Pinning of interfaces in random media

TL;DR

This paper proves the existence of stationary supersolutions for a curvature-sensitive interface model in random media, revealing persistent hysteresis effects even under slow loading conditions.

Contribution

It establishes the existence of stationary supersolutions in a curvature-sensitive interface model and its linearized form, demonstrating persistent hysteresis phenomena.

Findings

Existence of stationary positive supersolutions at non-zero load.

Hysteresis persists even under slow loading conditions.

Linear velocity in the model is proportional to the driving force.

Abstract

For a model for the propagation of a curvature sensitive interface in a time independent random medium, as well as for a linearized version which is commonly referred to as Quenched Edwards-Wilkinson equation, we prove existence of a stationary positive supersolution at non-vanishing applied load. This leads to the emergence of a hysteresis that does not vanish for slow loading, even though the local evolution law is viscous (in particular, the velocity of the interface in the model is linear in the driving force).