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The influence of elasticity on analysis: Modern developments. (English) Zbl 0533.73001

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.

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.

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.

Reviewer: H.Troger

MSC:
74-03Historical (mechanics of deformable solids)
74B20Nonlinear elasticity
58E07Abstract bifurcation theory
58E35Variational inequalities (global problems)
49M15Newton-type methods in calculus of variations
01A60Mathematics in the 20th century
35B32Bifurcation (PDE)
47J05Equations involving nonlinear operators (general)