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\(\mathcal{R}\)-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type
Robert Denk, University of Regensburg, Germany, Matthias Hieber, University of Darmstadt, Germany, and Jan Prüss, University of Halle, Germany

Memoirs of the American Mathematical Society
2003; 114 pp; softcover
Volume: 166
ISBN-10: 0-8218-3378-2
ISBN-13: 978-0-8218-3378-0
List Price: US$61
Individual Members: US$36.60
Institutional Members: US$48.80
Order Code: MEMO/166/788
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The property of maximal \(L_p\)-regularity for parabolic evolution equations is investigated via the concept of \(\mathcal R\)-sectorial operators and operator-valued Fourier multipliers. As application, we consider the \(L_q\)-realization of an elliptic boundary value problem of order \(2m\) with operator-valued coefficients subject to general boundary conditions. We show that there is maximal \(L_p\)-\(L_q\)-regularity for the solution of the associated Cauchy problem provided the top order coefficients are bounded and uniformly continuous.


Graduate students and research mathematicians interested in differential equations.

Table of Contents

  • Introduction
  • Notations and conventions
\(\mathcal R\)-Boundedness and Sectorial Operators
  • Sectorial operators
  • The classes \({\mathcal{BIP}}(X)\) and \(\mathcal H^\infty(X)\)
  • \(\mathcal R\)-bounded families of operators
  • \(\mathcal R\)-sectorial operators and maximal \(L_p\)-regularity
Elliptic and Parabolic Boundary Value Problems
  • Elliptic differential operators in \(L_p(\mathbb{R}^n;E)\)
  • Elliptic problems in a half space: General Banach spaces
  • Elliptic problems in a half space: Banach spaces of class \(\mathcal{HT}\)
  • Elliptic and parabolic problems in domains
  • Notes
  • References
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