$L^p$-estimates for the square root of elliptic systems with mixed boundary conditions II

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 $(1,\infty)$.