I need a pretty standard interior regularity result for a second order elliptic operator of the form $$ -\nabla^b \cdot (A(x) \nabla^b v)+c v=f, \qquad \nabla^b=\nabla+ib(x) $$ where $A(x)$ is a bounded uniformly strictly positive symmetric matrix, $b(x)$ takes values in $R^n$ and $c$ is real valued. If the coefficients are smooth it is trivial to prove that if $f\in L^2_{loc}$ then $v\in H^2_{loc}$.
The twist is that I need to assume as little regularity as possible on the coefficients, and precisely in the divergence form above. I would like to assume $b$ in the Lorentz space $L^{n,\infty}$ and $c$ in $L^{n/2,\infty}$, possibly with some smallness assumption on the negative part of $c$. Concerning $A(x)$ I think that continuity should be enough (see e.g. Theorem 17.1.1 in Hormander III which is pretty close to what I need), but I am ok with locally Lipschitz $A(x)$. Note that if you expand the divergence form you get some derivatives of $b$, but I would rather not assume anything on the derivatives of $b$.
Now, it should be straightforward to write a proof but I am pretty sure that a good reference exists where this problem is studied in detail with sharp results. Any suggestion?
