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.
A nonlocal approximation of the area in codimension two
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For $s\in (0,1)$ we introduce a notion of fractional $s$-mass on $(n-2)$-dimensional closed, orientable surfaces in $\R^n$. Moreover, we prove its $\Gamma$-convergence, with respect to the flat topology, and pointwise convergence to the $(n-2)$-dimensional area.
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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.