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.