Symmetrization and Applications
Series: Series in Analysis -- v.3; Series in analysis
Publisher: World Scientific Publishing Company
Published: 2006
Place of publication: Singapore
Language: eng
Format: [electronic resource]
Number of pages: 162
ISBN-13: 9789812773937
Description: 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 applicat.
Notes: Description based upon print version of record.
Table of contents: title: Preface; Notations; Contents; 1. Symmetrization; 1.1 The Decreasing Rearrangement; 1.2 Some Rearrangement Inequalities; 1.3 Schwarz Symmetrization; 1.4 Variations on the Theme; 2. Some Classical Inequalities; 2.1 The Isoperimetric Inequality; 2.2 The Co-area Formula; 2.3 The Polya; title: Szego Theorem; 2.4 Sobolev's Inequality; 3. Comparison Theorems; 3.1 Talenti's Theorem; 3.2 The Equality Case; 3.3 Sobolev Imbeddings; 3.4 The Obstacle Problem; 3.5 Electrostatic Capacity; 3.6 The Saint Venant Problem; 3.7 Comments; 4. Eigenvalue Problems; 4.1 The Faber; title: Krahn Inequality; title: 4.2 The Szego; title: Weinberger Inequality4.3 Chiti's Theorem; 4.4 The Payne; title: Polya; title: Weinberger Conjecture; 4.5 Rayleigh's Conjecture for Clamped Plates; 4.6 The Buckling Problem; 4.7 Comments; 5. Nonlinear Problems; 5.1 Payne; title: Rayner Type Inequalities; 5.2 A System of Semilinear Equations; 5.3 Comments; Bibliography; Index