Let us consider the basic linear elliptic PDE $$ \mathrm{div} (A\,\mathrm{grad}\,u) + bu = f, $$ with $f\in L^p,$ $A,b$ uniformly bounded. Do we have, for a weak solution $u\in W^{1,p}(\Omega')$, $$ (A\,\mathrm{grad}\,u)\in (W^{1,p})^d(\Omega),\ \mbox{ in }\Omega\subset\subset\Omega'\subset\mathbb{R}^d? $$ or at least do we have $$ \mathbf{n}\cdot(A\,\mathrm{grad}\,u)\in L^{p}(\partial\Omega)? $$ This seems to me true at least if $A$ is piecewise Lipschitz uniformly because across an interface where $A$ has a jump, the transmission condition $A\,\mathrm{grad}\,u$ still has matched traces.
My final goal is to know under what least conditions, we have Green's representation formula in the sense of Lebesgue integrals: $$ u(\mathbf{y})=\int_{\partial\Omega}\frac{\partial G(\cdot,\mathbf{y})} {\partial \mathbf{n}_A}u- \frac{\partial u}{\partial \mathbf{n}_A}G(\cdot,\mathbf{y})~dS(\mathbf{x}) +\int_{\Omega}fG(\cdot,\mathbf{y})~d\mathbf{x}, $$ for $\mathbf{y}\in\Omega$ and $\frac{\partial v}{\partial\mathbf{n}_A}=\mathbf{n}\cdot(A\,\mathrm{grad}\,v).$
Edit: I'm still waiting a counter-example to the above guess when $A$ is bounded only.