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.

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We propose and study two variants of the Ambrosio-Tortorelli functional where the firstorder penalization of the edge variable v is replaced by a second-order term depending on the Hessian or on the Laplacian of v, respectively. We show that both the variants as above provide an elliptic approximation of the Mumford-Shah functional in the sense of Γ-convergence.In particular the variant with the Laplacian penalization can be implemented without any difficulties compared to the standard Ambrosio-Tortorelli functional. The computational results indicate several advantages however. First of all, the diffuse approximation of the edge contours appears smoother and clearer for the minimizers of the second-order functional. Moreover, the convergence of alternating minimization algorithms seems improved for the new functional. We also illustrate the findings with several computational results.

The combined effect of fine heterogeneities and small gradient perturbations is analyzed by means of an asymptotic development by Γ-convergence for a family of energies related to (one-dimensional) phase transformations. We show that multi-scale effects add up to the usual sharp-interface limit, due to the homogenization of microscopic interfaces, internal and external boundary layers, optimal arrangements of microscopic oscillations, etc. Several regimes are analyzed depending on the "size" of the heterogeneity (small or large perturbations of a homogeneous situation) and their relative period as compared with the characteristic length of the phase transitions (slow or fast oscillations).

In the framework of the asymptotic analysis of thin structures, we prove that, up to an extraction, it is possible to decompose a sequence of 'scaled gradients' (del(alpha)u(epsilon)vertical bar(1)/(epsilon)del(beta)u(epsilon)) (where is the gradient in the k-dimensional 'thin variable' x(beta)) bounded in L-p (Omega:R-mxn(1 < p < + infinity) as a sum of a sequence (del(alpha)v(epsilon)vertical bar(1)/(epsilon)del(beta)nu(epsilon)) whose p-th power is equi-integrable on Omega and a 'rest' that converges to zero in measure. In particular, for k = 1 we recover a well-known result for thin films by Bocea and Fonseca (ESAIM: COCV 7:443-470; 2002)