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**Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems.
2nd rev. and exp. ed.**
*(English)*
Zbl 0864.49001

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 34. Berlin: Springer. xvi, 272 p. (1996).

This second edition of “Variational methods” has been substantially expanded with respect to the first one of 1990 (Zbl 0746.49010) to include some recent developments that have taken place in the meantime.

Chapter I is devoted to the so-called direct methods in the calculus of variations, from the classical lower semicontinuity approach to the more recent compensated compactness and concentration-compactness principles. A short discussion on Hardy space methods has been added in Section I.3, and there is a new section on minimization problems depending on parameters. Here the case of harmonic maps with singularities is illustrated, together with the Ginzburg-Landau approximation model.

Chapter II deals, as in the previous edition, with minimax methods; the most classical tools as the Palais-Smale condition, the mountain pass lemma, index theory, and Ljusternik-Schnirelman theory are developed. The existence result of periodic solutions of Hamiltonian systems is considerably improved in Section II.9 where the existence is shown for almost every energy level.

Also Chapter III, which deals with limit cases of the Palais-Smale condition, has been expanded; now the Yamabe problem, with the Ye result, is presented in detail in Section III.4, and some more discussions on the harmonic maps of Riemannian manifolds complete Section III.6.

The list of references is updated to include several papers which recently appeared in the field of variational problems. Even if the bibliography is not exhaustive, the reader can find here most of the articles (classic and recent) which made important developments in the theory. The selection of subjects and the very clear style of presentation, well balanced between general abstract results and applications to specific problems, make the book an excellent choice as a textbook for a course at the Ph. D. level, and for everyone who wants to approach the wide field of variational problems.

Chapter I is devoted to the so-called direct methods in the calculus of variations, from the classical lower semicontinuity approach to the more recent compensated compactness and concentration-compactness principles. A short discussion on Hardy space methods has been added in Section I.3, and there is a new section on minimization problems depending on parameters. Here the case of harmonic maps with singularities is illustrated, together with the Ginzburg-Landau approximation model.

Chapter II deals, as in the previous edition, with minimax methods; the most classical tools as the Palais-Smale condition, the mountain pass lemma, index theory, and Ljusternik-Schnirelman theory are developed. The existence result of periodic solutions of Hamiltonian systems is considerably improved in Section II.9 where the existence is shown for almost every energy level.

Also Chapter III, which deals with limit cases of the Palais-Smale condition, has been expanded; now the Yamabe problem, with the Ye result, is presented in detail in Section III.4, and some more discussions on the harmonic maps of Riemannian manifolds complete Section III.6.

The list of references is updated to include several papers which recently appeared in the field of variational problems. Even if the bibliography is not exhaustive, the reader can find here most of the articles (classic and recent) which made important developments in the theory. The selection of subjects and the very clear style of presentation, well balanced between general abstract results and applications to specific problems, make the book an excellent choice as a textbook for a course at the Ph. D. level, and for everyone who wants to approach the wide field of variational problems.

Reviewer: G.Buttazzo (Pisa)

### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

34C25 | Periodic solutions to ordinary differential equations |

35A15 | Variational methods applied to PDEs |

35F20 | Nonlinear first-order PDEs |

37J45 | Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) |

47J30 | Variational methods involving nonlinear operators |

49J10 | Existence theories for free problems in two or more independent variables |

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |