Regularity of Free Boundaries in Anisotropic Capillarity Problems and the Validity of Young’s Law

Abstract

Abstract. Local volume-constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a closed negligible set, showing in particular the validity of Young's law at almost every point of the free boundary. Our regularity results are not specific to capillarity problems, and actually apply to sets of finite perimeter (and thus to codimension one integer rectifiable currents) arising as minimizers in other variational problems with free boundaries.

Dimensional estimates for singular sets in geometric variational problems with free boundaries

“…By combining the results of [SSA77] for the interior situation with the ones of [DPM14] for the boundary situation, one sees that E ∩ A is (equivalent to) an open set, that A ∩ ∂E ∩ ∂H is a set of finite perimeter in ∂H, and that…”

“…In [DPM14], having in mind applications to capillarity problems and to relative isoperimetric problems, we studied the regularity of free boundaries in anisotropic geometric variational problems. The main result contained in [DPM14] asserts that free boundaries are regular outside closed sets of vanishing H n−2 -measure.…”

“…The main result contained in [DPM14] asserts that free boundaries are regular outside closed sets of vanishing H n−2 -measure. In this paper we improve upon this result by showing H n−3 -negligibility of singular sets, see Theorem 1.5 below.…”

“…The boundary case is addressed here by combining the set of ideas introduced in [SSA77] with the H n−2 -negligibility we have obtained in [DPM14] (see, in particular, Lemma 2.7 below).…”

Improved convergence theorems for bubble clusters. I. The planar case

“…For example, in [MM15] we use Theorem 3.1 and the free boundary regularity theory from [DPM14a] to obtain an improved convergence theorem for capillarity droplets in containers.…”