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We construct several examples related to the scaling limits of energyminimizers and gradient flows of surface energy functionals in heterogeneousmedia. These include both sharp and diffuse interface models. The focus is ontwo separate but related issues, the regularity of effective surface tensionsand the occurrence of zero mobility in the associated gradient flows. Onregularity we build on the theory of Goldman, Chambolle and Novaga to show thatgradient discontinuities in the surface tension are generic for sharp interfacemodels. In the diffuse interface case we only show that the laminations byplane-like solutions satisfying the strong Birkhoff property generically arenot foliations and do have gaps. On mobility we construct examples in both thesharp 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. Inthe sharp interface case, these are related to examples previously constructedby Novaga and Valdinoci for forced mean curvature flow.

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