Phase-field approximation of a vectorial, geometrically nonlinear cohesive fracture energy

We consider a family of vectorial models for cohesive fracture, which may incorporate $\mathrm{SO}(n)$-invariance. The deformation belongs to the space of generalized functions of bounded variation and the energy contains an (elastic) volume energy, an opening-dependent jump energy concentrated on the fractured surface, and a Cantor part representing diffuse damage. We show that this type of functional can be naturally obtained as $\Gamma$-limit of an appropriate phase-field model. The energy densities entering the limiting functional can be expressed, in a partially implicit way, in terms of those appearing in the phase-field approximation.


Introduction
In variational models of nonlinear elasticity a hyper-elastic body with reference configuration Ω ⊂ R n (n = 2, 3) undergoes a deformation u : Ω → R m , whose stored energy reads as Ω Ψ(∇u)dx. (1.1) External loads can be included, adding linear perturbations to this energy, and Dirichlet boundary conditions, restricting the set of admissible deformations u. The energy density Ψ : R m×n → [0, +∞), acting on the deformation gradient ∇u, is typically assumed to be minimized by matrices in the set of proper rotations SO(n) (with m = n) and to have p-growth at infinity, p > 1. Correspondingly, the natural space for the deformation u is a (subset of) the Sobolev space W 1,p (Ω; R m ). There is an extensive literature on the theory of existence of minimizers of this type of functionals, and in particular the key property of weak lower semicontinuity of (1.1) is closely related to the quasiconvexity of the energy density Ψ. Fracture phenomena, both brittle and cohesive, require a richer modeling framework. Physically, cohesive fracture is often understood as a gradual separation phenomenon: load-displacement curves usually exhibit an initial increase of the load up to a critical value, and a subsequent decrease to zero, which is the value indicating the complete separation [BFM08,Dug60,Bar62,FCO14]. See [dPT98,dPT01] for discussions on different load-displacement behaviours. Evolutionary models (prescribing the crack path) have been studied in [DMZ07, BFM08, Cag08, CT11, LS14, Alm17, ACFS17, NS17, TZ17, NV18, CLO18], see also references therein. See [DMG08,CCF20] for further results on the topic.
Variational models of fracture are typically formulated using the space (G)BV of (generalised) functions of bounded variation [FM98,BFM08] and energy functionals of the form Ω W (∇u)dx + Ω l(dD c u) + Ju g([u], ν u )dH n−1 . (1. 2) The deformation u ∈ (G)BV (Ω; R m ) may exhibit discontinuities along a (n−1)dimensional set J u . We denote by [u] and ν u the opening of the crack and the normal vector to the crack set J u , respectively, while D c u represents the Cantor derivative of u (see [AFP00] for the definition and the relevant properties of functions of bounded variation). Working within deformation theory, the functional (1.2) contains both energetic and dissipative terms, which are physically distinct but need not be separated for this variational modeling. The densities W , l, and g entering (1.2) need to satisfy suitable growth conditions. Lower semicontinuity of the functional imposes several restrictions, as for example that l is positively one-homogeneous and quasiconvex, W quasiconvex, and g subadditive. Furthermore, l needs to match, after appropriate scaling, both the behavior of W at infinity and the behavior of g near zero. These properties will be discussed in more detail below (see, for example, Proposition 3.11).
The qualitative properties of W , l and g are selected according to the specific model of interest. For instance, the brittle regime is modelled by a constant surface density g and a superlinear bulk energy density W . These choices in turn imply that l(ξ) = ∞ for ξ = 0, so that D c u necessarily vanishes. The functional setting of the problem is then provided by the space of (generalised) special functions with bounded variation (G)SBV (Ω). In contrast, in cohesive models g is usually assumed to be approximately linear for small amplitudes and bounded.
The direct numerical simulation of functionals of the type (1.2) is highly problematic, due to the difficulty of finding good discretizations for (G)BV functions and of differentiating the functional with respect to the coefficients entering the finite-dimensional approximation. Therefore a number of regularizations have been proposed, of which one of the most successful is given by phase-field functionals. These are energies depending on a pair of variables (u, v), having a Sobolev regularity, where u represents a regularization of a discontinuous displacement, while v ∈ [0, 1] can be interpreted as a damage parameter, indicating the amount of damage at each point of the body (where v = 1 corresponds to the undamaged material and v = 0 to the completely damaged material). The basic structure of a phase-field model is where ε > 0 is a small parameter, f ε is a damage coefficient acting on the damage variable v, increasing from 0 to 1, and Ψ is an elastic energy density, as in (1.1). The first term in (1.3) represents the stored elastic energy, the other two terms represent the stored energy and dissipation due to the damage. Finding a variational approximation of the fracture model (1.2) by phasefield models means to construct f ε and Ψ such that the functionals (1.3) converge, in the sense of Γ-convergence, to (1.2) as ε → 0. This is not an easy task in general. The brittle case (g constant) in an antiplane shear, linear, framework (m = 1, Ψ quadratic) was the first outcome of this type [AT90,AT92]. It has been extended in several directions for different aims, giving rise to a very vast literature of both theoretical results [Sha96, AFM01, Cha04, Cha05, HMCX14, ALRC13, DMI13, Iur13, FI14, Iur14, BEZ15, CFZ21] and numerical simulations [BC94a, BSK06, Bou07, BFM08, BOS10, BOS13, BB21] (for other regularizations, see also [AFP00,BDMG99,Bra98,Fus03,BG06] and references therein). In particular, the extension of the results in [AT92] to the vector-valued (nonlinear) brittle case has been provided in [Foc01]. The variational approximation of cohesive models is considerably more involved. The antiplane shear, linear, case was obtained through a double Γ-limit of energies with 1-growth in [ABS99], then generalized to the vector-valued case in [AF02]. A drawback of these results is the 1-growth with respect to ∇u, which makes the approximants mechanically less meaningful and numerically less helpful.
