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Symmetrization in analysis

Author: Albert Baernstein, II; D Drasin; Richard Laugesen
Publisher: Cambridge : Cambridge University Press, [2019]. ©2019
Series: New mathematical monographs, 36.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version: Baernstein, Albert, II, 1941-2014.
Symmetrization in analysis.
Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2019
(DLC) 2018043277
(OCoLC)1057241274
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Albert Baernstein, II; D Drasin; Richard Laugesen
ISBN: 9781139020244 1139020242 9781108615112 1108615112
OCLC Number: 1089758043
Description: 1 online resource (xviii, 473 pages).
Contents: Cover; Half-title; Series information; Title page; Copyright information; Contents; Notation; Foreword; Preface; Introduction; 1 Rearrangements; 1.1 The Distribution Function; 1.2 The Decreasing Rearrangement; 1.3 Induced Measures; 1.4 Measure Preserving Transformations; 1.5 Nonatomic Measure Spaces; 1.6 Symmetric Decreasing Rearrangement on R[sup(n)]; 1.7 Polarization on R[sup(n)]; 1.8 Convergence Theorems for Rearrangements; 1.9 Notes and Comments; 2 Main Inequalities on R[sup(n)]; 2.1 Convex and AL Functions; 2.2 Main Inequalities for Two-Point Symmetrization 3.4 Sobolev Spaces W[sup(1,p)](R[sup(n)])3.5 Weak Compactness; 3.6 Symmetrization Decreases the p-Dirichlet Integral in W[sup(1,p)](R[sup(n)]); 3.7 Continuity and Discontinuity of the Symmetric Decreasing Rearrangement Operator; 3.8 Notes and Comments; 4 Geometric Isoperimetric and Sharp Sobolev Inequalities; 4.1 Hausdorff Measures, Area Formula, and the Gauss-Green Theorem; 4.2 Functions of Bounded Variation in R[sup(n)]; 4.3 Isoperimetric Inequalities for Perimeter and Hausdorff Measure; 4.4 Isoperimetric Inequalities for Minkowski Content; 4.5 Coarea Formula 5.8 Notes and Comments6 Steiner Symmetrization; 6.1 Definition of Steiner Symmetrization; 6.2 Steiner Counterparts for Results in Chapter 1; 6.3 Steiner Analogues for Two Simple Polarization Results; 6.4 Certain Integral Functionals Increase or Decrease under Steiner Symmetrization; 6.5 Steiner Symmetrization Decreases the Modulus of Continuity; 6.6 Steiner Symmetrization Decreases Dirichlet Integrals; 6.7 Proof of Lemma 6.18; 6.8 Steiner Symmetrization Decreases p-Dirichlet Integrals in W[sup(1,p)](R[sup(n)]); 6.9 Steiner Symmetrization Decreases Surface Area
Series Title: New mathematical monographs, 36.
Responsibility: Albert Baernstein II ; with David Drasin, Richard Laugesen.

Abstract:

Many extremal problems in mathematics and physics have symmetric solutions whose analytic and geometric properties often have elegant formulations. This book develops and applies symmetrization  Read more...

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'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 Read more...

 
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