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.

The existence of crack evolutions based on critical points of the energy functional is proved, in the case of a cohesive zone model with prescribed crack path. It turns out that evolutions of this type satisfy a maximum stress criterion for the crack initiation. With an explicit example, it is shown that evolutions based on the absolute minimization of the energy functional do not enjoy this property.

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. The isoperimetric inequality for Steiner symmetrization of any codimension is investigated and the equality cases are characterized. Moreover, a quantitative version of this inequality is proven for convex sets.

Mathematics Subject Classification. 49K10, 49Q20.

A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on H 1 0 (Γ ), Γ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of Γ . A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.

We investigate large-time asymptotics for viscous Hamilton-Jacobi equations with possibly degenerate diffusion terms. We establish new results on the convergence, which are the first general ones concerning equations which are neither uniformly parabolic nor first order. Our method is based on the nonlinear adjoint method and the derivation of new estimates on long time averaging effects. It also extends to the case of weakly coupled systems.

Characterizations results for equality cases and for rigidity of equality cases in Steiner's perimeter inequality are presented. (By rigidity, we mean the situation when all equality cases are vertical translations of the Steiner's symmetral under consideration.) We achieve this through the introduction of a suitable measure-theoretic notion of connectedness and a fine analysis of barycenter functions for sets of finite perimeter having segments as orthogonal sections with respect to an hyperplane.( 1.8) conversely, one says that K does not essentially disconnect G if, for every non-trivial BorelFinally, G is essentially connected if ∅ does not essentially disconnect G.Remark 1.4. By a non-trivial Borel partition {G + , G − } of G modulo H m we mean thatMoreover, ∂ e G denotes the essential boundary of G, that is defined aswhere G (0) and G (1) denote the sets of points of density 0 and 1 of G; see section 2.1.Remark 1.5. If H m (G∆G ′ ) = 0 and H m−1 (K∆K ′ ) = 0, then K essentially disconnects G if and only if K ′ essentially disconnects G ′ ; see Figure 1.4. 5