Area-minimizing capillary cones

TL;DR

This paper constructs a family of unique, non-flat capillary cones ${\mathbf C}_{n,k,\theta}$ with bi-orthogonal symmetry in the upper half-space, for all dimensions $n\ge3$, $1\le k\le n-2$, and angles $\theta\in(0,\tfrac{\pi}{2}]$, interpolating between the one-phase cone and the half-Lawson cone as $\theta$ varies. Existence and uniqueness are reduced to solving a symmetric free-boundary ODE, with a monotone functional controlling the zero/ blow-up behavior and leading to a bijection between initial height and contact angle. The authors establish minimality in ambient dimension $8$ or higher for suitable $(n,k,\theta)$ when the angle is small, via foliations built from sub- and supersolutions; near $\theta=\tfrac{\pi}{2}$ they prove one-sided minimality and, in most cases, full minimality using capillary calibrations and barrier gluing. Consequently, the work completes the capillary regularity theory for angles near $\pi/2$ and connects to the Alt–Caffarelli one-phase problem, yielding new singular minimizing free boundaries. The results reveal a smooth interpolation of cone behavior across capillary angles and dimensions, with detailed asymptotics of the cone links and convergence to Lawson-type cones in the limit cases. Overall, the paper advances the understanding of singular minimizing capillary hypersurfaces and their relation to the one-phase free boundary problem.

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 $π/2$. 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.

Figures (3)

• Figure 1: The left graph shows solutions to \ref{['eqn:odeStar']} and the right graph shows solutions to \ref{['eqn:rescaledODE']} numerically computed. The black curve illustrates the Lawson solution, the solid lines correspond to capillary cones with angle $\theta < \tfrac{\pi}{2}$, and the dashed lines are solutions to the ODE that start above the Lawson height and do not generate geometric solutions to the capillary problem. The blue curve is close, after rescaling, to the normalized solution of the linear one-phase problem. We see that for greater $a$, the root of $f_a$ moves inward and each graph $f_a$ crosses lower graphs once. On the $\lambda$ side, as $\lambda$ interpolates from 1 to 0, the functions $f_\lambda$ foliate the space between the solutions corresponding to the Lawson cone and one-phase limit.
• Figure 2: The nested positive phases of the subsolution (blue), the solution (green), and the supersolution (gray). This picture corresponds to $(n,k) = (7,2)$, small angle $\theta$, and parameters $\alpha = -3, \beta = -2.5$, for which $(t_0, \tau, \bar{r}, A) = (0.688, 0.731, 0.932, 1.514)$. The subsolution, as a perturbation at infinity of the capillary solution, has its boundary asymptotic to the cone. The supersolution is defined piecewise, on a compact set near the singularity, and at infinity, where it is again asymptotic to the cone. Along the compact piece, the free boundary is given by $s = v(r)$, which corresponds to the curve $t = \frac{1-\frac{\bar{r}^2-r^2}{2A}}{\sqrt{\left(1-\frac{\bar{r}^{2}-r^{2}}{2A}\right)^{2}+r^{2}}}$ in the $(\rho,t)$ coordinates. The dashed lines illustrate the interface ${ |x| = \bar{r} }$ along which the two pieces of the supersolution are glued.
• Figure 3: We depict the cross-sections of the cone $\mathbf{C}$ and the upper and lower barrier surface caps $S^\pm_{{\rm cap}}$ when $|x|$ is equal to a (large) constant. The upper cap, drawn in blue, has smaller contact angle and the lower cap, drawn in green, is non-graphical and has greater contact angle. The $x$-axis represents the variable $s = |y|$, including a negative component for illustrative purposes.

Theorems & Definitions (119)

• Theorem 1.1
• Theorem 1.2
• Remark 1
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Conjecture 1.6
• Definition 1
• Definition 2
• Definition 3