To overcome these problems, in [CFI16] we proposed a different approximation of (1.2) in the antiplane shear case, with quadratic models of the form (1.3), based on a damage coefficient f ε of the type f ε (s) := 1 ∧ ε 1 /2 ℓs 1 − s s ∈ [0, 1], ℓ > 0 , (1.4) and obtained Γ-convergence to a model of the type (1.2) in the scalar (m = 1) case. We remark that f ε is equal to 1 when v ∼ 1 (elastic response) and to 0 when v ∼ 0 (brittle fracture response). Moreover, the first addend in the energy in (1.3) competes against the second term if v is less than but close to 1, and with all the terms of (1.3) otherwise (pre-fracture response). This phase-field approximation of this scalar cohesive fracture was investigated numerically in [FI17]. A 1D cohesive quasistatic evolution (not prescribing the crack path) is presented in [BCI21] and related to the phase-field models of [CFI16]. A different approximation of (1.2), still in the scalar-valued framework, is obtained in [DMOT16] using elasto-plastic models.
In this paper we study the approximation of vector-valued cohesive models of the type (1.2) via phase-field models of the type (1.3) with the damage coefficient (1.4), as proposed in [CFI16]. In particular, this permits to extend the results of [CFI16] to a geometrically nonlinear framework, we refer to (2.2)-(2.5) for the specific hypotheses on Ψ. The main result is given in Theorem 2.1, the precise assumptions are discussed in Section 2.1.
In order to illustrate our result, let us consider the simplest model for the energy density Ψ in finite kinematics and m = n, Ψ 2 (ξ) := dist 2 (ξ, SO(n)) = min R∈SO(n) |ξ − R| 2 . (1.5) With this choice, our main result Theorem 2.1 states that the phase-field energies (1.3) Γ-converge in the L 1 -topology as ε → 0 to the energy (1.2), with W (ξ) := (dist 2 (·, SO(n)) ∧ ℓ dist(·, SO(n))) qc (ξ) , (1.6) and l(ξ) := ℓ|ξ|, g(z, ν) := g scal (|z|) , for every ξ ∈ R m×n , z ∈ R m , ν ∈ S n−1 , where g scal is the surface energy density appearing in the scalar model (cf. formula (4.4) for the definition of g scal , item (iii) in Proposition 3.12 with W = h qc and l = h qc,∞ to justify the second equality, and Corollary 3.5 for the third equality). As remarked above, g coincides with l asymptotically for infinitesimal amplitudes. Even in this simple case, the expression for W is somewhat implicit, as it involves a quasiconvex envelope, which in most cases can only be approximately computed numerically. We remark that even Ψ 2 itself as defined in (1.5) is not quasiconvex, we refer to [Š01,Example 4.2] for an explicit formula for its quasiconvex envelope Ψ qc 2 in the two-dimensional case. We recall that in the scalar case several different choices for f ε are possible without changing the overall effect of the approximation (cf. [CFI16, Section 4]).
A negative power-law divergence at 1 however leads to a corresponding powerlaw behaviour of g close to 0 (cf. [CFI16,Theorem 7.4]). We expect these findings to have a natural generalization to the current vectorial setting, this requires additional technical ingredients that will be the object of future work [CFI22].
Let us now briefly discuss some aspects of the proof of Theorem 2.1. One of the main difficulties is to identify the correct limit densities W , g, and l, given the density Ψ and the damage coefficient f ε of the phase-field (1.3). We do not expect that the cohesive energies that arise in the limit of our approximation exhaust all possible energies of the form (1.2), with densities W , g, and l satisfying the growth conditions and matching properties specified above. Indeed, we prove that, even in the simplest case Ψ(ξ) := |ξ| 2 , W is not convex (see Lemma 2.5 below). Thus, at least in this case, the limit energy is not given by the relaxation of a functional defined on SBV (Ω) (cf. [BC94b, Remark 2.2]). Convex functions may be obtained as densities of the bulk term of the energy under more specific choices of the damage variable (see for example [BIR21], where the damage variable is a characteristic function).
The effective surface energy density g of the Γ-limit of the family (F ε ) is defined in an abstract fashion by an asymptotic minimization formula as the Γ-limit of a simpler family of functionals computed on functions jumping on a hyperplane (cf. (2.12)). Alternative characterizations of g useful along the proofs are provided both in Propositions 3.1 and 3.2, in which we show that the test sequences in the very definition of g can be assumed to be periodic in (n−1) mutually orthogonal directions and with L 2 integrability, and in Proposition 3.3, where g is represented in terms of an asymptotic homogenization formula. Finally, the energy density l of the Cantor part turns out to coincide with the recession function W ∞ of W . Furthermore, an explicit characterization of l in terms of Ψ is given in Proposition 3.10.
The proof of the lower bound in BV is based on the blow-up technique. Roughly, to get the local estimate for the diffuse part given (u ε , v ε ) → (u, v) in L 1 , we analyze the asymptotic behaviour of the phase-field energies F ε restricted on the δ-superlevel sets of v ε , δ ∈ (0, 1), and then let δ ↑ 1. More precisely, in Lemma 4.4 we bound from below F ε (u ε , v ε ) in (1.3) pointwise with a functional defined on (G)SBV , that is independent of v ε and that is computed on a truncation of u ε with the characteristic function of a suitable superlevel set of v ε (depending on δ). This is actually true up to an error related to the measure of the corresponding sublevel set of v ε , and up to prefactors depending on δ which are converging to 1 as δ ↑ 1 for the volume term and vanishing for the surface term. The lower semicontinuity in L 1 of the diffuse part of such a functional then implies the lower bound. In addition, a slight variation of this argument shows directly that (GBV (Ω)) m is the domain of the Γ-limit.
Instead, to prove the local estimate for the surface part we show that under a surface scaling assumption we may replace v ε by its truncation at the threshold γ ε , being γ ε the smallest z ∈ [0, 1] satisfying f ε (z) = 1. The mentioned asymptotic minimization formula defining g then provides a natural lower bound. The liminf inequality in GBV is finally obtained by a further truncation argument.
