$\Gamma$-convergence of the $p$-Dirichlet energy for manifold-valued maps

Abstract

We prove a $\Gamma$-convergence result for the $p$-Dirichlet energy functional defined on maps from a smooth bounded domain $\Omega \subseteq {\mathbb{R}}^{n+k}$ to $\mathcal{N}$, a $(k-2)$-connected and smooth closed Riemannian manifold with Abelian fundamental group, where $n$ and $k$ are integers, $n\ge 0$, $k\ge 2$. We focus on the regime $p\to k^{-}$ under Dirichlet boundary conditions. The result provides a description of the asymptotic behavior of the $\text{topological singular sets}$ for families of $\mathcal{N}$-valued Sobolev maps which satisfy suitable energy bounds. Such topological singular sets are $n$-dimensional flat chains with coefficients in ${\pi }_{k-1}(\mathcal{N})$ endowed with a suitable norm. As a consequence of our main result, it follows that the topological singular sets of energy minimizing $p$-harmonic maps converge to a $n$-dimensional flat chain $S$ with coefficients in ${\pi }_{k-1}(\mathcal{N})$ which has finite mass and solves the Plateau problem within the homology class associated to the boundary datum.