Coercivity and Gamma-convergence of the p-energy of sphere-valued Sobolev maps

We consider sequences of maps from an (n+m)-dimensional domain into the (n-1)-sphere, which satisfy a natural p-energy growth, as p approaches n from below. We prove that, up to subsequences, the Jacobians of such maps converge in the flat topology to an integral m-current, and that the p-energy Gamma-converges to the mass of the limit current. As a corollary, we deduce that the Jacobians of p-energy minimizing maps converge to an integral m-current that is area-minimizing in a suitable cobordism class, depending on the boundary datum. Moreover, we obtain new estimates for the minimal p-energy of maps with prescribed singularities.