Γ-convergence of free-discontinuity problems

Abstract

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.

Functionals Defined on Piecewise Rigid Functions: Integral Representation and $$\varGamma $$-Convergence

“…In this final section, we focus on the case L = R d×d skew and L = SO(d), with d = 2, 3, which is relevant from the point of view of the applications. We consider an additional assumption (H 6 ) (in the spirit of [25,49]) for the functionals F n , and use it to truncate piecewise rigid functions at a low energy expense. This will allow us to overcome the lack of coercivity of our functionals, and to deduce the lower bound in the inequality (2.8) (see Lemma 7.5).…”

A compactness result in $$GSBV^p$$ and applications to $$\varGamma $$-convergence for free discontinuity problems

“…As applications, we prove existence of minimizers for energies of the form (1) under Dirichlet boundary data. Moreover, we revisit the Γ -convergence result for free discontinuity problems established recently in [12]. We show convergence of minimum values and minimizers for a sequence of boundary value problems without any fidelity term.…”

“…Accordingly, the piecewise Korn inequality [33] is replaced by a suitable piecewise Poincaré inequality (see Section 3.3), which is based on a careful use of the coarea formula in BV (see [5,Theorem 3.40]). Let us note that the coarea formula has been largely employed to approximate BV functions by piecewise constant functions, particularly to prove lower semicontinuity [2] and Γ -convergence results [11,12] in SBV , as well as the existence of quasistatic evolutions [26,30,36]. Compared to [35], the passage to modifications is more delicate due to the more general energies which may depend explicitly on the crack opening.…”

Homogenization of High-contrast Mumford--Shah Energies

“…Our result shows that the volume and surface terms of F ε interact in the limit. This case is different from the analysis in [13], where the authors devise a list of assumptions ensuring that the volume and surface energies do not interact in the homogenization of free-discontinuity functionals. The functionals F ε in (1.1) are not covered by the analysis in [13] due to their degenerate growth conditions.…”

“…This case is different from the analysis in [13], where the authors devise a list of assumptions ensuring that the volume and surface energies do not interact in the homogenization of free-discontinuity functionals. The functionals F ε in (1.1) are not covered by the analysis in [13] due to their degenerate growth conditions. Such a degeneracy, and the consequent lack of coerciveness, however, is not sufficient to cause the interaction of the terms of the energy, since this was not the case for the functionals F ε .…”