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Introduction to the calculus of variations. (Introduction au calcul des variations.) (French) Zbl 0757.49001

Cahiers Mathématiques de l’École Polytechnique Fédérale de Lausanne. 3. Lausanne: Presses Polytechniques et Universitaires Romandes. xiii, 213 p. (1992).
The book has seven chapters: in the introduction various examples of problems of the calculus of variations are given, in chapter 1 the author introduces the function spaces used afterwards and some notions of convex analysis. In the second chapter he presents for the minimum of an integral functional the classical conditions of Euler-Lagrange, Hamilton- Jacobi, Weierstrass and Hilbert. In the following chapter the author exposes the direct method for the existence of a minimum in some Sobolev space for a convex and coercive integrand. The regularity results in particular for unidimensional integrals are exposed in chapter 4. In chapter 5 the author studies the problem of minimal surfaces, where he cannot use results of chapter 4: he examines in detail the bidimensional case, giving some generalities on surfaces and he exposes the Douglas theorem for the Plateau problem. The last chapter is reserved for isoperimetric inequalities in two or more dimensions. The work is a rigorous introduction to the calculus of variations for graduate students.
Reviewer: G.Bottaro (Genova)

MSC:

49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
49J52 Nonsmooth analysis
49Q05 Minimal surfaces and optimization
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