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Problèmes de Dirichlet variationnels non linéaires. Partie 1 des comptes rendus du cours d’éťe OTAN “Variational Methods in Nonlinear Problems”. (Nonlinear variational Dirichlet problems. Part I of the Proceedings of the NATO summer school “Variational Methods in Nonlinear Problems”). (French) Zbl 0644.49001

Séminaire de Mathématiques Supérieures. Séminaire Scientifique OTAN (NATO Advanced Study Institute), 104. Université de Montréal, Département de Mathématiques et de Statistique. Montréal (Québec), Canada: Les Presses de l’Université de Montréal. 168 p.; $ 22.00 (1987).
This interesting monography contains an introduction to critical point theory and applications to the semi-linear Dirichlet problem \[ (*)\quad \ddot u(x)=D_ uF(x,u(x)),\quad u(0)=u(\pi)=0. \] The solutions of the above problem are exactly the critical points of the functional \[ f(u)=\int^{\pi}_{0}[| \dot u(x)|^ 2/2+F(x,u(x))]dx, \] defined on the usual Sobolev space \(H^ 1_ 0\). The basic tools of critical point theory (minimization, dual action, Ekeland’s variational principle, minimax method, \({\mathbb{Z}}_ 2\)-index, Morse theory) are introduced with many heuristic and historic comments and applied to problem (*).
The topological tools (Krasnoselski genus, relative homology,...) are clearly described so that basic integration theory and linear functional analysis are adequate preparation for the reading of the book.
Each chapter contains references to the corresponding problems (and difficulties) for partial differential equations. For a longer treatment of critical point theory and applications to periodic solutions of Hamiltonian systems see the forthcoming book by the author and the reviewer “Critical point theory and Hamiltonian systems”, Springer Verlag.
Reviewer: M.Willem

MSC:

49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49J20 Existence theories for optimal control problems involving partial differential equations
49J35 Existence of solutions for minimax problems
49J45 Methods involving semicontinuity and convergence; relaxation
49N15 Duality theory (optimization)
35J65 Nonlinear boundary value problems for linear elliptic equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems