Let $Q$ be a domain in the half-space $\mathbb R^n\cap\{x\_n>0\}$ and part of its boundary
is a domain $S$ on the hyperplane $x\_n=0$. Let $u\in C(\bar Q)\cap C^2( Q)$ satisfy $\Delta u=0$ in $Q$ and for some fixed $\bar b=(b\_1,\ldots,b\_n)$, $b\_n\ne0$,
$$
\lim\_{x\_n\to0+}\frac{ \partial u}{\partial\bar b}=f\in C(S)\quad \hbox{a.e. on $S$}.
$$
Does it follow that $\frac{ \partial u}{\partial\bar b}$ is continuous up to $S$?


A special case: if
$$
\lim\_{x\_n\to0+}\frac{ \partial u}{\partial x\_n}=0\quad \hbox{a.e. on $S$},
$$
will $u$ be smooth on $Q\!\cup\! S$? It is also enough to show that $\frac{ \partial u}{\partial x\_n}$ is continuous up to $S$.


Posted earlier on [MSE](https://math.stackexchange.com/questions/1129492/oblique-derivative-smoothness-of-harmonic-functions), but no answer.



Oblique derivative smoothness of harmonic functions
Let n=2. On the x-axis, let u be equal to the Hilbert transform of the Cantor-Lebesgue function. Then $\partial u/\partial y$ is equal to zero on every interval on which the Cantor-Lebesgue function is constant.


