The square root problem for second order, divergence form operators with mixed boundary conditions on $L^p$

We show that, under general conditions, the operator $\bigl (-\nabla \cdot \mu \nabla +1\bigr)^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W_D^{1,p}(\mathrm{\Omega })$ and $L^p(\mathrm{\Omega })$, for $p \in {]1,2[}$ifonepresupposesthatthisisomorphismholdstruefor$p=2$. The domain $\mathrm{\Omega }$ is assumed to be bounded, the Dirichlet part $D$ of the boundary has to satisfy the well-known Ahlfors-David condition, whilst for the points from $\overline{\mathrm{\partial }\mathrm{\Omega }\setminus D}$ the existence of bi-Lipschitzian boundary charts is required.