Suppose $B_r\subset \mathbb{R}^2$ is a hemidisc, i.e., $x^2+y^2 \leq r^2, y\geq 0$. Is there a regularity result of the type $\Vert \psi \Vert_{W^{2,p}(B_{1/2})} \leq C (\Vert \psi \Vert_{L^p(B_{1})} + \Vert \Delta \psi \Vert_{L^p(B_1)}) $?
What about similar Schauder estimates ?
No, you need some information on what $\psi$ does on the real line. A counterexample for your estimate is given by the bounded harmonic function $$ \psi(x,y)=\arctan\frac xy, $$ which does not even belong to $W^{1,p}(B_{1/2})$ for $p\ge 2$.
To see that Schauder estimates fail, you can similarly use the counterexample $$\psi(z)=\mathrm{Re} \, (z^{\alpha})$$ for $0<\alpha<1$. Then $\psi$ is harmonic and $\|\psi\|_{C^\alpha(B_1)}<\infty$, but as $\psi$ is not even differentiable at $0$, $\psi\notin C^{2,\alpha}(B_{1/2})$.
