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.

37 pages

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