The upper bound in BV is proven through an integral representation argument. In particular, a direct computation provides a rough linear estimate from above, in fact optimal for the diffuse part. This allows to apply the representation result for linear functionals given in [BFM98]. The sharp estimate for the surface density is obtained using the aforementioned characterization of g involving periodic boundary conditions. The full upper bound in GBV follows by a truncation argument.
The paper is structured as follows. In Section 2.1 we present the model, introducing the main definitions and stating the Γ-convergence result in Theorem 2.1. In Section 2.2 we focus on a simplified model and we prove that in this case the limiting volume energy density W , obtained by quasiconvexification as in (1.6), is not convex (Lemma 2.5). In Section 3 several properties of the surface and Cantor densities are discussed. In particular, Propositions 3.1 and 3.2 deal with the change of boundary conditions within the minimum problem defining g. Proposition 3.3 provides an equivalent expression of g. Section 4 is devoted to the proof of the lower bound: Proposition 4.1 proves the surface estimate in BV . The lower bound in BV for the diffuse part is addressed in Proposition 4.2. Finally, in Theorem 4.9 the lower bound is extended to the full space GBV via a continuity argument (cf. Proposition 4.8). The proof of the upper bound is the object of Section 5, which concludes the proof of Theorem 2.1. Finally, Section 6 addresses the problems of compactness and convergence of minimizers.

General definitions
In the entire paper Ω ⊂ R n is a bounded, open set with Lipschitz boundary, A(Ω) denotes the family of open subsets of Ω and | · | denotes the Euclidean norm, |ξ| 2 := ij ξ 2 ij = Tr ξ T ξ for ξ ∈ R m×n . For all ε > 0 we consider the functional F ε : and ℓ > 0 is a parameter representing the critical yield stress. We write briefly F ε (u, v) := F ε (u, v; Ω), and analogously for all the functionals that shall be introduced in what follows. We assume that Ψ : R m×n → [0, ∞) is continuous and such that We assume the ensuing limit to exist and that it is uniform on the set of ξ with |ξ| = 1. This means that for every δ > 0 there is t δ > 0 such that |Ψ(tξ)/t 2 − Ψ ∞ (ξ)| ≤ δ for all t ≥ t δ and all ξ with |ξ| = 1, which is the same as (2.5) By scaling, Ψ ∞ (tξ) = t 2 Ψ ∞ (ξ) and in particular Ψ ∞ (0) = 0. Uniform convergence also implies Ψ ∞ ∈ C 0 (R m×n ).
and denote by h qc its quasiconvex envelope, where the latter quantity is defined as in (2.7)-(2.9). We remark that, at variance with the convex case, one cannot in general replace the lim sup in (2.9) by a limit [Mül92,Theorem 2]. For all open subsets A ⊆ R n , u ∈ W 1,2 (A; R m ) and v ∈ W 1,2 (A; [0, 1]) it is convenient to introduce the functional The first term is interpreted to be zero whenever ∇u = 0, even if v = 1. For any ν ∈ S n−1 we fix a cube Q ν with side length 1, centered in the origin, and with one side parallel to ν. We write Q ν r := rQ ν . We define g : (2.12) Here u j ∈ W 1,2 (Q ν ; R m ) and v j ∈ W 1,2 (Q ν ; [0, 1]); obviously one can restrict to sequences v j → 1 in L 1 (Q ν ). We refer to Section 3 for the discussion of several properties of g.
We will prove the following result.
Remark 2.2. One can imagine several natural generalizations of Theorem 2.1. For example, one could allow Ψ to take negative values, replacing (2.3) by Whereas in purely elastic models like (1.1) one can add a constant to the energy density without any change in the analysis, the presence of the prefactor f 2 ε (v) renders this modification nontrivial, and influences several steps in the proof. Indeed, the construction in Step 1 of the proof of Theorem 5.2 shows that the definition of h in (2.6) needs to be replaced by h(ξ) := Ψ(ξ) ∧ ℓΨ 1 /2 + (ξ) . Alternatively, one could replace the quadratic growth of Ψ in (2.3) by p-growth, p > 1. The requirement that the effective energy scales linearly for large strains leads to corresponding adaptations in the other parts of the functional.
For simplicity we only address here the growth condition in (2.3).

Notation. For
and correspondingly for the Γ-lim sup. We drop the dependence on the reference set A if A = Ω. We refer to Section 4.1 for the definition of the vector measure D c u if u ∈ (GBV (Ω)) m .

Simplified model
In this Section we consider the simplified case Ψ simp (ξ) := |ξ| 2 , the corresponding unrelaxed energy density h simp : (2.14) its quasiconvex envelope h qc simp as in (2.7), and its recession function h qc,∞ simp as in (2.9). These functions only depend on the space dimension and the single parameter ℓ > 0, which could be eliminated by scaling.
In this case it is possible to obtain simple closed-form expressions for several of the quantities defined above. However, an explicit characterization of the quasiconvex envelope in (2.7) remains difficult. Indeed, we show in Lemma 2.5(iii) below that even in this simplified setting the result is not convex. Since it has linear growth, lower bounds with polyconvexity cannot be used, and an explicit determination of h qc simp seems difficult. We believe this to be a strong indication that in most cases of interest the function h qc can only be approximated numerically, and not computed explicitly. Lemma 2.5 and this observation are not used in the proof of Theorem 2.1.
We next prove that the quasiconvex envelope h qc simp is not convex. For this we need a linear algebra statement that we present first.
and consider for ξ ∈ R m×n the linear map T : R m×n×n sym → R m×n×n of the form If rank ξ ≥ 2, then T is injective. In particular, it has an inverse S : Proof. It suffices to show that there is no Γ ∈ R m×n×n sym with T Γ = 0 and Γ = 0. We assume it exists and define v ∈ R n componentwise by (2.20) Then T Γ = 0 is equivalent to hence Γ ijk = ξ ij v k , for all i, j, and k. Moreover, Γ = 0 in turn implies that v = 0. From Γ ∈ R m×n×n sym we obtain As rank ξ ≥ 2 there is a vector w ∈ R n with v · w = 0 and ξw = 0. We take the scalar product of the previous equation with w and obtain which gives 0 = v j (ξw) i for all i and j. As v = 0 and ξw = 0, this is a contradiction.
