A new space of generalised functions with bounded variation motivated by fracture mechanics

We introduce a new space of generalised functions with bounded variation to prove the existence of a solution to a minimum problem that arises in the variational approach to fracture mechanics in elastoplastic materials. We study the fine properties of the functions belonging to this space and prove a compactness result. In order to use the Direct Method of the Calculus of Variations we prove a lower semicontinuity result for the functional occurring in this minimum problem. Moreover, we adapt a nontrivial argument introduced by Friedrich to show that every minimizing sequence can be modified to obtain a new minimizing sequence that satisfies the hypotheses of our compactness result.


Introduction
The variational approach to rate-independent evolution problems developed in [10] and [11] is based on a time discretization scheme, where the approximate solution at a given time is obtained by solving an incremental minimum problem which involves the solution at the previous time. The same approach was introduced independently in fracture mechanics in [8] (we refer also to [3] for further developments in this field).
In this framework, the study of crack growth in linearly elastic-perfectly plastic materials in the small strain regime leads to incremental minimization problems that involve the crack Γ as well as the elastic part e and the plastic part p of the strain. In the (generalised) antiplane case, the reference configuration is a bounded Lipschitz domain Ω ⊂ R d , the crack is a Borel set Γ ⊂ Ω, with H d−1 (Γ) < +∞, and the displacement is a function u : Ω \ Γ → R, whose gradient is additively decomposed as Du = e + p, where e, the elastic part, is an L 2 -function defined in Ω \ Γ and p, the plastic part, is a bounded Radon measure defined on Ω \ Γ.
Given a Borel set Γ 0 ⊂ Ω (the crack at the previous time), with H d−1 (Γ 0 ) < +∞, and a bounded Radon measure p 0 in Ω \ Γ 0 (the plastic strain at the previous time), the incremental minimum problem takes the form where the minimum is taken over all competing cracks Γ ⊃ Γ 0 (irreversibility condition) and all pairs (e, p) such that e is an L 2 -function, p is a Radon measure, and e+p = Du in Ω\Γ for some displacement u satisfying prescribed boundary conditions (see Sect. 4 for a precise formulation). The purpose of this paper is to prove the existence of a solution to problem (1.1). In [6] we considered the same problem only in the case d = 2, with the additional constraint that Γ and Γ 0 are compact and satisfy an a priori bound on the number of their connected components. In this case, given a minimizing sequence (Γ k , e k , p k ) k , we can extract a subsequence (not relabelled) such that Γ k → Γ in the Hausdorff metric, e k e weakly in L 2 , and p k * p locally weakly * as measures on Ω \ Γ. Therefore the existence of a solution to (1.1) follows from the Direct Method of the Calculus of Variations, since all terms in (1.1) are lower semicontinuous. This simplified approach cannot be applied when d > 2, nor when d = 2 without bounds on the number of connected components of Γ. For this reason we prefer to rewrite the minimum problem (1.1) in terms of the displacement u, considered as a function defined L d -a.e. in Ω.
Let u 0 be the displacement at the previous time, let e 0 be its elastic strain, so that Du 0 = e 0 + p 0 , and let f : R d → R be defined by f (ξ) = 1 2 |ξ| 2 , if |ξ| ≤ 1, and f (ξ) = |ξ| − 1  Unfortunately, there are boundary conditions for which the minimum problem (1.2) has no solution in the space of functions of bounded variation, as shown by the example provided in Proposition 6.5. The reason is that, while the first term in (1.2) controls ∇v and the second one controls D c v, the third term does not control the whole jump part of Dv, which is given by the integral of |[v]| on J v . Therefore, in order to prove the existence of a solution to the minimum problem (1.2), and hence (1.1), we consider a larger functional space for the admissible displacements, which we denote by GBV (Ω). This is a subset of the space GBV (Ω) of generalised functions of bounded variation introduced in [1, Sect. 1] (see also [2,Definition 4.26]). In Sect. 3 we study the fine properties of functions in GBV (Ω), as well as some structure properties of this space. In particular, we prove in Theorem 3.11 that, if (v k ) k is a minimizing sequence of (1.2) in GBV (Ω) and for some continuous function ψ with ψ(t) → +∞ as t → +∞, then a subsequence of (v k ) k converges pointwise L d -a.e. to a function v ∈ GBV (Ω). In Sect. 5 we prove that every minimizing sequence of problem (1.2) can be modified in order to obtain a new minimizing sequence which satisfies (1.3) for a suitable function ψ, depending on the sequence. The construction of ψ is not trivial and is achieved by adapting to GBV (Ω) the arguments introduced in [9] for GSBV p (Ω).
To use the Direct Method of the Calculus of Variations we prove in Theorem 6.1 that the functional considered in (1.2) is lower semicontinuous. If e 0 is constant we can easily reduce the problem to the case e 0 = 0, which was studied in [4]. The result can be easily extended to the piecewise constant case by a localization argument. The general case is obtained by approximation.
The existence of a minimizing sequence satisfying (1.3) and the semicontinuity result imply that there exists a solution to (1.2) in GBV (Ω), see Theorem 6.2. Since problem (1.1) is equivalent to problem (1.2) in GBV (Ω), see Lemmas 4.1 and 4.2, we conclude that problem (1.1) has a solution, see Corollary 6.3.

Preliminaries on BV -spaces
In this section we fix the general notation used in the paper and we recall the fine properties of BV and GBV functions that will be used in the sequel.
For every topological space X, for every Borel set Y ⊂ X, and for ev- For every Borel measure μ on U and every Borel set where B ρ (x) is the ball in R d of radius ρ and centre x.
Given an L d -measurable set E ⊂ U and an L d -measurable function u : E → R, we say that a ∈ R is the approximate limit of u(y) as y tends to a point x ∈ E (α) for some α > 0, in symbols It follows from the definition that if f : R → R is a continuous function and ap lim y→x u(y) exists, then Let u : U → R be an L d -measurable function. We define the jump set J u as the set of all points x ∈ U such that there exist u + (x), u − (x) ∈ R with u + (x) = u − (x) and a unit vector ν u (x) ∈ R d such that, setting It is easy to see that the triple (u + (x), u − (x), ν u (x)) is uniquely defined up to a swap of the first two terms and a change of sign in the third one. For every The following theorem gives a formula for the distributional gradient of the truncation of a BV function. Theorem 2.1. Let u ∈ BV (U ) and let m ∈ R + . Then u (m) ∈ BV (U ) and Proof. It is enough to apply [2,Theorem 3.99] to f (t) = t (m) .

Remark 2.2.
Let u ∈ BV (U ) and let m ∈ R + . It follows from Theorem 2.1 that Proof. The statement about γ E u is proved in [2,Theorem 3.77]. The properties concerning v can be easily deduced from [2, Theorem 3.84].
In the following proposition we summarize the fine properties of functions in GBV (U ).   (m) implies that |u (m) (y)−ũ(x)| ≤ |u(y)− u(x)| for every y ∈ U . The definition of ap lim then gives that ap lim y→x u (m) (y) = u(x). Hence u (m) (x) =ũ(x), and the same result holds for u (n) . The conclusion follows from Lemma 2.3 and Remark 2.5.
In the following theorem we show that, if u ∈ GBV (U ) satisfies condition (2.15) below, then we can define an R d -valued Radon measure that plays the role of the Cantor part of Du, even if the measure Du cannot be defined.
Therefore (2.20) and Proposition 2.6(d) give This concludes the proof.
Together with (2.24) this implies (2.22). The first equality in (2.23) follows from Proposition 2.6(d). To prove the other equality, for every Borel set B ⊂ U we define (2.25) Using the monotonicity with respect to m stated in Proposition 2.6(d) we can prove that ν is a Borel measure. Therefore, it is enough to prove (2.23) for every Borel set B ⊂ {|ũ| ≤ m} and for every Borel set B ⊂ {|ũ| = +∞} ∪ J u . The former follows from (2.16), while the latter follows from (2.17) and Remark 2.5, taking into account the fact that |D c u (m) |(J u ) = 0 by the general properties of the Cantor part of the gradient of a BV function.

The function space used in our problem
We now introduce the function space that will be used to formulate and solve problem (1.1). Throughout this section U is a bounded open set in R d .
