On the Lp-theory for second-order elliptic operators in divergence form with complex coefficients

Given a complex, elliptic coefficient function we investigate for which values of p the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly continuous semigroup on L-p (Omega). Additional properties like analyticity of the semigroup, H-infinity-calculus and maximal regularity are also discussed. Finally, we prove a perturbation result for real coefficients that gives the whole range of p's for small imaginary parts of the coefficients. Our results are based on the recent notion of p-ellipticity, reverse Holder inequalities and Gaussian estimates for the real coefficients.