$L^p$ bounds for square roots of elliptic systems on open sets

We show Lp-estimates for square roots of second order elliptic systems L in divergence form on open sets in Rd 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. Our range is optimal, and upper and lower endpoints are sharp if they do not coincide with 1 or $\infty$