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Singular minimisers in the calculus of variations: A degenerate form of cavitation. (English) Zbl 0769.49030
The paper refers to a class of variational problems whose minimizers are linked with the mathematical model of cavitation. If \(B=\{x\in R^ 3: | x|<1\}\) is the domain occupied by an elastic body and \(u: B\to R^ 3\) a deformation of \(B\), the energy integral \[ E_ \alpha=\int_ B \bigl\{\textstyle{{1\over 2}}|\nabla u|^ 2+\alpha \text{det }\nabla u\bigr\}dx,\quad\alpha>0 \] is considered. The minimizers of \(E_ \alpha\) in the class of radial maps are studied. Results regarding the existence of minimizers, the stability of the radial minimizers with respect to smooth variations and stability of the radial cavity maps with respect to variations in the hole shape are given.
Reviewer: C.Simionescu

MSC:
49N60 Regularity of solutions in optimal control
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