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 $bigl (- abla cdot mu abla +1bigr)^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W^{1,p}_D(Omega)$ and $L^p(Omega)$, for $p in {]1,2[}$ if one presupposes that this isomorphism holds true for $p=2$. The domain $Omega$ 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 $overline {partial Omega setminus D}$ the existence of bi-Lipschitzian boundary charts is required.