It follows from the definition of GBV (U ) that GBV (U ) ⊂ GBV (U ). Moreover, using Remark 2.5 and Proposition 2.6 it is easy to see that the supremum in (3.1) can be taken over m ∈ N.
For every u ∈ GBV (U ) we define

Remark 3.5. Let λ > 0 and u ∈ GBV (U ). Then
Ju The converse implication can be proved in the same way.
Given a function u ∈ GBV (U ), by Theorem 2.7 and Proposition 3.3 the measures ∇uL d and D c u are well-defined and belong to , we cannot use (2.5) and (2.7) to define a measure which plays the role of Du. However, this is possible on a suitable subset of U and leads to a measure which will be crucial in the sequel. Definition 3.6. Let u ∈ GBV (U ) and let Γ ⊂ U be a Borel set with Remark 3.7. If u ∈ BV (U ), using (2.5) and (2.7) we see that D Γ u coincides with the restriction of Du to U \ Γ.
x ∈ J u (see [7,Theorem 5.9.3]). These properties do not hold for an arbitrary function in GBV (U ). The following theorem shows that they hold for functions in GBV (U ).
Proof. It is enough to repeat the proof of [2, Theorem 4.40], replacing the hypothesis It is well-known that GBV (U ) is not a vector space (see [2,Remark 4.27]). The additional properties considered in the definition of GBV (U ) lead to the following result. Proof. It is obvious that, if u ∈ GBV (U ) and λ ∈ R, then λu ∈ GBV (U ). Taking into account the definition of GBV (U ), by (3.14), (3.15) there exists a constant C > 0, independent of m, such that Using the definition of GBV (U ) we see that there exists a constant C m > 0 such that To conclude the proof we have to show that and since A r U as r → +∞ (recall that u and v have finite values) we get (3.19).
It remains to prove (3.21). To this end we observe that by Proposition 3.3 there exists a constant C > 0 such that for every m. This concludes the proof.
Proof. Equalities (3.23) follow immediately from the definition of the approximate limit. The second equality in (3.24) follows from the definition of D c u. To prove the first equality, we set w := u + v and we fix m and s with 0 ≤ 2s ≤ m. For every r ≥ s we havew (m) =w =ũ +ṽ =ũ (r) +ṽ (r) H d−1 -a.e. inÃ s , whereÃ s , is defined in (3.22). Since w (m) ∈ BV (U ) by Theorem 3.9 and u (r) , v (r) ∈ BV (U ) by the definition of GBV (U ), using (2.2) and Lemma 2.3 for every Borel set B in U we obtain By Proposition 2.9 we can pass to the limit as r → +∞ and we get Taking the limit as m → +∞ and using Proposition 2.9 again we obtain Finally, arguing as in the proof of Theorem 3.9 we can pass to the limit as s → +∞ and obtain the first equality in (3.24). By (3.23) and (3.24) to prove the first equality in (3.25) it is enough to show that for every Borel set B ⊂ U \Γ. This follows easily from the linearity of the jump and the locality property of approximate tangent spaces (see, e.g., [ Then there exist a subsequence, not relabelled, and a function u ∈ GBV (U ) Proof. We claim that for every m ∈ N the truncated functions u (m) k are bounded in BV (U ). Indeed, by Proposition 2.6(b) we have that ∇u (3.28) As for the estimate on the jump part, we observe that, by Proposition 2.6(c), we have that J u (m) k ⊂ J u k up to a set of H d−1 -measure zero, By the compactness of the embedding of BV (U ) into L 1 loc (U ), using a diagonal argument we can extract a subsequence of (u k ) k , not relabelled, such that for every m ∈ N, the sequence (u (m) k ) k converges L d -a.e. in U to a function v m ∈ L ∞ (U ). Since the BV -norm is lower semicontinuous with respect to L 1convergence, we obtain that v m ∈ BV (U ).
We observe that This is an obvious consequence of the fact that (u Let E ∞ be the intersection of the sets {|v m | = m} for m ∈ N. We claim that To prove this property we observe that it is not restrictive to assume that the function ψ in (3.27) is increasing. For every m ∈ N by the Fatou Lemma we have where in the last inequality we used the monotonicity of ψ. Since, by assumption, ψ(m) → +∞ as m → +∞, from (3.27) we obtain (3.33).
and we observe that, by (3.32), the function u is well-defined on U \ E ∞ and hence u ∈ GBV (U ).

