Given an axially-symmetric, (n+1) -dimensional convex cone Ω 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 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 Σₓf of deformations of Σ associated with a compactly-supported, scalar deformation field f, which satisfies the key property Σₓf 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.
Leonardi et al. (Fri,) studied this question.