The study of isoperimetric inequalities involves a fascinating interplay of analysis, geometry and the theory of partial differential equations. Several conjectures have been made and while many have been resolved, a large number still remain open. One of the principal tools in the study of isoperimetric problems, especially when spherical symmetry is involved, is Schwarz symmetrization, which is also known as the spherically symmetric and decreasing rearrangement of functions. The aim of this book is to give an introduction to the theory of Schwarz symmetrization and study some of its applications. The book gives an modern and up-to-date treatment of the subject and includes several new results proved recently. Effort has been made to keep the exposition as simple and self-contained as possible. A knowledge of the existence theory of weak solutions of elliptic partial differential equations in Sobolev spaces is, however, assumed. Apart from this and a general mathematical maturity at the graduate level, there are no other prerequisites.
Author(s): S. Kesavan
Series: Series in Analysis
Publisher: World Scientific Publishing Company
Year: 2006
Language: English
Pages: 162
Contents ......Page 12
Preface ......Page 6
Notations ......Page 10
1.1 The Decreasing Rearrangement ......Page 14
1.2 Some Rearrangement Inequalities ......Page 21
1.3 Schwarz Symmetrization ......Page 26
1.4 Variations on the Theme ......Page 28
2.1 The Isoperimetric Inequality ......Page 32
2.2 The Co-area Formula ......Page 41
2.3 The Polya - Szego Theorem ......Page 48
2.4 Sobolev's Inequality ......Page 56
3.1 Talenti's Theorem ......Page 60
3.2 The Equality Case ......Page 66
3.3 Sobolev Imbeddings ......Page 76
3.4 The Obstacle Problem ......Page 79
3.5 Electrostatic Capacity ......Page 83
3.6 The Saint Venant Problem ......Page 86
3.7 Comments ......Page 93
4.1 The Faber - Krahn Inequality ......Page 96
4.2 The Szego - Weinberger Inequality ......Page 102
4.3 Chiti's Theorem ......Page 108
4.4 The Payne - Polya - Weinberger Conjecture ......Page 111
4.5 Rayleigh's Conjecture for Clamped Plates ......Page 118
4.6 The Buckling Problem ......Page 129
4.7 Comments ......Page 135
5.1 Payne - Rayner Type Inequalities ......Page 138
5.2 A System of Semilinear Equations ......Page 145
5.3 Comments ......Page 152
Bibliography ......Page 154
Index ......Page 160