The incremental minimum problem
In this section we present a precise formulation of the incremental minimum problem (1.1), which appears in the variational approach to the quasistatic crack growth in elastic-perfectly plastic materials. The reference configuration is a bounded open set Ω ⊂ R d with Lipschitz boundary. The crack in the reference configuration is represented by a Borel set Γ ⊂ Ω, with H d−1 (Γ) < +∞. The set Ω \ Γ represents the elasto-plastic part of the body.
Since we are dealing with the antiplane case, the displacement of each material point is described by a function u : Ω \ Γ → R. Regarding u as a function defined L d -a.e. in Ω, we assume that Here and in the rest of the paper the trace on ∂Ω of a function v ∈ GBV (Ω) is still denoted by v. The strain corresponding to the displacement u is given by the measure D Γ u ∈ M b (Ω \ Γ; R d ) introduced in Definition 3.6 with U replaced by Ω and Γ replaced by Γ ∩ Ω. The Dirichlet boundary condition is assigned using the trace on ∂Ω of a function w ∈ H 1 (Ω). The elastic part of the strain is denoted by e and the plastic part by p. We assume that To solve this problem we introduce the function f : The minimum in the definition of f (ξ) is attained for η = ξ if |ξ| ≤ 1 , ξ/|ξ| if |ξ| ≥ 1 . It is convenient to introduce the maps π 1 , π 2 : R d → R d defined by We note that ξ = π 1 (ξ) + π 2 (ξ) , |π 1 (ξ)| ≤ 1 , and f (ξ) = 1 2 |π 1 (ξ)| 2 + |π 2 (ξ)| . (4.13) For later use we observe that the definition (4.12) of f implies (4.14) To deal with the boundary condition (4.4) in (4.11) it is convenient to introduce a bounded open set Ω with Ω ⊂ Ω (4.15) and to extend w, w 0 , e 0 in such a way that w, w 0 ∈ H 1 (Ω ) and e 0 ∈ L 2 (Ω ; R d ) . (4.16) We now prove that problem (4.11) is equivalent to the following minimum problem Proof. Let Γ and (u, e, p) be a solution of (4.11). It is clear that H d−1 (Γ\Γ 0 ) < +∞, hence (4.8) implies that H d−1 (Γ) < +∞. Let v be as in the statement of the lemma. To prove that Γ and v solve (4.17) we fix a Borel setΓ, with Γ 0 ⊂Γ ⊂ Ω, andv ∈ GBV (Ω ), witĥ v = w − w 0 L d -a.e. in Ω \ Ω . (4.18) We want to show that where the last equality follows from (4.5) and (4.9). Thenê ∈ L 2 (Ω; R d ) and g ∈ L 1 (Ω; R d ). We now defineû :=v| Ω + u 0 and note thatû ∈ GBV (Ω) by Theorem 3.9. Moreover we definep ∈ M b (Ω \Γ; R d ) bŷ We remark that Jû\Γ |[û]|dH d−1 < +∞ and ∂Ω\Γ |û|dH d−1 < +∞ by (4.1), (4.9), (4.18), and (4.20). This shows that the definition ofp makes sense. We note that DΓû =ê +p in Ω \Γ andp = (w −û)ν Ω H d−1 on ∂Ω \Γ, hence (û,ê,p) ∈ A(Γ, w). Consequently, the minimality of Γ and (u, e, p) gives On the other hand by the definition ofp we have hence, by (4.13) and (4.21), where Similarly, using the definition (4.12) of f instead of (4.13), we obtain which shows that Γ and v solve (4.17). Conversely, assume that Γ and v solve (4.17). By (4.8) it is clear that H d−1 (Γ) < +∞. We observe that the triple (u, e, p) defined in the second part of the statement of the lemma belongs to A(Γ, w) and that To prove that Γ and (u, e, p) solve (4.11) we fix a Borel setΓ with Γ 0 ⊂Γ ⊂ Ω and a triple (û,ê,p) ∈ A(Γ, w). We want to show that Then, arguing as in the first part of the proof we obtain in this case (4.25) with an equality and (4.23) with the inequality ≤ . Then (4.27) follows from (4.22) and (4.26), which is an obvious consequence of (4.17).
