Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems

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Springer Science & Business Media, Nov 5, 2008 - Science - 302 pages
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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 Radó. 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.

 

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Contents

The Direct Methods in the Calculus of Variations
1
1 Lower Semicontinuity
2
2 Constraints
13
3 Compensated Compactness
25
4 The ConcentrationCompactness Principle
36
5 Ekelands Variational Principle
51
6 Duality
58
7 Minimization Problems Depending on Parameters
69
10 Critical Points of Mountain Pass Type
143
11 Nondifferentiable Functionals
150
12 LusternikSchnirelman Theory on Convex Sets
162
Limit Cases of the PalaisSmale Condition
169
1 Pohozaevs Nonexistence Result
170
2 The BrezisNirenberg Result
173
3 The Effect of Topology
183
4 The Yamabe Problem
194

Minimax Methods
74
2 The PalaisSmale Condition
77
3 A General Deformation Lemma
81
4 The Minimax Principle
87
5 Index Theory
94
6 The Mountain Pass Lemma and its Variants
108
7 Perturbation Theory
118
8 Linking
125
9 Parameter Dependence
137
5 The Dirichlet Problem for the Equation of Constant Mean Curvature
220
6 Harmonic Maps of Riemannian Surfaces
231
Appendix A
263
Appendix B
268
Appendix C
274
References
277
Index
301
Copyright

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About the author (2008)

Michael Struwe is full Professor of Mathematics at ETH Zurich.

Prof. Struwe was born on October 6, 1955 in Wuppertal, Germany. He studied mathematics at the University of Bonn. After receiving his doctorate in 1980, he was a member of the scientific staff in the special research sector 72 of the German research foundation and later an assistant at the Mathematical Institute of the University of Bonn. He spent extended research visits in Paris and at the ETH Zurich. In 1984 he was awarded the Felix Hausdorff Prize of the University of Bonn.

On April 1, 1986 Michael Struwe was appointed assistant professor, on October 1, 1990 associate professor and in 1993 he became full Professor of Mathematics at the ETH Zurich. From October, 2002 to September, 2004 he served as head of the ETH Mathematics Department. His research focuses on non-linear partial differential equations and the calculus of variations as well as their applications in mathematical physics and differential geometry.

In 2006 he received the Credit Suisse Award For Best Teaching.

He is editor of the series «Lectures in Mathematics, ETH Zürich» and co-editor of the series «Zurich Lectures in Advanced Mathematics»; moreover, he is co-editor of the journals «Calculus of Variations», «Duke Mathematical Journal», and «International Mathematical Research Notices».