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.
Rezensionen / Stimmen
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)
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ISBN-13
978-3-662-04194-9 (9783662041949)
DOI
10.1007/978-3-662-04194-9
Schweitzer Klassifikation
I. The Direct Methods in the Calculus of Variations.- 1. Lower Semi-Continuity.- 2. Constraints.- 3. Compensated Compactness.- 4. The Concentration-Compactness Principle.- 5. Ekeland's Variational Principle.- 6. Duality.- 7. Minimization Problems Depending on Parameters.- II. Minimax Methods.- 1. The Finite Dimensional Case.- 2. The Palais-Smale Condition.- 3. A General Deformation Lemma.- 4. The Minimax Principle.- 5. Index Theory.- 6. The Mountain Pass Lemma and its Variants.- 7. Perturbation Theory.- 8. Linking.- 9. Parameter Dependence.- 10. Critical Points of Mountain Pass Type.- 11. Non-Differentiable Functionals.- 12. Ljusternik-Schnirelman Theory on Convex Sets.- III. Limit Cases of the Palais-Smale Condition.- 1. Pohozaev's Non-Existence Result.- 2. The Brezis-Nirenberg Result.- 3. The Effect of Topology.- 4. The Yamabe Problem.- 5. The Dirichlet Problem for the Equation of Constant Mean Curvature.- 6. Harmonic Maps of Riemannian Surfaces.- Appendix A.- Sobolev Spaces.- Hölder Spaces.- Imbedding Theorems.- Density Theorem.- Trace and Extension Theorems.- Poincaré Inequality.- Appendix B.- Schauder Estimates.- Weak Solutions.- A Regularity Result.- Maximum Principle.- Weak Maximum Principle.- Application.- Appendix C.- Fréchet Differentiability.- Natural Growth Conditions.- References.
