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

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

Mathematical Notes

Paperback Published: 9th May 2004
ISBN: 9780691119533
Number Of Pages: 224
For Ages: 22+ years old

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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 Schrodinger 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.

"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

Background materialp. 1
Riemannian geometryp. 1
Basics in nonlinear analysisp. 7
The model equationsp. 13
Palais-Smale sequencesp. 14
Strong solutions of minimal energyp. 17
Strong solutions of high energiesp. 19
The case of the spherep. 23
Blow-up theory in Sobolev spacesp. 25
The H[subscript 1][superscript 2]-decomposition for Palais-Smale sequencesp. 26
Subtracting a bubble and nonnegative solutionsp. 32
The De Giogri-Nash-Moser iterative scheme for strong solutionsp. 45
Exhaustion and weak pointwise estimatesp. 51
Weak pointwise estimatesp. 52
Exhaustion of blow-up pointsp. 54
Asymptotics when the energy is of minimal typep. 67
Strong convergence and blow-upp. 68
Sharp pointwise estimatesp. 72
Asymptotics when the energy is arbitraryp. 83
A fundamental estimate : 1p. 88
A fundamental estimate : 2p. 143
Asymptotic behaviorp. 182
The Green's function on compact manifoldsp. 201
Coercivity is a necessary conditionp. 209
Bibliographyp. 213
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780691119533
ISBN-10: 0691119538
Series: Mathematical Notes
Audience: Tertiary; University or College
For Ages: 22+ years old
Format: Paperback
Language: English
Number Of Pages: 224
Published: 9th May 2004
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 1.63
Weight (kg): 0.32