A relaxation result for a second order energy of mappings into the sphere

A relaxation problem for maps from 3-dimensional domains into the unit 2-sphere is analyzed, the energy being given in the smooth case by the integral of the modulus of the Laplacean vector. For second order Sobolev maps, a complete explicit formula of the relaxed energy is obtained. Our proof is based on the following results: minimal energy computation of maps with fixed degree, dipole-like problems, lower semicontinuity of the extended energy, and a strong approximation result on Cartesian currents.