We give results for the Γ-limit of a scaled elastic energy of a film as the thickness h > 0 converges to zero. The elastic energy density models materials with multiple phases or variants and is thus non-convex. The model includes an interfacial energy that allows sharp interfaces between the phases and variants and is proportional to the total variation of the deformation gradient.

In this paper we make a survey of some recent developments of the theory of Sobolev spaces W (X, d,m), 1 < q < ∞, in metric measure spaces (X, d,m). In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on Γ-convergence; this result extends Cheeger’s work because no Poincaré inequality is needed and the measure-theoretic doubling property is weakene...

By looking at the Γ-convergence of some easy functionals in terms of underlying Young measures, we emphasize the connection of this issue with the Monge-Kantorovich mass transfer problem. After exploring this relationship, we study a typical example in periodic homogenization to realize the difference between the Monge-Kantorovich problem and the usual cell-problem. We also state a version of t...

We investigate the limiting description for a finite-difference approximation of a singularly perturbed Allen–Cahn type energy functional. The key issue is to understand the interaction between two small length-scales: the interfacial thickness ε and the mesh size of spatial discretization δ. Depending on their relative sizes, we obtain results in the framework of Γ-convergence for the (i) subc...

This paper considers a wavelet analogue of the classical Ginzburg-Landau energy, where the Hseminorm is replaced by the Besov seminorm defined via an arbitrary regular wavelet. We prove that functionals of this type converge, in the Γ-sense, to a weighted analogue of the TV functional on characteristic functions of finite-perimeter sets. The Γ-limiting functional is defined explicitly, in terms...

A homogenization problem arising in the gradient theory of fluid-fluid phase transitions is addressed in the vector-valued setting by means of Γ-convergence.