Eigendamage: an eigendeformation model for the variational approximation of cohesive fracture—a one-dimensional case study

We study an approximation scheme for a variational theory of cohesive fracture in a one-dimensional setting. Here, the energy functional is approximated by a family of functionals depending on a small parameter 0<ε≪1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < \varepsilon \ll 1$$\end{document} and on two fields: the elastic part of the displacement field and an eigendeformation field that describes the inelastic response of the material beyond the elastic regime. We measure the inelastic contributions of the latter in terms of a non-local energy functional. Our main result shows that, as ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document}, the approximate functionals Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma$$\end{document}-converge to a cohesive zone model.


Introduction
A tension test on a bar will typically show that small deformations are completely reversible (elastic regime) while large deformations lead to complete failure (fracture regime).Only for very brittle materials one observes a sharp transition between these two regimes (brittle fracture).By way of contrast, many materials exhibit an intermediate cohesive zone (damage regime) in which plastic flow occurs and a body shows gradually increasing damage before eventual rupture (ductile fracture).
Variational models have been extremely successfully applied to problems in fracture mechanics, cf., e.g., [2,7,8,18] and the references therein including, in particular, the seminal contribution of Francfort and Marigo [23].Here energy functionals are considered that act on deformations in the class of functions of bounded variation (or deformation).The derivatives of these functions are merely measures, and the singular part of such a measure is directly related to the inelastic behavior of the bar.The resulting variational problems are of free discontinuity type allowing for solutions with jump discontinuity (macroscopic cracks).Moreover, within a damage regime the strain can contain a diffuse singular part describing continuous deformations beyond the elastic regime that can be related to the occurrence of microcracks, see e.g.[1,12,19,21,22,24,25,35].In fact, when considering variational problems with stored energy functions of linear growth at infinity and surface energy contributions that scale linearly for small crack openings, all these contributions to the total strain interact, cp.[6], which renders the problem challenging, both from a theoretical and a computational point of view.
As free discontinuity problems are of great interest not only in fracture mechanics but also in image processing, several approximation schemes have been proposed with the aim to devise efficient numerical approaches to simulations.Most notably, the Ambrosio-Tortorelli approximation [3,4] has triggered a still continuing interest in phase field models in which a second field (the 'phase field') is introduced that can be interpreted as a damage indicator and whose value influences the elastic response of the material.
With a particular focus on cohesive zone damage models we refer to, e.g., [13,[15][16][17]26].A small parameter ε is introduced in such models that corresponds to an intrinsic length scale over which sharp interfaces of the phase field variable are smeared out.A different approach has been initiated by Braides and Dal Maso for the Mumford-Shah functional, and then extended to various generalized settings in, e.g., [8,11,14,[27][28][29][30]32] which involves a non-local approximation of the original field u in terms of convolution kernels with intrinsic length scale ε ≪ 1.
Our main motivation comes from the Eigenfracture approach to brittle materials that has been developed in [37] and further considered in [33,34,36].Our main aim is to extend this model to a ductile fracture regime with a significant damage zone.The variables of the model are the deformation field u ε and an eigendeformation field g ε , which induces a decomposition of the strain u ′ ε = (u ′ ε − g ε ) + g ε into an elastic and an inelastic part, the latter describing deformation modes that cost no local elastic energy.(We refer to [31] for more details on the concept of eigendeformations to describe inelastic deformations and, in particular, plastic deformations.)The energy associated to the formation and increase of damage is accordingly modeled in terms of a non-local functional acting on g ε , which replaces the non-local contribution defined in terms of a simple ε-neighborhood of the crack set in the original Eigenfracture model by a more general (and softer) convolution approximation.
We would like to point out that our set-up thus introduces a novel modeling aspect to damage functionals.Instead of an explicit dependence of the stored energy function on the damage as being encoded in a phase field, in our model the constitutive laws, i.e., the linear elastic energy | • | 2 and fracture contribution f (see below) remain unchanged.An increase of damage is rather related to a transition from the elastic deformation field to the eigendeformation field.In particular, plastic deformations at the onset of the inelastic regime need not immediately lead to softening of the material.With respect to non-local convolution approximation schemes of the deformation field u we remark that in our model such non-local contributions need to be evaluated only near the support of g ε but not on purely elastic regions.
In the present contribution we focus on the one-dimensional case.In this setting our analysis will benefit from the corresponding studys [29] of Lussardi and Vitali for pure convolution functional.Indeed, we will follow along the same path in order to adapt and extend their methods to our two field set-up.There are, however, a number of notable differences in our analysis which lead us, also in view of later extensions to higher dimensions [5], to provide a self-contained account of our results.A main difficulty stems from the fact that there is no pre-assigned functional relation between the eigendeformation fields g ε and the strain fields u ′ ε .Rather these quantities are merely 'coupled by regularity' in the sense that u ′ ε − g ε ∈ L 2 .As the limit of g ε needs to be studied in a rather weak space, this leads to technical difficulties when transferring asymptotic properties from u ε to g ε .
Our results also constitute the first step towards higher-dimensional models.In particular, the case of antiplane shear will be addressed in a forthcoming contribution [5].Here the lack of a direct relation between g ε and ∇u ε and hence the absence of an underlying gradient structure will pose severe additional challenges.

