Take

- $\Omega\subset R^n$ with smooth boundary (take a ball for example)
- a function $f\in L^{\infty}(\Omega)$ with support strictly contained in $\Omega$ and with $\int _{\Omega} f \; dx=0$
- a scalar coefficient $d\in L^{\infty}(\Omega)$ with $d(x)>\alpha >0 \ \forall x\in \Omega$

and consider the weak equation $-div (d\nabla u)=f$ with Neumann boundary condition $d\; \nabla \cdot n_{\Omega}=0 $.

What can be said about the regularity of $q=-d\; \nabla u$?

Does it belong just to $[L^2(\Omega)]^n$?

Elliptic Equations of the Second Order. $\endgroup$