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

TL;DR

This paper studies the $L^p$-theory for second-order elliptic operators with complex coefficients, focusing on semigroup generation, analyticity, and perturbation results, using recent concepts like $p$-ellipticity and Gaussian estimates.

Contribution

It extends the $L^p$-theory for elliptic operators with complex coefficients, providing conditions for semigroup generation and perturbation results based on $p$-ellipticity.

Findings

Characterization of $p$ for which the operator generates a strongly continuous semigroup

Analysis of analyticity, $H^$-calculus, and maximal regularity properties

Perturbation results for operators with small imaginary parts of 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 H\"older inequalities and Gaussian estimates for the real coefficients.