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Non convex integrals of the calculus of variations. (English) Zbl 0735.49002
Methods of nonconvex analysis, Lect. 1st Sess. CIME, Varenna/Italy 1989, Lect. Notes Math. 1446, 16-57 (1990).

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[For the entire collection see Zbl 0705.00022.]
These notes are a complete review on the variational problems related with integrals of the calculus of variation, whose integrand functions are non-convex. The author begins with the relation between the convexity of the integrand function and the lower semicontinuity of the functional. Then he considers the existence of minimizers for non-convex problems by presenting the principal methods and results giving many references on the subject. Moreover, he obtains some new results for a class of non- coercive and non-convex one-dimensional integrals. Finally, he considers non-convex functionals arising within non-linear elasticity, for which the existence of minimizers is still an open problem.
Reviewer: E.Mascolo

49J10 Existence theories for free problems in two or more independent variables
49J52 Nonsmooth analysis
49J45 Methods involving semicontinuity and convergence; relaxation