  We show $L^p$ estimates for square roots of second order complex elliptic
systems $L$ in divergence form on open sets in $\mathbb{R}^d$ subject to mixed
boundary conditions. The underlying set is supposed to be locally uniform near
the Neumann boundary part, and the Dirichlet boundary part is Ahlfors-David
regular. The lower endpoint for the interval where such estimates are available
is characterized by $p$-boundedness properties of the semigroup generated by
$-L$, and the upper endpoint by extrapolation properties of the Lax-Milgram
isomorphism. Also, we show that the extrapolation range is relatively open in
$(1,\infty)$.
