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

TL;DR

This paper establishes a $ ext{Gamma}$-convergence result for the $p$-Dirichlet energy of manifold-valued maps, describing the asymptotic behavior of topological singular sets as $p$ approaches $k$ from below, with implications for energy minimizers.

Contribution

It provides a novel $ ext{Gamma}$-convergence analysis for the $p$-Dirichlet energy on manifold-valued maps, characterizing the limit of topological singular sets as flat chains with finite mass.

Findings

Topological singular sets converge to flat chains with finite mass.

Energy minimizing $p$-harmonic maps' singular sets solve the Plateau problem.

The result describes the asymptotic behavior of singular sets as $p o k^-$.

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 $\mathscr{N}$, a $(k-2)$-connected and smooth closed Riemannian manifold with Abelian fundamental group, where $n$ and $k$ are integers, $n \geq 0$, $k \geq 2$. We focus on the regime $p \to~k^-$ under Dirichlet boundary conditions. The result provides a description of the asymptotic behavior of the $\textit{topological singular sets}$ for families of $\mathscr{N}$-valued Sobolev maps which satisfy suitable energy bounds. Such topological singular sets are $n$-dimensional flat chains with coefficients in $\pi_{k-1}(\mathscr{N})$ 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 $\pi_{k-1}(\mathscr{N})$ which has finite mass and solves the Plateau problem within the homology class associated to the boundary datum.