We now prove that (4.17) is equivalent to the minimum problem   Proof. Assume that Γ and v solve (4.17). Letv ∈ GBV (Ω ) be such that where we used the fact that H d−1 (Jv \ Ω) = 0 sincev ∈ H 1 (Ω \ Ω). Therefore, by the minimality of Γ and v we have which proves that v solves (4.28).
Conversely, assume now that v solves (4.28) and let Γ : we obtain that which shows that Γ and v solve (4.17).
The results of this section show that the existence of a solution to the minimum problem (4.11) can be obtained by proving that the minimum problem (4.28) has a solution. To this aim we shall use the Direct Method of the Calculus of Variations. Unfortunately, not every energy-bounded sequence for (4.28) is relatively compact. For instance, if w = w 0 = 0, Γ 0 = ∅, e 0 = 0, and v k = kχ E , where E is a set of finite perimeter with L d (E) > 0, then (v k ) k is energy-bounded for (4.28), but it has no subsequence which converges L d -a.e. to a finite-valued function.
The origin of this problem is the fact that, in general, an energy-bounded sequence does not satisfy (3.27). In the next section we shall construct a relatively compact minimizing sequence for problem (4.28), while in Sect. 6 we shall prove a lower semicontinuity result, which will allow us to obtain the existence of a minimizer.

Construction of a relatively compact minimizing sequence
In this section Ω is a bounded open subset of R d with Lipschitz boundary, c 1 , c 2 are constants with 0 < c 1 ≤ c 2 , and a 1 , a 2 ∈ L 1 (Ω). Given a Borel set Γ 0 ⊂ Ω, with H d−1 (Γ 0 ) < +∞, and a Borel measurable function g : Ω × R d → R, with for L d -a.e. x ∈ Ω and every ξ ∈ R d , we consider the functional G g Γ0 defined by for every u ∈ GBV (Ω). The aim of this section is to show that, if (u k ) k is a minimizing sequence for G g Γ0 , then we can modify it by means of piecewise constant translations obtaining a new minimizing sequence which satisfies the hypotheses of the compactness Theorem 3.11. The construction of the modified sequence follows the lines of [9] and requires several steps. We begin by constructing a suitable Caccioppoli partition (see [2,Definition 4.16]).
there exist a Caccioppoli partition (P j ) j of Ω and corresponding translations belongs to BV (Ω) ∩ L ∞ (Ω) and the following estimates hold: Proof. We may assume that For every i ∈ Z there exists t i ∈ (iA, (i + 1)A) such that the set {u > t i } has finite perimeter in Ω and Let us prove now that First of all, we claim that every x ∈ J 1 u belongs at most to two sets ∂ * E i . Indeed, it is known (see, e.g., [2, Theorem 3.61]) that for every 2 , which is clearly impossible. This proves our claim, which implies that A similar argument shows that every x ∈ ∂Ω belongs to at most one set (5.10) Therefore (5.7), (5.9), and (5.10) give (5.8), which shows that (E i ) i is a Caccioppoli partition of Ω.
Let us define v := u − i∈Z t i χ Ei . For every x ∈ E i we have 11) which shows that v L ∞ (Ω) ≤ 2M .
We show that v ∈ BV (Ω). To this end let us consider v k := |i|≤k z i , with z i := (u−t i )χ Ei . By (5.11) we have z i = (u (mi) −t i )χ Ei , with m i = 2M + |t i |. Since both u (mi) −t i and χ Ei belong to BV (Ω)∩L ∞ (Ω), by Lemma 2.4 we have z i ∈ BV (Ω). Recalling (5.11) we have [u (mi) ] = [u] and 0 ≤ [u (mi) − t i ] ≤ 2M on E (1) i ∩J u by the definition of E (1) , while the trace operator γ Ei defined in Lemma 2.4 satisfies |γ Ei (u mi −t i )| ≤ 2M H d−1 -a.e. on ∂ * E i . Using Lemma 2.4 again, from these properties we obtain where the last inequality follows from Proposition 2.6(c) and (2.23). Since the sets E whereM := 2M 2 + 3M + H d−1 (∂Ω) . Since the right-hand side is finite, we obtain that |Dv k |(Ω) is bounded uniformly with respect to k. On the other hand, since (E i ) i is a partition, inequality (5.12) implies that the sequence (v k ) k is bounded in L ∞ (Ω) and that v k → v strongly in L 1 (Ω). Therefore v ∈ BV (Ω).
