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 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,∞).