(iii) If rank ξ ≥ 2 and |ξ| > ℓ 2 , then h conv simp (ξ) < h qc simp (ξ). Proof. We work for ℓ = 1 (the general case can be reduced to this one by a rescaling), to shorten notation we write h for h simp .
In particular, for any j we have We integrate over (0, 1) n , take the limit j → ∞ and recall that g(∇ϕ j ) → 0 in L 1 by (2.22). We obtain lim sup for any ε ∈ (0, 1]. By (2.21) and Lemma 2.3(ii) the sequence ∇ϕ j is bounded in L 1 , and since ε was arbitrary we conclude that lim sup We next prove that (2.25) implies that ∇ϕ j converges to the constant ξ strongly in weak-L 1 . To do this we show that standard singular integral estimates imply rigidity. To simplify notation, we write u j (x) := ϕ j (x) − ξx and R j := ∇ϕ ⊥ j = ∇u ⊥ j , both extended by zero to the rest of R n , in the next steps. We observe that whereξ := ξ |ξ| . Taking a derivative, and writing components, we obtain with T obtained fromξ as in Lemma 2.4. Let S be the inverse operator. Then so that in particular ∆u j is given by a linear combination of the components of ∇R j , with coefficients which depend only on ξ. As u j (x) = 0 outside (0, 1) n , we obtain, denoting by N the fundamental solution of Laplace's equation in R n (which solves −∆N = δ 0 ), with c depending only on ξ. Recalling the definition of u j and R j as well as (2.25), To conclude the proof we choose z ∈ (h conv (ξ), h(ξ)) (here we use again that |ξ| > 1 2 ). By continuity of h, there is δ > 0 such that h(η) ≥ z for all η ∈ R m×n with |η − ξ| < δ. By definition of the weak-L 1 norm, This contradicts (2.21) and concludes the proof.

Energy densities of the surface and Cantor part
In this section we discuss several properties of the energy densities g and h qc,∞ . We warn the reader that while the results dealing with g contained in subsections 3.1 and 3.2 will be crucial in the proof of¸Theorem 2.1, those in subsection 3.3 will not be employed in that proof. Actually, Proposition 3.9 and Corollary 3.11 take advantage of Theorem 2.1 itself (in particular of the lower semicontinuity of Γ-limits).

Equivalent characterizations of g(z, ν)
We show below that we may reduce the test sequences in the definition of g(z, ν) in (2.12) to those converging in L 2 and satisfying periodic boundary conditions in (n−1) directions orthogonal to ν and mutually orthogonal to each other. This is the content of the next two propositions, which will be crucial in the proof of the upper bound for the surface part (Theorem 5.2 Step 2). The proof draws inspiration from that of [BF94, Lemma 4.2]. We fix a mollifier and To simplify the notation we write Next, we choose a sequence η j → 0 such that and set K j := ⌊η j /ε j ⌋, we can assume K j ≥ 4. We letR j k := Q ν 1−kεj \Q ν 1−(k+1)εj , where we write for brevity Q ν r := rQ ν for the scaled cube. We select k j ∈ {K j + 1, . . . , 2K j } such that, writing R j :=R j kj , j is more complex. In the interior part, it should match v j . In the exterior, V j . In the interpolation region, it should be not larger than v j and V j , but also not larger than 1 − η j . Therefore we first definê which coincides with 1−η j in the interpolation region R j , and with 1 at distance larger than η j ε j from it, then which coincides with V j outside Q ν 1−(kj +1)εj , and with 1 inside Q ν 1−(kj +3)εj as well as for |x · ν| ≥ 3ε j (cf. the definition of V j ), and finallỹ We then combine these three ingredients to obtain On ∂Q ν the first and the last term are equal to 1, hence v * j =V j = V j .
Step 2. Estimate of the elastic energy. By the definition of u * j , Integrating over R j and using (3.5) in the first term, (3.4) in the second one, Using first that the definition of K j implies lim j→∞ K j ε j /η j = 1 and then (3.3), Using again that the supports of ∇U j and V j are disjoint, we have Step 3. Estimate of the energy of the phase field. By the definition of v * j , From the definition of V j andV j , we see that Combining this with (3.9) concludes the proof.
We are now ready to perform the claimed reduction on the test sequences in the definition of g(·, ν) in (2.12). To this aim we fix a sequence (a k ) k ⊂ (0, ∞) such that a k < a k+1 , a k ↑ ∞, and such that there are functions and ∇T k L ∞ (R m ) ≤ 1. Following De Giorgi's averaging/slicing procedure on the codomain, the family T k will be used in several instances along the paper to obtain from a sequence converging in L 1 to a limit belonging to L ∞ , a sequence with the same L 1 limit which is in addition equi-bounded in L ∞ . Moreover, this substitution can be done up to paying an error in energy which can be made arbitrarily small.
Step 1. Reduction to an optimal sequence in (2.12) converging We are left with establishing (3.13). To this aim consider T k (u j ) and note that We estimate the second term in (3.14). The growth conditions on Ψ (cf. (2.3)) and (3.15) Collecting (3.14) and (3.15) and using F ∞ Let now M ∈ N, by averaging there exists k M,j ∈ {M + 1, . . . , 2M } such that i.e. (3.13).
Step 2. Conclusion. In view of Step 1 there is an optimal sequence for g(z, ν) in (2.12) converging in L 2 (Q ν ; R m+1 ). Let (ε k , u k , v k ) be the sequence from Proposition 3.1.
