Limit shapes of local minimizers for the Alt-Caffarelli energy functional in inhomogeneous media

TL;DR

This paper investigates the structure of local minimizers for the Alt-Caffarelli energy in periodic media, revealing the existence of pinned slopes, their regularity properties, and bounds on limit shapes, with implications for contact lines and surface phenomena.

Contribution

It introduces new characterizations of pinning intervals and limit shapes for the Alt-Caffarelli problem in inhomogeneous media, including regularity results and bounds in two and higher dimensions.

Findings

Existence of an interval of effective pinned slopes in each direction.

Regularity properties of the pinning interval depend on the normal direction.

Bounds on the class of limit shapes of local minimizers, especially in 2D.

Abstract

This paper considers the Alt-Caffarelli free boundary problem in a periodic medium. This is a convenient model for several interesting phenomena appearing in the study of contact lines on rough surfaces, pinning, hysteresis and the formation of facets. We show the existence of an interval of effective pinned slopes at each direction $e \in S^{d-1}$. In $d=2$ we characterize the regularity properties of the pinning interval in terms of the normal direction, including possible discontinuities at rational directions. These results require a careful study of the families of plane-like solutions available with a given slope. Using the same techniques we also obtain strong, in some cases optimal, bounds on the class of limit shapes of local minimizers in $d=2$, and preliminary results in $d \geq 3$.