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

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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.

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Acknowledgements

The authors wish to thank the referee for the suggestions and additional references.

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Authors and Affiliations

  1. Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    A. F. M. ter Elst

  2. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    R. Haller-Dintelmann

  3. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany

    J. Rehberg

  4. Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55128, Mainz, Germany

    P. Tolksdorf

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  1. A. F. M. ter Elst
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  2. R. Haller-Dintelmann
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  4. P. Tolksdorf
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Correspondence to A. F. M. ter Elst.

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Part of this work is supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand. P. Tolksdorf was partially supported by the project ANR INFAMIE (ANR-15-CE40-0011).

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ter Elst, A.F.M., Haller-Dintelmann, R., Rehberg, J. et al. On the \(\mathrm {L}^p\)-theory for second-order elliptic operators in divergence form with complex coefficients. J. Evol. Equ. 21, 3963–4003 (2021). https://doi.org/10.1007/s00028-021-00711-4

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