Variational Methods

Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems

By: Michael Struwe

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Hardcover | 13 June 2008 | Edition Number 4

At a Glance

Hardcover
324 Pages

Edition Type
Revised

Dimensions(cm)
23.5 x 15.24 x 2.54

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$229.00

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Hilbert's talk at the second International Congress of 1900 in Paris marked the beginning of a new era in the calculus of variations. A development began which, within a few decades, brought tremendous success, highlighted by the 1929 theorem of Ljusternik and Schnirelman on the existence of three distinct prime closed geodesics on any compact surface of genus zero, and the 1930/31 solution of Plateau's problem by Douglas and Rado. The book gives a concise introduction to variational methods and presents an overview of areas of current research in the field.

The fourth edition gives a survey on new developments in the field. In particular it includes the proof for the convergence of the Yamabe flow and a detailed treatment of the phenomenon of blow-up. Also the recently discovered results for backward bubbling in the heat flow for harmonic maps or surfaces are discussed. Aside from these more significant additions, a number of smaller changes throughout the text have been made and the references have been updated.

Industry Reviews

From the reviews of the fourth edition: "The fourth edition of Michael Struwe,s book Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems was published in 2008, 18 years after the first edition. ... The bibliography alone would make it a valuable reference as it contains nearly 500 references. ... Struwe,s book is addressed to researchers in differential geometry and partial differential equations." (John D. Cook, MAA Online, January, 2009) "This is the fourth edition of a standard reference work on direct methods in the calculus of variations. ... The book contains a wealth of important results that would otherwise be hard to find in one single place." (M. Kunzinger, Monatshefte fur Mathematik, Vol. 160 (4), July, 2010)

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The Direct Methods in the Calculus of Variations	p. 1
Lower Semi-continuity	p. 2
Constraints	p. 13
Compensated Compactness	p. 25
The Concentration-Compactness Principle	p. 36
Ekeland's Variational Principle	p. 51
Duality	p. 58
Minimization Problems Depending on Parameters	p. 69
Minimax Methods	p. 74
The Finite Dimensional Case	p. 74
The Palais-Smale Condition	p. 77
A General Deformation Lemma	p. 81
The Minimax Principle	p. 87
Index Theory	p. 94
The Mountain Pass Lemma and its Variants	p. 108
Perturbation Theory	p. 118
Linking	p. 125
Parameter Dependence	p. 137
Critical Points of Mountain Pass Type	p. 143
Non-differentiable Functionals	p. 150
Ljusternik-Schnirelman Theory on Convex Sets	p. 162
Limit Cases of the Palais-Smale Condition	p. 169
Pohozaev's Non-existence Result	p. 170
The Brezis-Nirenberg Result	p. 173
The Effect of Topology	p. 183
The Yamabe Problem	p. 194
The Dirichlet Problem for the Equation of Constant Mean Curvature	p. 220
Harmonic Maps of Riemannian Surfaces	p. 231
App. A	p. 263
App. B	p. 268
App. C	p. 274
References	p. 277
Index	p. 301
Table of Contents provided by Blackwell. All Rights Reserved.

Variational Methods

Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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