We investigate the continuity properties of the homogenized boundary data (g) over bar for oscillating Dirichlet boundary data problems. The homogenized boundary condition arises as the boundary layer tail of a problem set in a half-space. The continuity properties of this boundary layer tail depending on the normal direction of the half space play an important role in the homogenization process in general bounded domains. We show that, for a generic non-rotation-invariant operator and boundary data, (g) over bar is discontinuous at every rational direction. In particular this implies that the continuity condition of Choi and Kim [16] is essentially sharp. On the other hand, when the condition of [16] holds, we show a Holder modulus of continuity for (g) over bar. When the operator is linear we show that (g) over bar is Holder-1/d up to a logarithmic factor. The proofs are based on a new geometric observation on the limiting behavior of (g) over bar at rational directions, reducing to a class of two dimensional problems for projections of the homogenized operator.
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