On the $$\mathrm {L}^p$$-theory for second-order elliptic operators in divergence form with complex coefficients

Abstract

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 \(\mathrm {L}^p(\Omega )\). Additional properties like analyticity of the semigroup, \(\mathrm {H}^\infty \)-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 Hölder inequalities and Gaussian estimates for the real coefficients.