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Blow-up theory for elliptic PDEs in Riemannian geometry
Authors: --- ---
ISBN: 1282087231 1282935372 9786612935374 9786612087233 1400826160 9781400826162 0691119538 9780691119533 9781282087231 9781282935372 6612935375 6612087234 Year: 2004 Publisher: Princeton, N.J. Princeton University Press

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Abstract

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 1980's. 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.

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