Symmetrization And Applications

by Kesavan, S.

• ISBN13: 9789812567338
• ISBN10: 981256733X
• Format: Hardcover
• Copyright: 2006-06-30
• Publisher: World Scientific Pub Co Inc

Summary

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.

Table of Contents

Preface	p. v
Notations	p. ix
Symmetrization	p. 1
The Decreasing Rearrangement	p. 1
Some Rearrangement Inequalities	p. 8
Schwarz Symmetrization	p. 13
Variations on the Theme	p. 15
Some Classical Inequalities	p. 19
The Isoperimetric Inequality	p. 19
The Co-area Formula	p. 28
The Polya - Szego Theorem	p. 35
Sobolev's Inequality	p. 43
Comparison Theorems	p. 47
Talenti's Theorem	p. 47
The Equality Case	p. 53
Sobolev Imbeddings	p. 63
The Obstacle Problem	p. 66
Electrostatic Capacity	p. 70
The Saint Venant Problem	p. 73
Comments	p. 80
Eigenvalue Problems	p. 83
The Faber - Krahn Inequality	p. 83
The Szego - Weinberger Inequality	p. 89
Chiti's Theorem	p. 95
The Payne - Polya - Weinberger Conjecture	p. 98
Rayleigh's Conjecture for Clamped Plates	p. 105
The Buckling Problem	p. 116
Comments	p. 122
Nonlinear Problems	p. 125
Payne - Rayner Type Inequalities	p. 125
A System of Semilinear Equations	p. 132
Comments	p. 139
Bibliography	p. 141
Index	p. 147
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