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Free, publicly-accessible full text available February 13, 2025
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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
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.L^2 -
The
Ising model of statistical physics has served as a keystone example of phase transitions, thermodynamic limits, scaling laws, and many other phenomena and mathematical methods. We introduce and explore anIsing game , a variant of the Ising model that features competing agents influencing the behavior of the spins. With long-range interactions, we consider a mean-field limit resulting in a nonlocal potential game at the mesoscopic scale. This game exhibits a phase transition and multiple constant Nash-equilibria in the supercritical regime. Our analysis focuses on a sharp interface limit for which potential minimizing solutions to the Ising game concentrate on two of the constant Nash-equilibria. We show that the mesoscopic problem can be recast as a mixed local/nonlocal space-time Allen-Cahn type minimization problem. We prove, using a Γ-convergence argument, that the limiting interface minimizes a space-time anisotropic perimeter type energy functional. This macroscopic scale problem could also be viewed as a problem of optimal control of interface motion. Sharp interface limits of Allen-Cahn type functionals have been well studied. We build on that literature with new techniques to handle a mixture of local derivative terms and nonlocal interactions. The boundary conditions imposed by the game theoretic considerations also appear as novel terms and require special treatment. -
Abstract We apply new results on free boundary regularity to obtain a quantitative convergence rate for the shape optimizers of the first Dirichlet eigenvalue in periodic homogenization. We obtain a linear (with logarithmic factors) convergence rate for the optimizing eigenvalue. Large scale Lipschitz free boundary regularity of almost minimizers is used to apply the optimal homogenization theory in Lipschitz domains of Kenig et al. A key idea, to deal with the hard constraint on the volume, is a combination of a large scale almost dilation invariance with a selection principle argument.