Given an axially-symmetric, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(n+1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dimensional convex cone <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, we study the stability of the free-boundary minimal surface <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> obtained by intersecting <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> with a n-plane that contains the axis of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. In the case <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> is always unstable, as a special case of the vertex-skipping property that we recently proved in another article. Conversely, as soon as <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> has a sufficiently large aperture (depending on the dimension n), we show that <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> is strictly stable. For our stability analysis, we introduce a Lipschitz flow <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\Sigma _{t}[f]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Σ</mi> <mi>t</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of deformations of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> associated with a compactly-supported, scalar deformation field f, which satisfies the key property <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\partial \Sigma _{t}[f] \subset \partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <msub> <mi mathvariant="normal">Σ</mi> <mi>t</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> </mrow> <mo>⊂</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$t\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>. Then, we compute the lower-right second variation of the area of <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> 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.

Stability of axial free-boundary hyperplanes in circular cones

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2025年10月10日

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Gian Paolo Leonardi,
Giacomo Vianello

摘要

Given an axially-symmetric, $(n+1)$ $(n + 1)$ -dimensional convex cone $\Omega \subset \mathbb {R}^{n+1}$ $Ω \subset R^{n + 1}$ , we study the stability of the free-boundary minimal surface $\Sigma $ $Σ$ obtained by intersecting $\Omega $ $Ω$ with a n-plane that contains the axis of $\Omega $ $Ω$ . In the case $n=2$ $n = 2$ , $\Sigma $ $Σ$ is always unstable, as a special case of the vertex-skipping property that we recently proved in another article. Conversely, as soon as $n\ge 3$ $n \geq 3$ and $\Omega $ $Ω$ has a sufficiently large aperture (depending on the dimension n), we show that $\Sigma $ $Σ$ is strictly stable. For our stability analysis, we introduce a Lipschitz flow $\Sigma _{t}[f]$ $Σ_{t} [f]$ of deformations of $\Sigma $ $Σ$ associated with a compactly-supported, scalar deformation field f, which satisfies the key property $\partial \Sigma _{t}[f] \subset \partial \Omega $ $\partial Σ_{t} [f] \subset \partial Ω$ for all $t\in \mathbb {R}$ $t \in R$ . Then, we compute the lower-right second variation of the area of $\Sigma $ $Σ$ 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.