Injectivity and self-contact in nonlinear elasticity

Let Ω be a bounded open connected subset of ℝ³ with a sufficiently smooth boundary. The additional condition ∫ det ▽ψ dx ≦ vol ψ(Ω) is imposed on the admissible deformations ψ: -Ω → ℝ of a hyperelastic body whose reference configuration is -Ω. We show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate. We also extend J. Ball's theorems on existence to this case by establishing the existence of a minimizer of the energy in the space W^{1, p}(Ω;ℝ³), p > 3, that is injective almost everywhere. © 1987 Springer-Verlag GmbH & Co. KG.