Outline
We start by describing the setting of the problem and by stating the main results in Section 2. In Section 3 we remind some facts on functions of bounded variation and the flat topology.Section 4 is devoted to a compactness result.The Γ-lower limit for the eigendamage model is established in Section 5. To this end, we first derive the estimate from below of the jump part, subsequently the estimate from below of the volume term and the Cantor term, and the proof of the Γ-lim inf inequality is then completed by combining the previous results.In Section 6 we then address the estimate from above of the Γ-upper limit.Finally, in Section 7 the asymptotic behavior of the minimal energies with respect to the eigendeformation variable is studied.

Setting of the problem and main result
Suppose that a beam occupies the region (a, b) with 0 < a < b < ∞ and that a displacement u : (a, b) → R affects the beam.As cohesive energy associated with u we shall consider where c 0 is a fixed positive constant, and ψ, f : [0, ∞) → [0, ∞) are functions defined via The main ingredients in the energy (1) are a volume term, depending on the strain of the beam u ′ and corresponding to the stored energy, a surface term, depending on the crack opening [u] := u(• +) − u(• −) on the jump set J u and modeling the energy caused by cracks, and finally a diffuse damage term, depending on the Cantor derivative D c u and corresponding to the energy caused by microcracks.
The natural function space in order to study such functionals in one dimension is the space BV (a, b) of functions of bounded variation on (a, b).Notice that the distributional derivative of each function u ∈ BV (a, b) allows for a decomposition Du = u ′ L 1 + D s u into the absolutely continuous and the singular part with respect to the Lebesgue measure, and the singular part D s u = [u]H 0 J u + D c u in turn into the jump part and the Cantor part, which we have used in (1).We consider both models with an apriori bound u L ∞ (a,b) ≤ K, K < ∞, and unrestricted models with K = ∞.
We next introduce a functional depending on two fields ) with a non-local approximation of the the second variable γ, given as with ε > 0, I ε (x) := (x − ε, x + ε), and either K > 0 a fixed constant or K = ∞.We notice that E ε (u, γ) can only be finite if γ is absolutely continuous with respect to the Lebesgue measure, with density in L 1 (a, b).In this case u ′ represents the strain of the beam and γ is intended to compensate u ′ in regions where u ′ is above a certain strain level.Hence, u ′ −g is the elastic strain of the material, while γ describes the deformation of the material beyond the elastic regime, indicating that a permanent deformation is exhibited if In what follows, we are interested into the asymptotic behavior of the functionals {E ε } ε>0 as ε ց 0 (in the sense of Γ-convergence).Focussing first on the case K < ∞, it will be described by the energy functional E which for (u, γ) ∈ L 1 (a, b) × M(a, b) is defined as Let us notice that for a finite energy E(u, γ), the displacement field u and the eigendeformation field γ need to be linked in a very particular way.The singular part γ s of the measure γ with respect to the Lebesgue measure needs to coincide with the singular part D s u of the distributional derivative of u.The absolutely continuous part gL 1 of γ instead is not completely determined by the function u, but only the integrability restric- By a pointwise minimization of the integrand, the minimizer g * is explicitly given as For later purposes we notice that the eigendeformation field γ is completely described in terms of the function u as Moreover, the corresponding energy functional E(u, γ opt ) reduces to a one-field functional depending only on the displacement u ∈ BV (a, b), which under the additional restriction is precisely given by the energy F (u) introduced in (1).
In order to state our Γ-convergence result we need to endow L 1 (a, b) × M(a, b) with a topology.A natural choice for the first component is the strong topology on L 1 (a, b).One appropriate choice for the second component is the flat topology, that is the norm topology on the dual of the space of Lipschitz continuous functions with compact support, while an alternative choice is the topology induced by suitable negative W −1,q -Sobolev norms, see Section 3 for more details.Our main result is the following: Theorem 2.1.Let L 1 (a, b) be equipped with the strong topology and M(a, b) be equipped with the flat topology.Assume K < ∞.Then the family The associated compactness result is stated in Theorem 4.2, where we in fact establish for the second variable convergence in W −1,q (a, b) for all 1 < q < ∞.Therefore, we obtain as a direct consequence of Theorem 2.1 also Γ-convergence of {E ε } ε>0 to E in L 1 (a, b) × M(a, b), when M(a, b) is equipped with the stronger topology of convergence in W −1,q (a, b) for some 1 < q < ∞.
Remark 2.2.Our result can be seen as a two-field extension of the setting considered in [29].Indeed, introducing the constraints g = u ′ , respectively γ = Du, one is lead to functionals F ε (u) = E ε (u, u ′ L 1 ) and F (u) = E(u, Du) depending on u only.In this case the Γ-convergence of the sequence {F ε } ε to F has been obtained in [29].
The unrestricted problem is in fact strongly related.Indeed, even for K = ∞ an energy bound implies L ∞ bounds away from an asymptotically small exceptional set.The complement of the exceptional set can be chosen as a union of a bounded number of intervals, concentrating on the points of a finite partition a = x 0 < x 1 < . . .< x m = b of (a, b) in the limit ε → 0 such that {u ε } ε and {g ε } ε converge with respect to the L 1 norm, respectively the flat norm, locally on (a, b) \ {x 1 , . . ., x m−1 }.On the exceptional set, however, u ′ ε and g ε can assume extremely large values, spoiling their convergence even in a weak distributional sense.As a result, large jumps can develop in the limit and parts of u ε may elapse to ±∞.In order to account for such a possibility we consider limiting functions taking values in R = R∪{−∞, +∞}.More precisely, let P = (x 0 , . . ., x m ) : a = x 0 < x 1 < . . .< x m = b, m ∈ N and consider BV ∞,P (a, b) as consisting of functions u : (a, b) → R of the form with α i ∈ {−∞, 0, +∞}, i = 1, . . ., m, (x 0 , . . ., x m ) ∈ P and w ∈ BV (a, b).We denote the part where u is finite by We say that (u The corresponding compactness result for K = ∞ with respect to this particular convergence is stated in Theorem 4.3. Remark 2.4.In fact, E(u, γ) can be finite only if the restriction of u to F(u) is a BV function and not merely an element of GBV (F(u)).In particular, if E(u, γ) < ∞ and u ∈ L 1 (a, b), then u ∈ BV (a, b).
Remark 2.5.Our methods can easily be adapted to obtain an alternative asymptotic description by considering renormalized functionals: From the above partition P one can derive a coarser one (whose members are finite unions of intervals) so that on each such set one has an L ∞ bound on u ε modulo a single additive constant and the mutual distance of u ε on two different sets diverges.This allows for an asymptotic description also of those parts that escape to infinity.
Remark 2.6.Our results remain true if restricted to preassigned boundary values u As for a bounded energy sequence parts of the jump set could accumulate at the boundary {a, b}, a usual way to implement boundary conditions is to consider E ua,u b ε and E ua,u b on an extended interval (a−η, b+η), with η > 0 fixed, which are defined as E ε and E, respectively, before but with the additional constraints Indeed, the Γ-lim inf inequality and the compactness property are direct consequences of the case with free boundaries.The Γ-lim sup inequality for K < ∞ follows from the observation that the recovery sequence constructed in Proposition 6.1 indeed satisfies u ε = u near {a, b} and from Remark 6.4.The case K = ∞ is a direct consequence as there the recovery sequence is built as in the case K < ∞ near {a, b} since u a , u b ∈ R.
Remark 2.7.Our results can also be adapted to general continuous stored energy functions W leading to a general non-quadratic bulk contribution b a W (u ′ − g) dx, whenever W satisfies a p-growth condition of the form c|r| p − C ≤ W (r) ≤ C|r| p + C for suitable constants c, C > 0 and some p ∈ (1, ∞), and for convenience we also assume that min W = W (0) = 0.The first term in the limiting functional is then replaced by b a W * * (u ′ − g) dx, respectively, F (u) W * * (u ′ − g) dx, where W * * is the convex envelope of W .In fact, making use of the estimate W ≥ W * * , compactness and the Γ-lim inf inequality follow exactly as before with W * * instead of | • | 2 by taking account of the obvious adaptions such as replacing L 2 by L p and SBV 2 by SBV p .The Γ-lim sup inequality requires an extra relaxation step, which is detailed in Remark 6.2, and is otherwise straightforward as well.
For completeness we also give the corresponding approximation results for the minimal energies with respect to the second variable γ, which are defined as and Ẽ(u) := inf for u ∈ L 1 (a, b) and u ∈ L 0 ((a, b), R), respectively.As a direct consequence of the previous Γ-convergence result we obtain: Corollary 2.8.The family { Ẽε } ε>0 Γ-converges to Ẽ, on L 1 (a, b) equipped with the strong topology if K < ∞ and with respect to convergence a.e. on L 0 ((a, b), R) if K = ∞.

Preliminaries
In this section, we recall some basics on BV -functions, for simplicity on a one-dimensional interval (a, b) ⊂ R, and convergence of measures.where |Du|(a, b) is the total variation of Du.We here collect some basic facts from [2] for functions of bounded variation, which are relevant for our paper.We recall the notions of weak- * and strict convergence for sequences {u n } n in BV (a, b), which are useful for compactness properties and approximation arguments, respectively.We say that {u n } n converges weakly- * to u ∈ BV (a, b), denoted by We notice that every weakly- * converging sequence in BV (a, b) is norm-bounded by Banach-Steinhaus, while every norm-bounded sequence in BV (a, b) contains a weakly- * converging subsequence (see [2,Theorem 3.23]).We further say that b) with respect to the strict topology (see [2, Theorem 3.9]).
We next discuss approximate continuity and discontinuity properties of a function u ∈ L 1 loc (a, b).We say that u has an approximate limit We denote by S u the set, where this condition fails, and call it the approximate discontinuity set of u.It is L 1 -negligible, and u coincides L 1 -a.e. in (a, b) \ S u with u.Furthermore, we say that u has an approximate jump point at x ∈ S u if there exist (unique) We denote by J u the set of approximate jump points and call it the jump set of u.Notice that u(x+) and u(x−) can be considered as one-sided limits from the right and from the left, respectively.For u ∈ BV (a, b) the set J u coincides with S u and is at most countable.It is also convenient to work with the precise representative According to the Radon-Nikodým theorem the measure derivative Du = D a uL 1 + D s u can be decomposed into the absolutely continuous and the singular part with respect to the Lebesgue measure L 1 .We then define the jump and the Cantor part of Du as From the identifications D a u = u ′ L 1 with the approximate gradient u ′ for the absolutely continuous part and We can actually decompose the function u as for an absolutely continuous function u a ∈ W 1,1 (a, b) with Du a = D a u, a jump function u j with Du j = D j u, and a (continuous) Cantor function u c with Du c = D c u (notice that these functions are determined uniquely up to additive constants).Thus, the decomposition of Du is recovered from a corresponding decomposition of the function itself (which for BV -functions defined on open subsets of R d with d > 1 in general is not possible).
We finally mention the subspace SBV (a, b) of special functions of bounded variation, which contains all functions u ∈ BV (a, b) with D c u = 0.In addition, we define Convergence in negative Sobolev spaces and in the flat topology.The negative Sobolev spaces W −1,q (a, b) with 1 < q ≤ ∞ are defined as usual as the dual spaces of denoting the conjugate exponent to q with 1 q + 1 q ′ = 1), and correspondingly the norm is defined via the duality pairing as for every T ∈ W −1,q (a, b).Consequently, the spaces W −1,q (a, b) with 1 < q < ∞ are reflexive and separable.For later purposes, we mention two specific situations.
then, by the Hölder inequality and the continuous embedding (with constant C ′ ), we obtain T Dv , T w ∈ W −1,q (a, b) with Because of the continuous and dense embedding W 1,1 0 (a, b) ⊂ C 0 (a, b), the negative Sobolev norms can actually be considered on the space M(a, b) of all finite Radon measures on (a, b), for which the duality pairing reads as If we allow q ′ = ∞ in this expression, we obtain the flat norm Here we have the inequalities for all functions v ∈ BV (a, b) and w ∈ L 1 (a, b).Let us still notice that due to Schauder's theorem and the compact embedding W 1,∞ 0 (a, b) ⋐ C 0 (a, b), the flat topology metrizes weak- * convergence of (uniformly bounded) measures.Therefore, we have the following relations for the convergence of measure with respect to convergence in W −1,q (a, b), the flat norm and in the weak- * sense.
Lemma 3.1 (on convergence of measures).For a measure µ ∈ M(a, b) and a sequence {µ n } n∈N of measures in M(a, b), we have:

