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Some remarks on singular capillary cones with free boundary

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arxiv 2502.07697 v1 pith:2IXAH7RO submitted 2025-02-11 math.DG math.AP

Alberto Pacati , Giorgio Tortone , Bozhidar Velichkov This is my paper · ORCID
classification math.DGmath.AP
keywords capillaryconesboundaryminimizingcriteriondimensionsfreesingular
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abstract

We study minimizing singular cones with free boundary associated with the capillarity problem. Precisely, we provide a stability criterion $\`a$ la Jerison-Savin for capillary hypersurfaces and show that, in dimensions up to $4$, minimizing cones with non-sign-changing mean curvature are flat. We apply this criterion to minimizing capillary drops and, additionally, establish the instability of non-trivial axially symmetric cones in dimensions up to $6$. The main results are based on a Simons-type inequality for a class of convex, homogeneous, symmetric functions of the principal curvatures, combined with a boundary condition specific to the capillary setting.

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  1. Weiss monotonicity and capillary hypersurfaces

    math.AP 2025-06 accept novelty 7.0 of 10

    Renormalized capillary area density converges to the Weiss energy, giving angle-independent curvature estimates and a Bernstein theorem for capillary minimizers.

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