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.