Continuity and Discontinuity of the Boundary Layer Tail
Continuity and Discontinuity of the Boundary Layer Tail
复制标题
边界层尾部的连续性和不连续性
DOI：
10.24033/asens.2338
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Inwon C. Kim
中科院分区：
文献类型：
--
作者：
W. M. Feldman;Inwon C. Kim
我们研究了振荡狄利克雷边界数据问题的均质边界数据 $\overline{g}$ 的连续性特性。我们证明，对于通用的非旋转不变算子和边界数据，$\overline{g}$ 在每个有理方向上都是不连续的。特别是，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，我们显示 $\overline{g}$ 的 H\"{o}lder 连续模量。当算子是线性时，我们显示 $\overline{g}$ 是 H\"{o}lder-$\frac{1}{d}$ 直至对数因子。证明基于对 $\overline{g}$ 在有理方向上的极限行为的新几何观察，简化为均匀算子投影的一类二维问题。
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 H\"{o}lder modulus of continuity for $\overline{g}$. When the operator is linear we show that $\overline{g}$ 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 $\overline{g}$ at rational directions, reducing to a class of two dimensional problems for projections of the homogenized operator.