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An existence theory for nonlinear elasticity that allows for cavitation. (English) Zbl 0836.73025
In this long paper of mathematical style the existence of minimizers in nonlinear elasticity is proved under assumptions on the stored energy that permit the formation of new holes in the body – cavitation. The corresponding fully three-dimensional problem is considered, and an additional, physically motivated, energy term proportional to the area of the boundary of the deformed body is included. This extends some work of J. Ball [Philos. Trans. R. Soc. Lond., Ser. A 306, No. 2, 557-611 (1982; Zbl 0513.73020)]and makes use of some ideas of V. Sverak [Arch. Ration. Mech. Anal. 100, No. 2, 105-127 (1988; Zbl 0659.73038)]. The minimizers obtained lie in a subclass of maps in \(W^{1,p}\), \(2 < p < 3\), that are one-to-one everywhere and preserve orientation.
Reviewer: G.A.Maugin (Paris)

MSC:
74B20 Nonlinear elasticity
49J20 Existence theories for optimal control problems involving partial differential equations
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