$Γ$-convergence of the $p$-Dirichlet energy for manifold-valued maps
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arXiv
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2025
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| author | Canevari, Giacomo Le, Van Phu Cuong Oliver-Bonafoux, Ramon Orlandi, Giandomenico |
| author_facet | Canevari, Giacomo Le, Van Phu Cuong Oliver-Bonafoux, Ramon Orlandi, Giandomenico |
| contents | We prove a $Γ$-convergence result for the $p$-Dirichlet energy functional defined on maps from a smooth bounded domain $Ω\subseteq \mathbb{R}^{n+k}$ 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 $π_{k-1}(\mathscr{N})$ 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 $π_{k-1}(\mathscr{N})$ which has finite mass and solves the Plateau problem within the homology class associated to the boundary datum. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_21257 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $Γ$-convergence of the $p$-Dirichlet energy for manifold-valued maps Canevari, Giacomo Le, Van Phu Cuong Oliver-Bonafoux, Ramon Orlandi, Giandomenico Analysis of PDEs 49Q15, 49Q20, 58E12, 58E20 We prove a $Γ$-convergence result for the $p$-Dirichlet energy functional defined on maps from a smooth bounded domain $Ω\subseteq \mathbb{R}^{n+k}$ 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 $π_{k-1}(\mathscr{N})$ 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 $π_{k-1}(\mathscr{N})$ which has finite mass and solves the Plateau problem within the homology class associated to the boundary datum. |
| title | $Γ$-convergence of the $p$-Dirichlet energy for manifold-valued maps |
| topic | Analysis of PDEs 49Q15, 49Q20, 58E12, 58E20 |
| url | https://arxiv.org/abs/2505.21257 |