Compactness
In this section we establish a compactness result for sequences in L 1 (a, b) × M(a, b) with bounded energy E ε .This result together with a Γ-convergence result implies the convergence of minimizers and the corresponding minimum values.In order to bound suitable norms of the two fields in terms of the energy, we first prove the following technical lemma: |g| dt dx.
Proof.We proceed analogously to [29, proof of Lemma 4 The application of [10, Lemma 4.2] (with η = 2ε), which is a consequence of the mean value theorem for integrals, shows that holds for a suitable x ε ∈ R. By non-negativity of f , the choice of the cut-off function φ ε and the definition of G ε , we hence have If we now set then the claim follows directly from (10), after rewriting the right-hand side via the definition of f as We can now address the aforementioned compactness results.
, b) for all 1 < q < ∞ and in particular in the flat norm.Moreover, there holds γ s = D s u and u ′ − g ∈ L 2 (a, b).
Proof.We first observe from the finiteness of E ε (u ε , γ ε ) that we necessarily have ) and consequently contains a subsequence, which converges weakly- * in W −1,∞ (a, b) to some γ ∈ W −1,∞ (a, b).We next study the convergence of the sequence {u ε } ε .To this end, we consider the function v ε defined by As a consequence of Lemma 4.1 and the definition of the energy E ε , we deduce i.e., that #J vε is bounded independently of ε.By the Cauchy-Schwarz inequality and again by Lemma 4.1, we additionally have . By the Rellich-Kondrachov theorem, there exists a subsequence that converges in L q (a, b) to some u ∈ BV (a, b) with u L ∞ (a,b) ≤ K, for any 1 ≤ q < ∞.To identify u as the limit in L q (a, b) of (the same subsequence of) It remains to show convergence of (the subsequence of) {γ ε } ε in W −1,q (a, b) for any 1 < q < ∞ and the claimed relations between γ and Du.In view of (8), we have with α i ∈ {−∞, 0, +∞}, i = 1, . . ., m and w ∈ BV ((a, b); R), and a measure γ ∈ M(a, b) such that, up to subsequences, {u ε } ε converges to u a.e. and in Proof.We define G ′ ε exactly as in Lemma 4.1 and denote by J 1 , . . ., J mε the connected components of {I ε (x α ) : x α ∈ G ′ ε }.We then notice that the arguments in the preceding proof show that for some constant C we have m ε ≤ C and while L 1 ((a, b) \ (J 1 ∪ . . .∪ J mε )) ≤ Cε.We set x ε,i = sup J i and α ε,i = − J i u ε dx, i = 1, . . ., m ε .Note that (11) implies Passing to a subsequence we may assume that m ε = m for some m independent of ε and ) such that {x 0 , . . ., x m } = {x 0 , x1 , . . ., x m} (with x0 := a and, by construction, x m = b) and set ) and the uniform bounds in (11) and (12) (12).In this case we define w ∈ BV (x i−1 , x i ) as the weak* limit in BV loc   Remark 4.6.In principle, with the compactness result of Theorem 4.2 at hand, we could infer our Γ-convergence result in Theorem 2.1 from known results, by separate considerations of the elastic and inelastic contributions.To this end, given . This allows to express as the sum of the standard Dirichlet energy for W ε and a non-local energy involving only G ε considered by Lussardi and Vitali in [29].However, the understanding of the coupling between u ε and γ ε is essential for the extension to higher dimensions addressed in [5], which cannot be traced back to the one-dimensional case via the slicing technique.
For this reason, we prefer to give a self-contained proof of Theorem 2.1.
5 Estimate from below of the Γ-lower limit Next, we start with the proof of the Γ-lim inf inequality.Except for the very last paragraph we assume K < ∞ in the whole section.To show this inequality it is useful to introduce the localized version of the functionals {E ε } ε and E. They are defined for (u, γ) ) and every open subset A of (a, b) via and We denote the Γ-lower and Γ-upper limit of {E ε } ε by respectively.For the Γ-lower limit we also need a localized version, for which we adapt the notation and write E ′ (•, •, A) for every open subset A of (a, b).(i) The properties that the set functions A → E ε (u, γ, A) are increasing and superadditive (on disjoint sets) for each ε carry immediately over to A → E ′ (u, γ, A).In this section we prove the Γ-lim inf inequality, where in view of Remark 4.5 it is sufficient to consider (u, γ) ∈ BV (a, b) × M(a, b) with u L ∞ (a,b) ≤ K, γ = D s u + gL 1 and u ′ − g ∈ L 2 (a, b).The basic idea is to derive three separate estimates for the jump part, the volume term and the Cantor term, respectively, and to infer the desired estimate then from a combination of these estimates by means of measure theory.

