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The stored-energy for some discontinuous deformations in nonlinear elasticity. (English) Zbl 0679.73006
Partial differential equations and the calculus of variations. Essays in Honor of Ennio De Giorgi, 767-786 (1989).
[For the entire collection see Zbl 0671.00007.]
In this work an idea, which Ennio De Giorgi demonstrated to be fruitful in various contexts, for example in the study of minimal surfaces or of \(\Gamma\)-convergence [Boll. Un. Mat. Ital., IV. Ser. 8, Suppl. to 2, 80-88 (1973; Zbl 0289.49042); Celocisl. Potoki minimal. Poverhn. 113-120 (1973; Zbl 0276.49031)], is applied to nonlinear elastic deformations which are subsets of \(R^ n\), \(n\geq 2\) (sic!). It concerns “estensione dello spazio ambiente”: just as it is efficacious first to seek surfaces of minimum area in the class BV of functions of bounded variation, rather than among the functions of class \(C^ 1\), so in problems of \(\Gamma\)-convergence or existence of a minimizer in the calculus of variations it is efficacious to extend the function space \(C^ 1\) to the Sobolev space \(H^{1,p}\), or even to \(L^ p\). (From author’s summary in Italian.)
The specific problem treated is that of cavitation produced at the centre of a unit ball due to its dilatation. Care is given to the extension of the domain of a specific stored-energy function defined on \(C^ 1\) to the larger space.
Reviewer: J.Dunwoody

74B20 Nonlinear elasticity
74S30 Other numerical methods in solid mechanics (MSC2010)
49Q05 Minimal surfaces and optimization