Area-minimizing capillary cones

TL;DR

This paper constructs new non-flat minimal capillary cones with symmetry, demonstrating their existence, uniqueness, and minimizing properties in higher dimensions, and linking them to singular free boundary solutions.

Contribution

It introduces a novel family of minimal capillary cones with symmetry, expanding the understanding of singularities in capillary hypersurfaces and free boundary problems.

Findings

Constructed non-flat minimal capillary cones with symmetry in all dimensions.

Proved existence and uniqueness of these cones via nonlinear free boundary equations.

Showed cones are minimizing in dimensions 8 and higher, revealing singularities in capillary regularity theory.

Abstract

We construct non-flat minimal capillary cones with bi-orthogonal symmetry groups for any dimension and contact angle. These cones interpolate between rescalings of a singular solution to the one-phase problem and the free-boundary cone obtained by halving a Lawson cone along a hyperplane of symmetry. The existence and uniqueness of such cones is proved by solving a nonlinear free boundary equation parametrized by the contact angle and obtaining monotonicity properties for the solutions. The constructed cones are minimizing in ambient dimension $8$ or higher, for appropriate contact angles, demonstrating that the regularity theory for minimizing capillary hypersurfaces can have singularities in codimension $7$ and completing the capillary regularity theory for contact angles near $\pi/2$. We further develop the connection between capillary hypersurfaces and solutions of the one-phase problem, consequently producing new examples of singular minimizing free boundaries for the Alt-Caffarelli functional.