Rigidity of critical points of hydrophobic capillary functionals

2025/09/26 by Antonio De Rosa, De Rosa, Antonio, Robin Neumayer +3 · 1 citation
Engineering · #49Q10 #49Q20 #53A10 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Field-Flow Fractionation Techniques

paper · pdf · doi:10.48550/arxiv.2509.22532

openalex publication_date 2025/09/26 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

We prove the rigidity, among sets of finite perimeter, of volume-preserving critical points of the capillary energy in the half space, in the case where the prescribed interior contact angle is between 90^∘ and 120^∘. No structural or regularity assumption is required on the finite perimeter sets. Assuming that the ``tangential'' part of the capillary boundary is Hⁿ-null, this rigidity theorem extends to the full hydrophobic regime of interior contact angles between 90^∘ and 180^∘. Furthermore, we establish the anisotropic counterpart of this theorem under the assumption of lower density bounds.

Citations

• Existence and regularity of min-max anisotropic minimal hypersurfaces
• Regularity of minimal surfaces with capillary boundary conditions
• Allard-Type Regularity for Varifolds with Prescribed Contact Angle
• Local Minimizers of the Anisotropic Isoperimetric Problem on Closed Manifolds
• Alexandrov-type theorem for singular capillary CMC hypersurfaces in the half-space
• Rigidity and compactness with constant mean curvature in warped product manifolds
• The double and triple bubble problem for stationary varifolds: the convex case
• Heintze-Karcher inequality and capillary hypersurfaces in a wedge
• The anisotropic Min-Max theory: Existence of anisotropic minimal and CMC surfaces
• The Area Blow Up set for bounded mean curvature submanifolds with\n respect to elliptic surface energy functionals
• Minimization of anisotropic energies in classes of rectifiable varifolds
• Rectifiability of varifolds with locally bounded first variation with\n respect to anisotropic surface energies
• Minkowski formulae and Alexandrov theorems in spacetime
• Regularity of free boundaries in anisotropic capillarity problems and\n the validity of Young's law
• Constant mean curvature surfaces in warped product manifolds
• Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

Cited by

• Variational properties of the total inverse mean curvature in the plane under boundary constraints

Related

• Weiss monotonicity and capillary hypersurfaces
• The isoperimetric inequality for the capillary energy outside convex sets
• Capillary surfaces: stability, index and curvature estimates
• The isoperimetric inequality for the capillary energy outside convex cylinders
• Some rigidity results on compact hypersurfaces with capillary boundary in Hyperbolic space