To conclude the proof it is enough to take P j = E σ(j) and b j = t σ(j) where σ : N → Z is bijective.
In the following lemma the Caccioppoli partition is finite and we provide a precise estimate on the translations.

Lemma 5.2. (Piecewise Poincaré inequality)
Let α ≥ 1 and let 0 < θ < 1. Then there exist positive constants C Ω and C θ,α,d such that for every u ∈ GBV (Ω) there exist a finite Caccioppoli partition Ω = J j=1 P j ∪ R 1 ∪ R 2 , a finite family of translations (b j ) J j=1 ⊂ R, and a constant λ ∈ [1, C θ,α,d ], depending on u, satisfying the following estimates: Proof. It is enough to repeat the proof of [9, Lemma 3.5] replacing the space The following theorem shows that we can modify a function u by means of piecewise constant translations, with a precise control on the value taken by the functional G g Γ0 defined in (5.2) on the modified function.
there exist a finite Caccioppoli partition Ω = J j=1 P j ∪ R and a finite family of translations (t j ) J j=1 ⊂ R such that the function
It remains to prove (5.24). More precisely, we shall prove that which gives (5.26).
The previous result can be extended to the case of functions satisfying prescribed boundary conditions in the usual BV sense considered in (4.17) and (4.28). To this aim we introduce a bounded open set Ω ⊂ R d with Lipschitz boundary and containing Ω.
is defined as in (5.2), with Ω replaced by Ω , and a = |a 1 | + |a 2 |. Moreover, we can choose (P j ) J j=1 and (t j ) J j=1 so that the following additional property holds: Then v ∈ BV (Ω ) and satisfies (5.29), (5.30), and (5.31). It remains to prove (5.35). More precisely, we shall prove that To prove (5.40) we observe that ∇v = ∇u L d -a.e. on Ω \ R, while ∇v = ∇h L d -a.e. on R, so that which concludes the proof.
We are now in a position to prove the main result of this section.
Then there exist a subsequence of (u k ) k , not relabelled, modifications y k ∈ GBV (Ω ) of u k , with y k = h on Ω \ Ω, and a continuous function ψ : Remark 5.6. By (5.44), if (u k ) k is a minimizing sequence for the functional G g Γ0 with u k = h L d -a.e. in Ω \ Ω, then the same is true for (y k ) k . Inequalities (5.43) and (5.44), together with (5.1), imply that (y k ) k satisfies (3.26), while (5.45) guarantees that (3.27) also holds. Hence, by Theorem 3.11 there exists a subsequence of (y k ) k , not relabelled, and a function u ∈ GBV (Ω ) such that y k → u L d -a.e. in Ω .
