Global weak solutions for the inverse mean curvature flow in the Heisenberg group
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| Format: | Preprint |
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2024
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| _version_ | 1866911333949112320 |
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| author | Pisante, Adriano Vecchi, Eugenio |
| author_facet | Pisante, Adriano Vecchi, Eugenio |
| contents | We consider the inverse mean curvature flow (IMCF) in the Heisenberg group $(\He^n, d_\varepsilon)$, where $d_\varepsilon$ is distance associated to either $| \cdot |_\varepsilon$, $\varepsilon>0$, the natural family of left-invariant Riemannian metrics, or with their sub-Riemannian counterparts for $\varepsilon=0$. For $Ω\subseteq \He^n$ an open set with smooth boundary $Σ_0=\partial Ω$ satisfying a uniform exterior gauge-ball condition and bounded complement we show existence of a global weak IMCF of generalized hypersurfaces $\{Σ^\varepsilon_s\}_{s \geq 0} \subseteq \mathbb{H}^n$ which are level sets of a proper globally Lipschitz function with logarithmic growth at infinity. Here, both in the Riemannian and in the sub-Riemannian setting, we adopt the weak formulation introduced by Huisken and Ilmanen in \cite{HuiskenIlmanen}, following the approach in \cite{Moser} due to Moser and based on the the link between IMCF and $p$-harmonic functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_15123 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global weak solutions for the inverse mean curvature flow in the Heisenberg group Pisante, Adriano Vecchi, Eugenio Analysis of PDEs Primary 35R03, 53E10, Secondary 35B45, 35B50, 35J92 We consider the inverse mean curvature flow (IMCF) in the Heisenberg group $(\He^n, d_\varepsilon)$, where $d_\varepsilon$ is distance associated to either $| \cdot |_\varepsilon$, $\varepsilon>0$, the natural family of left-invariant Riemannian metrics, or with their sub-Riemannian counterparts for $\varepsilon=0$. For $Ω\subseteq \He^n$ an open set with smooth boundary $Σ_0=\partial Ω$ satisfying a uniform exterior gauge-ball condition and bounded complement we show existence of a global weak IMCF of generalized hypersurfaces $\{Σ^\varepsilon_s\}_{s \geq 0} \subseteq \mathbb{H}^n$ which are level sets of a proper globally Lipschitz function with logarithmic growth at infinity. Here, both in the Riemannian and in the sub-Riemannian setting, we adopt the weak formulation introduced by Huisken and Ilmanen in \cite{HuiskenIlmanen}, following the approach in \cite{Moser} due to Moser and based on the the link between IMCF and $p$-harmonic functions. |
| title | Global weak solutions for the inverse mean curvature flow in the Heisenberg group |
| topic | Analysis of PDEs Primary 35R03, 53E10, Secondary 35B45, 35B50, 35J92 |
| url | https://arxiv.org/abs/2406.15123 |