Consider $\;-\nabla\cdot(a_i\nabla u_i)=f$, $I=1,2$ on a sufficiently smooth domain $\Omega$. We have that $\;f\in L_2$, $\;a_i$ are smooth and $\;a_i(x)\leq a_0 <0$.
I would like to prove the stability with respect to $a_i$ in the form $$\|\nabla(u_1-u_2)\|\leq\frac{C}{a_0^2}\max_{x\in\Omega}{|a_1(x)-a_2(x)|}\|f\|.$$
I would very much appreciate any hint or link to a similar problem!
Test the difference of the defining equations for $u_1$ and $u_2$ with $u_1-u_2$ to obtain $$ \int_\Omega a_1 |\nabla (u_1-u_2)|^2 = - \int_\Omega (a_1-a_2)\nabla u_2\cdot \nabla (u_1-u_2). $$ You should be able to proceed from here.