We study the Γ-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u.We obtain three main results: compactness with respect to Γ-convergence, representation of the Γ-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.MSC 2010: 49J45, 49Q20, 74Q05.for suitable, k-independent constants 0 < c1 ≤ c2, c4 ≤ c5 < +∞. Note that g k in (1.3) is independent of ζ, which, together with the restriction m = 1, introduces lots of simplifications in the analysis. In particular, these simplifications guarantee that sequences (u k ) with bounded energy E k are bounded in BV , up to a truncation, and hence also in [25] it is natural to study the Γ-convergence of E k in L 1 . By using the abstract integral representation result in [7], it is shown in [25] that the Γ-limit of E k is a free-discontinuity functional of the same type, and that also in this case no interaction occurs between the bulk and the surface part of the functionals in the Γ-convergence process.Therefore, the volume and surface terms decouple in the limit both in the periodic case -for vectorvalued u and with dependence of the surface densities on [u], under strong coercivity assumptions -and in the non-periodic case -for scalar u and with no dependence on [u]. This raises the question of determining general assumptions for f k and g k guaranteeing the decoupling.

In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.

Abstract. In this paper we rigorously derive a line-tension model for plasticity as the Γ-limit of a nonlinear mesoscopic dislocation energy, without resorting to the introduction of an ad hoc cut-off radius. The Γ-limit we obtain as the length of the Burgers vector tends to zero has the same form as the Γ-limit obtained by starting from a linear, semi-discrete dislocation energy. The nonlinearity, however, creates several mathematical difficulties, which we tackled by proving suitable versions of the Rigidity Estimate in non-simply-connected domains and by performing a rigorous two-scale linearisation of the energy around an equilibrium configuration.

We provide a homogenisation result for the energy-functional associated with a purely brittle composite whose microstructure is characterised by soft periodic inclusions embedded in a stiffer matrix. We show that the two constituents as above can be suitably arranged on a microscopic scale ε to obtain, in the limit as ε tends to zero, a homogeneous macroscopic energy-functional explicitly depending on the opening of the crack.

Motivated by applications to image reconstruction, in this paper we analyse a finite-difference discretisation of the Ambrosio–Tortorelli functional. Denoted by \varepsilon the elliptic-approximation parameter and by \delta the discretisation step-size, we fully describe the relative impact of \varepsilon and \delta in terms of \Gamma -limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when \varepsilon and \delta are of the same order, the underlying lattice structure affects the \Gamma -limit which turns out to be an anisotropic free-discontinuity functional.