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

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 2014 theory of Chambolle, Goldman, 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, that is, there is pinning at every direction. In the sharp interface case, these are related to examples previously constructed for forced mean curvature flow in Novaga and Valdinoci's 2011 paper.