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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.28668 |
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| _version_ | 1866916056352686080 |
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| author | Martino, Dorian Mazowiecka, Katarzyna Rodiac, Rémy |
| author_facet | Martino, Dorian Mazowiecka, Katarzyna Rodiac, Rémy |
| contents | Let $n\geq 3$ and let $Ω\subset \mathbb{R}^n$ be a $\mathcal{C}^1$ bounded domain which is diffeomorphic to a ball. We investigate here the problem of finding critical points of the $n$-energy in the space $\mathcal{I}=\{v\in W^{1,n}(Ω,\mathbb{R}^n) ; \ |\mathrm{tr}_{|\partial Ω}v|=1\}$. Maps in $\mathcal{I}$ have a well-defined topological degree on $\partial Ω$ but this degree is not continuous for the weak convergence in $W^{1,n}$. Hence finding critical points with prescribed degrees results in a problem of lack of compactness. We first prove that minimizers of the $n$-energy exist only when $Ω$ is a round ball and when the prescribed degree is $-1,0$ or $1$. We then develop a mountain pass approach for the $(n+α)$-energies and study the convergence, when $α$ goes to zero, of the resulting critical points via a bubbling analysis. We exclude the existence of bubbles in the case where $Ω$ is close to a ball by proving an energy gap result for free boundary $n$-harmonic maps from $\mathbb{B}^n$ to $\mathbb{B}^n$. We thus obtain the existence of critical points of the $n$-energy with prescribed degree $1$ when $Ω$ is close to a ball. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28668 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Min-max $n$-harmonic maps of degree 1 with free-boundary into $\mathbb{S}^{n-1}$ in almost round balls Martino, Dorian Mazowiecka, Katarzyna Rodiac, Rémy Analysis of PDEs Let $n\geq 3$ and let $Ω\subset \mathbb{R}^n$ be a $\mathcal{C}^1$ bounded domain which is diffeomorphic to a ball. We investigate here the problem of finding critical points of the $n$-energy in the space $\mathcal{I}=\{v\in W^{1,n}(Ω,\mathbb{R}^n) ; \ |\mathrm{tr}_{|\partial Ω}v|=1\}$. Maps in $\mathcal{I}$ have a well-defined topological degree on $\partial Ω$ but this degree is not continuous for the weak convergence in $W^{1,n}$. Hence finding critical points with prescribed degrees results in a problem of lack of compactness. We first prove that minimizers of the $n$-energy exist only when $Ω$ is a round ball and when the prescribed degree is $-1,0$ or $1$. We then develop a mountain pass approach for the $(n+α)$-energies and study the convergence, when $α$ goes to zero, of the resulting critical points via a bubbling analysis. We exclude the existence of bubbles in the case where $Ω$ is close to a ball by proving an energy gap result for free boundary $n$-harmonic maps from $\mathbb{B}^n$ to $\mathbb{B}^n$. We thus obtain the existence of critical points of the $n$-energy with prescribed degree $1$ when $Ω$ is close to a ball. |
| title | Min-max $n$-harmonic maps of degree 1 with free-boundary into $\mathbb{S}^{n-1}$ in almost round balls |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.28668 |