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This paper establishes bounds on the homogenized surface tension for a heterogeneous Allen-Cahn energy functional in a periodic medium. The approach is based on relating the homogenized energy to a purely geometric variational problem involving the large scale behaviour of the signed distance function to a hyperplane in periodic media. Motivated by this, a homogenization result for the signed distance function to a hyperplane in both periodic and almost periodic media is proven. 

An overview of recent analytical developments in the study of epitaxial growth is presented. Quasistatic equilibrium is established, regularity of solutions is addressed, and the evolution of epitaxially strained elastic films is treated using minimizing movements. 

We consider a homogenization problem associated with quasi-crystalline multiple inte- grals of the form uε ∈ Lp(Ω;Rd) 7→ ˆΩ fR x, xε,uε(x) dx, where uε is subject to constant-coefficient linear partial differential constraints. The quasi-crystalline structure of the underlying composite is encoded in the dependence on the second variable of the Lagrangian, fR, and is modeled via the cut-and-project scheme that interprets the heterogeneous microstructure to be homogenized as an irrational subspace of a higher-dimensional space. A key step in our analysis is the characterization of the quasi-crystalline two-scale limits of sequences of the vector fields uε that are in the kernel of a given constant-coefficient linear partial differential operator, A, that is, Auε = 0. Our results provide a generalization of related ones in the literature concerning the A = curl case to more general differential operators A with constant coefficients, and without coercivity assumptions on the Lagrangian fR.