Stability of axial free-boundary hyperplanes in circular cones

Given an axially-symmetric, $(n+1)$-dimensional convex cone Ω⊂ Rⁿ⁺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 ≥ 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 Σₜ[f] of deformations of $Σ$ associated with a compactly-supported, scalar deformation field f, which satisfies the key property ∂ Σₜ[f] ⊂ ∂ Ω for all t∈ 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.