I am interested in the regularity of ellitpic equations like

$$ -\Delta u(x) +a(x) \cdot \nabla u(x) + C(x) u(x) =f(x) \quad \Omega$$ with $ \partial_\nu u =0$ on $ \partial \Omega$ where $ \Omega=(0,1) \times (0,1)$.

Assuming $a,C$ and $f$ are smooth, how much regularity can one expect for $u$. Lets assume apriori we know $u$ is bounded and say in $H^1(\Omega)$ (say its a variational solution). How about if we weaken $ f \in L^p$ for $1<p<\infty$. What are the $W^{2,p}$ (and more importantly) what Holder spaces can i expect $u$ to be in. \

On a side comment, for the dirichlet problem on a cone one can consider $v(r,\theta)= r^\alpha \sin( \beta \theta)$ to get a feel for the expected regularity. My question here is two fold. Do these little examples for the Dirichlet case really give the correct regularity one should expect?

Secondly, are there little examples like this for the Neumann problem to give one some intuition. thanks