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Zusatztext "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 Informationen zum Autor 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. Klappentext 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. Zusammenfassung 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. Inhaltsverzeichnis Preface vii Chapter 1. Background Material 1 1.1 Riemannian Geometry 1 1.2 Basics in Nonlinear Analysis 7 Chapter 2. The Model Equations 13 2.1 Palais-Smale Sequences 14 2.2 Strong Solutions of Minimal Energy 17 2.3 Strong Solutions of High Energies 19 2.4 The Case of the Sphere 23 Chapter 3. Blow-up Theory in Sobolev Spaces 25 3.1 The H 2/1-Decomposition for Palais-Smale Sequences 26 3.2 Subtracting a Bubble and Nonnegative Solutions 32 3.3 The De Giorgi-Nash-Moser Iterative Scheme for Strong Solution...
