Interfacial energies on quasicrystals

Pantographic lattices with non-orthogonal fibres: Experiments and their numerical simulations

“…While the numerical simulations show a surprising agreement with experimental evidence, we feel that a rigorous basis on the homogenization results presented in [2] needs to be firmly established. We expect that Γ-convergence results can be now confidently formulated and conjectured, see [13]. Moreover we expect that the methods exploited in [14] could be adapted to get also a priori error estimates in the replacement process involved when passing from discrete to continuum models.…”

Continuum limit and stochastic homogenization of discrete ferromagnetic thin films

“…To deal with convergence of sequences u ε : εL → S, we adopt the idea of [15]. We will see in Section 6 that this notion of convergence is indeed meaningful for variational problems in a random environment.…”

“…We say that L is an admissible set of points if the following two requirements are satisfied: (i) there exists r > 0 such that |x − y| ≥ r for all x = y, x, y ∈ L, (ii) there exists R > 0 such that dist(x, L) ≤ R for all x ∈ R k . Within this definition we may include 'slices' of periodic lattices [3], and also aperiodic geometries [15]. Given a probability space (Ω, F, P), a random variable L : Ω → (R d ) N is called an admissible stochastic lattice if, uniformly with respect to ω ∈ Ω, L(ω) is an admissible set of points.…”

Asymptotic analysis of a ferromagnetic Ising system with “diffuse” interfacial energy

“…for general f (and u i taking values in a more general set), showing that the limit is a bulk integral g * * (u) dx (8) (g * * denotes the convex envelope of g). Note however that in the ferromagnetic spin case g is trivial, and can be interpreted as a double-well potential with minima in ±1.…”