The occurrence of surface tension gradient discontinuities and zero mobility for Allen-Cahn and curvature flows in periodic media

TL;DR

This paper investigates the regularity of effective surface tensions and the occurrence of zero mobility in gradient flows within heterogeneous media, revealing generic discontinuities and pinning phenomena in both sharp and diffuse interface models.

Contribution

It demonstrates that gradient discontinuities are typical in sharp interface models and shows the existence of zero mobility scenarios in both sharp and diffuse models, extending previous work.

Findings

Gradient discontinuities are generic for sharp interface models.

Laminate solutions in diffuse models often are not foliations and have gaps.

Examples of zero mobility and pinning at every direction are constructed.

Abstract

We construct several examples related to the scaling limits of energy minimizers and gradient flows of surface energy functionals in heterogeneous media. These include both sharp and diffuse interface models. The focus is on two separate but related issues, the regularity of effective surface tensions and the occurrence of zero mobility in the associated gradient flows. On regularity we build on the theory of Goldman, Chambolle and Novaga to show that gradient discontinuities in the surface tension are generic for sharp interface models. In the diffuse interface case we only show that the laminations by plane-like solutions satisfying the strong Birkhoff property generically are not foliations and do have gaps. On mobility we construct examples in both the sharp and diffuse interface case where the homogenization scaling limit of the $L^2$ gradient flow is trivial, i.e. there is pinning at every direction. In the sharp interface case, these are related to examples previously constructed by Novaga and Valdinoci for forced mean curvature flow.