Some remarks on singular capillary cones with free boundary
Differential Geometry
2025-02-12 v1 Analysis of PDEs
Abstract
We study minimizing singular cones with free boundary associated with the capillarity problem. Precisely, we provide a stability criterion la Jerison-Savin for capillary hypersurfaces and show that, in dimensions up to , 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 . 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.
Keywords
Cite
@article{arxiv.2502.07697,
title = {Some remarks on singular capillary cones with free boundary},
author = {Alberto Pacati and Giorgio Tortone and Bozhidar Velichkov},
journal= {arXiv preprint arXiv:2502.07697},
year = {2025}
}