Book contents
- Frontmatter
- Contents
- Notation
- Foreword
- Preface
- Introduction
- 1 Rearrangements
- 2 Main Inequalities on Rn3
- 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
3 - Dirichlet Integral Inequalities
Published online by Cambridge University Press: 22 February 2019
Book contents
- Frontmatter
- Contents
- Notation
- Foreword
- Preface
- Introduction
- 1 Rearrangements
- 2 Main Inequalities on Rn3
- 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
Summary
Chapter 3develops the basic Dirichlet integral inequalities for symmetric decreasing rearrangement. The main result is the decrease of the integral of the p-th power of the gradient(or p-Dirichlet integral) of a function under symmetric decreasing rearrangement. Background material on Sobolev spaces and functional analysis is included as needed to study the continuity of the symmetric decreasing rearrangement in various Sobolev spaces.
- Type
- Chapter
- Information
- Symmetrization in Analysis , pp. 92 - 118Publisher: Cambridge University PressPrint publication year: 2019