On p-elliptic divergence form operators and holomorphic semigroups

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.