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Erratum and addendum to “Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity”. (English) Zbl 0712.73009
Summary: [Concerns the authors’ article, ibid. 106, No.2, 97-159 (1989; Zbl 0677.73014).]
Our proof of Theorem 1 on p. 107 is not correct since the claim (iii) of Proposition 1 is false. Moreover it is not clear whether the space \(Cart(\Omega,{\mathbb{R}}^ N)\) in Definition 1 is closed. Because of this, to minimize functionals in \(Cart(\Omega,{\mathbb{R}}^ N)\) becomes unreasonable.
Our idea was to consider the smallest closed set with respect to weak convergence in \({\mathcal C}\) containing the graphs of regular mappings. According to that, the definition of \(Cart(\Omega,{\mathbb{R}}^ N)\) has to be changed.

MSC:
74B20 Nonlinear elasticity
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49Q20 Variational problems in a geometric measure-theoretic setting
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
Citations:
Zbl 0677.73014
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References:
[1] S. Banach, Théorie des opérations linéaires, Warsawa, 1932.
[2] P. J. Ciarlet & J. Nečas, Unilateral problems in nonlinear, three-dimensional elasticity, Arch. Rational Mech. Anal. 97 (1987) 171–188. · Zbl 0628.73043
[3] M. Giaquinta, G. Modica, & J. Souček, Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity, Archive for Rational Mech. Anal. 106 (1989) 97–159. · Zbl 0677.73014
[4] I. P. Natanson, Theory of functions of a real variable, Vol. 2, New York, 1964. · Zbl 0156.29202
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