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  3. Symmetrization in Analysis
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    • Publisher:
      Cambridge University Press
      Publication date:
      February 2019
      March 2019
      ISBN:
      9781139020244
      9780521830478
      DOI:
      https://doi.org/10.1017/9781139020244
      Dimensions:
      (228 x 152 mm)
      Weight & Pages:
      0.82kg, 490 Pages
      Dimensions:
      Weight & Pages:
      • Subjects:
      Mathematics (general), Mathematics, Differential and Integral Equations, Dynamical Systems and Control Theory, Abstract Analysis
      Series:
      New Mathematical Monographs (36)
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    Subjects:
    Mathematics (general), Mathematics, Differential and Integral Equations, Dynamical Systems and Control Theory, Abstract Analysis
    Series:
    New Mathematical Monographs (36)
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    Book description

    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.

    Reviews

    '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 Source: MathsSciNet

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