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.



Let $\mathrm{div}\,(A\,\mathrm{grad}\,u) + b u = f$. Is $(A\,\mathrm{grad}\,u)$ weakly differentiable?
As far as I know, there is no such results if you only assume $A\in L^\infty(\Omega)$.


But if you assume that the domain $\Omega$ is partitioned into two subdomains $\Omega\_1\cup\Omega\_2$ by a smooth interface $\Gamma$ inside $\Omega$, and in each subdomain you assume that $\,A\in W^{1,\infty}(\Omega\_1)\cap W^{1,\infty}(\Omega\_2)$ ($\,A$ may be discontinuous across the interface), then the solution of
$$
\left\{\begin{array}{ll}
\nabla\cdot(A\nabla u)=f &\mbox{in}~~\Omega,\\
u=0 &\mbox{on}~~\Omega,
\end{array}\right.
$$
with $f\in L^p(\Omega)$ satisfies 
$$
\|u\|\_{ W^{2,p}(\Omega\_1)}+\|u\|\_{W^{2,p}(\Omega\_2)}\leq C\|f\|\_{L^p(\Omega)} 
$$ 
and the trace of ${\bf n}\cdot A\nabla u$ on the two sides of the interface coincide, with ${\bf n}\cdot A\nabla u\in L^p(\partial\Omega)$.


If $A\in C^{1+\alpha}(\overline\Omega\_1)\cap C^{1+\alpha}(\overline\Omega\_2)$, then
$$
\|u\|\_{ C^{2+\alpha}(\overline\Omega\_1)}+\|u\|\_{C^{2+\alpha}(\overline\Omega\_2)}\leq C(\|f\|\_{C^\alpha(\overline\Omega\_1)}+\|f\|\_{C^\alpha(\overline\Omega\_2)}) .
$$ 


These classical results probably can be found in the following papers


"Gradient estimates for the perfect conductivity problem" by Ellen Shiting Bao, Yanyan Li and Biao Yin


"Gradient Estimates for the Perfect and Insulated Conductivity Problems with Multiple Inclusions" by Ellen Shiting Bao, Yanyan Li and Biao Yin


