0
$\begingroup$

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, but no answer.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Thanks! Is the Hilbert transform of the Cantor-Lebesgue function continuous? $\endgroup$ – Andrew Feb 13 '15 at 19:56
  • $\begingroup$ The Cantor-Lebesgue function is Hoelder continuous, and Hilbert transform preserves Hoelder continuity. $\endgroup$ – Michael Renardy Feb 13 '15 at 20:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.