Stability of axial free-boundary hyperplanes in circular cones
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908536268652544 |
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| author | Leonardi, Gian Paolo Vianello, Giacomo |
| author_facet | Leonardi, Gian Paolo Vianello, Giacomo |
| contents | Given an axially-symmetric, $(n+1)$-dimensional convex cone $Ω\subset \mathbb{R}^{n+1}$, we study the stability of the free-boundary minimal surface $Σ$ obtained by intersecting $Ω$ with a $n$-plane that contains the axis of $Ω$. In the case $n=2$, $Σ$ is always unstable, as a special case of the vertex-skipping property that we recently proved in another article. Conversely, as soon as $n \ge 3$ and $Ω$ has a sufficiently large aperture (depending on the dimension $n$), we show that $Σ$ is strictly stable. For our stability analysis, we introduce a Lipschitz flow $Σ_{t}[f]$ of deformations of $Σ$ associated with a compactly-supported, scalar deformation field $f$, which satisfies the key property $\partial Σ_{t}[f] \subset \partial Ω$ for all $t\in \mathbb{R}$. Then, we compute the lower-right second variation of the area of $Σ$ along the flow, and ultimately show that it is positive by exploiting its connection with a functional inequality studied in the context of reaction-diffusion problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_21205 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability of axial free-boundary hyperplanes in circular cones Leonardi, Gian Paolo Vianello, Giacomo Analysis of PDEs Classical Analysis and ODEs Given an axially-symmetric, $(n+1)$-dimensional convex cone $Ω\subset \mathbb{R}^{n+1}$, we study the stability of the free-boundary minimal surface $Σ$ obtained by intersecting $Ω$ with a $n$-plane that contains the axis of $Ω$. In the case $n=2$, $Σ$ is always unstable, as a special case of the vertex-skipping property that we recently proved in another article. Conversely, as soon as $n \ge 3$ and $Ω$ has a sufficiently large aperture (depending on the dimension $n$), we show that $Σ$ is strictly stable. For our stability analysis, we introduce a Lipschitz flow $Σ_{t}[f]$ of deformations of $Σ$ associated with a compactly-supported, scalar deformation field $f$, which satisfies the key property $\partial Σ_{t}[f] \subset \partial Ω$ for all $t\in \mathbb{R}$. Then, we compute the lower-right second variation of the area of $Σ$ along the flow, and ultimately show that it is positive by exploiting its connection with a functional inequality studied in the context of reaction-diffusion problems. |
| title | Stability of axial free-boundary hyperplanes in circular cones |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2502.21205 |