Since lim k→∞ lim j→0 ε * j /ε k = 0, we can select a nondecreasing sequence k(j) → ∞ such that λ j := ε * j /ε k(j) → 0. We where U * j and V * j are defined as in (3.2) using ε * j . One easily verifies that U * j (x) = U k(j) ( x−y λj ) for all y ∈ ν ⊥ , and the same for V . By the boundary conditions (3.1), these functions are continuous and therefore in W 1,2 (Q ν ; R m+1 ). We further estimate Taking j → ∞, and recalling that lim In what follows we provide an equivalent characterization for the surface energy g in the spirit of [CFI16, Proposition 4.3].
Proof. For every (z, ν) ∈ R m × S n−1 and T > 0 set We first prove that lim sup Indeed, if T j ↑ ∞ is a sequence achieving the superior limit on the left-hand side above, thanks to Proposition 3.2 we may consider ( Then, define (ũ j (y),ṽ j (y)) := u j ( y Tj ), v j ( y Tj ) for y ∈ Q ν Tj , and note that by a change of variable it is true that and that (ũ j ,ṽ j ) ∈ U Tj z,ν in view of (3.18). Then, by (3.19), the choice of T j and the definition of g T (z, ν) we conclude straightforwardly (3.17).
In order to prove the converse inequality we assume for the sake of notational simplicity ν = e n . We then fix ρ > 0 and take T > 6, depending on ρ, and (u T , v T ) ∈ U T z,en such that 1 Let ε j → 0 and set by the choice of (u T , v T ) and T (cf. (3.21)). As ρ → 0 we get (3.20). Estimates (3.17) and (3.20) yield the existence of the limit of g T (z, ν) as T ↑ ∞ and equality (3.16), as well.
With this representation of g at hand we can obtain a version of Proposition 3.2 which also accounts for a regularization term of the form η ε Ψ(∇u)dx.
Proposition 3.4. For any ε j ↓ 0 and η j ↓ 0 with η j /ε j → 0, and any (z, Proof. We use the same construction as above (without loss of generality, explicitly written only for ν = e n ), and compute similarly To conclude the proof it suffices to choose T j → ∞ so slow that η j C Tj /ε j → 0.
For an equivalent definition of g scal see equation (4.4) below and [CFI16, Proof. By [CFI16, Proposition 4.3] or by Proposition 3.3, the following characterization holds for g scal : belong to U T |z| and satisfy by Fubini's theorem Taking the infimum over (u, v) ∈ U T z,ν and passing to the limit T → ∞ we get (3.22).

Structural properties of g(z, ν)
We next deduce the coercivity properties of g.
Lemma 3.7. There is c > 0 such that, for all z, ν ∈ R m × S n−1 , We provide here a direct proof of the lemma. Alternatively, these bounds may be derived estimating F ε by its 1D counterpart (as in (4.2) below) and recalling the bounds holding for g scal , see [CFI16,Prop. 4.1].
Proof. We start with the lower bound. Let z ∈ R m , ν ∈ S n−1 , and fix sequences has measure at least 2 3 and, using (2.3) to estimate 1 also has measure at least 2 3 . Therefore we can fix y j such that both inequalities hold. If g(z, ν) < ∞, then necessarily v * j → 1 in L 2 ((− 1 2 , 1 2 )), and it has a continuous representative. We can therefore assume that sup v * j ≥ 3 4 for large j. If inf v * j ≤ 1 2 then Otherwise, v * j ≥ 1 2 pointwise and We turn to the upper bound. We define u j (x) := u * j (x · ν), v j (x) := v * j (x · ν), where, denoting by AI the affine interpolation between the boundary data in the relevant segments, if |t| ∈ (ε j , 2ε j ).
If ℓ|z| < 1, then the upper bound in (2.3) leads to If instead ℓ|z| ≥ 1 the first term vanishes, and We prove next the subadditivity and continuity of g.
Proof. (i): Fix z 1 , z 2 ∈ R m , ν ∈ S n−1 . Let (u i j , v i j ) be the sequences from Proposition 3.2 corresponding to ε j := 1/j and the pair (ν, z i ), for i = 1, 2. We implicitly extend both periodically in the directions of ν ⊥ ∩ Q ν , and constant in the direction ν. In particular, for {x · ν ≥ 1 2 } we have u i j = z i and v i j = 1; for {x · ν ≤ − 1 2 } we have u i j = 0 and v i j = 1 for i ∈ {1, 2} and all j. We use a rescaling similar to the one of Proposition 3.2. We fix a sequence M j ∈ N, M j → ∞, and define (u j , v j ) ∈ W 1,2 (R n ; R m × [0, 1]) by and, correspondingly, By the periodicity of (u i j , v i j ) in the directions of ν ⊥ ∩ Q ν , these maps belong to W 1,2 (Q ν ; R m ). Furthermore, u j = 0 and v j = 1 if x · ν ≤ − 1 Mj , u j = z 1 + z 2 and v j = 1 if x · ν ≥ 1 Mj , and (u j , v j ) is 1 Mj -periodic in the directions of ν ⊥ ∩ Q ν . Therefore, by changing variables we find Arguing similarly, we infer The conclusion follows taking the limit j → ∞.
