Continuity and Discontinuity of the Boundary Layer Tail

TL;DR

This paper studies the continuity properties of homogenized boundary data in oscillating boundary problems, revealing generic discontinuity at rational directions and establishing conditions for Hölder continuity.

Contribution

It demonstrates that the homogenized boundary data is generically discontinuous at rational directions and provides Hölder continuity results under certain conditions, especially for linear operators.

Findings

Homogenized boundary data is discontinuous at every rational direction for generic operators.

Hölder continuity of the boundary data is established when a specific condition holds.

For linear operators, the boundary data is Hölder-1/d up to a logarithmic factor.

Abstract

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.