Estimate from below of the jump term
Proof.
Step 1: For every x ∈ J u ∩ A we have Since only finite energy approximations are of interest, we consider a sequence . By definition of (u(x+), u(x−)) and since u ε → u in measure, we readily find points x− ε ∈ (x − δ, x) and x+ ε ∈ (x, x + δ) such that, for sufficiently small ε, (also cp.[29, Lemma 5.1]).Using the monotonicity of A → E ε (u ε , γ ε , A) and applying the estimate (10) with (a, b) replaced by (x − 2δ, x + 2δ) on an associated grid of points Ḡε , we obtain from the subadditivity and non-negativity of f for all ε < δ (notice the inclusion For the argument on the right-hand side, we observe from the Cauchy-Schwarz inequality, the inequalities in (14) and from the energy bound .
Using once again the fact that f is increasing, we can continue to estimate (15) from below via Now, passing to the lim inf as ε → 0 and then letting δ → 0 + , we obtain (13).
Step 2. For an arbitrary M ∈ N with M ≤ #(J u ∩ A) we select a set {x 1 , . . ., x M } containing M points of J u ∩ A and pairwise disjoint open intervals I 1 , . . ., I M in A such that x i ∈ I i for all i = 1, . . ., M .First, we apply the monotonicity and superadditivity of E ′ (u, γ, • ) (see Remark 5.1) and then the estimate of Step 1 for I i instead of A. This yields Since M is arbitrary and J u is at most countable, the claim of the proposition follows.

