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Symmetrization in Analysis


Part of New Mathematical Monographs

Walter Hayman, David Drasin, Richard S. Laugesen
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  • Date Published: March 2019
  • availability: In stock
  • format: Hardback
  • isbn: 9780521830478

£ 134.00

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About the Authors
  • Symmetrization is a rich area of mathematical analysis whose history reaches back to antiquity. This book presents many aspects of the theory, including symmetric decreasing rearrangement and circular and Steiner symmetrization in Euclidean spaces, spheres and hyperbolic spaces. Many energies, frequencies, capacities, eigenvalues, perimeters and function norms are shown to either decrease or increase under symmetrization. The book begins by focusing on Euclidean space, building up from two-point polarization with respect to hyperplanes. Background material in geometric measure theory and analysis is carefully developed, yielding self-contained proofs of all the major theorems. This leads to the analysis of functions defined on spheres and hyperbolic spaces, and then to convolutions, multiple integrals and hypercontractivity of the Poisson semigroup. The author's 'star function' method, which preserves subharmonicity, is developed with applications to semilinear PDEs. The book concludes with a thorough self-contained account of the star function's role in complex analysis, covering value distribution theory, conformal mapping and the hyperbolic metric.

    • A self-contained treatment of all major theorems in symmetrization
    • Graduate students and researchers will benefit from this unified treatment of the major symmetrization theorems
    • Covers all the major application areas: geometry, partial differential equations, real analysis, and complex analysis
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    Reviews & endorsements

    'The book itself is a comprehensive and detailed study of the notion of symmetrization and is a welcome addition to existing literature on the subject. This book is a remarkable text collecting a variety of ideas in one unified framework; historical notes put the results in perspective. This book is very well written and will be useful to people working in a wide variety of fields.' Stefan Steinerberger, MathsSciNet

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    Product details

    • Date Published: March 2019
    • format: Hardback
    • isbn: 9780521830478
    • length: 490 pages
    • dimensions: 234 x 156 x 30 mm
    • weight: 0.82kg
    • contains: 9 b/w illus.
    • availability: In stock
  • Table of Contents

    Foreword Walter Hayman
    Preface David Drasin and Richard S. Laugesen
    1. Rearrangements
    2. Main inequalities on Rn
    3. Dirichlet integral inequalities
    4. Geometric isoperimetric and sharp Sobolev inequalities
    5. Isoperimetric inequalities for physical quantities
    6. Steiner symmetrization
    7. Symmetrization on spheres, and hyperbolic and Gauss spaces
    8. Convolution and beyond
    9. The *-function
    10. Comparison principles for semilinear Poisson PDEs
    11. The *-function in complex analysis

  • Author

    Albert Baernstein II, Washington University, St Louis
    Albert Baernstein, II was Professor in the Department of Mathematics at Washington University, St Louis before his death in 2014. He gained international renown for innovative solutions to extremal problems in complex and harmonic analysis. His invention of the 'star function' method in the 1970s prompted an invitation to the International Congress of Mathematicians held in Helsinki in 1978, and during the 1980s and 90s he substantially extended the breadth and applications of this method.


    Walter Hayman, David Drasin, Richard S. Laugesen

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