Continuity and Discontinuity of the Boundary Layer Tail

2015/02/03 by Feldman, William M., Kim, Inwon C. · 1 citation
#35B27 #35J57 #35J60 #76F40 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1502.00966

Abstract

We investigate the continuity properties of the homogenized boundary data g for oscillating Dirichlet boundary data problems. We show that, for a generic non-rotation-invariant operator and boundary data, 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 Hölder modulus of continuity for g. When the operator is linear we show that g is Hölder-(1)/(d) up to a logarithmic factor. The proofs are based on a new geometric observation on the limiting behavior of g at rational directions, reducing to a class of two dimensional problems for projections of the homogenized operator.

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• First-order homogenization

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