Estimate from below of the volume and Cantor terms
We basically follow the idea of Lussardi and Vitali from [29, Lemma 4.3 and Lemma 4.4].We start by proving that approximation sequences in W 1,1 (a, b)× M(a, b) can be modified in such a way that in the limit we additionally have the optimal L ∞ -estimate.
Proof.We follow the outline of the proof for [29, Lemma 4.3], which, however, needs some modifications due to the additional variable γ.In what follows, we may assume that the precise representatives of u and u ε for each ε are considered.The function u can be decomposed as u a + u j + u c , see (7), where u j is a jump function with jump discontinuities at any point of J u and where u a + u c is uniformly continuous in (a, b).We set and first claim that for every n ∈ N there exists δ n ∈ (0, 1  n ] such that In fact, there are only finitely many points x1 , . . ., xm(n) in J u that have to be excluded to deduce that By choosing δ n > 0 sufficiently small we can guarantee ( 16) by (18) and the definition of σ provided that each interval of length δ n contains at most one xj , and we can further ensure (17), by the uniform continuity of u a + u c .We then consider a partition P n of (a, b), i.e., (where the dependence of the points on n is not written explicitly) such that the mesh size is less than δ n , i.e., x i+1 − x i < δ n for all i ∈ {0, . . ., k}, x i / ∈ J u and u ε (x i ) → u(x i ) as ε → 0 for every i ∈ {1, . . ., k}, which is possible by the pointwise convergence u ε → u a.e. in (a, b).Since by construction of δ n at most one of the points x1 , . . ., xm(n) ∈ J u , where a large jump of u j occurs, may belong to the interval [x i , x i+1 ], we necessarily have |u j (x) − u j (y)| < 1 n for y = x i or y = x i+1 such that as a consequence of ( 17) there holds for all x ∈ [x i , x i+1 ] and every i ∈ {1, . . ., k − 1}.After having fixed the partitions P n , we can now start with the construction of the sequence {ū ε } ε .Since u ε → u in measure and u ε (x i ) → u(x i ) for every i ∈ {1, . . ., k}, we can fix a "level" εn for each n ∈ N such that for every i = 1, . . ., k and all ε < εn .
Notice that we can choose {ε n } n strictly decreasing and such that εn → 0 + as n → ∞.
For ε > ε1 we then set ūε := u ε and γε := γ ε .Otherwise, if ε ∈ (0, ε1 ], we first determine the unique n = n(ε) ∈ N with ε ∈ (ε n+1 , εn ].On the first and the last interval of P n we set ūε equal to u ε (x 1 ) and u ε (x k ), respectively.On an arbitrary interior interval [α, β] of the form [x i , x i+1 ] for some i ∈ {1, . . ., k − 1} we define, after assuming without loss of generality for every i ∈ {1, . . ., k} and u ε ∈ W 1,1 (a, b), we clearly have ūε ∈ W 1,1 (a, b).Moreover, the L ∞ bound on u ε with constant K directly transfers to ūε .We next study the asymptotic behavior of the sequence {ū ε } ε .From the definition of ūε and with (21), we observe for all ε ≤ ε1 ūε which, via ( 16) and ( 17), implies for all x ∈ [x i , x i+1 ] and every i ∈ {1, . . ., k − 1}.Since the latter estimate is also satisfied for the first and the last interval of the partition, we then infer from n = n(ε) → ∞ as ε → 0 the estimate lim sup In order to show the convergence claims of {ū ε } ε , we again consider an arbitrary interior interval [α, β] of the partition P n .If we denote the pointwise projection of as we know u(x) ∈ [u ε (α) − 3/n, u ε (β) + 3/n] due to (19) and (21).Since the length of the first and last interval of P n vanish in the limit n → ∞ and hence for ε → 0, this implies the pointwise convergence of {ū ε } ε to u a.e. on (a, b).In addition, as {ū ε } ε is bounded in L ∞ (a, b), convergence in L q (a, b) for all 1 ≤ q < ∞ follows from the dominated convergence theorem.For later purposes, we notice from the definition of ūε and the previous inclusion for u that for every ε we have We next define the sequence {γ ε } ε in M(a, b) by setting for every ε ḡε (x) := g ε (x) if u ε (x) = ūε (x), 0 otherwise, and γε := ḡε L 1 .Since there holds ū′ In view of ( 9) and the Cauchy-Schwarz inequality we then notice for If we consider the limit ε → 0 on the right-hand side, the first term disappears, since we have the convergence ūε − u ε → 0 in L 1 (a, b), while the second term disappears by (22) combined with (20) and the fact that the length of the intervals in P n vanish for ε → 0. Therefore, we have γε − γ ε → 0 in the flat norm.Since by assumption there holds γ ε → γ in the flat norm, we conclude that we also have γε → γ in the flat norm, which completes the proof of the lemma.
For a localization procedure, we further need the following statement on the relation between E ′ (u, γ, I) and the Γ-lower limit E ′ (u I , γ I ), where I ⊂ (a, b) is an open interval, u I is the extension of u| I to (a, b) with inner traces and γ I is the restriction of γ to I.
and γ I := γ I there holds Proof.We proceed analogously to the proof of [29,Lemma 4.4].By definition of E ′ as the Γ-lower limit of {E ε } ε , there exists a sequence Without loss of generality, we may also assume pointwise convergence u ε → u a.e. in (a, b).For an arbitrary η ∈ (0, (β − α)/2) we then pick points α η ∈ (α, α + η) and β η ∈ (β − η, β) such that on the one hand and on the other hand For I η := (α η , β η ) ⊂ (α, β) we now consider the functions (u Iη , γ Iη ) and the sequence We next observe u Iη → u I in L 1 (a, b) and γ Iη → γ I in the flat norm as η → 0, from (24), respectively, Lemma 3.1 since γ Iη * ⇀ γ I in M(a, b) by dominated convergence (as we have pointwise convergence ½ Iη → ½ I on (a, b)).By the lower semicontinuity of E ′ we then arrive at the claim Now, we finally turn to the estimate from below for the volume and the Cantor terms.
Proposition 5.5.Let A be an open subset of (a, b).For every Proof.
Step 1: With σ := sup x∈Ju |u(x+) − u(x−)|, there holds the preliminary estimate By definition of E ′ as the Γ-lower limit of {E ε } ε , there exists a sequence After assuming without loss of generality E ′ (u, γ) < ∞ and passing to a subsequence (not relabeled) and a possible modification of the sequence via Lemma 5.3 we may further suppose for a positive constant C 0 as well as lim sup Let η > 0 be fixed.We may assume that u ε − u L ∞ (a,b) ≤ σ + η holds for all ε.Analogously as in the proof of Lemma 5.3 (cf.( 16) and ( 17)), there exists δ η > 0 such that |u(x) − u(y)| < σ + η for all x, y ∈ (a, b) with |x − y| < δ η .Thus, there holds for all such ε.Next, we apply Lemma 4.1 with u = u ε and γ = γ ε .In this way we find a uniform grid G ε in the interval (a, b) with grid size 2ε such that and then extended to (a, b) by the constant values ṽε (a + * ) and ṽε (b − * ), respectively, for all ε.As ṽε is bounded with ṽε L ∞ (a,b) ≤ K and coincides with u ε on the set {I ε (x α ) : by ( 26) and ( 28), we observe ṽε → u in L q (a, b) for all 1 ≤ q < ∞.Moreover, we have ṽε provided that ε is sufficiently small, i.e., 4ε < δ η (such that ( 27) is satisfied).Since the number of jumps of ṽε is bounded via (28) and the definition of E ε by we end up with the estimate for the size of D s ṽε in terms of the energy for all ε.In order to show that {γ ε + D s ṽε } ε is an approximating sequence of γ, we first notice from the definition of ṽε and γε = gε L 1 in ( 29) and ( 32), with ṽ′ With (9) and the Cauchy-Schwarz inequality we can then continue to estimate We now study the terms on the right-hand side.From ( 26) and the definition of Together with (30) and taking into account also the strong convergences u ε → u and ṽε → u in L 1 (a, b), we then arrive at For the latter conclusion we have also used the fact that 28) and the estimate (31) (recall also the bound (26) on the energies).
After having discussed the convergence properties of the sequence {(ṽ ε , γε )} ε , we can finally turn to the proof of the estimate (25).From the definition of E ε we obtain via ( 28) and ( 31) By the choice of the sequence {(u ε , γ ε )} ε with (26) it follows that  33) and the lower semicontinuity of the total variation with respect to weak- * convergence.By the arbitrariness of η > 0 we conclude from the previous inequality the desired estimate (25).

