Interfacial Energies on Penrose Lattices

Abstract

In this paper we prove a homogenization theorem for interfacial discrete energies defined on an a-periodic Penrose tiling in ℝ2. A general result on the homogenization of surface energies cannot be directly adapted to this case; the existence of the limit interfacial energy is therefore proved by showing some refined "quasi-periodic" properties of the tilings.

Motion of discrete interfaces in periodic media

“…Hence, this defines a meaningful convergence with respect to which compute the Γ-limit of P α,β ε as ε → 0. A general theory for the homogenization of energies (5), in a more general context, has been developed in [11] (see also [9,12,13]), where it is shown that the Γ-limit's domain is precisely the family of sets of finite perimeter and its general form is…”

Interfacial energies on quasicrystals

“…A key point is the description of the surface tension ϕ, which is characterized by an asymptotic formula involving discrete optimal transition problems on large sets in Σ, corresponding to large cubes in R n , similar to those introduced in [7] to deal with homogenization problems on continuous thin films. The main technical issue of the paper is precisely the proof of the existence of such ϕ, for which we adapt the quasiperiodic arguments of [11].…”

“…We note that, contrary to the case of Penrose lattices considered in [11] where one can essentially reason directly on the projection of the higher-dimensional lattice on Π n , here the description of the system as a discrete thin film is exploited both in the proof of the coerciveness of the energies and in the control over translations.…”

Domain Formation in Magnetic Polymer Composites: An Approach Via Stochastic Homogenization

“…We recall here that the variational analysis of the properties of ground states of Ising-like systems can be traced back to the pioneering paper by Caffarelli and de la Llave [18] and that derivation of continuum theories from atomistic spin-like ones in the framework of Γ-convergence is not new and has been initiated by Braides and collaborators in [2]. Since then it has been developed by many authors in connection with the theory of surfactants, nematic elastomers, dislocations in plasticity, superfluids or frustrated magnetic chains, to cite a few (see for instance [3,5,6,14,16,17,20,26]).…”