A polyconvex relaxation of three-dimensional registration model based on Saint Venant-Kirchhoff type stored energy

In this work we introduce a variational registration model admitting large deformations. The regularizer is described by the energy function of the Saint Venant-Kirchhoff material which is known to be isotropic, homogeneous and hyperelastic. The additional term controls the change of the areas and the change of the volumes. We compute the polyconvex envelope of the resulted energy, derive a relaxed problem associated to the original one and prove the existence of minimizers and Γ-convergence of the relaxed problem.