This, the fourth edition of Stuwe's book on the calculus of variations, surveys new developments in this exciting field. It also gives a concise introduction to variational methods. In particular it includes the proof for the convergence of the Yamabe flow and a detailed treatment of the phenomenon of blow-up. Recently discovered results for backward bubbling in the heat flow for harmonic maps or surfaces are discussed. A number of changes have been made throughout the text.

Autorentext

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».

Zusammenfassung

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.

Inhalt
The Direct Methods in the Calculus of Variations.- Lower Semi-Continuity.- Constraints.- Compensated Compactness.- The Concentration-Compactness Principle.- Ekeland's Variational Principle.- Duality.- Minimization Problems Depending on Parameters.- Minimax Methods.- The Finite Dimensional Case.- The Palais-Smale Condition.- A General Deformation Lemma.- The Minimax Principle.- Index Theory.- The Mountain Pass Lemma and its Variants.- Perturbation Theory.- Linking.- Parameter Dependence.- Critical Points of Mountain Pass Type.- Non-Differentiable Functionals.- Ljusternik-Schnirelman Theory on Convex Sets.- Limit Cases of the Palais-Smale Condition.- Pohozaev's Non-Existence Result.- The Brezis-Nierenberg Result.- The Effect of Topology.- The Yamabe Problem.- The Dirichlet Problem for the Equation of Constant Mean Curvature.- Harmonic Maps of Riemannian Surfaces.- Appendix A.- Appendix B.- Appendix C.- References.- Index.