Let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $\sigma_1,\sigma_2:\Omega\to(c^{-1},c)$ measurable (for some constant $1<c<\infty$). Let $f\in H^{1/2}(\partial\Omega)=H^1(\Omega)/H^1_0(\Omega)$. Let $u_i\in H^1(\Omega)$, $i=1,2$, be the solution of $$ \begin{cases} \operatorname{div}(\sigma_i\nabla u_i)=0 & \text{in }\Omega\\ u_i=f & \text{on }\partial\Omega. \end{cases} $$ Is there an estimate like $$ \|\nabla u_1-\nabla u_2\|_{L^p} \leq \|f\|_{H^{1/2}(\partial\Omega)} C(c,\Omega,n,p)\omega(\|\sigma_1-\sigma_2\|_{?}) $$ where $C$ is a constant depending on the given parameters, $\omega$ is a modulus of continuity (depending on the same parameters) and the question mark norm is some norm? What I'm most interested in is the choice of the norm and shape of $\omega$ (linear or worse). I would like to have $p$ as large as possible ($p=\infty$ would be great), but $p=2$ would also be good.

If an estimate like this is not known (or is known not to exist), are there results in the same spirit for the stability of a solution in perturbations of the equation? It is no problem if you need to assume more regularity of $\sigma$; smoothness is ok, but analyticity would be too much.