Proof of Theorem 5.5. We repeat the proof of [9, Theorem 3.8] with some modifications. By (5.43) we have , where a = |a 1 |+|a 2 |. We define θ := 2 − and apply Corollary 5.4. Let us remark that, since we will pass to subsequences (not relabelled), we will eventually have only the inequality Step 1 (Application of Corollary 5.4) We apply Corollary 5.4 to the functions u k and the boundary data h with parameters θ and M := M1 min{c1,1} . We find finite Caccioppoli partitions Ω = ∪ j≥1 P k, j ∪ R k and piecewise translated functions v k ∈ BV (Ω ) defined by where (t k, j ) j≥1 are suitable finite families of translations. For notational convenience we shall also use the notation P k, 0 = R k so that (P k, j ) j≥0 is a partition of Ω . By Corollary 5.4 we have for every k, there is at most one j with L d (P k, (5.54) By (5.51) there exists a decreasing sequence η converging to zero such that which together with (5.54) gives For later use we recall that for every family (t k, belong to BV (Ω ) and satisfy Step 2 (Limiting objects for each ) By (5.43), (5.50), and (5.55) we obtain that for every the sequence (v k ) k is bounded in BV (Ω ). Indeed, arguing as in the proof of (5.46), by (5.43) and (5.55) we have that Together with the previous bounds this implies that |Dv k |(Ω ) is bounded uniformly with respect to k. Since v k = h on Ω \ Ω, by the Poincaré inequality we deduce that (v k ) k is bounded in BV (Ω ). Using a diagonal argument we obtain a subsequence of (k) k (not relabelled) such that for every there exist a function v ∈ BV (Ω ) and a constant L ∈ [0, C M,θ ,Ω ] (see (5.50 (5.57) By the semicontinuity of the L ∞ -norm we obtain Arguing as in Step 2 of the proof of [9, Theorem 3.8] we find Caccioppoli partitions (P j ) j≥0 and (P j ) j≥0 such that after extracting (not relabelled) subsequences in and k, we get Step 3 (Conclusion of the proof ) If (L ) does not tend to +∞ as → +∞, by (5.58) there exists a subsequence, not relabelled, such that (v − h) is bounded in L ∞ (Ω ). Then (v ) is bounded in L 1 (Ω ) and we can take ψ(t) = t to obtain sup Ω ψ(|v |)dx < +∞ . (5.60) The conclusion can now be obtained by repeating Step 5 of the proof of [9, Theorem 3.8] replacingv ,v k , and E k by v , v k , and G g Γ0 (·, Ω ), respectively. If L → +∞, passing to a subsequence, not relabelled, we may assume that L < L +1 . By the definition of L , for every we can find an increasing Hence we can repeat the argument leading to (5.57) and we obtain a subsequence of (k) k (not relabelled) and, for every , a functionv ∈ BV (Ω ) such thatv The conclusion can now be obtained by repeating Steps 4 and 5 in the proof of [9, Theorem 3.8] with E k replaced by G g Γ0 (·, Ω ).

Existence result
In this section we shall prove that the minimum problem (4.28) has a solution.
As observed at the end of Sect. 4, this will lead to the proof of the existence of a solution to problem (4.11).    Since F ξ,U (v) = F 0,U (v + ξ ), where ξ (x) := ξ · x, we deduce that F ξ,U is lower semicontinuous on GBV (U ) with respect to the convergence in measure on U .
To prove a similar result for F Φ Γ0 we fix a sequence (v k ) k ⊂ GBV (Ω ) which converges in measure on Ω to a function v ∈ GBV (Ω ), and an increasing sequence (K j ) j of compact subsets of Γ 0 such that H d−1 (Γ 0 \K j ) → 0. It is easy to construct a sequence (Φ j ) j of piecewise constant functions converging to Φ in L 1 (Ω ; R d ) such that for every j there exists a partition U 1 j , . . . , U ij j , N j of Ω \ K j , with U i j open and H d−1 (N j ) < +∞, such that Φ j = ξ i j in U i j for suitable constant vectors ξ i j ∈ R d . It is not restrictive to assume also that H d−1 (J v ∩ N j ) = 0. By the previous step of the proof, for every j we have Since f is Lipschitz continuous with constant 1, for every u ∈ GBV (Ω ) we have Passing to the limit as j → ∞ we obtain the lower semicontinuity inequality along the sequence (v k ) k .
We are now ready to prove the existence of a solution to the minimum problem (4.28). Proof. Since F Φ Γ0 coincides with the functional G g Γ0 introduced in (5.2), with g(x, ξ) := f (ξ+Φ(x)), and by (4.14) g satisfies (5.1), we can apply Theorem 5.5 and obtain that there exist a minimizing sequence (u k ) k ⊂ GBV (Ω ), with u k = w L d -a.e. in Ω \ Ω, and a continuous function ψ : R + → R + , with ψ(t) → +∞ as t → +∞, such that (3.26) and (3.27) hold. Then by the Compactness Theorem 3.11 there exist a subsequence, not relabelled, and a function u ∈ GBV (Ω ) such that u k → u L d -a.e. in Ω . By the Semicontinuity Theorem 6.1 we obtain that Since (u k ) k is a minimizing sequence and u = w L d -a.e. in Ω \ Ω we conclude that u is a solution of the minimum problem (6.3).
We now show that the minimum problem (4.11) has a solution.
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