边界层尾部的连续性和不连续性
Continuity and Discontinuity of the Boundary Layer Tail
摘要：
研究了振荡Dirichlet边界数据问题的均匀化边界数据$\overline{g}$的连续性.我们表明，对于一个通用的非旋转不变的运营商和边界数据，$\overline{g}$是不连续的，在每个合理的方向。特别地，这意味着Choi和Kim的连续性条件本质上是尖锐的。另一方面，当这个条件成立时，我们证明了$\overline{g}$的Hölder连续模。当算子是线性算子时，我们证明了$\overline{g}$是Hölder- $\frac{1}{d}$的对数因子.证明是基于一个新的几何观察的限制行为的$\overline{g}$在合理的方向，减少到一类二维问题的投影的均匀化运营商。
英文摘要：
We investigate the continuity properties of the homogenized boundary data $\overline{g}$ for oscillating Dirichlet boundary data problems. We show that, for a generic non-rotation-invariant operator and boundary data, $\overline{g}$ 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 $\overline{g}$. When the operator is linear we show that $\overline{g}$ is Hölder-$\frac{1}{d}$ up to a logarithmic factor. The proofs are based on a new geometric observation on the limiting behavior of $\overline{g}$ at rational directions, reducing to a class of two dimensional problems for projections of the homogenized operator.
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