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.

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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 f�r Mathematik, Vol. 160 (4), July, 2010)

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