2021/08/01 by William M. Feldman, Feldman, William M, Peter S. Morfe +1 · 1 citation
Computer Science · Materials Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Solidification and crystal growth phenomena #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2108.00558
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 L2 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.