Abstract
1. Introduction. It might be said that the use of weak compactness in the calculus of variations is implicitly contained in the classical method of proving existence theorems by selecting weakly convergent subsequences. In recent years the use of weak topology in the calculus of variations has become a good deal more explicit and systematic. It is based on the following facts:
I. In any topological space E a real valued function f defined on a compact subset A takes a (absolute) minimum in some point of A if it is lower semi-continuous, and also a maximum if it is continuous.
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Bibliography
T. Ando, On gradient mappings in Banach spaces, Proc. Amer. Math. Soc. 12(1961), 297–299.
N. Bourbaki, Eléments de mathématique, livre V, Espaces vecto-riels topologiques, Paris, Hermann and Co., 1953.
F.E. Browder, Variational methods for nonlinear elliptic eigenvalue problems. Bull. Am. Math. Soc. 71(1965), 176é183.
F.E. Browder, Remarks on the direct methods of the calculus of variations, Archive for Rational Mechanics and Analysis, 20(1965), 251–258.
N. Dunford and J. T. Schwartz, Linear operators Part I, General theory, Interscience Publishers, New York, 1958.
G. Fichera, Semicontinuity of multiple integrals in ordinary form, Archive for Rational Mechanics and Analysis, 17(1964), 339–352.
J. Gil de Lamadrid, On finite dimensional approximations of mappings in Banach spaces, Proc. Am. Math. Soc. 13(1962), 163–168.
J. Leray and J. Schauder, Topologie et équations fonctionnelles, An-nales Scientifiques de l'Ecole normale supérieure (3), 51(1934), 45–78.
E. H. Rothe, Gradient mappings and extrema in Banach spaces, Duke Math. J., 15(1948), 421–431.
A note on the Banach spaces of Calkin and Morrey, Pacific J. Math., 3(1953), 493é499.
An existence theorem in the calculus of variations based on Sobolev's imbedding theorems, Archive for Rational Mechanics and Analysis 21(1966), 151–162.
V.I. Smirnov, A course of higher mathematics, vol. V, Pergamon Press, 1964 (translated from the Russian edition of 1960).
S.L. Sobolev, Applications of functional analysis in mathematical physics, Amer. Math. Soc. 1963 (translated from the Russian edition of 1950).
C. de la Valine Poussin, Intégrales de Lebesgue, fonctions d'ensemble, classes de Baire, Paris, Gauthiers-Villars, 1916.
M. M. Vainberg, Variational methods for the study of non-linear operators, Holden-Day, 1964(translated from the Russian edition of 1956). The University of Michigan Ann Arbor, Michigan
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Rothe, E.H. (2010). Weak Topology and Calculus of Variations. In: Conti, R. (eds) Calculus of Variations, Classical and Modern. C.I.M.E. Summer Schools, vol 39. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11042-9_6
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DOI: https://doi.org/10.1007/978-3-642-11042-9_6
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