Regularity of free boundaries in anisotropic capillarity problems and the validity of Young's law

TL;DR

This paper proves the regularity of free boundaries in anisotropic capillarity problems, establishing Young's law at most boundary points and extending the results to general variational problems with free boundaries.

Contribution

It demonstrates the regularity of free boundaries in anisotropic capillarity problems and confirms Young's law at almost all boundary points, with broader applicability to variational minimizers.

Findings

Free boundaries are regular outside a negligible set.

Young's law holds at almost every free boundary point.

Results extend to sets of finite perimeter in variational problems.

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.