When investigaing the regularity of certain functions, I encountered this problem:
if $u\in H^{3/2}([0,1]\times [0,1])$, what can we say about the continuity of $\nabla u\cdot\overrightarrow{n}$ across a line segment inside the domain, where $\overrightarrow{n}$ is the unit normal ?
Without loss of generality, we formulate the problem as below:
Let $\Omega$ be the unit square in two dimensions and let $\gamma$ denote a vertical line segment inside $\Omega$.
If $u\in H^{3/2}(\Omega)$, can we deduce that $$\frac{\partial u}{\partial x}|_{\gamma^+}=\frac{\partial u}{\partial x}|_{\gamma^-} \;\;\; ?$$
The subscripts $\gamma^+$ and $\gamma^-$ denote traces from left and right sides of $\gamma$, respectively.
Here we assume that $$ \frac{\partial u}{\partial x}|_{\gamma^+}\in L^2(\gamma),\quad \frac{\partial u}{\partial x}|_{\gamma^-}\in L^2(\gamma).$$ Note that this assumption is not redundant because trace theorem does not hold for $H^{1/2}(\Omega)$ with index $1/2$.
What if the assumption above is weakened into: $$ \frac{\partial u}{\partial x}|_{\gamma^+}\in H^{-1/2}(\gamma),\quad \frac{\partial u}{\partial x}|_{\gamma^-}\in H^{-1/2}(\gamma) \;\;\; ?$$
The question was asked in MSE last year but there is still no response, so I plan to ask here.