Since Ψ ∞ is continuous and positive on the compact set S nm−1 ⊆ R m×n , there is a monotone modulus of continuity ω : This implies that Ψ ∞ (η) ≤ (1 + ω |R−Id| )Ψ ∞ (ηR) for any η ∈ R m×n , R ∈ O(n) (3.24) (it suffices to insert η/|η| and ηR/|η| in the above expression). Fix ν ∈ S n−1 , a sequence ε j → 0, and let (u j , v j ) be as in Proposition 3.2, extended periodically in the directions of ν ⊥ ∩ Q ν and constant along ν, as in the proof of (i). Letν ∈ S n−1 ,ν = ν, and choose R ∈ O(n) such that ν = Rν and |R − Id| ≤ c|ν −ν| (for example, R can be the identity on vectors orthogonal to both ν andν, and map (ν,ν ⊥ ) to (ν, ν ⊥ ) in this two-dimensional subspace). We fix a sequence M j → ∞ (for example, M j := j) and definẽ Inserting in the definition of F ∞ εj (ũ j ,ṽ j ; Qν) and using a change of variables leads to We observe that, although Rν = ν, we cannot in general expect RQν = Q ν . However, as (u j , v j ) are periodic in the directions orthogonal to ν, the (n − 1)dimensional square ν ⊥ ∩ M j RQν can be covered by at most M n−1 j + cM n−2 j disjoint translated copies of the (n− 1)-dimensional unit square ν ⊥ ∩Q ν . Therefore

Density of the Cantor part
We study now the behaviour of the surface energy density g at small jump amplitudes. The next result is probably well known to experts. Despite this, we give a self-contained proof since we have not found a precise reference in the literature. Similar constructions are performed in [AFP00, Proposition 5.1] for isotropic functionals defined on vector-valued measures. The L 1 lower semicontinuity of F 0 is assumed to hold in Proposition 3.9 below, as already mentioned at the beginning of Section 3. Such a property follows, for instance, from the validity of Theorem 2.1. We stress again that Proposition 3.9 is not used in the proof of Theorem 2.1, rather it provides a further piece of information on g showing its linear behavior at small amplitudes.
Proof. With fixed ν ∈ S n−1 , let x 0 ∈ Ω and ρ > 0 be such that Q ν ρ (x 0 ) ⊂ Ω. Upon translating and scaling, it is not restrictive to assume x 0 = 0 and ρ = 1. For every z ∈ R m consider the sequence where ϕ(t) := (t ∧ 1) ∨ 0 for every t ∈ R. Clearly, w j → u z (x) := zχ {x·ν≥0} in L 1 (Q ν ; R m ), and thus by the L 1 (Q ν ; R m ) lower semicontinuity of F 0 we conclude that On the other hand, given z ∈ R m and any couple of sequences z j → z and t j → 0 + , denote by M j the integer part of t −1 j and define for every k ∈ N, k ≥ 3, We show that u j,k converges, as j → ∞, to w k as defined in (3.25) for every k ≥ 3. Indeed, for s : Therefore, by the L 1 (Q ν ; R m ) lower semicontinuity of F 0 we conclude that As this holds for every sequence, this implies (3.27) Indeed, the superior limit in the definition of h qc,∞ is actually a limit on rank-1 directions being h qc,∞ convex on those directions. Let now z j → 0 be a sequence for which .
Upon setting z j := zj | zj | , up to subsequences we may assume that z j → z ∞ ∈ S n−1 . In addition, t j := | z j | → 0. Therefore, being h qc,∞ one-homogeneous we have that .
We now identify h qc,∞ explicitly as stated in (2.10).
Therefore, being h ≥ 0 (cf. again (2.3)) from (3.30) we infer that from which we conclude that From Propositions 3.9 and 3.10 we deduce straightforwardly the ensuing statement.
We conclude this section by proving that, under our hypotheses, the superior limit in the definition of Ψ 1 /2 is in fact a limit and that the operations of quasiconvexification and of recession for Ψ 1 /2 commute.
Proof. The second equality in (i) follows immediately from (2.4). Then, the first is a consequence of the very definition of recession function. Alternatively, by (2.5) we infer that, for all δ > 0, there is C δ > 0 satisfying This, together with the definition of recession function, implies (i).

Lower bound 4.1 Domain of the limits
In order to characterize the compactness properties and the space in which the limit is finite it is useful to consider the scalar simplification of functional, (4.2) with the same constant c ≥ 1 as in (2.3). In particular, [CFI16,Prop. 6 where U t := {α, β ∈ W 1,2 ((0, 1)) : α(0) = 0, α(1) = t, 0 ≤ β ≤ 1, β(0) = β(1) = 1}. In particular, g scal satisfies (i) g scal is subadditive: g scal (t 1 + t 2 ) ≤ g scal (t 1 ) + g scal (t 2 ) for every t 1 , t 2 ∈ [0, ∞), Precisely, [AF02, Lemma 2.10] implies that for every u ∈ (GBV (Ω)) m for which |D c u| is a finite measure on Ω, one can construct a vector measure on Ω with total variation coinciding exactly with |D c u|(B) for every Borel subset B of Ω. For this reason such a vector measure, is denoted by D c u. Let us briefly recall the construction of D c u. To this aim, the family of truncations T k defined in (3.11) is employed. Indeed, for every u ∈ (GBV (Ω)) m such that |D c u| is a finite measure on Ω, it is possible to show that the following limit exists for every Borel subset B of Ω λ(B) := lim k→∞ D c (T k (u))(B) . (4.5) In addition, λ is actually independent from the chosen family of truncations. The set function λ turns out to be a vector Radon measure on Ω, and moreover equality |λ|(B) = |D c u|(B) is true for every B as above.
Finally, for functions u ∈ (GBV (Ω)) m satisfying estimate (4.3) it is also true that

Surface energy in BV
We prove below the lower bound in BV for the surface term. We recall that the definition of the surface energy density g has been given in (2.12).
Up to subsequences and with a small abuse of notation, we can assume that the previous lower limit is in fact a limit. Let us define the measures µ ε ∈ M + b (A) Extracting a further subsequence, we can assume that Equation (4.7) will follow once we have proved that (4.9) We will prove the last inequality for points where ν := ν u (x 0 ) and Q ν ρ (x 0 ):= x 0 + ρQ ν is the cube centred in x 0 , with side length ρ, and one face orthogonal to ν. We remark that such conditions define a set of full measure in J u ∩ A.

Diffuse part in BV
where h qc and h qc,∞ have been defined in (2.6)-(2.9).
We remark that this statement can be proven using the lower-semicontinuity result by Fonseca Then for any u ∈ BV (Ω; R m ) we have where φ ∞ (ξ) := lim sup t→∞ φ(tξ)/t. In particular the latter functional is lower semicontinuous with respect to the strong L 1 (Ω; R m ) convergence.
