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Operator-valued Symbols for Elliptic and Parabolic Problems on Wedges

  • Jan Prüss
  • Gieri Simonett
Part of the Operator Theory: Advances and Applications book series (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.

Keywords

Parabolic Problem Free Boundary Problem Sectorial Operator Maximal Regularity Kernel Operator 
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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Jan Prüss
    • 1
  • Gieri Simonett
    • 2
  1. 1.Fachbereich Mathematik und InformatikMartin-Luther-Universität Halle-WittenbergHalleGermany
  2. 2.Department of MathematicsVanderbilt UniversityNashvilleUSA

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