Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) / / Olivier Druet, Frédéric Robert, Emmanuel Hebey.

• Superior document: Title is part of eBook package: De Gruyter Princeton Mathematical Notes eBook-Package 1970-2016
• VerfasserIn: Druet, Olivier,; Hebey, Emmanuel,; Robert, Frédéric,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2009]; ©2004
• Year of Publication: 2009
• Edition: Course Book
• Language: English
• Series: Mathematical Notes ; 45
• Physical Description: 1 online resource (224 p.)

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.

Table of Contents

Frontmatter -- Contents -- Preface -- Chapter 1. Background Material -- Chapter 2. The Model Equations -- Chapter 3. Blow-up Theory in Sobolev Spaces -- Chapter 4. Exhaustion and Weak Pointwise Estimates -- Chapter 5. Asymptotics When the Energy Is of Minimal Type -- Chapter 6. Asymptotics When the Energy Is Arbitrary -- Appendix A. The Green's Function on Compact Manifolds -- Appendix B. Coercivity Is a Necessary Condition -- Bibliography