We start with a truncation result.
Proof of Proposition 4.2.
The lower bound in BV follows at once from the lower bounds for the surface and the diffuse parts.

Lower bound in GBV
In this section we extend the validity of the lower bound Theorem 4.6 to every u ∈ (GBV (Ω)) m . We first prove that the functional F 0 is continuous under truncations.
Proof. We prove the convergence of the volume, Cantor and surface terms separately. It is useful to recall for the rest of the proof that ∇T k L ∞ (R m ) ≤ 1.
For the volume part, we observe that (2.8) implies |∇u| ∈ L 1 (Ω). We have ∇(T k (u)) = ∇u for L n -a.e. x ∈ Ω k := {|u| ≤ a k }, therefore in view of (2.8) we get For the surface term we recall that J T k (u) ⊆ J u for every k ∈ N with ν T k (u) = ν u for H n−1 -a.e. x ∈ J T k (u) . Then, thanks to (4.6) we infer that (T k (u)) ± → u ± , χ J T k (u) → χ Ju and |[T k (u)]| ≤ |[u]| H n−1 -a.e. in J u , and then we conclude , ν u )dH n−1 thanks to Lemmata 3.7 and 3.8 (ii) and to the Dominated Convergence Theorem.
For what the Cantor part of the energy is concerned, by (2.8) we have that 0 ≤ h qc,∞ (ξ) ≤ c|ξ|. Further, the definitions of T k and of D c u outlined in (4.5) yield in particular which concludes the proof.
We are ready to prove the lower bound for generalized functions of bounded variations.  where F ε and F 0 have been defined in (2.1) and (2.13).
Recalling the definition of the truncation T k in (3.11), we have that T k (u ε ) → T k (u) in L 1 (Ω; R m ) for any k and that T k (u) ∈ BV (Ω; R m ), being F 0 (u, 1) < ∞. Hence, we can apply Theorem 4.6 to say that (4.34) We claim that for all M ∈ N there is k M ∈ {M + 1, . . . , 2M } independent of ε such that after extracting a further subsequence for some c > 0 independent of ε and of M . Given this for granted, we get by (4.34), (4.35) and by the convergence u ε → u in measure lim sup Finally, using the continuity under truncations for F 0 established in Proposition 4.8, we obtain and hence (4.33). It remains to prove (4.35). To this aim we argue as in Proposition 3.2 using De Giorgi's averaging-slicing method on the range. First, for all k ∈ N we split the energy contributions By (2.3) and the definition of T k , the last but one term in the previous expression can be estimated as for some c > 0. Summing (4.36) and (4.37) and averaging, we conclude that there exists k M,ε ∈ {M + 1, . . . , 2M } such that for some c > 0. As ε → 0, there exists a subsequence of {k M,ε } that is independent of ε. This yields (4.35) and concludes the proof.

Upper bound
In this Section we prove the Γ − lim sup inequality in Theorem 2.1. In order to be able to obtain existence of minimizers for the perturbed functionals (see Section 6), we consider a perturbed version of the functional which includes an additional uniformly coercive term, and prove the upper bound directly for the modified functional. We fix a function η : where F ε has been defined in (2.1).
One key ingredient in the proof of the upper bound is that the Γ-limit of F η ε satisfies the hypotheses of [BFM98, Theorem 3.12], so that it can be represented as an integral functional. Its diffuse and surface densities will be identified by a direct computation.
In order to prove that Γ-lim ε→0 F η ε (u, 1; ·) is a Borel measure, we first check the weak subadditivity of the Γ-upper limit of F η ε .
where F η ε has been defined in (5.2).
Proof. To simplify the notation let us set F ′′ := Γ(L 1 )-lim sup ε→0 F η ε . It is not restrictive to assume that the right-hand side of (5.3) is finite, so that for J ∈ {A, B}.
Step 1. Estimate (5.3) is valid if u ∈ BV ∩L 2 (A∪B; R m ) and (5.5) holds for two sequences converging to u in L 2 (Ω; R m ). For δ := dist(A ′ , ∂A) > 0 and some M ∈ N, we set for all i ∈ {1, . . . , M } Let now M ∈ N, by summing up the latter inequality for both A and B and by averaging, there exists k ε,M ∈ {M + 1, . . . , 2M } such that Up to a subsequence, we may take the index k ε,M = k M , i.e. to be independent of ε. Therefore, passing to the limit as ε → 0, the convergence u J ε → u in measure for J ∈ {A, B}, (5.4), (5.5), (5.10) and Step 1 yield Eventually, since T kM (u) → u in L 1 (Ω; R m ) as M ↑ ∞, by the lower semicontinuity of F ′′ for the L 1 (Ω; R m ) convergence we conclude (5.3).
We are now ready to prove the upper bound inequality.
Theorem 5.2. Let F η ε and F 0 be defined in (5.2) and (2.13), respectively. For every (u, v) ∈ L 1 (Ω; R m+1 ) it holds Proof. Given a subsequence (F η ε k ) of (F η ε ), there exists a further subsequence, not relabeled, which Γ-converges to some functional F , that is, where F ′ and F ′′ denote here the Γ(L 1 )-lower and upper limits of F η ε k and where the subscript − denotes the inner regular envelope of the relevant functional ([Dal93, Definition 16.2 and Theorem 16.9]).
We remark that F (u, v; ·) is the restriction of a Borel measure to open sets by [Dal93,Theorem 14.23]. Indeed, F (u, v; ·) is increasing and inner regular by definition; additivity follows from (5.14), once one checks that (F ′ ) − is superadditive and (F ′′ ) − is subadditive. The former condition is a direct consequence of the additivity of We divide the proof of (5.13) into several steps. First note that it is sufficient to prove it for v = 1 L n -a.e. on Ω.
