Weiss monotonicity and capillary hypersurfaces

TL;DR

This paper demonstrates that as the capillary angle approaches zero, the capillary area-density converges to Weiss energy density, leading to angle-independent curvature estimates and regularity of capillary minimizers.

Contribution

It establishes the convergence of capillary area-density to Weiss energy density in the zero-angle limit and derives regularity results for capillary minimizers.

Findings

Capillary area-density converges to Weiss energy density as the capillary angle tends to zero.

Angle-independent curvature estimates are obtained for capillary minimizers.

Regularity results for capillary minimizers are established in the zero-angle limit.

Abstract

Previous work of the authors established the rigorous limiting behavior of minimizing capillary surfaces to minimizers of the Alt--Caffarelli functional as the capillary angle tends to zero. We prove here that in this limit, the capillary area-density converges to the Weiss energy density. We apply this to obtain angle-independent curvature estimates and regularity results for capillary minimizers.