Let $\Omega=\mathbb{D}\cap\{ (x,y)\, \vert\, y>0\}$, $I=(-1,1)\times \{0\}$ and $A=\partial\Omega\setminus I$. Let $Q\in L^1(\Omega)$, and $R\in C^\infty\_{loc}(I)$.


I am looking to the following problem


$$
 \left\{
 \begin{aligned}
 \Delta \psi = Q & \hbox{ in } \Omega \\
 \psi = 0 & \hbox{ on } A \\
 \partial\_{\nu}\psi = R & \hbox{ on } I \\
 \end{aligned}
 \right.
$$


What is the best regularity I can expect on $\psi$? is it at least continuous? This case doesn't seem so classical, I found nothing precise in Gilbarg & Trudinger for instance, and perhaps the fact my domain get corners could be problematic.... any precise reference would be welcome.


Thanks in advance


%%%%%%%%%%%%%%


If Needed, but the main assumption are above:


I know a bit more on $Q$ and $R$, for instance I have
$$ \vert Q(z)\vert \leq \frac{C}{(1-\vert z\vert)^2}$$
and
$$ \vert R(z)\vert \leq \frac{C}{(1-\vert z\vert)}$$


but of cours no $L^\infty$ bound.



Existence en regularity of elliptic PDE with mixed boundary
The continuity of $\psi$ up to $\partial \Omega$ is false without more control on $R$. 


Consider for example the harmonic function that is $1$ on the upper half-circle and $-1$ on the lower half-circle. A model for the behavior near e.g. the lower left corner is the (zero-homogeneous) angle function $\frac{2}{\pi} \Im\, \log(z + 1)$, which on $I$ has normal derivative blowing up like inverse of distance to the corner. One can compute the exact solution using complex analysis:
$$\psi = \frac{2}{\pi}\Im\,\log\left(\frac{1+z}{1-z}\right) = \frac{2}{\pi}\tan^{-1}\left(\frac{2y}{1-|z|^2}\right),$$
which satisfies
$$Q= 0, \quad R = \psi\_y(x,\,0) = \frac{4}{\pi(1-x^2)} \leq \frac{C}{1-|x|}.$$


