# elliptic regularity of Neumann problem on Square

I asked a similar question the other day, but I will be more precise now.
Consider $\Omega:=(0,1 ) \times (0,1)$ and consider
$$- u_{xx}(x,y) - u_{yy}(x,y) + a(x) u_x + b(y) u_y + u = f(x,y) \mbox{ in } \Omega$$ with $\partial_\nu u=0$ on $\partial \Omega$ (assume one has some sort of variational solution $u \in H^1(\Omega)$. Suppose $a(x),b(y)$ are smooth in $\Omega$ and assume $f$ is Holder continuous on $\Omega$. \

By looking at the even extension of $u$ (across the various boundaries and which then makes one look at odd extensions of $a$ and $b$ which have jump discontinuities) i seem to be able to prove that $u \in C^{2,\alpha}( \overline{\Omega})$. I am curious whether this result is true or ? So my question is: does this regularity seem correct or is it known one does not have this regularity at the corners? (the result suprised me a bit but I am also not familiar with what to expect with Neumann boundary conditions).
thanks a bunch for any comments.