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Peer-reviewedCC BYJournal of Geometric Analysis

Some Remarks on Singular Capillary Cones with Free Boundary

  • Alberto Pacati1
  • Giorgio Tortone2
  • Bozhidar Velichkov3
  1. 1Department of Mathematics, ETH Zurich, Ramistrasse 101, 8092, Zurich, Switzerland
  2. 2Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Via Carlo Alberto 10, 10124, Turin, Italy
  3. 3Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5, 56127, Pisa, Italy
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Abstract

We study singular minimizing capillary cones with free boundary. We provide a stability criterion à la Jerison–Savin and use it to prove that a minimizing capillary cone is flat when its free boundary mean curvature is non-negative and n≤ 4 n ≤ 4 , or non-positive and n≤ 6 n ≤ 6 . We also show that minimizing cones with axially symmetric free boundary are flat in dimensions up to 6 6 , and derive improved regularity results for graphical minimizing capillary hypersurfaces. The proofs rely on Simons-type inequalities for convex, homogeneous, symmetric functions of the principal curvatures, coupled with a boundary identity intrinsic to the capillary setting.

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Cite

BibTeXRIS
BibTeX
@article{pacati2026some,
  title = {{Some Remarks on Singular Capillary Cones with Free Boundary}},
  author = {Alberto Pacati and Giorgio Tortone and Bozhidar Velichkov},
  journal = {Journal of Geometric Analysis},
  year = {2026},
  month = sep,
  doi = {10.1007/s12220-026-02616-5},
  url = {https://doi.org/10.1007/s12220-026-02616-5},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Some Remarks on Singular Capillary Cones with Free Boundary
AU  - Alberto Pacati
AU  - Giorgio Tortone
AU  - Bozhidar Velichkov
PY  - 2026
DA  - 2026/09/22
JO  - Journal of Geometric Analysis
DO  - 10.1007/s12220-026-02616-5
UR  - https://doi.org/10.1007/s12220-026-02616-5
AB  - We study singular minimizing capillary cones with free boundary. We provide a stability criterion à la Jerison–Savin and use it to prove that a minimizing capillary cone is flat when its free boundary mean curvature is non-negative and n≤ 4 n ≤ 4 , or non-positive and n≤ 6 n ≤ 6 . We also show that minimizing cones with axially symmetric free boundary are flat in dimensions up to 6 6 , and derive improved regularity results for graphical minimizing capillary hypersurfaces. The proofs rely on Simons-type inequalities for convex, homogeneous, symmetric functions of the principal curvatures, coupled with a boundary identity intrinsic to the capillary setting.
SN  - 1050-6926
SN  - 1559-002X
ER  -