Harmonic Dipoles and the Relaxation of the Neo-Hookean Energy in 3D Elasticity

We consider the problem of minimizing the neo-Hookean energy in 3D. The difficulty of this problem is that the space of maps without cavitation is not compact, as shown by Conti & De Lellis with a pathological example involving a dipole. In order to rule out this behaviour we consider the relaxation of the neo-Hookean energy in the space of axisymmetric maps without cavitation. We propose a minimization space and a new explicit energy penalizing the creation of dipoles. This new energy, which is a lower bound of the relaxation of the original energy, bears strong similarities with the relaxed energy of Bethuel-Brezis-Helein in the context of harmonic maps into the sphere.