WIAS Preprint No. 2590, (2019)

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


  • ter Elst, A. F. M.
  • Haller-Dintelmann, Robert
  • Rehberg, Joachim
  • Tolksdorf, Patrick
    ORCID: 0000-0002-8395-4111

2010 Mathematics Subject Classification

  • 35J15 47D06 47B44


  • Divergence form operators on open sets, p-ellipticity, sectorial, operators, analytic semigroups, maximal regularity, reverse Hölder inequalities, Gaussian estimates, De Giorgi estimates




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 Lp(Ω). Additional properties like analyticity of the semigroup, H-calculus and maximal regularity arealso 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.

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