Step 1. Estimate on the diffuse part for u ∈ BV (Ω; R m ). We first prove a global rough estimate for F ′′ which actually turns out to be sharp for the diffuse part if u ∈ BV (Ω; R m ). To this aim we set H :  follows for every u ∈ BV (Ω; R m ) and A ∈ A(Ω).
To prove (5.16), assume first that u is an affine function, say u(x) = ξx + b, with ξ ∈ R m×n , b ∈ R m . Then, the pair Instead, ifū Therefore, we conclude (5.16) for every affine function u in view of the last two estimates.
Assume now that u ∈ C 0 (Ω; R m ) is a piecewise affine function, say u( and Ω i ∈ A(Ω) disjoint and with Lipschitz boundary, and such that L n (Ω \ ∪ N i=1 Ω i ) = 0. Then, set and {ϕ i } 1≤i≤N is a partition of unity subordinated to the covering Then, a straightforward computation shows that where c depends on ℓ, Ψ, and ξ 1 , . . . , ξ N . Therefore we conclude (5.16) when u is piecewise affine, namely as δ → 0 in the latter inequality we have If u ∈ W 1,1 (Ω; R m ), we consider an extension of u itself (still denoted by u for convenience) to W 1,1 0 (Ω ′ ; R m ), for some open and bounded Ω ′ ⊃⊃ Ω (recall that Ω is assumed to be Lipschitz regular). Then, we use a classical density result [ET99, Proposition 2.1 in Chapter X] to find u k ∈ W 1,1 0 (Ω ′ ; R m ) piecewise affine such that u k → u in W 1,1 (Ω ′ ; R m ). The continuity of H for the W 1,1 (Ω; R m ) convergence, and the lower semicontinuity of F ′′ for the L 1 (Ω; R m+1 ) convergence finally imply (5.16).
Step 3. Integral representation of the Γ(L 1 )-limit on BV (Ω; R m ) × {1}. We now would like to represent F as an integral functional through [BFM98, Theorem 3.12] and to estimate its diffuse and surface densities. In order to satisfy the coercivity hypothesis [BFM98, Eq. (2.3')], we introduce an auxiliary functional for all u ∈ BV (Ω; R m ) and for some c > 0. Note that F λ also satisfies the continuity hypothesis [BFM98, Eq. (2.4)], since , for all (u, v) ∈ W 1,2 (Ω; R m+1 ), z, b ∈ R m , A ∈ A(Ω), and analogous properties then hold for F .
This, together with the lower bound Theorem 4.6 allows to identify uniquely the Γ-limit of the subsequence F η ε k . Finally, Urysohn's property ([Dal93, Proposition 8.3]) extends the result to the whole family F η ε .
The conclusion then follows by the L 1 -lower semicontinuity of F ′′ and by Proposition 4.8.
We are ready to prove Theorem 2.1.
Proof of Theorem 2.1. The lower bound has been proven in Theorem 4.9. The upper bound follows by Theorem 5.2 with η ε = 0.
Next theorem establishes the compactness of sequences equibounded in energy and in L 1 .
Proof. This follows arguing componentwise, that is, estimating F ε with its one-dimensional counterpart evaluated in a component, and applying the onedimensional compactness result obtained in [CFI16, Theorem 3.3] as done in subsection 4.1 (see also the argument in Remark 4.7).
Instead, the addition of the term η ε Ψ(∇w) is instrumental to guarantee the existence of a minimizer for G ε , provided that Ψ is quasiconvex. In general, the coercivity of G ε only ensures existence of minimizing sequences (u j ε ) j converging weakly in W 1,2 (Ω; R m ) to someū ε minimizing the relaxation of G ε . Since existence at fixed ε does not interact with the Γ-convergence, we state our result for asymptotically minimizing sequences.
Moreover, m ε tends to the minimum value of G 0 .
Proof. The proof of the corollary will be divided in three steps.
Step 1. Γ-limit of F η ε in L q × L 1 . We check that passing from the L 1 × L 1 to the L q × L 1 topology, the expression of the Γ-limit of F η ε remains the same Γ(L q × L 1 )-lim ε→0 F η ε (u, v) = F 0 (u, v).
Fix M ∈ N large enough such that a M > u ∞ (see (3.11) for the definition of a M ) and, for every ε > 0, choose k ε,M ∈ {M + 1, . . . , 2M } such that This implies with T kε,M (u ε ) uniformly bounded in L ∞ , T kε,M being defined in (3.11). This argument has been used several times throughout the paper, see for example Theorem 4.9. Passing to a further subsequence in ε, we can take k ε,M = k M independent of ε. Since (T kM (u ε )) ε is uniformly bounded in L ∞ and M is large, we get T kM (u ε ) → T kM (u) = u in L q (Ω; R m ) and in particular L n ({a M+1 < |u ε |}) → 0 as ε → 0, hence lim sup ε→0 F η ε (T kM (u ε ), v ε ) ≤ 1 + C M F 0 (u, 1) .
Diagonalizing with respect to M and recalling the lower estimate, we conclude that every subsequence of {F η ε } ε has a subsequence that Γ(L q × L 1 )-converges to F 0 in L ∞ (Ω; R m ) × L 1 (Ω). Finally Urysohn's lemma gives the convergence of the entire sequence in the same space.
Let us consider now the general case u ∈ (GBV ∩ L q (Ω)) m . Then T k (u) ∈ (BV ∩ L ∞ (Ω)) m , with T k again defined by (3.11), and Γ(L q × L 1 )-lim sup ε→0 F η ε (T k (u), 1) ≤ F 0 (T k (u), 1), by the first part of the proof. As k → ∞ we have T k (u) → u in L q (Ω; R m ) and we conclude by the lower semicontinuity of the Γ-limsup and the continuity of F 0 (see Proposition 4.8).
As for the upper bound, from Step 1 we know that for all u ∈ (GBV (Ω) ∩ L q ) m there exists a recovery sequence for F η ε in L q × L 1 . This is in particular a recovery sequence for G ε in L 1 × L 1 , which gives the conclusion.