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

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 Schr?dinger 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. 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. Intended To Be As Self-contained As Possible, This Accessible Book Will Interest Graduate Students And Researchers In A Range Of Mathematical Fields.--book Jacket. Ch. 1. Background Material -- 1.1. Riemannian Geometry -- 1.2. Basics In Nonlinear Analysis -- Ch. 2. Model Equations -- 2.1. Palais-smale Sequences -- 2.2. Strong Solutions Of Minimal Energy -- 2.3. Strong Solutions Of High Energies -- 2.4. Case Of The Sphere -- Ch. 3. Blow-up Theory In Sobolev Spaces -- 3.1. H[subscript 1][superscript 2]-decomposition For Palais-smale Sequences -- 3.2. Subtracting A Bubble And Nonnegative Solutions -- 3.3. De Giogri-nash-moser Iterative Scheme For Strong Solutions -- Ch. 4. Exhaustion And Weak Pointwise Estimates -- 4.1. Weak Pointwise Estimates -- 4.2. Exhaustion Of Blow-up Points -- Ch. 5. Asymptotics When The Energy Is Of Minimal Type -- 5.1. Strong Convergence And Blow-up -- 5.2. Sharp Pointwise Estimates -- Ch. 6. Asymptotics When The Energy Is Arbitrary -- 6.1. Fundamental Estimate : 1 -- 6.2. Fundamental Estimate : 2 -- 6.3. Asymptotic Behavior -- App. A. Green's Function On Compact Manifolds -- App. B. Coercivity Is A Necessary Condition. Olivier Druet, Emmanuel Hebey, Frédéric Robert. Includes Bibliographical References (p. [213]-218). This work develops critical new ideas and methods for the analysis of elliptic PDEs on compact Riemannian manifolds, especially in the framework of the Yamabe equation, critical Sobolev embedding and blow-up techniques (asymptotic analysis).