar $\bar{g}$ can be discontinuous even if the operator is rotation/reflection invariant.