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Zbl 0533.73001
Antman, Stuart S.
The influence of elasticity on analysis: Modern developments. (English)
[J] Bull. Am. Math. Soc., New Ser. 9, 267-291 (1983). ISSN 0273-0979; ISSN 1088-9485/e

The author shows how surprisingly many areas of modern analysis have been influenced from problems arising in elasticity. However, only the development since World War II is taken into account as according to the author only since then the interaction of nonlinear elasticity with nonlinear analysis has become of real importance. \par We only want to touch two of the many subjects mentioned in the paper. One are local and global bifurcation problems. Here on the one hand the development of singularity and catastrophe theory has provided a much deeper insight into the physical phenomena. On the other hand global problems initiated a wide range of investigations on variational methods based on functional analytic and topological techniques. The second field to be mentioned followed from the study of elastic rods and plates by Galerkin. He generalized the Rayleigh-Ritz method to obtain approximate solutions. His investigations stimulated big advances in the theory of numerical and functional analysis. Indeed, the finite element method is a version of Galerkin's method. \par This important paper clearly shows that it certainly is worthwhile for mathematicians to study elasticity and on the other hand it also shows physicists and engineers what a great variety of mathematical tools are available to help them to solve their problems.
[H.Troger]

MSC 2000:: *74-03 Historical (mechanics of deformable solids)
74B20 Nonlinear elasticity
58E07 Abstract bifurcation theory
58E35 Variational inequalities (global problems)
49M15 Methods of Newton-Raphson, Galerkin and Ritz types
01A60 Mathematics in the 20th century
35B32 Bifurcation (PDE)
47J05 Equations involving nonlinear operators (general)

Keywords: influence of elasticity upon development of mathematical analysis; variational inequalities; calculus of variations; weak convergence; development since World War II

Cited in: Zbl 0766.58002 Zbl 0555.73030 Zbl 0553.73002

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