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Über dieses Buch

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 third edition gives a survey on new developments in the field. References have been updated and a small number of mistakes have been rectified.



Chapter I. The Direct Methods in the Calculus of Variations

Many problems in analysis can be cast into the form of functional equations F(u) = 0, the solution u being sought among a class of admissible functions belonging to some Banach space V.
Michael Struwe

Chapter II. Minimax Methods

In the preceding chapter we have seen that (weak sequential) lower semi-continuity and (weak sequential) compactness of the sub-level sets of a functional E on a Banach space V suffice to guarantee the existence of a minimizer of E.
Michael Struwe

Chapter III. Limit Cases of the Palais-Smale Condition

Condition (P.-S.) may seem rather restrictive. Actually, as Hildebrandt [4; p. 324] records, for quite a while many mathematicians felt convinced that inspite of its success in dealing with one-dimensional variational problems like geodesics (see Birkhoff’s Theorem I.4.4, for example, or Palais’ [3] work on closed geodesics), the Palais-Smale condition could never play a role in the solution of “interesting” variational problems in higher dimensions.
Michael Struwe


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