I have not the densely defined operator $A_D:L_2\rightarrow L_2$ acting on $A_D^{-1}$ but the differential expression (with not-smooth coefficients under differentiation): $A:H^1\rightarrow H^{-1}$. I want to say that $A$ coincides with $A_D$ on the range of $A_D^{-1}$, but I not see how to say it since I cannot define the operator by the differential expression with the Dirichlet boundary condition in the case of non-smooth coefficients.