Symmetrization in Analysis

Develops the modern theory of symmetrization including applications to geometry, PDEs, and real and complex analysis.

Albert Baernstein II (Author)

9780521830478, Cambridge University Press

Hardback, published 14 March 2019

490 pages, 9 b/w illus.
23.4 x 15.6 x 3 cm, 0.82 kg

'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

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.

Foreword Walter Hayman
Preface David Drasin and Richard S. Laugesen
Introduction
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
References
Index.

Subject Areas: Geometry [PBM], Calculus & mathematical analysis [PBK]