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Partial Differential Equations and Functional Analysis pp 189–208Cite as

Operator-valued Symbols for Elliptic and Parabolic Problems on Wedges

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 168))

Abstract

We study evolution problems of the type e sx ∂ t u+h(∂ x)u = f where h is a holomorphic function on a vertical strip around the imaginary axis, and s > 0. If P is a second-order polynomial we give a complete characterization of the spectrum of the parameter-dependent operator λe sx + P(∂ x) in L p(ℝ). We show the surprising result that the spectrum is independent of λ whenever |arg λ| < π. Moreover, we also characterize the spectrum of ∂ t e sx + P(∂ x), and we show that this operator admits a bounded H ∞-calculus. Finally, we describe applications to free boundary problems with moving contact lines, and we study the diffusion equation in an angle or a wedge domain with dynamic boundary conditions. Our approach relies on the H ∞-calculus for sectorial operators, the concept of R-boundedness, and recent results for the sum of non-commuting operators.

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Author information

Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, Martin-Luther-Universität Halle-Wittenberg, D-60120, Halle, Germany

    Jan Prüss

  2. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Gieri Simonett

Authors
  1. Jan Prüss
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  2. Gieri Simonett
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Editor information

Editors and Affiliations

  1. Technische Universiteit Delft DIAM, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    Erik Koelink , Jan van Neerven , Ben de Pagter  & Guido Sweers , ,  & 

  2. Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstrasse 8-10 / 101, 1040, Wien, Austria

    Annemarie Luger  & Harald Woracek  & 

Additional information

Dedicated to Philippe Clément

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© 2006 Birkhäuser Verlag Basel/Switzerland

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Prüss, J., Simonett, G. (2006). Operator-valued Symbols for Elliptic and Parabolic Problems on Wedges. In: Koelink, E., van Neerven, J., de Pagter, B., Sweers, G., Luger, A., Woracek, H. (eds) Partial Differential Equations and Functional Analysis. Operator Theory: Advances and Applications, vol 168. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7601-5_12

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