Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45)

Olivier Druet; Emmanuel Hebey; Frédéric Robert

Published by Princeton University Press 2004

Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45)

Olivier Druet , Emmanuel Hebey and Frédéric Robert

Volume 45 in the series Mathematical Notes

https://doi.org/10.1515/9781400826162

Overview
Contents

About this book

Elliptic equations of critical Sobolev growth have been the target of investigation for decades because they have proved to be of great importance in analysis, geometry, and physics. The equations studied here are of the well-known Yamabe type. They involve Schrödinger operators on the left hand side and a critical nonlinearity on the right hand side.

A significant development in the study of such equations occurred in the 1980s. It was discovered that the sequence splits into a solution of the limit equation--a finite sum of bubbles--and a rest that converges strongly to zero in the Sobolev space consisting of square integrable functions whose gradient is also square integrable. This splitting is known as the integral theory for blow-up. In this book, the authors develop the pointwise theory for blow-up. They introduce new ideas and methods that lead to sharp pointwise estimates. These estimates have important applications when dealing with sharp constant problems (a case where the energy is minimal) and compactness results (a case where the energy is arbitrarily large). The authors carefully and thoroughly describe pointwise behavior when the energy is arbitrary.

Intended to be as self-contained as possible, this accessible book will interest graduate students and researchers in a range of mathematical fields.

Author / Editor information

Olivier Druet is Researcher at CNRS, Ecole Normale Supérieure de Lyon. Emmanuel Hebey is Professor at Université de Cergy-Pontoise. Frédéric Robert is Associate Professor at Université de Nice Sophia-Antipolis.

Reviews

"This is an important and original work. It develops critical new ideas and methods for the analysis of elliptic PDEs on compact manifolds, especially in the framework of the Yamabe equation, critical Sobolev embedding, and blow-up techniques. This volume will have an important influence on current research."—William Beckner, University of Texas at Austin

Topics

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	v
Preface Requires Authentication Unlicensed Licensed Download PDF	vii
Chapter 1. Background Material Requires Authentication Unlicensed Licensed Download PDF	1
Chapter 2. The Model Equations Requires Authentication Unlicensed Licensed Download PDF	13
Chapter 3. Blow-up Theory in Sobolev Spaces Requires Authentication Unlicensed Licensed Download PDF	25
Chapter 4. Exhaustion and Weak Pointwise Estimates Requires Authentication Unlicensed Licensed Download PDF	51
Chapter 5. Asymptotics When the Energy Is of Minimal Type Requires Authentication Unlicensed Licensed Download PDF	67
Chapter 6. Asymptotics When the Energy Is Arbitrary Requires Authentication Unlicensed Licensed Download PDF	83
Appendix A. The Green’s Function on Compact Manifolds Requires Authentication Unlicensed Licensed Download PDF	201
Appendix B. Coercivity Is a Necessary Condition Requires Authentication Unlicensed Licensed Download PDF	209
Bibliography Requires Authentication Unlicensed Licensed Download PDF	213

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