Existence theorems for boundary-value problems of hyperelasticity

The variational formulation of the boundary-value problem of elastostatics for hyperelastic materials are considered. The existence of a solution on the space $W^{1,p}(\Omega,\mathbb R^3)$, $p>1$, is proved for standard outside influences under the most general assumptions on the potential with superlinear growth in the modulus of the matrix argument. Counterexamples are given showing that the condition of coercivity is best possible. In the proof of the existence theorem the weak convergence of the determinants of the gradients of the maps for the minimizing sequence is not used. This enable us to generalize significantly Ball's results. The condition of preservation of orientation (or of incompressibility) almost everywhere in the domain for a global minimizer is proved directly.