Existence of minimizers for a polyconvex energy in a crystal with dislocations

We provide existence theorems in nonlinear elasticity for minimum problems modeling the deformations of a crystal with a given dislocation. A key technical difficulty is that due to the presence of a the dislocation the elastic deformation gradient cannot be in L ². Thus one needs to consider elastic energies with slow growth, to which the original results of Ball cannot be applied directly.