Conclusion and proof of the Γ-lim inf inequalities
we have proved so far in Propositions 5.2 and 5.5 the following lower bounds for the volume, the Cantor and the jump part: for every open subset A of (a, b).These are now combined to prove the estimate from below of the Γ-lower limit which shows the first part of Theorem 2.1.
Next, we define The Γ-lim inf in the case K = ∞ is a direct consequence of Theorem 5.6.
Proof.Assuming without loss of generality E ε (u ε , γ ε ) < C 0 for a positive constant C 0 , by Theorem 4.3 we have that u ∈ BV ∞,P (a, b) and there is a partition a = x 0 < x 1 . . .< x m = b such that J u ⊂ {x 1 , . . ., x m−1 } ∪ F(u) and for From Remark 4.4 and Theorem 5.6 (applied for a suitable K) we then get The assertion follows in the limit δ ց 0 from the monotone convergence theorem.
6 Estimate from above of the Γ-upper limit We now turn to the estimate from above of the Γ-upper limit E ′′ .Except for the very last paragraph we assume K < ∞ in the whole section.We again restrict ourselves to pairs (u, , the jump set is finite, i.e., J u = {x 1 , . . ., x N −1 } for some N ∈ N, and we may further assume by the Sobolev embedding theorem that u is a piecewise continuous function with one-sided limits u(x±) for all x ∈ (a, b).Thus, γ is of the form with g ∈ L 2 (a, b).Let x 0 = a and x N = b.We then choose ε small enough such that We first define u ε ∈ W 1,1 (a, b) nearby the jumps of u by linear interpolation via (see the figure below).By construction, we have u ε L ∞ (a,b) ≤ K for all ε, and taking advantage of the fact that u is only modified on the intervals (x i − ε 2 − 2ε, x i + ε 2 + 2ε) for i ∈ {1, . . ., N − 1}, we also have We next define γ ε ∈ M(a, b) as γ ε = g ε L 1 , where g ε ∈ L 2 (a, b) is given by Therefore, we observe from the definition of u ′ ε that for every function ϕ This shows the convergence of {γ ε } ε to γ in the flat norm.It only remains to establish the energy estimate.From the construction of (u ε , γ ε ), we clearly have Due to the monotonicity of f and f (t) ≤ c 0 t for all t ≥ 0, we estimate the non-local energy term by With the continuity of u outside of the jump set J u we can pass to the limit ε → 0 on the right-hand side.In this way, we finally arrive at Remark 6.2.For a general stored energy function W as described in Remark 2.7 the above argument can be augmented with a standard relaxation step by adding to u ε a function So also in this case we have We next address the modification of γ.We extend the absolutely continuous part g outside of (a, b) by 0 and set g a,h := g * ψ h for all h > 0. Then we have g a,h ∈ C ∞ (R) for all h > 0 and Now, we set , ( 36), ( 37) and (38), implies lim sup This does not yet show (34), since u h L ∞ (a,b) ≤ K might not be satisfied for all h > 0.
We resolve this problem in two steps.With u L ∞ (a,b) ≤ K and u h → u in L 1 (a, b), we can fix a sequence {η h } h in R + with η h → 0 + as h → 0 and We next define the truncated versions ũh (x) := min{max{u h (x), −K − η h }, K + η h } for all h > 0.
We then have ũh → u in L 1 (a, b), Dũ h → Du in the flat norm and, in addition, also ũh L ∞ (a,b) ≤ K + η h for all h > 0. Correspondingly, we set γh := D j ũh + g h ½ {ũ h =u h } L 1 for all h > 0.
By using (9) and by applying subsequently the Cauchy-Schwarz inequality, we get If we pass to the limit h → 0 on the right-hand side, the first term vanishes because of the uniform boundedness of u where the last two terms on the right-hand side vanish as h → ∞ by construction.
Again, the case K = ∞ is a direct consequence.

Γ-convergence for the minimal energies with respect to γ
We finally prove the Γ-convergence result in Corollary 2.8 for the minimal energies with respect to the second variable γ, i.e., we consider the energies Ẽε and Ẽ from ( 5) and ( 6), respectively.We only treat the case K < ∞, the necessary modifications for K = ∞ are straightforward.Notice that, as a direct consequence of the fact that the function g * from (3) solves the optimization problem in (2), for every u ∈ BV (a, b) there holds Ẽ(u) = E(u, D s u + g * L 1 ) = E(u, γ opt ). (41) For completeness we state also the corresponding compactness result.We first show the Γ-lim inf inequality.We consider an arbitrary sequence {u ε } ε in L 1 (a, b) with u ε → u in L 1 (a, b), for which we may assume Ẽε (u ε ) ≤ C 0 for some positive constant C 0 and all ε.We then select a low energy sequence {γ ε } ε in M(a, b) with E ε (u ε , γ ε ) ≤ Ẽε (u ε ) + ε for every ε > 0.
By passing to a subsequence if necessary, we may assume that lim inf a, b) is required.A particularly interesting choice of g for a given function u ∈ BV (a, b) constitutes the unique minimizer g * of the optimization problem to minimize b a |u ′ − g| 2 dx + b a c 0 |g| dx among all g ∈ L 1 (a, b).
e. on (a − η, a) and u = u b a.e. on (b, b + η) and γ ((a − η, a) ∪ (b, b + η)) = 0.So if u does not satisfy the given boundary values on (a, b) in the limiting problem, this leads to an extra energy cost:

