  Second order divergence form operators are studied on an open set with
various boundary conditions. It is shown that the p-ellipticity condition of
Carbonaro-Dragicevic and Dindos-Pipher implies extrapolation to a holomorphic
semigroup on Lebesgue spaces in a p-dependent range of exponents that extends
the maximal range for general strictly elliptic coefficients. Results have
immediate consequences for the harmonic analysis of such operators, including
Hoo-calculi and Riesz transforms.
