On sequences of maps with finite energies in trace spaces between manifolds

We deal with mappings defined between Riemannian manifolds that belong to trace spaces of Sobolev functions. Such mappings are equipped with a natural energy, equivalent to the fractional norm. We study the class of Cartesian currents that arise as weak limits of sequences of mappings with equibounded energies. Under suitable topological assumptions on the domain and target manifolds, we prove a density property of graphs of smooth maps. As a consequence, we discuss the corresponding relaxed energy. For mappings with values into the sphere, an explicit formula for the relaxed energy is obtained.