The Direct Methods in the Calculus of Variations.- Minimax Methods.- Limit Cases of the Palais-Smale Condition.

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.

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 Â'Lectures in Mathematics, ETH ZÿrichÂ' and co-editor of the series Â'Zurich Lectures in Advanced MathematicsÂ'; moreover, he is co-editor of the journals Â'Calculus of VariationsÂ', Â'Duke Mathematical JournalÂ', and Â'International Mathematical Research NoticesÂ'.