Continuity and Discontinuity of the Boundary Layer Tail

We investigate the continuity properties of the homogenized boundary data $\bar{g}$ for oscillating Dirichlet boundary data problems. We show that, for a generic non-rotation-invariant operator and boundary data, $\bar{g}$ is discontinuous at every rational direction. In particular this implies that the continuity condition of Choi and Kim is essentially sharp. On the other hand, when this condition holds, we show a H"{o}lder modulus of continuity for $\bar{g}$. When the operator is linear we show that $\bar{g}$ is H"{o}lder-$\frac{1}{d}$ up to a logarithmic factor. The proofs are based on a new geometric observation on the limiting behavior of $\bar{g}$ at rational directions, reducing to a class of two dimensional problems for projections of the homogenized operator.