On the history of variational methods of non-linear equations investigations and the contribution of Soviet scientists (1920s-1950s).

Egor Mikhailovich Bogatov

Antiquitates Mathematicae, Tome 14 (2020), pp. 3-38.

Résumé

Variational methods for solving nonlinear equations (differential, integral, etc.) are perhaps the most common methods at the present time. However, the history of their origin and development, both in the USSR and in the whole world, has not been studied enough. The author attempts to fill this gap, limiting himself mainly to the 1920s-1950s, starting with Hilbert’s works on the justification of the Dirichlet principle and with Poincaré’s work on the study of closed geodesics on convex surfaces. The article includes an analysis of the results of Soviet mathematicians - L.А. Lyusternik, L.G. Shnirelman, S.L. Sobolev, M.A. Krasnoselskii, M.M. Vainberg et al., obtained both in the field of solvability and direct methods and in the field of qualitative analysis. Their achievements are gauged in the context of the development of this topic in the works of their foreign colleagues - L. Tonelli, W. Ritz, L. Lichtenstein, G.D. Birkhoff, M. Morse, A. Hammerstein, M. Golomb, etc. Along the way, the question of the origin of Sobolev spaces and the mutual influence of functional analysis and variational methods for solving operator equations is investigated. In conclusion, the author gives an example of applying the qualitative theory of variational methods to the problems of nonlinear mechanics that were realized by the Soviet scientist I.I. Vorovich.

Détail

DOI : 10.14708/am.v14i1.6418

Classification : 01A60
Mots-clés : history of nonlinear functional analysis, variational methods, Dirichlet principle, Hammerstein equation, Ritz method, potential operators, Lyusternik-Shnirelman theory, Sobolev space, semi-continuous functionals, justification of linearization in the bif