Let $O\subset\mathbb{R}^d$
 be a bounded domain of the class $C^{1,1}$ 
 (or $C^2$
 for simplicity). Let the operator $A_D$
 be formally given by the differential expression $A=-\mathrm{div}g(x)\nabla$ 
 with the Dirichlet boundary condition. ($A$
 means the expression, and $A_D$
 is the corresponding operator.) Here $g(x)>0$, $g,g^{−1}\in L_\infty$. 
The precise definition is given via the quadratic form on $H^1_0(O)$. 


If the matrix of coefficients $g$
 and the boundary are smooth enough, we can define $A_D$
 by the differential expression $-\mathrm{div}g(x)\nabla$ 
 on $H^2(O)\cap H^1_0(O)=\mathrm{Ran}A_D^{-1}$, so for the expression $A$
 acting on the resolvent $A^{−1}_D$
 we have
$$\Vert A A_D^{-1}\Vert _{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}=\Vert A_D A_D^{-1}\Vert _{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}=1.$$

**Question:**

Is there are any hope to prove the estimate 
$$\Vert A A_D^{-1}\Vert _{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant \mathrm{const}$$
for the operator with non-smooth coefficients in the domain of class $C^{1,1}$
 (or $C^2$
)? (Here the expression $A$ is initially considered as operator from $H^1$
 to $H^{−1}$
.) E. g., by using of mollification of coefficients.

(I am also interested in this question for strongly elliptic systems in a divergent form: $A=b(D)^∗g(x)b(D)$
, $b=\sum _{l=1}^d b_lD_l$
, $b_l$
 are constant matrices, the symbol of $b(D)$
 has maximal rank.)