Blow-up Theory for Elliptic Pdes In Riemannian Geometry (mn-45) (ebook)

Argumento de Blow-up Theory for Elliptic Pdes In Riemannian Geometry (mn-45) (ebook)

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.0Preface 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 Solutions 45
Chapter 4. Exhaustion and Weak Pointwise Estimates 51
4.1 Weak Pointwise Estimates 52
4.2 Exhaustion of Blow-up Points 54
Chapter 5. Asymptotics When the Energy Is of Minimal Type 67
5.1 Strong Convergence and Blow-up 68
5.2 Sharp Pointwise Estimates 72
Chapter 6. Asymptotics When the Energy Is Arbitrary 83
6.1 A Fundamental Estimate: 1 88
6.2 A Fundamental Estimate: 2 143
6.3 Asymptotic Behavior 182
Appendix A. The Green's Function on Compact Manifolds 201
Appendix B. Coercivity Is a Necessary Condition 209
Bibliography 213