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Coercivity and Gamma-convergence of the $p$-energy of sphere-valued Sobolev maps
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abstract
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.
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$\Gamma$-convergence of the $p$-Dirichlet energy for manifold-valued maps
As p approaches k from below, the rescaled p-Dirichlet energies of maps into a manifold Gamma-converge to the mass of the n-dimensional flat chain that solves the homological Plateau problem for the boundary datum.
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