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We show that, under general conditions, the operator $\bigl (-\nabla \cdot\mu \nabla +1\bigr)^{1/2}$ with mixed boundary conditions provides atopological 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 $p=2$. Thedomain $\Omega$ is assumed to be bounded, the Dirichlet part $D$ of theboundary has to satisfy the well-known Ahlfors-David condition, whilst for thepoints from $\overline {\partial \Omega \setminus D}$ the existence ofbi-Lipschitzian boundary charts is required.

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