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Quasi-L p-contractive analytic semigroups generated by elliptic operators with complex unbounded coefficients on arbitrary domains

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Abstract

Let Ω be a domain in ℝN and consider a second order linear partial differential operator A in divergence form on Ω which is not required to be uniformly elliptic and whose coefficients are allowed to be complex, unbounded and measurable. Under rather general conditions on the growth of the coefficients we construct a quasi-contractive analytic semigroup \((e^{-t A_{V}})_{t\geqslant0}\) on L 2(Ω,dx), whose generator A V gives an operator realization of A under general boundary conditions. Under suitable additional conditions on the imaginary parts of the diffusion coefficients, we prove that for a wide class of boundary conditions, the semigroup \((e^{-t A_{V}})_{t\geqslant0}\) is quasi-L p-contractive for 1<p<∞. Similar results hold for second order nondivergence form operators whose coefficients satisfy conditions similar to those on the coefficients of the operator A, except for some further requirements on the diffusion coefficients. Some examples where our results can be applied are provided.

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Notes

  1. Without loss of generality, we will assume in what follows that \(\operatorname{ess\,inf}(\operatorname{Re}a_{0}+\omega)\geqslant1\) and \(\operatorname{ess\,inf}|\beta_{k}|\geqslant1\).

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Acknowledgements

The authors are greatly indebted to a very conscientious referee for a careful reading of this paper and a number of very useful comments. The first named author expresses his sincere gratitude to Professor El-Maati Ouhabaz for helpful discussions. He also would like to thank Professor Vincenzo Vespri for kindly sending him the preprint version of [12], as well as Professor E. Brian Davies for a helpful conversation on the subject of boundary conditions.

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  1. Department of Mathematics, Faculty of Sciences of Tunis, Campus Universitaire, 1060, Tunis, Tunisia

    Sami Mourou & Mohamed Selmi

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  1. Sami Mourou
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  2. Mohamed Selmi
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Correspondence to Sami Mourou.

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Communicated by Jerome A. Goldstein.

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Mourou, S., Selmi, M. Quasi-L p-contractive analytic semigroups generated by elliptic operators with complex unbounded coefficients on arbitrary domains. Semigroup Forum 85, 5–36 (2012). https://doi.org/10.1007/s00233-012-9402-6

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