H∞-Calculus for cylindrical boundary value problems

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2012
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Saal, Juergen
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Advances in Differential Equations. Khayyam Publishing. 2012, 17(7/8), pp. 767-800. ISSN 1079-9389
Zusammenfassung

In this note an R-bounded H∞-calculus for linear operators associated to cylindrical boundary value problems is proved. The obtained results are based on an abstract result on operator-valued functional calculus by N. Kalton and L. Weis; cf. [28]. Cylindrical in this context means that both domain and differential operator possess a certain cylindrical structure. In comparison to standard methods (e.g. localization procedures), our approach appears less technical and provides short proofs. Besides, we are even able to deal with some classes of equations on rough domains. For instance, we can extend the well-known (and in general sharp) range for p such that the (weak) Dirichlet Laplacian admits an H∞-calculus on Lp(Ω), from (3+ε)′<p<3+ε to (4+ε)′<p<4+ε for three-dimensional bounded or unbounded Lipschitz cylinders Ω. Our approach even admits mixed Dirichlet Neumann boundary conditions in this situation.

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ISO 690NAU, Tobias, Juergen SAAL, 2012. H∞-Calculus for cylindrical boundary value problems. In: Advances in Differential Equations. Khayyam Publishing. 2012, 17(7/8), pp. 767-800. ISSN 1079-9389
BibTex
@article{Nau2012HCalc-49220,
  year={2012},
  title={H∞-Calculus for cylindrical boundary value problems},
  url={https://projecteuclid.org/euclid.ade/1355702976},
  number={7/8},
  volume={17},
  issn={1079-9389},
  journal={Advances in Differential Equations},
  pages={767--800},
  author={Nau, Tobias and Saal, Juergen}
}
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