Functions of bounded
variation.A function u ∈ L 1 (a, b) is said to belong to the space BV (a, b) of functions of bounded variation if its distributional derivative is a finite Radon measure, i.e, if the integration-by-parts formula b a uϕ ′ dx = − b a ϕ dDu for every ϕ ∈ C 1 c (a, b) is valid for a (unique) measure Du ∈ M(a, b).The space BV (a, b) is a Banach space endowed with the norm u BV (a,b) := u L 1 (a,b) + |Du|(a, b), for the jump part (see [2, Theorem 3.83 and formula (3.90)]) we arrive at the decomposition , b) and thus, via Lemma 3.1, also in the flat norm.This shows γ = Du−wL 1 ∈ M(a, b), which in turn, by the Radon-Nikodým theorem, yields γ s = D s u and w = u ′ − g ∈ L 2 (a, b).

Theorem
is bounded independently of I by the uniform energy bound, we have indeed w ′ − g ∈ L 2 (a, b).

Remark 4 . 4 .
For later use we notice that Lemma 4.1 and the proof of Theorem 4.3 show that, under the assumptions of Theorem 4.3, for any open A ⋐ (a, b) \ {x 1 , . . ., x m−1 }

Remark 4 . 5 .
According to the compactness results of Theorems 4.2 and 4.3 we may restrict ourselves to pairs (u, γ)∈ BV (a, b) × M(a, b) with u L ∞ (a,b) ≤ K and (u, γ) ∈ BV ∞,P (a, b) × M(a, b), respectively, and such that the measure Du − γ, is absolutely continuous with respect to the Lebesgue measure with density in L 2 (F(u)).The statements of Theorems 2.1 and 2.3 are trivial otherwise.

(
ii) A direct consequence of (i) is that lower bounds for A → E ′ (u, γ, A) transfer from intervals in (a, b) to arbitrary open subsets of (a, b), i.e., if for a positive Borel measure λ an estimate of the form E ′ (u, γ, A) ≥ λ(A) holds for all intervals A ⊂ (a, b), then the estimate actually holds for any open subset A ⊂ (a, b), cp.[29, Remark 4.6] for a similar statement.

Proposition 5 . 2 .
Let A be an open subset of (a, b).For every (u, γ) ∈ BV (a, b)×M(a, b) with γ = D s u + gL 1 and u ′ − g ∈ L 2 (a, b) we have the flat norm and Lemma 3.1, we then conclude γε + D s ṽε → γ in the flat norm and γε + D s ṽε * ⇀ γ in M(a, b).

1 →
ε | dx + c 0 |D s ṽε |(a, b) ≥ b a |u ′ − g| 2 dx + c 0 b a |g| dx + c 0 |D s u|(a, b) ≥ b a |u ′ − g| 2 dx + c 0 b a |g| dx + c 0 |D c u|(a, b).Let us comment on the second-last inequality.For the first term we first deduce from the boundedness of the sequence {u ′ ε − g ε } ε in L 2 (a, b) combined with the convergences u ′ ε L Du and γ ε → γ = D s u + gL 1 in the flat norm that u ′ ε − g ε ⇀ u ′ − g in L 2 (a, b) and then employ the lower semicontinuity of the L 2 -norm with respect to weak convergence in L 2 (a, b).For the second and third term we use the weak- * convergence γε + D s ṽε * ⇀ γ = D s u + gL 1 in M(a, b) from (

Proposition 6 . 1 .
γ) ∈ BV (a, b) × M(a, b) with u L ∞ (a,b) ≤ K, γ = D s u + gL 1 and u ′ − g ∈ L 2 (a, b) since the estimates are trivial otherwise.We first show the result for the particular case u ∈ SBV 2 (a, b) and then deduce the general result by approximation.For every (u, γ) ∈ SBV 2 (a, b) × M(a, b) with u L ∞ (a,b) ≤ K, γ = D s u + gL 1 and u ′ − g ∈ L 2 (a, b) we have

Fig. :
Fig.: Construction of the recovery sequence for a piecewise affine function with jump discontinuities.

Corollary 7 . 1 (
Compactness of the minimal energies with respect to γ).Let {u ε } ε be a sequence in L 1 (a, b) with Ẽε (u ε ) ≤ C 0 for a positive constant C 0 and all ε > 0. There exists a function u∈ BV (a, b) with u L ∞ (a,b) ≤ K such that, up to a subsequence, {u ε } ε converges to u in L 1 (a, b).Proof.We choose a low energy sequence{γ ε } ε in M(a, b) with E ε (u ε , γ ε ) ≤ Ẽε (u ε ) + 1 for all ε.Since there holds E ε (u ε , γ ε ) ≤ C 0 + 1 for all ε, according to Theorem 4.2 there exists a function u ∈ BV (a, b) with u L ∞ (a,b) ≤ K such that u ε → u in L 1 (a, b).Proof of Corollary 2.8.It is again sufficient to establish the Γ-lim inf inequality and the Γ-lim sup-inequality only for u ∈ BV (a, b) with u L ∞ (a,b) ≤ K since the estimates are trivial otherwise.
e. and γ ε → γ in the flat norm locally in F(u) on the complement of a finite set, i.e., γ ε A → γ A on each open set A ⋐ F(u) \ {x 1 , . . ., x m−1 } for some (x 0 , . . ., x m ) ∈ P. With this notion of convergence we have: