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

Veronika Auer-Volkmann***Universität Augsburg, Institut für Mathematik, 86135
Augsburg, Germany<EMAIL_ADDRESS>Lisa
<EMAIL_ADDRESS>and Bernd Schmidt*
<EMAIL_ADDRESS>
###### Abstract

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 $0<\varepsilon\ll 1$ 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 $\varepsilon\to 0$, the approximate functionals $\Gamma$-converge to a
cohesive zone model.

Keywords. Cohesive fracture, eigendeformation, two-field approximation,
$\Gamma$-convergence.

Mathematics Subject Classification. 74A45, 74R20, 74C05, 49J45

##  1 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., [8, 2, 7, 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. [22, 24, 21, 25, 19, 12, 35, 1]. 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 [4, 3] 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., [16,
26, 13, 17, 15]. A small parameter $\varepsilon$ 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., [11, 8, 29, 14, 30, 32, 27,
28] which involves a non-local approximation of the original field $u$ in
terms of convolution kernels with intrinsic length scale $\varepsilon\ll 1$.

Our main motivation comes from the Eigenfracture approach to brittle materials
that has been developed in [37] and further considered in [34, 33, 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
$u_{\varepsilon}$ and an eigendeformation field $g_{\varepsilon}$, which
induces a decomposition of the strain
$u_{\varepsilon}^{\prime}=(u_{\varepsilon}^{\prime}-g_{\varepsilon})+g_{\varepsilon}$
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 $g_{\varepsilon}$, which replaces the non-local contribution defined
in terms of a simple $\varepsilon$-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 $|\cdot|^{2}$ 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 $g_{\varepsilon}$ 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 $g_{\varepsilon}$ and the strain fields
$u_{\varepsilon}^{\prime}$. Rather these quantities are merely ‘coupled by
regularity’ in the sense that $u_{\varepsilon}^{\prime}-g_{\varepsilon}\in
L^{2}$. As the limit of $g_{\varepsilon}$ needs to be studied in a rather weak
space, this leads to technical difficulties when transferring asymptotic
properties from $u_{\varepsilon}$ to $g_{\varepsilon}$.

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 $g_{\varepsilon}$
and $\nabla u_{\varepsilon}$ 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 $\Gamma$-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 $\Gamma\text{-}\liminf$ inequality is then
completed by combining the previous results. In Section 6 we then address the
estimate from above of the $\Gamma$-upper limit. Finally, in Section 7 the
asymptotic behavior of the minimal energies with respect to the
eigendeformation variable is studied.

##  2 Setting of the problem and main result

Suppose that a beam occupies the region $(a,b)$ with $0<a<b<\infty$ and that a
displacement $u\colon(a,b)\to{\mathbb{R}}$ affects the beam. As cohesive
energy associated with $u$ we shall consider

$\displaystyle F(u)$
$\displaystyle=\int_{a}^{b}{\psi(|u^{\prime}|)}\,\textup{d}x+2\sum_{x\in
J_{u}}f\left(\frac{1}{2}|[u](x)|\right)+c_{0}|D^{c}u|(a,b),$ (1)

where $c_{0}$ is a fixed positive constant, and
$\psi,f\colon[0,\infty)\to[0,\infty)$ are functions defined via

$\displaystyle\psi(t)$ $\displaystyle=\begin{cases}t^{2}&\text{if
}t<\frac{c_{0}}{2},\\\ c_{0}\,t-\frac{c_{0}^{2}}{4}&\text{if
}t\geq\frac{c_{0}}{2},\end{cases}$ and $\displaystyle f(t)$
$\displaystyle=\begin{cases}c_{0}\,t&\quad\text{if }t<1,\\\
c_{0}&\quad\text{if }t\geq 1.\end{cases}$

$t$$1$$c_{0}$$f$$\psi$

Note that $f$ is the simplest continuous function such that $f(0)=0$,

$\lim_{t\to
0^{+}}\frac{f(t)}{t}=c_{0}\quad\text{and}\quad\lim_{t\to\infty}f(t)=c_{0}.$

The main ingredients in the energy (1) are a volume term, depending on the
strain of the beam $u^{\prime}$ and corresponding to the stored energy, a
surface term, depending on the crack opening $[u]\coloneqq
u(\cdot\,+)-u(\cdot\,-)$ 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 $u\in BV(a,b)$ allows for
a decomposition $Du=u^{\prime}\mathcal{L}^{1}+D^{s}u$ into the absolutely
continuous and the singular part with respect to the Lebesgue measure, and the
singular part $D^{s}u=[u]\mathcal{H}^{0}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}J_{u}+D^{c}u$ in turn into the
jump part and the Cantor part, which we have used in (1). We consider both
models with an apriori bound $\left\lVert
u\right\rVert_{{L^{\infty}}(a,b)}\leq K$, $K<\infty$, and unrestricted models
with $K=\infty$.

We next introduce a functional depending on two fields $u\in L^{1}(a,b)$ (in
case $K<\infty$), respectively, $u\in L^{0}((a,b),\overline{{\mathbb{R}}})$
(in case $K=\infty$) and $\gamma\in\mathcal{M}(a,b)$ with a non-local
approximation of the the second variable $\gamma$, given as

$\displaystyle
E_{\varepsilon}(u,\gamma)\coloneqq\begin{cases}\int_{a}^{b}{|u^{\prime}-g|^{2}\,\textup{d}x}&\quad\text{if
}u\in{W^{1,1}}(a,b),\,\left\lVert u\right\rVert_{{L^{\infty}}(a,b)}\leq K,\\\
+\frac{1}{\varepsilon}\int_{a}^{b}{f\big{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|g|}\,\textup{d}t}\big{)}\,\textup{d}x&\quad\phantom{if}\gamma=g\mathcal{L}^{1},\,g\in
L^{1}(a,b),\\\ &\quad\phantom{if}\text{and }u^{\prime}-g\in
L^{2}(a,b),\\\\[5.69046pt] \infty&\quad\text{otherwise},\end{cases}$

with $\varepsilon>0$,
$I_{\varepsilon}(x)\coloneqq(x-\varepsilon,x+\varepsilon)$, and either $K>0$ a
fixed constant or $K=\infty$. We notice that $E_{\varepsilon}(u,\gamma)$ can
only be finite if $\gamma$ is absolutely continuous with respect to the
Lebesgue measure, with density in $L^{1}(a,b)$. In this case $u^{\prime}$
represents the strain of the beam and $\gamma$ is intended to compensate
$u^{\prime}$ in regions where $u^{\prime}$ is above a certain strain level.
Hence, $u^{\prime}-g$ is the elastic strain of the material, while $\gamma$
describes the deformation of the material beyond the elastic regime,
indicating that a permanent deformation is exhibited if $\gamma\neq 0$. In
what follows, we are interested into the asymptotic behavior of the
functionals $\\{E_{\varepsilon}\\}_{\varepsilon>0}$ as $\varepsilon\searrow 0$
(in the sense of $\Gamma$-convergence). Focussing first on the case
$K<\infty$, it will be described by the energy functional $E$ which for
$(u,\gamma)\in L^{1}(a,b)\times\mathcal{M}(a,b)$ is defined as

$E(u,\gamma)\coloneqq\begin{cases}\int_{a}^{b}{|u^{\prime}-g|^{2}\,\textup{d}x}+c_{0}\int_{a}^{b}{|g|\,\textup{d}x}&\quad\text{if
}u\in BV(a,b),\,\left\lVert u\right\rVert_{{L^{\infty}}(a,b)}\leq K,\\\
\quad+2\,\sum_{x\in
J_{u}}f\left(\frac{1}{2}|[u](x)|\right)&\quad\quad\gamma=D^{s}u+g\mathcal{L}^{1},\,g\in
L^{1}(a,b),\\\ \quad+c_{0}|D^{c}u|(a,b)&\quad\phantom{if}\text{and
}u^{\prime}-g\in L^{2}(a,b),\\\\[5.69046pt]
\infty&\quad\text{otherwise}.\end{cases}$

Let us notice that for a finite energy $E(u,\gamma)$, the displacement field
$u$ and the eigendeformation field $\gamma$ need to be linked in a very
particular way. The singular part $\gamma^{s}$ of the measure $\gamma$ 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 $g\mathcal{L}^{1}$ of $\gamma$ instead is not completely determined by
the function $u$, but only the integrability restriction $u^{\prime}-g\in
L^{2}(a,b)$ is required. A particularly interesting choice of $g$ for a given
function $u\in BV(a,b)$ constitutes the unique minimizer $g^{\ast}$ of the
optimization problem

$\text{to minimize
}\int_{a}^{b}{|u^{\prime}-g|^{2}\,\textup{d}x}+\int_{a}^{b}{c_{0}|g|\,\textup{d}x}\quad\text{among
all }g\in L^{1}(a,b).$ (2)

By a pointwise minimization of the integrand, the minimizer $g^{\ast}$ is
explicitly given as

$g^{\ast}=\begin{cases}u^{\prime}-\operatorname{sign}(u^{\prime})\frac{c_{0}}{2}&\text{if
}|u^{\prime}|>\frac{c_{0}}{2},\\\ 0&\text{if
}|u^{\prime}|\leq\frac{c_{0}}{2}.\end{cases}$ (3)

For later purposes we notice that the eigendeformation field $\gamma$ is
completely described in terms of the function $u$ as

$\gamma^{\textup{opt}}\coloneqq D^{s}u+g^{\ast}\mathcal{L}^{1}.$

Moreover, the corresponding energy functional $E(u,\gamma^{\textup{opt}})$
reduces to a one-field functional depending only on the displacement $u\in
BV(a,b)$, which under the additional restriction $\left\lVert
u\right\rVert_{{L^{\infty}}(a,b)}\leq K$ (if $K<\infty$) is precisely given by
the energy $F(u)$ introduced in (1).

In order to state our $\Gamma$-convergence result we need to endow
$L^{1}(a,b)\times\mathcal{M}(a,b)$ 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 $\mathcal{M}(a,b)$
be equipped with the flat topology. Assume $K<\infty$. Then the family
$\\{E_{\varepsilon}\\}_{\varepsilon>0}$ $\Gamma$-converges to $E$ in
$L^{1}(a,b)\times\mathcal{M}(a,b)$, i.e., we have

  1. (i)

_( $\liminf$ inequality)_ For every sequence
$\\{(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$ in
${L^{1}}(a,b)\times\mathcal{M}(a,b)$ converging to $(u,\gamma)\in
L^{1}(a,b)\times\mathcal{M}(a,b)$, i.e., $u_{\varepsilon}\to u$ in
$L^{1}(a,b)$ and $\gamma_{\varepsilon}\to\gamma$ in the flat norm, we have

$\liminf_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\geq E(u,\gamma).$

  2. (ii)

_( $\limsup$ inequality)_ For every $(u,\gamma)\in
L^{1}(a,b)\times\mathcal{M}(a,b)$ there exists a sequence
$\\{(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$ in
${L^{1}}(a,b)\times\mathcal{M}(a,b)$ such that $u_{\varepsilon}\to u$ in
$L^{1}(a,b)$, $\gamma_{\varepsilon}\to\gamma$ in the flat norm, and

$\limsup_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\leq E(u,\gamma).$

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
$1<q<\infty$. Therefore, we obtain as a direct consequence of Theorem 2.1 also
$\Gamma$-convergence of $\\{E_{\varepsilon}\\}_{\varepsilon>0}$ to $E$ in
$L^{1}(a,b)\times\mathcal{M}(a,b)$, when $\mathcal{M}(a,b)$ is equipped with
the stronger topology of convergence in ${W^{-1,q}}(a,b)$ for some
$1<q<\infty$.

######  Remark 2.2.

Our result can be seen as a two-field extension of the setting considered in
[29]. Indeed, introducing the constraints $g=u^{\prime}$, respectively
$\gamma=Du$, one is lead to functionals
$F_{\varepsilon}(u)=E_{\varepsilon}(u,u^{\prime}\mathcal{L}^{1})$ and
$F(u)=E(u,Du)$ depending on $u$ only. In this case the $\Gamma$-convergence of
the sequence $\\{F_{\varepsilon}\\}_{\varepsilon}$ to $F$ has been obtained in
[29].

The unrestricted problem is in fact strongly related. Indeed, even for
$K=\infty$ an energy bound implies $L^{\infty}$ 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 $a=x_{0}<x_{1}<\ldots<x_{m}=b$ of $(a,b)$ in
the limit $\varepsilon\to 0$ such that $\\{u_{\varepsilon}\\}_{\varepsilon}$
and $\\{g_{\varepsilon}\\}_{\varepsilon}$ converge with respect to the $L^{1}$
norm, respectively the flat norm, locally on
$(a,b)\setminus\\{x_{1},\ldots,x_{m-1}\\}$. On the exceptional set, however,
$u_{\varepsilon}^{\prime}$ and $g_{\varepsilon}$ 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 $u_{\varepsilon}$
may elapse to $\pm\infty$. In order to account for such a possibility we
consider limiting functions taking values in
$\overline{\mathbb{R}}=\mathbb{R}\cup\\{-\infty,+\infty\\}$. More precisely,
let $\mathcal{P}=\big{\\{}(x_{0},\ldots,x_{m})\colon
a=x_{0}<x_{1}<\ldots<x_{m}=b,\ m\in\mathbb{N}\big{\\}}$ and consider
$BV_{\infty,\mathcal{P}}(a,b)$ as consisting of functions
$u\colon(a,b)\to\overline{{\mathbb{R}}}$ of the form

$u=w+\sum_{i=1}^{m}\alpha_{i}\chi_{(x_{i-1},x_{i})}$ (4)

with $\alpha_{i}\in\\{-\infty,0,+\infty\\}$, $i=1,\ldots,m$,
$(x_{0},\ldots,x_{m})\in\mathcal{P}$ and $w\in BV(a,b)$. We denote the part
where $u$ is finite by
$\mathcal{F}(u)=\big{(}\bigcup_{i:\alpha_{i}=0}[x_{i-1},x_{i}]\big{)}^{\circ}$,
set
$J_{u}=\\{x\in(a,b)\colon[u](x)=u(x+)-u(x-)\in\overline{{\mathbb{R}}}\setminus\\{0\\}\\}$
and read $f\left(\infty\right)=c_{0}$. We then extend $E$ to
$L^{0}((a,b),\overline{{\mathbb{R}}})\times\mathcal{M}(a,b)$ by setting

$E(u,\gamma)\coloneqq\begin{cases}\int_{\mathcal{F}(u)}{|u^{\prime}-g|^{2}\,\textup{d}x}+c_{0}\int_{\mathcal{F}(u)}{|g|\,\textup{d}x}&\quad\text{if
}u\in BV_{\infty,\mathcal{P}}(a,b),\,g\in L^{1}(\mathcal{F}(u)),\\\
\quad+2\,\sum_{x\in
J_{u}}f\left(\frac{1}{2}|[u](x)|\right)&\quad\quad\gamma\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathcal{F}(u)=(D^{s}u+g\mathcal{L}^{1})\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathcal{F}(u),\\\
\quad+c_{0}|D^{c}u|(\mathcal{F}(u))&\quad\phantom{if}\text{and
}u^{\prime}-g\in L^{2}(\mathcal{F}(u)),\\\\[5.69046pt]
\infty&\quad\text{otherwise}.\end{cases}$

We say that $(u_{\varepsilon},\gamma_{\varepsilon})\to(u,\gamma)$ in
$L^{0}((a,b),\overline{{\mathbb{R}}})\times\mathcal{M}(a,b)$ if
$u_{\varepsilon}\to u$ a.e. and $\gamma_{\varepsilon}\to\gamma$ in the flat
norm locally in $\mathcal{F}(u)$ on the complement of a finite set, i.e.,
$\gamma_{\varepsilon}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}A\to\gamma\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}A$ on each open set
$A\Subset\mathcal{F}(u)\setminus\\{x_{1},\ldots,x_{m-1}\\}$ for some
$(x_{0},\ldots,x_{m})\in\mathcal{P}$. With this notion of convergence we have:

######  Theorem 2.3.

Assume $K=\infty$. Then the family $\\{E_{\varepsilon}\\}_{\varepsilon>0}$
$\Gamma$-converges to $E$ in
$L^{0}((a,b),\overline{{\mathbb{R}}})\times\mathcal{M}(a,b)$.

The corresponding compactness result for $K=\infty$ with respect to this
particular convergence is stated in Theorem 4.3.

######  Remark 2.4.

In fact, $E(u,\gamma)$ can be finite only if the restriction of $u$ to
$\mathcal{F}(u)$ is a $BV$ function and not merely an element of
$GBV(\mathcal{F}(u))$. In particular, if $E(u,\gamma)<\infty$ and $u\in
L^{1}(a,b)$, then $u\in BV(a,b)$.

######  Remark 2.5.

Our methods can easily be adapted to obtain an alternative asymptotic
description by considering renormalized functionals: From the above partition
$\mathcal{P}$ one can derive a coarser one (whose members are finite unions of
intervals) so that on each such set one has an $L^{\infty}$ bound on
$u_{\varepsilon}$ modulo a single additive constant and the mutual distance of
$u_{\varepsilon}$ 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
$u(a)=u_{a}$, $u(b)=u_{b}$, where $u_{a},u_{b}\in{\mathbb{R}}$ such that
$-K\leq u_{a},u_{b}\leq K$. 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 $E^{u_{a},u_{b}}_{\varepsilon}$ and
$E^{u_{a},u_{b}}$ on an extended interval $(a-\eta,b+\eta)$, with $\eta>0$
fixed, which are defined as $E_{\varepsilon}$ and $E$, respectively, before
but with the additional constraints
$E^{u_{a},u_{b}}_{\varepsilon}(u,\gamma)=E^{u_{a},u_{b}}(u,\gamma)=\infty$ if
not $u=u_{a}$ a.e. on $(a-\eta,a)$ and $u=u_{b}$ a.e. on $(b,b+\eta)$ and
$\gamma\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}((a-\eta,a)\cup(b,b+\eta))=0$. So
if $u$ does not satisfy the given boundary values on $(a,b)$ in the limiting
problem, this leads to an extra energy cost:

$E^{u_{a},u_{b}}(u,\gamma)=E\big{(}u|_{(a,b)},\gamma\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(a,b)\big{)}+2f\left(\frac{1}{2}|u(a+)-u_{a}|\right)+2f\left(\frac{1}{2}|u(b-)-u_{b}|\right).$

Indeed, the $\Gamma$-$\liminf$ inequality and the compactness property are
direct consequences of the case with free boundaries. The $\Gamma$-$\limsup$
inequality for $K<\infty$ follows from the observation that the recovery
sequence constructed in Proposition 6.1 indeed satisfies $u_{\varepsilon}=u$
near $\\{a,b\\}$ and from Remark 6.4. The case $K=\infty$ is a direct
consequence as there the recovery sequence is built as in the case $K<\infty$
near $\\{a,b\\}$ since $u_{a},u_{b}\in{\mathbb{R}}$.

######  Remark 2.7.

Our results can also be adapted to general continuous stored energy functions
$W$ leading to a general non-quadratic bulk contribution
$\int_{a}^{b}W(u^{\prime}-g)\,\textup{d}x$, whenever $W$ satisfies a
$p$-growth condition of the form $c|r|^{p}-C\leq W(r)\leq C|r|^{p}+C$ for
suitable constants $c,C>0$ and some $p\in(1,\infty)$, and for convenience we
also assume that $\min W=W(0)=0$. The first term in the limiting functional is
then replaced by $\int_{a}^{b}W^{**}(u^{\prime}-g)\,\textup{d}x$,
respectively, $\int_{\mathcal{F}(u)}W^{**}(u^{\prime}-g)\,\textup{d}x$, where
$W^{**}$ is the convex envelope of $W$. In fact, making use of the estimate
$W\geq W^{**}$, compactness and the $\Gamma$-$\liminf$ inequality follow
exactly as before with $W^{**}$ instead of $|\cdot|^{2}$ by taking account of
the obvious adaptions such as replacing $L^{2}$ by $L^{p}$ and $SBV^{2}$ by
$SBV^{p}$. The $\Gamma$-$\limsup$ 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 $\gamma$, which are
defined as

$\tilde{E}_{\varepsilon}(u)\coloneqq\inf_{\gamma\in\mathcal{M}(a,b)}E_{\varepsilon}(u,\gamma)=\inf_{g\in
L^{1}(a,b)}E_{\varepsilon}(u,g\mathcal{L}^{1})$ (5)

and

$\tilde{E}(u)\coloneqq\inf_{g\in L^{1}(a,b)}E(u,D^{s}u+g\mathcal{L}^{1}),$ (6)

for $u\in L^{1}(a,b)$ and $u\in L^{0}((a,b),\overline{{\mathbb{R}}})$,
respectively. As a direct consequence of the previous $\Gamma$-convergence
result we obtain:

######  Corollary 2.8.

The family $\\{\tilde{E}_{\varepsilon}\\}_{\varepsilon>0}$ $\Gamma$-converges
to $\tilde{E}$, on $L^{1}(a,b)$ equipped with the strong topology if
$K<\infty$ and with respect to convergence a.e. on
$L^{0}((a,b),\overline{{\mathbb{R}}})$ if $K=\infty$.

##  3 Preliminaries

In this section, we recall some basics on $BV$-functions, for simplicity on a
one-dimensional interval $(a,b)\subset{\mathbb{R}}$, and convergence of
measures.

#### Functions of bounded variation.

A function $u\in L^{1}(a,b)$ 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

$\int_{a}^{b}u\varphi^{\prime}\,\textup{d}x=-\int_{a}^{b}\varphi\,\textup{d}Du\quad\text{
for every }\varphi\in C_{c}^{1}(a,b)$

is valid for a (unique) measure $Du\in\mathcal{M}(a,b)$. The space $BV(a,b)$
is a Banach space endowed with the norm

$\left\lVert u\right\rVert_{BV(a,b)}\coloneqq\left\lVert
u\right\rVert_{L^{1}(a,b)}+|Du|(a,b),$

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 $u\in BV(a,b)$, denoted by
$u_{n}\overset{*}{\rightharpoonup}u$, if $u_{n}\to u$ in $L^{1}(a,b)$ and
$Du_{n}\overset{*}{\rightharpoonup}Du$ in $\mathcal{M}(a,b)$. 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
$\\{u_{n}\\}_{n}$ _converges strictly_ to $u\in BV(a,b)$ if $u_{n}\to u$ in
$L^{1}(a,b)$ and $|Du_{n}|(a,b)\to|Du|(a,b)$. As a matter of fact, the space
$C^{\infty}(a,b)$ is dense in $BV(a,b)$ with respect to the strict topology
(see [2, Theorem 3.9]).

We next discuss approximate continuity and discontinuity properties of a
function $u\in L^{1}_{\textnormal{loc}}(a,b)$. We say that $u$ has an
_approximate limit_ at $x\in(a,b)$ if there exists a (unique)
$\widetilde{u}(x)\in{\mathbb{R}}$ such that

$\lim_{\rho\to 0^{+}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{(x-\rho,x+\rho)\cap(a,b)}|u(y)-\widetilde{u}(x)|\,\textup{d}y=0.$

We denote by $S_{u}$ the set, where this condition fails, and call it the
_approximate discontinuity set_ of $u$. It is $\mathcal{L}^{1}$-negligible,
and $\widetilde{u}$ coincides $\mathcal{L}^{1}$-a.e. in $(a,b)\setminus S_{u}$
with $u$. Furthermore, we say that $u$ has an _approximate jump point_ at
$x\in S_{u}$ if there exist (unique) $u(x+),u(x-)\in{\mathbb{R}}$ with
$u(x+)\neq u(x-)$ such that

$\lim_{\rho\to 0^{+}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{(x,x+\rho)}|u(y)-u(x+)|\,\textup{d}y=0\quad\text{and}\quad\lim_{\rho\to
0^{+}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{(x-\rho,x)}|u(y)-u(x-)|\,\textup{d}y=0.$

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 $u\in BV(a,b)$ the set $J_{u}$ coincides with $S_{u}$ and is at most
countable. It is also convenient to work with the precise representative

$u^{\ast}(x)\coloneqq\begin{cases}\widetilde{u}(x)&\quad\text{if
}x\in(a,b)\setminus J_{u},\\\ [u](x)/2&\quad\text{if }x\in J_{u}.\end{cases}$

According to the Radon–Nikodým theorem the measure derivative
$Du=D^{a}u\mathcal{L}^{1}+D^{s}u$ can be decomposed into the absolutely
continuous and the singular part with respect to the Lebesgue measure
$\mathcal{L}^{1}$. We then define the jump and the Cantor part of $Du$ as

$D^{j}u\coloneqq D^{s}u\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}J_{u}\quad\text{and}\quad
D^{c}u\coloneqq D^{s}u\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}((a,b)\setminus J_{u}).$

From the identifications $D^{a}u=u^{\prime}\mathcal{L}^{1}$ with the
approximate gradient $u^{\prime}$ for the absolutely continuous part and
$D^{j}u=[u]\mathcal{H}^{0}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}J_{u}$ with $[u]\coloneqq
u(\cdot\,+)-u(\cdot\,-)$ for the jump part (see [2, Theorem 3.83 and formula
(3.90)]) we arrive at the decomposition

$Du=u^{\prime}\mathcal{L}^{1}+[u]\mathcal{H}^{0}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}J_{u}+D^{c}u.$

We can actually decompose the function $u$ as

$u=u_{a}+u_{j}+u_{c}$ (7)

for an absolutely continuous function $u_{a}\in W^{1,1}(a,b)$ 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
${\mathbb{R}}^{d}$ 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 $u\in BV(a,b)$ with $D^{c}u=0$. In
addition, we define

$SBV^{2}(a,b)\coloneqq\big{\\{}u\in SBV(a,b)\colon u^{\prime}\in
L^{2}(a,b)\text{ and }\mathcal{H}^{0}(J_{u})<\infty\big{\\}}.$

#### Convergence in negative Sobolev spaces and in the flat topology.

The negative Sobolev spaces $W^{-1,q}(a,b)$ with $1<q\leq\infty$ are defined
as usual as the dual spaces of $W^{1,q^{\prime}}_{0}(a,b)$ (with
$q^{\prime}\in[1,\infty)$ denoting the conjugate exponent to $q$ with
$\tfrac{1}{q}+\tfrac{1}{q^{\prime}}=1$), and correspondingly the norm is
defined via the duality pairing as

$\|T\|_{W^{-1,q}(a,b)}\coloneqq\sup\Big{\\{}T(\varphi)\colon\varphi\in
W^{1,q^{\prime}}_{0}(a,b)\text{ with
}\left\lVert\varphi\right\rVert_{W^{1,q^{\prime}}(a,b)}\leq 1\Big{\\}},$

for every $T\in W^{-1,q}(a,b)$. Consequently, the spaces $W^{-1,q}(a,b)$ with
$1<q<\infty$ are reflexive and separable. For later purposes, we mention two
specific situations. Let $v\in L^{q}(a,b)$ and $w\in L^{r}(a,b)$ with $1\leq
r\leq\infty$ such that the embedding $W^{1,q^{\prime}}_{0}(a,b)\subset
L^{r^{\prime}}(a,b)$ is continuous. If we set

$T_{Dv}(\varphi)\coloneqq-\int_{a}^{b}v\varphi^{\prime}\mathop{}\\!\mathrm{d}x\quad\text{and}\quad
T_{w}(\varphi)\coloneqq\int_{a}^{b}w\varphi\mathop{}\\!\mathrm{d}x\quad\text{for
all }\varphi\in W^{1,q^{\prime}}_{0}(a,b),$

then, by the Hölder inequality and the continuous embedding (with constant
$C^{\prime}$), we obtain $T_{Dv},T_{w}\in W^{-1,q}(a,b)$ with

$\|T_{Dv}\|_{W^{-1,q}(a,b)}\leq\left\lVert
v\right\rVert_{L^{q}(a,b)}\quad\text{and}\quad\|T_{w}\|_{W^{-1,q}(a,b)}\leq
C^{\prime}\left\lVert w\right\rVert_{L^{r}(a,b)}.$ (8)

Because of the continuous and dense embedding $W^{1,1}_{0}(a,b)\subset
C_{0}(a,b)$, the negative Sobolev norms can actually be considered on the
space $\mathcal{M}(a,b)$ of all finite Radon measures on $(a,b)$, for which
the duality pairing reads as

$\|\mu\|_{W^{-1,q}(a,b)}=\sup\bigg{\\{}\int_{a}^{b}\varphi\,\textup{d}\mu\colon\varphi\in
W^{1,q^{\prime}}_{0}(a,b)\text{ with
}\left\lVert\varphi\right\rVert_{W^{1,q^{\prime}}_{0}(a,b)}\leq 1\bigg{\\}}$

for $\mu\in\mathcal{M}(a,b)$. If we allow $q^{\prime}=\infty$ in this
expression, we obtain the flat norm

$\left\lVert\mu\right\rVert_{\text{flat}}\coloneqq\sup\bigg{\\{}\int_{a}^{b}\varphi\,\textup{d}\mu\colon\varphi\in
W^{1,\infty}_{0}(a,b)\text{ with
}\left\lVert\varphi\right\rVert_{W^{1,\infty}_{0}(a,b)}\leq 1\bigg{\\}},$

for $\mu\in\mathcal{M}(a,b)$. Here we have the inequalities

$\left\lVert Dv\right\rVert_{\text{flat}}\leq\left\lVert
v\right\rVert_{L^{1}(a,b)}\quad\text{and}\quad\|w\mathcal{L}^{1}\|_{\text{flat}}\leq\left\lVert
w\right\rVert_{L^{1}(a,b)}$ (9)

for all functions $v\in BV(a,b)$ and $w\in L^{1}(a,b)$. Let us still notice
that due to Schauder’s theorem and the compact embedding
$W^{1,\infty}_{0}(a,b)\Subset C_{0}(a,b)$, the flat topology metrizes
weak-$\ast$ 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-$\ast$ sense.

######  Lemma 3.1 (on convergence of measures).

For a measure $\mu\in\mathcal{M}(a,b)$ and a sequence
$\\{\mu_{n}\\}_{n\in{\mathbb{N}}}$ of measures in $\mathcal{M}(a,b)$, we have:

  1. (i)

If $\mu_{n}\to\mu$ in $W^{-1,q}(a,b)$ for some $1<q\leq\infty$, then
$\mu_{n}\to\mu$ in the flat norm.

  2. (ii)

If $\mu_{n}\overset{*}{\rightharpoonup}\mu$ in $\mathcal{M}(a,b)$, then
$\mu_{n}\to\mu$ in the flat norm.

  3. (iii)

If $\mu_{n}\to\mu$ in the flat norm and
$\sup_{n\in{\mathbb{N}}}|\mu_{n}|(a,b)<\infty$, then
$\mu_{n}\overset{*}{\rightharpoonup}\mu$ in $\mathcal{M}(a,b)$.

##  4 Compactness

In this section we establish a compactness result for sequences in
$L^{1}(a,b)\times\mathcal{M}(a,b)$ with bounded energy $E_{\varepsilon}$. This
result together with a ${\Gamma\text{-convergence}}$ 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:

######  Lemma 4.1.

Let $g\in L^{1}(a,b)$. For $\varepsilon>0$ there exists
$x_{\varepsilon}\in{\mathbb{R}}$ such that the grid

$\mathscr{G}_{\varepsilon}\coloneqq\big{\\{}x_{\alpha}\coloneqq
x_{\varepsilon}+2\varepsilon\alpha\colon\alpha\in{\mathbb{Z}}\text{ and
}x_{\alpha}\in(a+\varepsilon,b-\varepsilon)\big{\\}}$

contains a subset
$\mathscr{G}^{\prime}_{\varepsilon}\subset\mathscr{G}_{\varepsilon}$ with

$\int_{\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}}|g|\mathop{}\\!\mathrm{d}t+2\\#(\mathscr{G}_{\varepsilon}\setminus\mathscr{G}^{\prime}_{\varepsilon})\leq\frac{1}{c_{0}\varepsilon}\int_{a}^{b}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|g|\mathop{}\\!\mathrm{d}t}\bigg{)}\,\textup{d}x.$

###### Proof.

We proceed analogously to [29, proof of Lemma 4.2]. Let $\phi_{\varepsilon}\in
C^{\infty}_{0}(a,b)$ be a cut-off function with $0\leq\phi_{\varepsilon}\leq
1$ in $(a,b)$ and $\phi_{\varepsilon}\equiv 1$ in
$(a+\varepsilon,b-\varepsilon)$. We then consider $\psi_{\varepsilon}\in
C_{0}({\mathbb{R}})$ defined via

$\psi_{\varepsilon}(x)\coloneqq\phi_{\varepsilon}(x)\,f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|g|\mathop{}\\!\mathrm{d}t}\bigg{)}$

for $x\in(a,b)$ and $\psi_{\varepsilon}(x)\coloneqq 0$ for
$x\in{\mathbb{R}}\setminus(a,b)$. The application of [10, Lemma 4.2] (with
$\eta=2\varepsilon$), which is a consequence of the mean value theorem for
integrals, shows that

$\int_{{\mathbb{R}}}\psi_{\varepsilon}\,\textup{d}x=2\varepsilon\sum_{\alpha\in{\mathbb{Z}}}\psi_{\varepsilon}(x_{\varepsilon}+2\varepsilon\alpha)$

holds for a suitable $x_{\varepsilon}\in{\mathbb{R}}$. By non-negativity of
$f$, the choice of the cut-off function $\phi_{\varepsilon}$ and the
definition of $\mathscr{G}_{\varepsilon}$, we hence have

$\displaystyle\frac{1}{\varepsilon}\int_{a}^{b}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|g|\mathop{}\\!\mathrm{d}t}\bigg{)}\,\textup{d}x$
$\displaystyle\geq\frac{1}{\varepsilon}\int_{\mathbb{R}}\psi_{\varepsilon}(x)\mathop{}\\!\mathrm{d}x$
$\displaystyle\geq
2\sum_{x_{\alpha}\in\mathscr{G}_{\varepsilon}}\psi_{\varepsilon}(x_{\alpha})=2\sum_{x_{\alpha}\in\mathscr{G}_{\varepsilon}}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x_{\alpha})}{|g|\mathop{}\\!\mathrm{d}t}\bigg{)}.$
(10)

If we now set

$\mathscr{G}^{\prime}_{\varepsilon}\coloneqq\bigg{\\{}x_{\alpha}\in\mathscr{G}_{\varepsilon}\colon\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x_{\alpha})}|g|\mathop{}\\!\mathrm{d}t<1\bigg{\\}},$

then the claim follows directly from (10), after rewriting the right-hand side
via the definition of $f$ as

$\displaystyle
2\sum_{x_{\alpha}\in\mathscr{G}_{\varepsilon}}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x_{\alpha})}{|g|\mathop{}\\!\mathrm{d}t}\bigg{)}$
$\displaystyle=2c_{0}\sum_{x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x_{\alpha})}{|g|\mathop{}\\!\mathrm{d}t}+2\sum_{x_{\alpha}\in\mathscr{G}_{\varepsilon}\setminus\mathscr{G}^{\prime}_{\varepsilon}}c_{0}$
$\displaystyle=c_{0}\sum_{x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}}\int_{I_{\varepsilon}(x_{\alpha})}|g|\mathop{}\\!\mathrm{d}t+2c_{0}\\#(\mathscr{G}_{\varepsilon}\setminus\mathscr{G}^{\prime}_{\varepsilon}).\qed$

We can now address the aforementioned compactness results.

######  Theorem 4.2 (Compactness for $K<\infty$).

Assume $K<\infty$. Let
$\\{(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$ be a sequence in
${L^{1}}(a,b)\times\mathcal{M}(a,b)$ with

$E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\leq C_{0}$

for a positive constant $C_{0}$ and all $\varepsilon>0$. There exist a
function $u\in BV(a,b)$ with $\left\lVert
u\right\rVert_{{L^{\infty}}(a,b)}\leq K$ and a measure
$\gamma\in\mathcal{M}(a,b)$ such that, up to subsequences,
$\\{u_{\varepsilon}\\}_{\varepsilon}$ converges to $u$ in $L^{1}(a,b)$ and
$\\{\gamma_{\varepsilon}\\}_{\varepsilon}$ converges to
$\gamma=\gamma^{s}+g\mathcal{L}^{1}$ in ${W^{-1,q}}(a,b)$ for all $1<q<\infty$
and in particular in the flat norm. Moreover, there holds $\gamma^{s}=D^{s}u$
and $u^{\prime}-g\in L^{2}(a,b)$.

###### Proof.

We first observe from the finiteness of
$E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})$ that we necessarily
have $u_{\varepsilon}\in{W^{1,1}}(a,b)$ with $\left\lVert
u_{\varepsilon}\right\rVert_{L^{\infty}(a,b)}\leq K$ and
$\gamma_{\varepsilon}=g_{\varepsilon}\mathcal{L}^{1}$ for some functions
$g_{\varepsilon}\in L^{1}(a,b)$ for all $\varepsilon>0$. By the uniform
boundedness of $\left\lVert
u_{\varepsilon}^{\prime}-g_{\varepsilon}\right\rVert_{L^{2}(a,b)}$ and
$\left\lVert u_{\varepsilon}\right\rVert_{L^{\infty}(a,b)}$ we deduce from (8)

$\displaystyle\left\lVert\gamma_{\varepsilon}\right\rVert_{W^{-1,\infty}(a,b)}$
$\displaystyle\leq\left\lVert
u_{\varepsilon}^{\prime}\mathcal{L}^{1}\right\rVert_{W^{-1,\infty}(a,b)}+\left\lVert\gamma_{\varepsilon}-u_{\varepsilon}^{\prime}\mathcal{L}^{1}\right\rVert_{W^{-1,\infty}(a,b)}$
$\displaystyle\leq\left\lVert
u_{\varepsilon}\right\rVert_{L^{\infty}(a,b)}+C^{\prime}\left\lVert
g_{\varepsilon}-u_{\varepsilon}^{\prime}\right\rVert_{L^{2}(a,b)}\leq
C(a,b,K),$

independently of $\varepsilon$. Therefore,
$\\{\gamma_{\varepsilon}\\}_{\varepsilon}$ is bounded in $W^{-1,\infty}(a,b)$
and consequently contains a subsequence, which converges weakly-$\ast$ in
$W^{-1,\infty}(a,b)$ to some $\gamma\in W^{-1,\infty}(a,b)$. We next study the
convergence of the sequence $\\{u_{\varepsilon}\\}_{\varepsilon}$. To this
end, we consider the function $v_{\varepsilon}$ defined by

$v_{\varepsilon}(x)\coloneqq\begin{cases}u_{\varepsilon}(x)&\quad
x\in\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\},\\\
0&\quad\text{otherwise}.\end{cases}$

Since $u_{\varepsilon}\in W^{1,1}(a,b)$ with $\left\lVert
u_{\varepsilon}\right\rVert_{L^{\infty}(a,b)}\leq K$ is assumed, we clearly
have ${v_{\varepsilon}\in SBV(a,b)}$ with $\left\lVert
v_{\varepsilon}\right\rVert_{L^{\infty}(a,b)}\leq K$, for every $\varepsilon$.
Moreover, jump discontinuities of $v_{\varepsilon}$ can only occur at points
$x_{\alpha}\pm\varepsilon$ for
$x_{\alpha}\in\mathscr{G}_{\varepsilon}\setminus\mathscr{G}^{\prime}_{\varepsilon}$
and close to the boundary at $\min\mathscr{G}^{\prime}_{\varepsilon}$ or at
$\max\mathscr{G}^{\prime}_{\varepsilon}$. As a consequence of Lemma 4.1 and
the definition of the energy $E_{\varepsilon}$, we deduce

$\\#J_{v_{\varepsilon}}\leq
2\\#(\mathscr{G}_{\varepsilon}\setminus\mathscr{G}^{\prime}_{\varepsilon})+2\leq\frac{1}{c_{0}\varepsilon}\int_{a}^{b}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|g|\mathop{}\\!\mathrm{d}t}\bigg{)}\,\textup{d}x+2\leq\frac{C_{0}}{c_{0}}+2,$

i.e., that $\\#J_{v_{\varepsilon}}$ is bounded independently of $\varepsilon$.
By the Cauchy–Schwarz inequality and again by Lemma 4.1, we additionally have

$\displaystyle\int_{a}^{b}{|v^{\prime}_{\varepsilon}(x)|\mathop{}\\!\mathrm{d}x}$
$\displaystyle\leq\int_{\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}}|u^{\prime}_{\varepsilon}(x)-g_{\varepsilon}(x)|\mathop{}\\!\mathrm{d}x+\int_{\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}}|g_{\varepsilon}(x)|\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq|b-a|^{\frac{1}{2}}\left(\int_{a}^{b}|u^{\prime}_{\varepsilon}(x)-g_{\varepsilon}(x)|^{2}\mathop{}\\!\mathrm{d}x\right)^{\frac{1}{2}}+\frac{1}{c_{0}\varepsilon}\int_{a}^{b}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|g_{\varepsilon}|\mathop{}\\!\mathrm{d}t}\bigg{)}\,\textup{d}x$
$\displaystyle\leq|b-a|^{\frac{1}{2}}C_{0}^{\frac{1}{2}}+\frac{C_{0}}{c_{0}},$

independently of $\varepsilon$. Hence, $\\{v_{\varepsilon}\\}_{\varepsilon}$
is bounded in $BV(a,b)$. By the Rellich–Kondrachov theorem, there exists a
subsequence that converges in $L^{q}(a,b)$ to some $u\in BV(a,b)$ with
$\left\lVert u\right\rVert_{{L^{\infty}}(a,b)}\leq K$, for any $1\leq
q<\infty$. To identify $u$ as the limit in $L^{q}(a,b)$ of (the same
subsequence of) $\\{u_{\varepsilon}\\}_{\varepsilon}$ we notice from the
definition of $v_{\varepsilon}$

$\left\lVert
v_{\varepsilon}-u_{\varepsilon}\right\rVert^{q}_{L^{q}(a,b)}\leq\int_{(a,b)\setminus\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}}K^{q}\,\textup{d}x\leq
2\varepsilon(\\#(\mathscr{G}_{\varepsilon}\setminus\mathscr{G}^{\prime}_{\varepsilon})+2)K^{q}.$

Since
$\\#(\mathscr{G}_{\varepsilon}\setminus\mathscr{G}^{\prime}_{\varepsilon})$ is
bounded uniformly in $\varepsilon$, this allows to conclude the convergence of
$\\{u_{\varepsilon}\\}_{\varepsilon}$ to $u$ in $L^{q}(a,b)$, for any $1\leq
q<\infty$.

It remains to show convergence of (the subsequence of)
$\\{\gamma_{\varepsilon}\\}_{\varepsilon}$ in $W^{-1,q}(a,b)$ for any
$1<q<\infty$ and the claimed relations between $\gamma$ and $Du$. In view of
(8), we have $u^{\prime}_{\varepsilon}\mathcal{L}^{1}\to
Du=u^{\prime}\mathcal{L}^{1}+D^{s}u$ in $W^{-1,q}(a,b)$ for every
$1<q<\infty$. Furthermore, by the boundedness of
$\\{u^{\prime}_{\varepsilon}-g_{\varepsilon}\\}_{\varepsilon}$ in
$L^{2}(a,b)$, we can extract a subsequence which converges weakly in
$L^{2}(a,b)$ to some $w\in L^{2}(a,b)$. Since $L^{2}(a,b)$ is compactly
embedded in $W^{-1,q}(a,b)$ for all ${1<q<\infty}$, we obtain
$\gamma_{\varepsilon}\to Du-w\mathcal{L}^{1}$ in $W^{-1,q}(a,b)$ and thus, via
Lemma 3.1, also in the flat norm. This shows
$\gamma=Du-w\mathcal{L}^{1}\in\mathcal{M}(a,b)$, which in turn, by the
Radon–Nikodým theorem, yields $\gamma^{s}=D^{s}u$ and $w=u^{\prime}-g\in
L^{2}(a,b)$. ∎

######  Theorem 4.3 (Compactness for $K=\infty$).

Assume $K=\infty$. Suppose
$\\{(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$ is a sequence in
$L^{0}((a,b);\overline{{\mathbb{R}}})\times\mathcal{M}(a,b)$ with

$E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\leq C_{0}$

for a positive constant $C_{0}$ and all $\varepsilon>0$. There is a partition
$a=x_{0}<x_{1}<\ldots<x_{m}=b$, a function $u\in
BV_{\infty,\mathcal{P}}(a,b)$, say

$u=w+\sum_{i=1}^{m}\alpha_{i}\chi_{(x_{i-1},x_{i})}$

with $\alpha_{i}\in\\{-\infty,0,+\infty\\}$, $i=1,\ldots,m$ and $w\in
BV((a,b);\mathbb{R})$, and a measure $\gamma\in\mathcal{M}(a,b)$ such that, up
to subsequences, $\\{u_{\varepsilon}\\}_{\varepsilon}$ converges to $u$ a.e.
and in $L^{1}_{\rm loc}(\mathcal{F}(u)\setminus\\{x_{1},\ldots,x_{m}\\})$ and
$\\{g_{\varepsilon}\mathcal{L}^{1}\\}_{\varepsilon}$ converges to
$\gamma=D^{s}w+g\mathcal{L}^{1}$ in $W^{-1,q}_{\rm
loc}((a,b)\setminus\\{x_{1},\ldots,x_{m}\\})$ for all $1<q<\infty$ and in
particular in the flat norm on each open $I$ that is compactly contained in
$(a,b)\setminus\\{x_{1},\ldots,x_{m}\\}$. Moreover, $w^{\prime}-g\in
L^{2}(a,b)$. In particular,
$(u_{\varepsilon},\gamma_{\varepsilon})\to(u,\gamma)$ in
$L^{0}((a,b);\overline{{\mathbb{R}}})\times\mathcal{M}(a,b)$.

###### Proof.

We define $\mathcal{G}^{\prime}_{\varepsilon}$ exactly as in Lemma 4.1 and
denote by $J_{1},\ldots,J_{m_{\varepsilon}}$ the connected components of
$\bigcup\\{\overline{I_{\varepsilon}(x_{\alpha})}:x_{\alpha}\in\mathcal{G}^{\prime}_{\varepsilon}\\}$.
We then notice that the arguments in the preceding proof show that for some
constant $C$ we have $m_{\varepsilon}\leq C$ and

$\displaystyle\int_{J_{1}\cup\ldots\cup
J_{m_{\varepsilon}}}|u_{\varepsilon}^{\prime}(x)|\,\mathrm{d}x\leq C,$ (11)

while $\mathcal{L}^{1}((a,b)\setminus(J_{1}\cup\ldots\cup
J_{m_{\varepsilon}}))\leq C\varepsilon$.

We set $x_{\varepsilon,i}=\sup J_{i}$ and
$\alpha_{\varepsilon,i}=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{J_{i}}u_{\varepsilon}\,\mathrm{d}x$,
$i=1,\ldots,m_{\varepsilon}$. Note that (11) implies

$\displaystyle\|u_{\varepsilon}-\alpha_{\varepsilon,i}\|_{L^{\infty}(J_{i})}\leq
C.$ (12)

Passing to a subsequence we may assume that $m_{\varepsilon}=\tilde{m}$ for
some $\tilde{m}$ independent of $\varepsilon$ and
$x_{\varepsilon,i}\to\tilde{x}_{i}\in[a,b]$ as well as
$\alpha_{\varepsilon,i}\to\tilde{\alpha}_{i}\in\overline{\mathbb{R}}=\mathbb{R}\cup\\{-\infty,+\infty\\}$,
for $i=1,\ldots,\tilde{m}$. Choose $a=x_{0}<x_{1}<\ldots<x_{m}=b$
($m\leq\tilde{m}$) such that
$\\{x_{0},\ldots,x_{m}\\}=\\{\tilde{x}_{0},\tilde{x}_{1},\ldots,\tilde{x}_{\tilde{m}}\\}$
(with $\tilde{x}_{0}:=a$ and, by construction, $\tilde{x}_{\tilde{m}}=b$) and
set $\alpha_{i}=\tilde{\alpha}_{j}$ if $\tilde{x}_{j-1}<\tilde{x}_{j}=x_{i}$.

If $|\alpha_{i}|<\infty$, then $\\{u_{\varepsilon}\\}_{\varepsilon}$ converges
to a function $w$ weakly* in $BV_{\rm loc}(x_{i-1},x_{i})$ and the uniform
bounds in (11) and (12) imply
$\|w\|_{L^{\infty}(x_{i-1},x_{i})}+Dw((x_{i-1},x_{i}))\leq C$, in particular,
$w\in BV(x_{i-1},x_{i})$. If $\alpha_{i}=\pm\infty$, then
$u_{\varepsilon}\to\pm\infty$ a.e. on $(x_{i-1},x_{i})$ by (12). In this case
we define $w\in BV(x_{i-1},x_{i})$ as the weak* limit in $BV_{\rm
loc}(x_{i-1},x_{i})$ of
$\\{u_{\varepsilon}-\alpha_{i,\varepsilon}\\}_{\varepsilon}$. The convergence
of $\\{\gamma_{\varepsilon}\\}_{\varepsilon}$ follows directly by applying
Theorem 4.2 on compact intervals $I$ of $(x_{i-1},x_{i})$ to
$\\{(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$ and
$\\{(u_{\varepsilon}-\alpha_{i,\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$,
respectively. As $\|w^{\prime}-g\|_{L^{2}(I)}$ is bounded independently of $I$
by the uniform energy bound, we have indeed $w^{\prime}-g\in L^{2}(a,b)$. ∎

######  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
$A\Subset(a,b)\setminus\\{x_{1},\ldots,x_{m-1}\\}$

$\displaystyle\int_{a}^{b}{|u_{\varepsilon}^{\prime}-g_{\varepsilon}|^{2}\,\textup{d}x}+\frac{1}{\varepsilon}\int_{a}^{b}{f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|g_{\varepsilon}|}\,\textup{d}t}\bigg{)}\,\textup{d}x$
$\displaystyle\quad\geq
2c_{0}(m-1)+\int_{A}{|u_{\varepsilon}^{\prime}-g_{\varepsilon}|^{2}\,\textup{d}x}+\frac{1}{\varepsilon}\int_{A}{f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|g_{\varepsilon}|}\,\textup{d}t}\bigg{)}\,\textup{d}x$

for sufficiently small $\varepsilon$ since
$\\#(\mathscr{G}_{\varepsilon}\setminus\mathscr{G}^{\prime}_{\varepsilon})\geq\tilde{m}-1\geq
m-1$.

######  Remark 4.5.

According to the compactness results of Theorems 4.2 and 4.3 we may restrict
ourselves to pairs $(u,\gamma)\in BV(a,b)\times\mathcal{M}(a,b)$ with
$\left\lVert u\right\rVert_{L^{\infty}(a,b)}\leq K$ and $(u,\gamma)\in
BV_{\infty,\mathcal{P}}(a,b)\times\mathcal{M}(a,b)$, respectively, and such
that the measure $Du-\gamma$, is absolutely continuous with respect to the
Lebesgue measure with density in $L^{2}(\mathcal{F}(u))$. The statements of
Theorems 2.1 and 2.3 are trivial otherwise.

######  Remark 4.6.

In principle, with the compactness result of Theorem 4.2 at hand, we could
infer our $\Gamma$-convergence result in Theorem 2.1 from known results, by
separate considerations of the elastic and inelastic contributions. To this
end, given $(u_{\varepsilon},\gamma_{\varepsilon})\in
W^{1,1}(a,b)\times\mathcal{M}(a,b)$ with
$\gamma_{\varepsilon}=g_{\varepsilon}\mathcal{L}^{1}$ and
$u_{\varepsilon}^{\prime}-g_{\varepsilon}\in L^{2}(a,b)$, we define
$(W_{\varepsilon},G_{\varepsilon})\in W^{1,2}(a,b)\times W^{1,1}(a,b)$ via
$W_{\varepsilon}(a)=G_{\varepsilon}(a)=0$ and
$W_{\varepsilon}^{\prime}=u_{\varepsilon}-g_{\varepsilon}$,
$G_{\varepsilon}^{\prime}=g_{\varepsilon}$ on $(a,b)$. This allows to express

$E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})=\int_{a}^{b}|W_{\varepsilon}^{\prime}|^{2}\,\textup{d}x+\frac{1}{\varepsilon}\int_{a}^{b}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|G_{\varepsilon}^{\prime}|\mathop{}\\!\mathrm{d}t}\bigg{)}\,\textup{d}x$

as the sum of the standard Dirichlet energy for $W_{\varepsilon}$ and a non-
local energy involving only $G_{\varepsilon}$ considered by Lussardi and
Vitali in [29]. However, the understanding of the coupling between
$u_{\varepsilon}$ and $\gamma_{\varepsilon}$ 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 $\Gamma$-lower limit

Next, we start with the proof of the $\Gamma$-$\liminf$ inequality. Except for
the very last paragraph we assume $K<\infty$ in the whole section. To show
this inequality it is useful to introduce the localized version of the
functionals $\\{E_{\varepsilon}\\}_{\varepsilon}$ and $E$. They are defined
for $(u,\gamma)\in L^{1}(a,b)\times\mathcal{M}(a,b)$ and every open subset $A$
of $(a,b)$ via

$\displaystyle
E_{\varepsilon}(u,\gamma,A)\coloneqq\begin{cases}\int_{A}{|u^{\prime}-g|^{2}\,\textup{d}x}&\quad\text{if
}u\in{W^{1,1}}(a,b),\,\left\lVert u\right\rVert_{{L^{\infty}}(a,b)}\leq K,\\\
+\frac{1}{\varepsilon}\int_{A}{f\big{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|g|}\,\textup{d}t}\big{)}\,\textup{d}x&\quad\phantom{if}\gamma=g\mathcal{L}^{1},\,g\in
L^{1}(a,b),\\\ &\quad\phantom{if}\text{and }u^{\prime}-g\in
L^{2}(a,b),\\\\[5.69046pt] \infty&\quad\text{otherwise},\end{cases}$

and

$E(u,\gamma,A)\coloneqq\begin{cases}\int_{A}{|u^{\prime}-g|^{2}\,\textup{d}x}+\int_{A}{c_{0}|g|\,\textup{d}x}&\quad\text{if
}u\in BV(a,b),\,\left\lVert u\right\rVert_{{L^{\infty}}(a,b)}\leq K,\\\
+2\,\sum_{x\in J_{u}\cap
A}f\left(\frac{1}{2}|[u](x)|\right)&\quad\phantom{if}\gamma=D^{s}u+g\mathcal{L}^{1},\,g\in
L^{1}(a,b),\\\ +c_{0}|D^{c}u|(A)&\quad\phantom{if}\text{and }u^{\prime}-g\in
L^{2}(a,b),\\\\[5.69046pt] \infty&\quad\text{otherwise}.\end{cases}$

We denote the $\Gamma$-lower and $\Gamma$-upper limit of
$\\{E_{\varepsilon}\\}_{\varepsilon}$ by

$\displaystyle E^{\prime}(u,\gamma)$
$\displaystyle\coloneqq\inf\big{\\{}\liminf_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\colon
u_{\varepsilon}\to u\text{ in
}L^{1}(a,b),\,\gamma_{\varepsilon}\to\gamma\text{ in the flat norm}\big{\\}},$
$\displaystyle E^{\prime\prime}(u,\gamma)$
$\displaystyle\coloneqq\inf\big{\\{}\limsup_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\colon
u_{\varepsilon}\to u\text{ in
}L^{1}(a,b),\,\gamma_{\varepsilon}\to\gamma\text{ in the flat norm}\big{\\}},$

respectively. For the $\Gamma$-lower limit we also need a localized version,
for which we adapt the notation and write $E^{\prime}(\cdot,\cdot,A)$ for
every open subset $A$ of $(a,b)$.

######  Remark 5.1 (Properties of the localized $\Gamma$-lower limit).

Let $(u,\gamma)\in{L^{1}}(a,b)\times\mathcal{M}(a,b)$.

  1. (i)

The properties that the set functions ${A\mapsto E_{\varepsilon}(u,\gamma,A)}$
are increasing and superadditive (on disjoint sets) for each $\varepsilon$
carry immediately over to ${A\mapsto E^{\prime}(u,\gamma,A)}$.

  2. (ii)

A direct consequence of (i) is that lower bounds for $A\mapsto
E^{\prime}(u,\gamma,A)$ transfer from intervals in $(a,b)$ to arbitrary open
subsets of $(a,b)$, i.e., if for a positive Borel measure $\lambda$ an
estimate of the form

$E^{\prime}(u,\gamma,A)\geq\lambda(A)$

holds for all intervals $A\subset(a,b)$, then the estimate actually holds for
any open subset $A\subset(a,b)$, cp. [29, Remark 4.6] for a similar statement.

In this section we prove the $\Gamma$-$\liminf$ inequality, where in view of
Remark 4.5 it is sufficient to consider $(u,\gamma)\in
BV(a,b)\times\mathcal{M}(a,b)$ with $\left\lVert
u\right\rVert_{L^{\infty}(a,b)}\leq K$, $\gamma=D^{s}u+g\mathcal{L}^{1}$ and
$u^{\prime}-g\in L^{2}(a,b)$. 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.

###  5.1 Estimate from below of the jump term

######  Proposition 5.2.

Let $A$ be an open subset of $(a,b)$. For every $(u,\gamma)\in
BV(a,b)\times\mathcal{M}(a,b)$ with $\gamma=D^{s}u+g\mathcal{L}^{1}$ and
$u^{\prime}-g\in L^{2}(a,b)$ we have

$E^{\prime}(u,\gamma,A)\geq 2\,\sum_{x\in J_{u}\cap
A}f\bigg{(}\frac{1}{2}|[u](x)|\bigg{)}.$

###### Proof.

_Step 1: For every ${\bar{x}}\in J_{u}\cap A$ we have_

$E^{\prime}(u,\gamma,A)\geq 2f\bigg{(}\frac{1}{2}|[u](\bar{x})|\bigg{)}.$ (13)

Since only finite energy approximations are of interest, we consider a
sequence $\\{(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$ in
${W^{1,1}}(a,b)\times\mathcal{M}(a,b)$ with
$\gamma_{\varepsilon}=g_{\varepsilon}\mathcal{L}^{1}$ for some
$g_{\varepsilon}\in L^{1}(a,b)$ and
$E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\leq C_{0}$ for some
uniform constant $C_{0}$, for all $\varepsilon>0$, such that
$u_{\varepsilon}\to u$ in $L^{1}(a,b)$ and $\gamma_{\varepsilon}\to\gamma$ in
the flat norm. We fix an arbitrary $\delta>0$ such that
$(\bar{x}-2\delta,\bar{x}+2\delta)\subset A$ (recall that $A\subset(a,b)$ is
open). By definition of $(u(\bar{x}+),u(\bar{x}-))$ and since
$u_{\varepsilon}\to u$ in measure, we readily find points
$\bar{x}_{\varepsilon}^{-}\in(\bar{x}-\delta,\bar{x})$ and
$\bar{x}_{\varepsilon}^{+}\in(\bar{x},\bar{x}+\delta)$ such that, for
sufficiently small $\varepsilon$,

$|u_{\varepsilon}(\bar{x}^{+}_{\varepsilon})-u(\bar{x}+)|<\delta\quad\text{and}\quad|u_{\varepsilon}(\bar{x}^{-}_{\varepsilon})-u(\bar{x}-)|<\delta$
(14)

(also cp. [29, Lemma 5.1]). Using the monotonicity of $A\mapsto
E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon},A)$ and applying the
estimate (10) with $(a,b)$ replaced by $(\bar{x}-2\delta,\bar{x}+2\delta)$ on
an associated grid of points $\bar{\mathscr{G}}_{\varepsilon}$, we obtain from
the subadditivity and non-negativity of $f$

$\displaystyle E_{\varepsilon}({u_{\varepsilon}},{\gamma_{\varepsilon}},A)$
$\displaystyle\geq
2\sum_{x_{\alpha}\in\bar{\mathscr{G}}_{\varepsilon}}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x_{\alpha})}{|g_{\varepsilon}|\mathop{}\\!\mathrm{d}t}\bigg{)}$
$\displaystyle\geq
2f\bigg{(}\sum_{x_{\alpha}\in\bar{\mathscr{G}}_{\varepsilon}}\frac{1}{2}\int_{I_{\varepsilon}(x_{\alpha})}{|{g_{\varepsilon}}|\,\textup{d}x}\bigg{)}\geq
2f\bigg{(}\frac{1}{2}\int_{(\bar{x}_{\varepsilon}^{-},\bar{x}_{\varepsilon}^{+})}{|g_{\varepsilon}|\,\textup{d}x}\bigg{)},$
(15)

for all $\varepsilon<\delta$ (notice the inclusion
$(\bar{x}_{\varepsilon}^{-},\bar{x}_{\varepsilon}^{+})\subset\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\bar{\mathscr{G}}_{\varepsilon}\\}$). For the argument on the
right-hand side, we observe from the Cauchy–Schwarz inequality, the
inequalities in (14) and from the energy bound

$\displaystyle\int_{(\bar{x}_{\varepsilon}^{-},\bar{x}_{\varepsilon}^{+})}{|g_{\varepsilon}|\,\textup{d}x}$
$\displaystyle\geq\Bigg{[}\int_{(\bar{x}_{\varepsilon}^{-},\bar{x}_{\varepsilon}^{+})}{|{u}^{\prime}_{\varepsilon}|\,\textup{d}x}-\int_{(\bar{x}_{\varepsilon}^{-},\bar{x}_{\varepsilon}^{+})}{|{g_{\varepsilon}}-{u}^{\prime}_{\varepsilon}|\,\textup{d}x}\Bigg{]}_{+}$
$\displaystyle\geq\Big{[}|{u}_{\varepsilon}(\bar{x}_{\varepsilon}^{+})-u_{\varepsilon}(\bar{x}_{\varepsilon}^{-})|-|\bar{x}_{\varepsilon}^{+}-\bar{x}_{\varepsilon}^{-}|^{\frac{1}{2}}\left\lVert
g_{\varepsilon}-u_{\varepsilon}^{\prime}\right\rVert_{L^{2}(a,b)}\Big{]}_{+}$
$\displaystyle\geq\Big{[}|u(\bar{x}+)-u(\bar{x}-)|-2\delta-(2\delta)^{\frac{1}{2}}C_{0}^{\frac{1}{2}}\Big{]}_{+}.$

Using once again the fact that $f$ is increasing, we can continue to estimate
(15) from below via

$E_{\varepsilon}({u_{\varepsilon}},{\gamma_{\varepsilon}},A)\geq
2f\left(\frac{1}{2}\Big{[}|[u](\bar{x})|-2\delta-
C_{0}^{\frac{1}{2}}\,(2\delta)^{\frac{1}{2}}\Big{]}_{+}\right).$

Now, passing to the $\liminf$ as $\varepsilon\to 0$ and then letting
$\delta\to 0^{+}$, we obtain (13).

_Step 2._ For an arbitrary $M\in{\mathbb{N}}$ with $M\leq\\#(J_{u}\cap A)$ we
select a set $\\{x_{1},\dots,x_{M}\\}$ containing $M$ points of $J_{u}\cap A$
and pairwise disjoint open intervals $I_{1},\dots,I_{M}$ in $A$ such that
$x_{i}\in I_{i}$ for all $i=1,\dots,M$. First, we apply the monotonicity and
superadditivity of $E^{\prime}(u,\gamma,\,\cdot\,)$ (see Remark 5.1) and then
the estimate of Step 1 for $I_{i}$ instead of $A$. This yields

$E^{\prime}(u,\gamma,A)\geq\sum_{i=1}^{M}E^{\prime}(u,\gamma,I_{i})\geq\sum_{i=1}^{M}2f\left(\frac{1}{2}|[u](x_{i})|\right).$

Since $M$ is arbitrary and $J_{u}$ is at most countable, the claim of the
proposition follows. ∎

###  5.2 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
$W^{1,1}(a,b)\times\mathcal{M}(a,b)$ can be modified in such a way that in the
limit we additionally have the optimal $L^{\infty}$-estimate.

######  Lemma 5.3.

Let $(u,\gamma)\in BV(a,b)\times\mathcal{M}(a,b)$ with $\left\lVert
u\right\rVert_{L^{\infty}(a,b)}\leq K$, $\gamma=D^{s}u+g\mathcal{L}^{1}$ and
$u^{\prime}-g\in L^{2}(a,b)$. Furthermore, let
$\\{(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$ be a sequence in
$W^{1,1}(a,b)\times\mathcal{M}(a,b)$ with $\left\lVert
u_{\varepsilon}\right\rVert_{L^{\infty}(a,b)}\leq K$,
$\gamma_{\varepsilon}=g_{\varepsilon}\mathcal{L}^{1}$ and
$u_{\varepsilon}^{\prime}-g_{\varepsilon}\in L^{2}(a,b)$ for all
$\varepsilon>0$ such that $\\{u_{\varepsilon}\\}_{\varepsilon}$ converges to
$u$ a.e. in $(a,b)$ and in $L^{1}(a,b)$ and such that
$\\{\gamma_{\varepsilon}\\}_{\varepsilon}$ converges to $\gamma$ in the flat
norm. There exists a sequence
$\\{(\bar{u}_{\varepsilon},\bar{\gamma}_{\varepsilon})\\}_{\varepsilon}$ in
$W^{1,1}(a,b)\times\mathcal{M}(a,b)$ with
$\left\lVert\bar{u}_{\varepsilon}\right\rVert_{L^{\infty}(a,b)}\leq K$,
$\bar{\gamma}_{\varepsilon}=\bar{g}_{\varepsilon}\mathcal{L}^{1}$ for some
$\bar{g}_{\varepsilon}\in L^{1}(a,b)$ and
$E_{\varepsilon}(\bar{u}_{\varepsilon},\bar{\gamma}_{\varepsilon})\leq
E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})$ for all $\varepsilon$
such that $\\{\bar{u}_{\varepsilon}\\}_{\varepsilon}$ converges to $u$ a.e. in
$(a,b)$ and in $L^{q}(a,b)$ for all $1\leq q<\infty$ and

$\limsup_{\varepsilon\to
0}\left\lVert\bar{u}_{\varepsilon}-u\right\rVert_{L^{\infty}(a,b)}\leq\sup\big{\\{}|[u](x)|\colon
x\in J_{u}\big{\\}}.$

If, in addition, the energies
$\\{E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$
are bounded, then $\\{\bar{\gamma}_{\varepsilon}\\}_{\varepsilon}$ converges
to $\gamma$ in the flat norm.

###### Proof.

We follow the outline of the proof for [29, Lemma 4.3], which, however, needs
some modifications due to the additional variable $\gamma$. In what follows,
we may assume that the precise representatives of $u$ and $u_{\varepsilon}$
for each $\varepsilon$ 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

$\sigma\coloneqq\sup\big{\\{}|[u_{j}](x)|\colon x\in J_{u}\big{\\}}$

and first claim that for every $n\in{\mathbb{N}}$ there exists
$\delta_{n}\in(0,\tfrac{1}{n}]$ such that

$\displaystyle x,y\in(a,b)\text{ with }|x-y|<\delta_{n}\quad\Rightarrow\quad$
$\displaystyle|u_{j}(x)-u_{j}(y)|<\sigma+\frac{1}{n},$ (16)
$\displaystyle|u_{a}(x)+u_{c}(x)-u_{a}(y)-u_{c}(y)|<\frac{1}{n}.$ (17)

In fact, there are only finitely many points
$\bar{x}_{1},\dots,\bar{x}_{m(n)}$ in $J_{u}$ that have to be excluded to
deduce that

$\displaystyle\sum_{z\in
J_{u}\setminus\\{\bar{x}_{1},\dots,\bar{x}_{m(n)}\\}}|[u_{j}](z)|<\frac{1}{n}.$
(18)

By choosing $\delta_{n}>0$ sufficiently small we can guarantee (16) by (18)
and the definition of $\sigma$ provided that each interval of length
$\delta_{n}$ contains at most one $\bar{x}_{j}$, 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.,

$a=x_{0}<x_{1}<\ldots<x_{k}<x_{k+1}=b$

(where the dependence of the points on $n$ is not written explicitly) such
that the mesh size is less than $\delta_{n}$, i.e., $x_{i+1}-x_{i}<\delta_{n}$
for all $i\in\\{0,\ldots,k\\}$, $x_{i}\notin J_{u}$ and
$u_{\varepsilon}(x_{i})\to u(x_{i})$ as $\varepsilon\to 0$ for every
$i\in\\{1,\dots,k\\}$, which is possible by the pointwise convergence
$u_{\varepsilon}\to u$ a.e. in $(a,b)$. Since by construction of $\delta_{n}$
at most one of the points $\bar{x}_{1},\dots,\bar{x}_{m(n)}\in J_{u}$, where a
large jump of $u_{j}$ occurs, may belong to the interval $[x_{i},x_{i+1}]$, we
necessarily have $|u_{j}(x)-u_{j}(y)|<\tfrac{1}{n}$ for $y=x_{i}$ or
$y=x_{i+1}$ such that as a consequence of (17) there holds

$u(x)\in\left[\min\\{u(x_{i}),u(x_{i+1})\\}-\frac{2}{n},\max\\{u(x_{i}),u(x_{i+1})\\}+\frac{2}{n}\right]$
(19)

for all $x\in[x_{i},x_{i+1}]$ and every $i\in\\{1,\ldots,k-1\\}$.

After having fixed the partitions $P_{n}$, we can now start with the
construction of the sequence $\\{\bar{u}_{\varepsilon}\\}_{\varepsilon}$.
Since $u_{\varepsilon}\to u$ in measure and $u_{\varepsilon}(x_{i})\to
u(x_{i})$ for every $i\in\\{1,\dots,k\\}$, we can fix a “level”
$\bar{\varepsilon}_{n}$ for each $n\in{\mathbb{N}}$ such that

$\Big{|}\Big{\\{}x\in(a,b)\colon|u_{\varepsilon}(x)-u(x)|\geq\frac{1}{n}\Big{\\}}\Big{|}\leq\frac{1}{n}\quad\text{for
all }\varepsilon<\bar{\varepsilon}_{n}$ (20)

and

$|u_{\varepsilon}(x_{i})-u(x_{i})|<\frac{1}{n}\quad\text{for every
}i=1,\dots,k\text{ and all }\varepsilon<\bar{\varepsilon}_{n}.$ (21)

Notice that we can choose $\\{\bar{\varepsilon}_{n}\\}_{n}$ strictly
decreasing and such that $\bar{\varepsilon}_{n}\to 0^{+}$ as $n\to\infty$. For
$\varepsilon>\bar{\varepsilon}_{1}$ we then set
$\bar{u}_{\varepsilon}\coloneqq u_{\varepsilon}$ and
$\bar{\gamma}_{\varepsilon}\coloneqq\gamma_{\varepsilon}$. Otherwise, if
$\varepsilon\in(0,\bar{\varepsilon}_{1}]$, we first determine the unique
$n=n(\varepsilon)\in{\mathbb{N}}$ with
$\varepsilon\in(\bar{\varepsilon}_{n+1},\bar{\varepsilon}_{n}]$. On the first
and the last interval of $P_{n}$ we set $\bar{u}_{\varepsilon}$ equal to
$u_{\varepsilon}(x_{1})$ and $u_{\varepsilon}(x_{k})$, respectively. On an
arbitrary interior interval $[\alpha,\beta]$ of the form $[x_{i},x_{i+1}]$ for
some $i\in\\{1,\dots,k-1\\}$ we define, after assuming without loss of
generality $u_{\varepsilon}(\alpha)\leq u_{\varepsilon}(\beta)$,

$\bar{u}_{\varepsilon}(x):=\min\left\\{\max\left\\{u_{\varepsilon}(x),u_{\varepsilon}(\alpha)-\frac{4}{n}\right\\},u_{\varepsilon}(\beta)+\frac{4}{n}\right\\}$

for every $x\in[\alpha,\beta]$, which is the projection of $u_{\varepsilon}$
onto $[u_{\varepsilon}(\alpha)-4/n,u_{\varepsilon}(\beta)+4/n]$. Because of
$\bar{u}_{\varepsilon}(x_{i})=u_{\varepsilon}(x_{i})$ for every
$i\in\\{1,\dots,k\\}$ and $u_{\varepsilon}\in W^{1,1}(a,b)$, we clearly have
$\bar{u}_{\varepsilon}\in W^{1,1}(a,b)$. Moreover, the $L^{\infty}$ bound on
$u_{\varepsilon}$ with constant $K$ directly transfers to
$\bar{u}_{\varepsilon}$.

We next study the asymptotic behavior of the sequence
$\\{\bar{u}_{\varepsilon}\\}_{\varepsilon}$. From the definition of
$\bar{u}_{\varepsilon}$ and with (21), we observe for all
$\varepsilon\leq\bar{\varepsilon}_{1}$

$\bar{u}_{\varepsilon}(x)\in\left[\min\\{u(x_{i}),u(x_{i+1})\\}-\frac{5}{n},\max\\{u(x_{i}),u(x_{i+1})\\}+\frac{5}{n}\right]$

which, via (16) and (17), implies

$|\bar{u}_{\varepsilon}(x)-u(x)|\leq\sigma+\frac{7}{n},$

for all $x\in[x_{i},x_{i+1}]$ and every $i\in\\{1,\ldots,k-1\\}$. Since the
latter estimate is also satisfied for the first and the last interval of the
partition, we then infer from $n=n(\varepsilon)\to\infty$ as $\varepsilon\to
0$ the estimate

$\limsup_{\varepsilon\to
0}\left\lVert\bar{u}_{\varepsilon}-u\right\rVert_{L^{\infty}(a,b)}\leq\sigma=\sup\big{\\{}|[u](x)|\colon
x\in J_{u}\big{\\}}.$

In order to show the convergence claims of
$\\{\bar{u}_{\varepsilon}\\}_{\varepsilon}$, we again consider an arbitrary
interior interval $[\alpha,\beta]$ of the partition $P_{n}$. If we denote the
pointwise projection of $u_{\varepsilon}$ onto
$[u_{\varepsilon}(\alpha)-3/n,u_{\varepsilon}(\beta)+3/n]$ by
$u_{\varepsilon}^{*}$, we have

$|\bar{u}_{\varepsilon}-u|\leq|\bar{u}_{\varepsilon}-u_{\varepsilon}^{*}|+|u_{\varepsilon}^{*}-u|\leq\frac{1}{n}+|u_{\varepsilon}-u|\quad\text{
in }[\alpha,\beta],$

as we know $u(x)\in[u_{\varepsilon}(\alpha)-3/n,u_{\varepsilon}(\beta)+3/n]$
due to (19) and (21). Since the length of the first and last interval of
$P_{n}$ vanish in the limit $n\to\infty$ and hence for $\varepsilon\to 0$,
this implies the pointwise convergence of
$\\{\bar{u}_{\varepsilon}\\}_{\varepsilon}$ to $u$ a.e. on $(a,b)$. In
addition, as $\\{\bar{u}_{\varepsilon}\\}_{\varepsilon}$ is bounded in
$L^{\infty}(a,b)$, convergence in $L^{q}(a,b)$ for all $1\leq q<\infty$
follows from the dominated convergence theorem. For later purposes, we notice
from the definition of $\bar{u}_{\varepsilon}$ and the previous inclusion for
$u$ that for every $\varepsilon$ we have

$\\{x\in(a,b)\colon u_{\varepsilon}(x)\neq\bar{u}_{\varepsilon}(x)\\}\\\
\subset\Big{\\{}x\in(a,b)\colon|u_{\varepsilon}(x)-u(x)|\geq\frac{1}{n}\Big{\\}}\cup(x_{0},x_{1})\cup(x_{k},x_{k+1}).$
(22)

We next define the sequence $\\{\bar{\gamma}_{\varepsilon}\\}_{\varepsilon}$
in $\mathcal{M}(a,b)$ by setting for every $\varepsilon$

$\displaystyle\bar{g}_{\varepsilon}(x):=\left\\{\begin{array}[]{l
l}g_{\varepsilon}(x)&\quad\text{if
}u_{\varepsilon}(x)=\bar{u}_{\varepsilon}(x),\\\
0&\quad\text{otherwise},\end{array}\right.$

and $\bar{\gamma}_{\varepsilon}\coloneqq\bar{g}_{\varepsilon}\mathcal{L}^{1}$.
Since there holds $\bar{u}_{\varepsilon}^{\prime}=0$ on
$\\{x\in(a,b)\colon\bar{u}_{\varepsilon}(x)\neq u_{\varepsilon}(x)\\}$ and
$\bar{u}_{\varepsilon}^{\prime}={u}_{\varepsilon}^{\prime}$ on
$\\{x\in(a,b)\colon\bar{u}_{\varepsilon}(x)=u_{\varepsilon}(x)\\}$, it follows
that

$\displaystyle
E_{\varepsilon}(\bar{u}_{\varepsilon},\bar{\gamma}_{\varepsilon})$
$\displaystyle=\int_{a}^{b}{|\bar{u}_{\varepsilon}^{\prime}(x)-\bar{g}_{\varepsilon}(x)|^{2}\mathop{}\\!\mathrm{d}x}+\frac{1}{\varepsilon}\int_{a}^{b}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|\bar{g}_{\varepsilon}(t)|\mathop{}\\!\mathrm{d}t}\bigg{)}\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\int_{a}^{b}{|{u}_{\varepsilon}^{\prime}(x)-{g}_{\varepsilon}(x)|^{2}\mathop{}\\!\mathrm{d}x}+\frac{1}{\varepsilon}\int_{a}^{b}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|g_{\varepsilon}(t)|\mathop{}\\!\mathrm{d}t}\bigg{)}\mathop{}\\!\mathrm{d}x=E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon}).$

If, in addition,
$\\{E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$ is
a bounded sequence, then
$\\{\bar{u}_{\varepsilon}^{\prime}-\bar{g}_{\varepsilon}\\}_{\varepsilon}$ is
a bounded sequence in $L^{2}(a,b)$. In view of (9) and the Cauchy–Schwarz
inequality we then notice for $1<q<\infty$

$\displaystyle\left\lVert\bar{\gamma}_{\varepsilon}-\gamma_{\varepsilon}\right\rVert_{\text{flat}}$
$\displaystyle\leq\left\lVert
D\bar{u}_{\varepsilon}-Du_{\varepsilon}\right\rVert_{\text{flat}}+\left\lVert(\bar{\gamma}_{\varepsilon}-D\bar{u}_{\varepsilon})-(\gamma_{\varepsilon}-Du_{\varepsilon})\right\rVert_{\text{flat}}$
$\displaystyle\leq\left\lVert\bar{u}_{\varepsilon}-u_{\varepsilon}\right\rVert_{L^{1}(a,b)}+\left\lVert(g_{\varepsilon}-u_{\varepsilon}^{\prime})\mathbbm{1}_{\\{u_{\varepsilon}\neq\bar{u}_{\varepsilon}\\}}\mathcal{L}^{1}\right\rVert_{\text{flat}}$
$\displaystyle\leq\left\lVert\bar{u}_{\varepsilon}-u_{\varepsilon}\right\rVert_{L^{1}(a,b)}+\left\lVert(g_{\varepsilon}-u_{\varepsilon}^{\prime})\mathbbm{1}_{\\{u_{\varepsilon}\neq\bar{u}_{\varepsilon}\\}}\right\rVert_{L^{1}(a,b)}$
$\displaystyle\leq\left\lVert\bar{u}_{\varepsilon}-u_{\varepsilon}\right\rVert_{L^{1}(a,b)}+\left\lVert
g_{\varepsilon}-u_{\varepsilon}^{\prime}\right\rVert_{L^{2}(a,b)}|\\{x\in(a,b)\colon
u_{\varepsilon}(x)\neq\bar{u}_{\varepsilon}(x)\\}|^{\frac{1}{2}}.$

If we consider the limit $\varepsilon\to 0$ on the right-hand side, the first
term disappears, since we have the convergence
$\bar{u}_{\varepsilon}-u_{\varepsilon}\to 0$ 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 $\varepsilon\to 0$. Therefore, we have
$\bar{\gamma}_{\varepsilon}-\gamma_{\varepsilon}\to 0$ in the flat norm. Since
by assumption there holds $\gamma_{\varepsilon}\to\gamma$ in the flat norm, we
conclude that we also have $\bar{\gamma}_{\varepsilon}\to\gamma$ 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 $E^{\prime}(u,\gamma,I)$ and the $\Gamma$-lower limit
$E^{\prime}(u^{I},\gamma^{I})$, where $I\subset(a,b)$ is an open interval,
$u^{I}$ is the extension of $u|_{I}$ to $(a,b)$ with inner traces and
$\gamma^{I}$ is the restriction of $\gamma$ to $I$.

######  Lemma 5.4.

Let $I=(\alpha,\beta)$ be an open interval in $(a,b)$. Let $(u,\gamma)\in
BV(a,b)\times\mathcal{M}(a,b)$ with $\left\lVert
u\right\rVert_{L^{\infty}(a,b)}\leq K$, $\gamma=D^{s}u+g\mathcal{L}^{1}$ and
$u^{\prime}-g\in L^{2}(a,b)$. Then for $(u^{I},\gamma^{I})\in
BV(a,b)\times\mathcal{M}(a,b)$ defined via

$\displaystyle u^{I}(x):=\left\\{\begin{array}[]{l l}u(\alpha+)&\quad
x\in(a,\alpha),\\\ u(x)&\quad x\in[\alpha,\beta],\\\ u(\beta-)&\quad
x\in(\beta,b),\end{array}\right.$

and $\gamma^{I}:=\gamma\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}I$ there holds

$E^{\prime}(u^{I},\gamma^{I})\leq E^{\prime}(u,\gamma,I).$

###### Proof.

We proceed analogously to the proof of [29, Lemma 4.4]. By definition of
$E^{\prime}$ as the $\Gamma$-lower limit of
$\\{E_{\varepsilon}\\}_{\varepsilon}$, there exists a sequence
$\\{(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$ in
${W^{1,1}}(a,b)\times\mathcal{M}(a,b)$ with $\left\lVert
u_{\varepsilon}\right\rVert_{L^{\infty}(a,b)}\leq K$ and
$\gamma_{\varepsilon}=g_{\varepsilon}\mathcal{L}^{1}$ such that
$u_{\varepsilon}\to u$ in $L^{1}(a,b)$, $\gamma_{\varepsilon}\to\gamma$ in the
flat norm and $\liminf_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon},I)=E^{\prime}(u,\gamma,I)$.
Without loss of generality, we may also assume pointwise convergence
$u_{\varepsilon}\to u$ a.e. in $(a,b)$. For an arbitrary
$\eta\in(0,(\beta-\alpha)/2)$ we then pick points
$\alpha_{\eta}\in(\alpha,\alpha+\eta)$ and $\beta_{\eta}\in(\beta-\eta,\beta)$
such that on the one hand

$u_{\varepsilon}(\alpha_{\eta})\to u(\alpha_{\eta})\quad\text{and}\quad
u_{\varepsilon}(\beta_{\eta})\to u(\beta_{\eta})\quad\text{as }\varepsilon\to
0,$ (23)

and on the other hand

$|u(\alpha_{\eta})-u(\alpha+)|+|u(\beta_{\eta})-u(\beta-)|<\eta.$ (24)

For $I_{\eta}\coloneqq(\alpha_{\eta},\beta_{\eta})\subset(\alpha,\beta)$ we
now consider the functions $(u^{I_{\eta}},\gamma^{I_{\eta}})$ and the sequence
$\\{(u^{I_{\eta}}_{\varepsilon},\gamma^{I_{\eta}}_{\varepsilon})\\}_{\varepsilon}$
in ${W^{1,1}}(a,b)\times\mathcal{M}(a,b)$ defined analogously to
$(u^{I},\gamma^{I})$. The condition (23) guarantees
$u^{I_{\eta}}_{\varepsilon}\to u^{I_{\eta}}$ in $L^{1}(a,b)$, while
$\gamma^{I_{\eta}}_{\varepsilon}\to\gamma^{I_{\eta}}$ in the flat norm is
trivially satisfied. Moreover, we notice
$E_{\varepsilon}(u^{I_{\eta}}_{\varepsilon},\gamma^{I_{\eta}}_{\varepsilon})=E_{\varepsilon}(u^{I_{\eta}}_{\varepsilon},\gamma^{I_{\eta}}_{\varepsilon},I)$
for all $\varepsilon<\min\\{\alpha_{\eta}-\alpha,\beta-\beta_{\eta}\\}$.
Therefore, we conclude from the definition of $E_{\varepsilon}$ that

$\displaystyle
E^{\prime}(u^{I_{\eta}},\gamma^{I_{\eta}})\leq\liminf_{\varepsilon\to
0}E_{\varepsilon}(u^{I_{\eta}}_{\varepsilon},\gamma^{I_{\eta}}_{\varepsilon})$
$\displaystyle=\liminf_{\varepsilon\to
0}E_{\varepsilon}(u^{I_{\eta}}_{\varepsilon},\gamma^{I_{\eta}}_{\varepsilon},I)$
$\displaystyle\leq\liminf_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon},I)=E^{\prime}(u,\gamma,I).$

We next observe $u^{I_{\eta}}\to u^{I}$ in $L^{1}(a,b)$ and
$\gamma^{I_{\eta}}\to\gamma^{I}$ in the flat norm as $\eta\to 0$, from (24),
respectively, Lemma 3.1 since
$\gamma^{I_{\eta}}\overset{*}{\rightharpoonup}\gamma^{I}$ in
$\mathcal{M}(a,b)$ by dominated convergence (as we have pointwise convergence
$\mathbbm{1}_{I_{\eta}}\to\mathbbm{1}_{I}$ on $(a,b)$). By the lower
semicontinuity of $E^{\prime}$ we then arrive at the claim

$E^{\prime}(u^{I},\gamma^{I})\leq\liminf_{\eta\to
0}E^{\prime}(u^{I_{\eta}},\gamma^{I_{\eta}})\leq E^{\prime}(u,\gamma,I).\qed$

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 $(u,\gamma)\in
BV(a,b)\times\mathcal{M}(a,b)$ with $\left\lVert
u\right\rVert_{L^{\infty}(a,b)}\leq K$, $\gamma=D^{s}u+g\mathcal{L}^{1}$ and
$u^{\prime}-g\in L^{2}(a,b)$ we have

$E^{\prime}(u,\gamma,A)\geq\int_{A}{|u^{\prime}-g|^{2}\,\textup{d}x}+c_{0}\int_{A}{|g|\,\textup{d}x}+c_{0}|D^{c}u|(A).$

###### Proof.

_Step 1: With $\sigma\coloneqq\sup_{x\in J_{u}}|u(x+)-u(x-)|$, there holds the
preliminary estimate_

$E^{\prime}(u,\gamma)\left(1+3\sigma\right)\geq\int_{a}^{b}|u^{\prime}-g|^{2}\,\textup{d}x+c_{0}\,\int_{a}^{b}{|g|}\,\textup{d}x+c_{0}|D^{c}u|(a,b).$
(25)

By definition of $E^{\prime}$ as the $\Gamma$-lower limit of
$\\{E_{\varepsilon}\\}_{\varepsilon}$, there exists a sequence
$\\{(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$ in
${W^{1,1}}(a,b)\times\mathcal{M}(a,b)$ with $\left\lVert
u_{\varepsilon}\right\rVert_{L^{\infty}(a,b)}\leq K$ and
$\gamma_{\varepsilon}=g_{\varepsilon}\mathcal{L}^{1}$ for some
$g_{\varepsilon}\in L^{1}(a,b)$ for every $\varepsilon>0$ such that
$u_{\varepsilon}\to u$ in $L^{1}(a,b)$, $\gamma_{\varepsilon}\to\gamma$ in the
flat norm and $\liminf_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})=E^{\prime}(u,\gamma)$.
After assuming without loss of generality $E^{\prime}(u,\gamma)<\infty$ and
passing to a subsequence (not relabeled) and a possible modification of the
sequence via Lemma 5.3 we may further suppose

$\lim_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})=E^{\prime}(u,\gamma),\quad\text{
in particular }\quad
E_{\varepsilon}({u}_{\varepsilon},{\gamma}_{\varepsilon})\leq
C_{0}\quad\text{for all }\varepsilon$ (26)

for a positive constant $C_{0}$ as well as

$\limsup_{\varepsilon\to 0}\left\lVert
u_{\varepsilon}-u\right\rVert_{L^{\infty}(a,b)}\leq\sigma.$

Let $\eta>0$ be fixed. We may assume that $\left\lVert
u_{\varepsilon}-u\right\rVert_{L^{\infty}(a,b)}\leq\sigma+\eta$ holds for all
$\varepsilon$. Analogously as in the proof of Lemma 5.3 (cf. (16) and (17)),
there exists $\delta_{\eta}>0$ such that $|u(x)-u(y)|<\sigma+\eta$ for all
$x,y\in(a,b)$ with $|x-y|<\delta_{\eta}$. Thus, there holds

$x,y\in(a,b)\text{ with
}|x-y|<\delta_{\eta}\quad\Rightarrow\quad|u_{\varepsilon}(x)-u_{\varepsilon}(y)|<3(\sigma+\eta)$
(27)

for all such $\varepsilon$. Next, we apply Lemma 4.1 with $u=u_{\varepsilon}$
and $\gamma=\gamma_{\varepsilon}$. In this way we find a uniform grid
$\mathscr{G}_{\varepsilon}$ in the interval $(a,b)$ with grid size
$2\varepsilon$ such that

$\int_{\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}}|g_{\varepsilon}|\mathop{}\\!\mathrm{d}t+2\\#(\mathscr{G}_{\varepsilon}\setminus\mathscr{G}^{\prime}_{\varepsilon})\\\
\leq\frac{1}{c_{0}\varepsilon}\int_{a}^{b}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|g_{\varepsilon}|\mathop{}\\!\mathrm{d}t}\bigg{)}\,\textup{d}x.$
(28)

Let $a_{\ast}=\min\mathscr{G}_{\varepsilon}-\varepsilon$ and
$b_{\ast}=\max\mathscr{G}_{\varepsilon}+\varepsilon$. We then consider a
sequence $\\{\tilde{v}_{\varepsilon}\\}_{\varepsilon}$ of functions in
$L^{\infty}(a,b)$, which is defined a.e. in $(a_{\ast},b_{\ast})$ by

$\displaystyle\tilde{v}_{\varepsilon}(x):=\left\\{\begin{array}[]{l
l}u_{\varepsilon}(x)&\quad x\in\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\},\\\
\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x_{\alpha})}u_{\varepsilon}(z)\,\textup{d}z&\quad
x\in I_{\varepsilon}(x_{\alpha})\text{ for some
}x_{\alpha}\in\mathscr{G}_{\varepsilon}\setminus\mathscr{G}^{\prime}_{\varepsilon},\end{array}\right.$
(31)

and then extended to $(a,b)$ by the constant values
$\tilde{v}_{\varepsilon}(a_{\ast}^{+})$ and
$\tilde{v}_{\varepsilon}(b_{\ast}^{-})$, respectively, for all $\varepsilon$.
As $\tilde{v}_{\varepsilon}$ is bounded with
$\left\lVert\tilde{v}_{\varepsilon}\right\rVert_{L^{\infty}(a,b)}\leq K$ and
coincides with $u_{\varepsilon}$ on the set
$\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}$, where

$\displaystyle|(a,b)\setminus\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}|\leq
2\varepsilon(2+\\#(\mathscr{G}_{\varepsilon}\setminus\mathscr{G}^{\prime}_{\varepsilon}))\leq
4\varepsilon+\varepsilon\frac{C_{0}}{c_{0}}$ (32)

by (26) and (28), we observe $\tilde{v}_{\varepsilon}\to u$ in $L^{q}(a,b)$
for all $1\leq q<\infty$. Moreover, we have ${\tilde{v}_{\varepsilon}\in
SBV(a,b)}$ with

$|[\tilde{v}_{\varepsilon}](x)|\leq 3(\sigma+\eta)\quad\text{for all }x\in
J_{\tilde{v}_{\varepsilon}},$

provided that $\varepsilon$ is sufficiently small, i.e.,
$4\varepsilon<\delta_{\eta}$ (such that (27) is satisfied). Since the number
of jumps of $\tilde{v}_{\varepsilon}$ is bounded via (28) and the definition
of $E_{\varepsilon}$ by

$\\#J_{\tilde{v}_{\varepsilon}}\leq
2\\#(\mathscr{G}_{\varepsilon}\setminus\mathscr{G}^{\prime}_{\varepsilon})\leq\frac{1}{c_{0}}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon}),$

we end up with the estimate

$|D^{s}\tilde{v}_{\varepsilon}|(a,b)\leq\frac{3(\sigma+\eta)}{c_{0}}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})$
(33)

for the size of $D^{s}\tilde{v}_{\varepsilon}$ in terms of the energy
$E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})$. Next, we introduce a
sequence $\\{\tilde{\gamma}_{\varepsilon}\\}_{\varepsilon}$ of measures in
$\mathcal{M}(a,b)$, by setting
$\tilde{\gamma}_{\varepsilon}\coloneqq\tilde{g}_{\varepsilon}\mathcal{L}^{1}$
with $g_{\varepsilon}\in L^{1}(a,b)$ defined as

$\displaystyle\tilde{g}_{\varepsilon}(x)\coloneqq\left\\{\begin{array}[]{l
l}g_{\varepsilon}(x)&\quad x\in\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\},\\\
0&\quad\text{otherwise},\end{array}\right.$ (36)

for all $\varepsilon$. In order to show that
$\\{\tilde{\gamma}_{\varepsilon}+D^{s}\tilde{v}_{\varepsilon}\\}_{\varepsilon}$
is an approximating sequence of $\gamma$, we first notice from the definition
of $\tilde{v}_{\varepsilon}$ and
$\tilde{\gamma}_{\varepsilon}=\tilde{g}_{\varepsilon}\mathcal{L}^{1}$ in (31)
and (36), with $\tilde{v}_{\varepsilon}^{\prime}=u_{\varepsilon}^{\prime}$ on
$\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}$ and
$\tilde{v}_{\varepsilon}^{\prime}=0$ on the remaining set of $(a,b)$, that

$\displaystyle\left\lVert\gamma_{\varepsilon}-(\tilde{\gamma}_{\varepsilon}+D^{s}\tilde{v}_{\varepsilon})\right\rVert_{\text{flat}}$
$\displaystyle=\left\lVert
g_{\varepsilon}\mathbbm{1}_{(a,b)\setminus\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}}\mathcal{L}^{1}-D^{s}\tilde{v}_{\varepsilon}\right\rVert_{\text{flat}}$
$\displaystyle=\left\lVert(g_{\varepsilon}-u_{\varepsilon}^{\prime})\mathbbm{1}_{(a,b)\setminus\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}}\mathcal{L}^{1}+u_{\varepsilon}^{\prime}\mathcal{L}^{1}-\tilde{v}_{\varepsilon}^{\prime}\mathcal{L}^{1}-D^{s}\tilde{v}_{\varepsilon}\right\rVert_{\text{flat}}$
$\displaystyle\leq\left\lVert(g_{\varepsilon}-u_{\varepsilon}^{\prime})\mathbbm{1}_{(a,b)\setminus\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}}\mathcal{L}^{1}\right\rVert_{\text{flat}}+\left\lVert
Du_{\varepsilon}-D\tilde{v}_{\varepsilon}\right\rVert_{\text{flat}}.$

With (9) and the Cauchy–Schwarz inequality we can then continue to estimate

$\displaystyle\left\lVert\gamma_{\varepsilon}-(\tilde{\gamma}_{\varepsilon}+D^{s}\tilde{v}_{\varepsilon})\right\rVert_{\text{flat}}$
$\displaystyle\leq\left\lVert(g_{\varepsilon}-u_{\varepsilon}^{\prime})\mathbbm{1}_{(a,b)\setminus\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}}\right\rVert_{L^{1}(a,b)}+\left\lVert
u_{\varepsilon}-\tilde{v}_{\varepsilon}\right\rVert_{L^{1}(a,b)}$
$\displaystyle\leq\left\lVert
g_{\varepsilon}-u_{\varepsilon}^{\prime}\right\rVert_{L^{2}(a,b)}|(a,b)\setminus\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}|^{\frac{1}{2}}+\left\lVert
u_{\varepsilon}-\tilde{v}_{\varepsilon}\right\rVert_{L^{1}(a,b)}.$

We now study the terms on the right-hand side. From (26) and the definition of
$E_{\varepsilon}$ we notice that
$\\{u_{\varepsilon}^{\prime}-g_{\varepsilon}\\}_{\varepsilon}$ is a bounded
sequence in $L^{2}(a,b)$. Together with (32) and taking into account also the
strong convergences $u_{\varepsilon}\to u$ and $\tilde{v}_{\varepsilon}\to u$
in $L^{1}(a,b)$, we then arrive at

$\left\lVert\gamma_{\varepsilon}-(\tilde{\gamma}_{\varepsilon}+D^{s}\tilde{v}_{\varepsilon})\right\rVert_{\text{flat}}\to
0\quad\text{as }\varepsilon\to 0.$

With $\gamma_{\varepsilon}\to\gamma=D^{s}u+g\mathcal{L}^{1}$ in the flat norm
and Lemma 3.1, we then conclude

$\tilde{\gamma}_{\varepsilon}+D^{s}\tilde{v}_{\varepsilon}\to\gamma\quad\text{in
the flat
norm}\quad\text{and}\quad\tilde{\gamma}_{\varepsilon}+D^{s}\tilde{v}_{\varepsilon}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\gamma\quad\text{in
}\mathcal{M}(a,b).$ (37)

For the latter conclusion we have also used the fact that
$\\{|\tilde{\gamma}_{\varepsilon}+D^{s}\tilde{v}_{\varepsilon}|(a,b)\\}_{\varepsilon}$
with

$|\tilde{\gamma}_{\varepsilon}+D^{s}\tilde{v}_{\varepsilon}|(a,b)=\int_{a}^{b}|\tilde{g}_{\varepsilon}|\,\textup{d}x+|D^{s}\tilde{v}_{\varepsilon}|(a,b)\quad\text{for
every }\varepsilon>0$

is a bounded sequence, which is a consequence of the boundedness of
$\\{\tilde{g}_{\varepsilon}\\}_{\varepsilon}$ in $L^{1}(a,b)$ via (28) and the
estimate (33) (recall also the bound (26) on the energies).

After having discussed the convergence properties of the sequence
$\\{(\tilde{v}_{\varepsilon},\tilde{\gamma}_{\varepsilon})\\}_{\varepsilon}$,
we can finally turn to the proof of the estimate (25). From the definition of
$E_{\varepsilon}$ we obtain via (28) and (33)

$\displaystyle
E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\left(1+3(\sigma+\eta)\right)$
$\displaystyle\geq\int_{a}^{b}|u_{\varepsilon}^{\prime}-g_{\varepsilon}|^{2}\,\textup{d}x+c_{0}\int_{\bigcup\\{I_{\varepsilon}(x_{\alpha})\colon
x_{\alpha}\in\mathscr{G}^{\prime}_{\varepsilon}\\}}|g_{\varepsilon}|\,\textup{d}x+c_{0}|D^{s}\tilde{v}_{\varepsilon}|(a,b)$
$\displaystyle=\int_{a}^{b}|u_{\varepsilon}^{\prime}-g_{\varepsilon}|^{2}\,\textup{d}x+c_{0}\int_{a}^{b}|\tilde{g}_{\varepsilon}|\,\textup{d}x+c_{0}|D^{s}\tilde{v}_{\varepsilon}|(a,b).$

By the choice of the sequence
$\\{(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$ with (26) it
follows that

$\displaystyle E^{\prime}(u,\gamma)\left(1+3(\sigma+\eta)\right)$
$\displaystyle\geq\liminf_{\varepsilon\to
0}\int_{a}^{b}|u_{\varepsilon}^{\prime}-g_{\varepsilon}|^{2}\,\textup{d}x+\liminf_{\varepsilon\to
0}\left[c_{0}\int_{a}^{b}|\tilde{g}_{\varepsilon}|\,\textup{d}x+c_{0}|D^{s}\tilde{v}_{\varepsilon}|(a,b)\right]$
$\displaystyle\geq\int_{a}^{b}|u^{\prime}-g|^{2}\,\textup{d}x+c_{0}\,\int_{a}^{b}{|g|\,\textup{d}x}+c_{0}|D^{s}u|(a,b)$
$\displaystyle\geq\int_{a}^{b}|u^{\prime}-g|^{2}\,\textup{d}x+c_{0}\,\int_{a}^{b}{|g|\,\textup{d}x}+c_{0}|D^{c}u|(a,b).$

Let us comment on the second-last inequality. For the first term we first
deduce from the boundedness of the sequence
$\\{u_{\varepsilon}^{\prime}-g_{\varepsilon}\\}_{\varepsilon}$ in $L^{2}(a,b)$
combined with the convergences $u_{\varepsilon}^{\prime}\mathcal{L}^{1}\to Du$
and $\gamma_{\varepsilon}\to\gamma=D^{s}u+g\mathcal{L}^{1}$ in the flat norm
that $u_{\varepsilon}^{\prime}-g_{\varepsilon}\rightharpoonup u^{\prime}-g$ 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-$\ast$ convergence
$\tilde{\gamma}_{\varepsilon}+D^{s}\tilde{v}_{\varepsilon}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\gamma=D^{s}u+g\mathcal{L}^{1}$
in $\mathcal{M}(a,b)$ from (37) and the lower semicontinuity of the total
variation with respect to weak-$\ast$ convergence. By the arbitrariness of
$\eta>0$ we conclude from the previous inequality the desired estimate (25).

_Step 2: Localization._ We fix an arbitrary $\bar{\sigma}>0$ and consider the
finite set of points $\\{x_{1},\ldots,x_{N-1}\\}\subset J_{u}$ such that
$|[u](x_{i})|>\bar{\sigma}$ for $i=1,\dots,N-1$. Let $x_{0}=a$ and $x_{N}=b$.
Then we have

$\sup_{x\in J_{u}\cap(x_{i},x_{i+1})}|[u](x)|\leq\bar{\sigma}\quad\text{for
every }i\in\\{0,\dots,N-1\\}.$

For every open subinterval $(\alpha,\beta)$ of $(a,b)$ we consider pairs
$(u^{(\alpha,\beta)},\gamma^{(\alpha,\beta)})\in
BV(a,b)\times\mathcal{M}(a,b)$ defined as in Lemma 5.4 as

$\displaystyle u^{(\alpha,\beta)}(x)=\left\\{\begin{array}[]{l
l}u(\alpha+)&\quad x\in(a,\alpha),\\\ u(x)&\quad x\in[\alpha,\beta],\\\
u(\beta-)&\quad x\in(\beta,b),\end{array}\right.$

and $\gamma^{(\alpha,\beta)}=\gamma\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(\alpha,\beta).$ By Lemma 5.4
(with $I=(x_{i},x_{i+1})$) and by Step 1 (applied with $u=u^{(x_{i},x_{i+1})}$
and $\gamma=\gamma^{(x_{i},x_{i+1})})$, we obtain

$\displaystyle E^{\prime}(u,\gamma,(x_{i},x_{i+1}))(1+3\bar{\sigma})$
$\displaystyle\geq
E^{\prime}(u^{(x_{i},x_{i+1})},\gamma^{(x_{i},x_{i+1})})(1+3\bar{\sigma})$
$\displaystyle\geq\int_{x_{i}}^{x_{i+1}}|u^{\prime}-g|^{2}\,\textup{d}x+c_{0}\int_{x_{i}}^{x_{i+1}}|g|\,\textup{d}x+c_{0}|D^{c}u|(x_{i},x_{i+1})$

for every $i\in\\{0,\dots,N-1\\}$. With the superadditivity of $A\mapsto
E^{\prime}(u,\gamma,A)$ from Remark 5.1 (i) we then deduce

$E^{\prime}(u,\gamma)\left(1+3\bar{\sigma}\right)\geq\int_{a}^{b}|u^{\prime}-g|^{2}\,\textup{d}x+c_{0}\int_{a}^{b}|g|\,\textup{d}x+c_{0}|D^{c}u|(a,b),$

which, by the arbitrariness of $\bar{\sigma}>0$, leads to

$\displaystyle E^{\prime}(u,\gamma)$
$\displaystyle\geq\int_{a}^{b}|u^{\prime}-g|^{2}\,\textup{d}x+c_{0}\int_{a}^{b}{|g|\,\textup{d}x}+c_{0}|D^{c}u|(a,b).$

Applying this estimate to $(u^{I},\gamma^{I})$, from Lemma 5.4 we infer that

$\displaystyle E^{\prime}(u,\gamma,I)\geq
E^{\prime}(u^{I},\gamma^{I})\geq\lambda(I)$

for every open interval $I\subset(a,b)$, where $\lambda$ denotes the positive
Borel measure on $(a,b)$ that is given by

$\lambda(B)\coloneqq\int_{B}|u^{\prime}-g|^{2}\,\textup{d}x+c_{0}\,\int_{B}{|g|}\,\textup{d}x+c_{0}|D^{c}u|(B)$

for every Borel subset $B$ of $(a,b)$. Therefore, the claim of the proposition
follows from Remark 5.1 (ii). ∎

###  5.3 Conclusion and proof of the $\Gamma$-$\liminf$ inequalities

For $(u,\gamma)\in BV(a,b)\times\mathcal{M}(a,b)$ with $\left\lVert
u\right\rVert_{L^{\infty}(a,b)}\leq K$, $\gamma=D^{s}u+g\mathcal{L}^{1}$ and
$u^{\prime}-g\in L^{2}(a,b)$ 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:

  1. 1.

$E^{\prime}(u,\gamma,A)\geq\int_{A}|u^{\prime}-g|^{2}\,\textup{d}x+c_{0}\int_{A}|g|\,\textup{d}x,$

  2. 2.

$E^{\prime}(u,\gamma,A)\geq c_{0}|D^{c}u|(A),$

  3. 3.

$E^{\prime}(u,\gamma,A)\geq 2\sum_{x\in J_{u}\cap
A}f\left(\frac{1}{2}|[u](x)|\right),$

for every open subset $A$ of $(a,b)$. These are now combined to prove the
estimate from below of the $\Gamma$-lower limit which shows the first part of
Theorem 2.1.

######  Theorem 5.6.

For every $(u,\gamma)\in BV(a,b)\times\mathcal{M}(a,b)$ with $\left\lVert
u\right\rVert_{L^{\infty}(a,b)}\leq K$, $\gamma=D^{s}u+g\mathcal{L}^{1}$ and
$u^{\prime}-g\in L^{2}(a,b)$ we have

$E^{\prime}(u,\gamma)\geq E(u,\gamma).$

###### Proof.

We consider the Radon measure $\kappa$ defined by

$\kappa(B)\coloneqq\mathcal{L}^{1}(B)+\\#(J_{u}\cap B)+|D^{c}u|(B)$

for every Borel subset $B$ of $(a,b)$. Let $C$ be a Borel subset of
$(a,b)\setminus J_{u}$ with $|C|=0$ such that $|D^{c}u|((a,b)\setminus C)=0$.
Then, we obtain

$E^{\prime}(u,\gamma,A)\geq\int_{A}\psi_{i}\mathop{}\\!\mathrm{d}\kappa$

for $i\in\\{1,2,3\\}$ and for every open set $A\subset(a,b)$, where

$\displaystyle\psi_{1}$
$\displaystyle\coloneqq\big{(}|u^{\prime}-g|^{2}+c_{0}|g|\big{)}\mathbbm{1}_{(a,b)\setminus(J_{u}\cup
C)},$ $\displaystyle\psi_{2}$ $\displaystyle\coloneqq
2f\left(\tfrac{1}{2}|[u]|\right)\mathbbm{1}_{J_{u}},$ $\displaystyle\psi_{3}$
$\displaystyle\coloneqq c_{0}\mathbbm{1}_{C}.$

Next, we define

$\displaystyle\psi(x)\coloneqq\sup_{i}\psi_{i}(x)=\left\\{\begin{array}[]{l
l}|u^{\prime}(x)-g(x)|^{2}+c_{0}|g(x)|&\quad\text{if
}x\in(a,b)\setminus(J_{u}\cup C),\\\
2\,f\left(\frac{1}{2}|[u](x)|\right)&\quad\text{if }x\in J_{u},\\\
c_{0}&\quad\text{if }x\in C.\end{array}\right.$

By a measure theoretic result (see e.g. [9, Lemma 15.2] applied with the set
function $\mu(\cdot)\coloneqq E^{\prime}(u,\gamma,\cdot)$) we conclude that

$E^{\prime}(u,\gamma,A)\geq\int_{A}\sup_{i}\psi_{i}\,\textup{d}\kappa=\int_{A}\psi\mathop{}\\!\mathrm{d}\kappa=E(u,\gamma,A)$

for every open subset $A$ of $(a,b)$. With $A=(a,b)$, this proves the theorem.
∎

The $\Gamma$-$\liminf$ in the case $K=\infty$ is a direct consequence of
Theorem 5.6.

######  Corollary 5.7.

Let $K=\infty$. If $(u_{\varepsilon},\gamma_{\varepsilon})\to(u,\gamma)$ in
$L^{0}((a,b);\overline{{\mathbb{R}}})\times\mathcal{M}(a,b)$, then

$\liminf_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\geq E(u,\gamma).$

###### Proof.

Assuming without loss of generality
$E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})<C_{0}$ for a positive
constant $C_{0}$, by Theorem 4.3 we have that $u\in
BV_{\infty,\mathcal{P}}(a,b)$ and there is a partition
$a=x_{0}<x_{1}\ldots<x_{m}=b$ such that
$J_{u}\subset\\{x_{1},\ldots,x_{m-1}\\}\cup\mathcal{F}(u)$ and for
$A_{\delta}\coloneqq\mathcal{F}(u)\cap\bigcup_{i}(x_{i-1}+\delta,x_{i}-\delta)$,
with $\delta>0$, we have
$\chi_{A_{\delta}}u_{\varepsilon}\to\chi_{A_{\delta}}u$ in $L^{1}(a,b)$ and
$\gamma_{\varepsilon}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}A_{\delta}\to\gamma\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}A_{\delta}$ in the flat norm.
From Remark 4.4 and Theorem 5.6 (applied for a suitable $K$) we then get

$\liminf_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\geq
2c_{0}(m-1)+E(u,\gamma,A_{\delta}).$

The assertion follows in the limit $\delta\searrow 0$ from the monotone
convergence theorem. ∎

##  6 Estimate from above of the $\Gamma$-upper limit

We now turn to the estimate from above of the $\Gamma$-upper limit
$E^{\prime\prime}$. Except for the very last paragraph we assume $K<\infty$ in
the whole section. We again restrict ourselves to pairs $(u,\gamma)\in
BV(a,b)\times\mathcal{M}(a,b)$ with $\left\lVert
u\right\rVert_{{L^{\infty}}(a,b)}\leq K$, $\gamma=D^{s}u+g\mathcal{L}^{1}$ and
$u^{\prime}-g\in L^{2}(a,b)$ since the estimates are trivial otherwise. We
first show the result for the particular case $u\in SBV^{2}(a,b)$ and then
deduce the general result by approximation.

######  Proposition 6.1.

For every $(u,\gamma)\in SBV^{2}(a,b)\times\mathcal{M}(a,b)$ with $\left\lVert
u\right\rVert_{L^{\infty}(a,b)}\leq K$, $\gamma=D^{s}u+g\mathcal{L}^{1}$ and
$u^{\prime}-g\in L^{2}(a,b)$ we have

$E^{\prime\prime}(u,\gamma)\leq E(u,\gamma).$

###### Proof.

Since $u\in SBV^{2}(a,b)\cap L^{\infty}(a,b)$, the jump set is finite, i.e.,
$J_{u}=\\{x_{1},\ldots,x_{N-1}\\}$ for some $N\in{\mathbb{N}}$, and we may
further assume by the Sobolev embedding theorem that $u$ is a piecewise
continuous function with one-sided limits $u(x\pm)$ for all $x\in(a,b)$. Thus,
$\gamma$ is of the form

$\gamma=\sum_{i=1}^{N-1}[u](x_{i})\delta_{x_{i}}+g\mathcal{L}^{1},$

with $g\in L^{2}(a,b)$. Let $x_{0}=a$ and $x_{N}=b$. We then choose
$\varepsilon$ small enough such that

$|x_{i+1}-x_{i}|>2\varepsilon^{2}+4\varepsilon\quad\text{for every
}i\in\\{0,\dots,N-1\\}.$

We first define $u_{\varepsilon}\in W^{1,1}(a,b)$ nearby the jumps of $u$ by
linear interpolation via

$\begin{array}[]{lll}&u_{\varepsilon}\coloneqq u&\text{on
}(a,b)\setminus\bigcup_{i=1}^{N-1}(x_{i}-\varepsilon^{2}-2\varepsilon,x_{i}+\varepsilon^{2}+2\varepsilon),\\\
&u^{\prime}_{\varepsilon}\coloneqq\frac{u(x_{i}+\varepsilon^{2}+2\varepsilon)-u(x_{i}-\varepsilon^{2}-2\varepsilon)}{2\varepsilon^{2}}&\text{on
}(x_{i}-\varepsilon^{2},x_{i}+\varepsilon^{2})\text{ for
}i\in\\{1,\ldots,N-1\\},\\\\[2.84544pt] &u^{\prime}_{\varepsilon}\coloneqq
0&\text{otherwise}\end{array}$

(see the figure below). By construction, we have $\left\lVert
u_{\varepsilon}\right\rVert_{L^{\infty}(a,b)}\leq K$ for all $\varepsilon$,
and taking advantage of the fact that $u$ is only modified on the intervals
$(x_{i}-\varepsilon^{2}-2\varepsilon,x_{i}+\varepsilon^{2}+2\varepsilon)$ for
$i\in\\{1,\ldots,N-1\\}$, we also have

$\left\lVert
u_{\varepsilon}-u\right\rVert_{L^{1}(a,b)}\leq\int_{\bigcup_{i=1}^{N-1}(x_{i}-\varepsilon^{2}-2\varepsilon,x_{i}+\varepsilon^{2}+2\varepsilon)}|u_{\varepsilon}-u|\,\textup{d}x\leq
4K(N-1)(\varepsilon^{2}+2\varepsilon),$

which shows strong convergence of $\\{u_{\varepsilon}\\}_{\varepsilon}$ to $u$
in $L^{1}(a,b)$.

$x$$u$$x$$u_{\varepsilon}$$2\varepsilon^{2}$$2\varepsilon^{2}$$x$$g_{\varepsilon}$$2\varepsilon^{2}+4\varepsilon$$2\varepsilon^{2}+4\varepsilon$$\gamma^{a}$$\frac{u(x_{i}+\varepsilon^{2}+2\varepsilon)-u(x_{i}-\varepsilon^{2}-2\varepsilon)}{2\varepsilon^{2}}$
Fig.: Construction of the recovery sequence for a piecewise affine function
with jump discontinuities.

We next define $\gamma_{\varepsilon}\in\mathcal{M}(a,b)$ as
$\gamma_{\varepsilon}=g_{\varepsilon}\mathcal{L}^{1}$, where
$g_{\varepsilon}\in L^{2}(a,b)$ is given by

$g_{\varepsilon}:=\begin{cases}g&\text{on
}(a,b)\setminus\bigcup_{i=1}^{N-1}(x_{i}-\varepsilon^{2}-2\varepsilon,x_{i}+\varepsilon^{2}+2\varepsilon),\\\
u_{\varepsilon}^{\prime}&\text{on
}(x_{i}-\varepsilon^{2},x_{i}+\varepsilon^{2})\text{ for
}i\in\\{1,\ldots,N-1\\},\\\ 0&\text{otherwise}.\end{cases}$

We notice that

$\gamma_{\varepsilon}-\gamma=\sum_{i=1}^{N-1}\Big{(}u_{\varepsilon}^{\prime}\mathbbm{1}_{(x_{i}-\varepsilon^{2},x_{i}+\varepsilon^{2})}\mathcal{L}^{1}-[u](x_{i})\delta_{x_{i}}-g\mathbbm{1}_{(x_{i}-\varepsilon^{2}-2\varepsilon,x_{i}+\varepsilon^{2}+2\varepsilon)}\mathcal{L}^{1}\Big{)}.$

Therefore, we observe from the definition of $u_{\varepsilon}^{\prime}$ that
for every function $\varphi\in W^{1,\infty}_{0}(a,b)$ with
$\left\lVert\varphi\right\rVert_{W^{1,\infty}_{0}(a,b)}\leq 1$ there holds

$\displaystyle\int_{x_{i}-\varepsilon^{2}-2\varepsilon}^{x_{i}+\varepsilon^{2}+2\varepsilon}\varphi\mathop{}\\!\mathrm{d}(\gamma_{\varepsilon}-\gamma)$
$\displaystyle=\big{[}u(x_{i}+\varepsilon^{2}+2\varepsilon)-u(x_{i}-\varepsilon^{2}-2\varepsilon)\big{]}\bigg{(}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{x_{i}-\varepsilon^{2}}^{x_{i}+\varepsilon^{2}}\varphi\mathop{}\\!\mathrm{d}x-\varphi(x_{i})\bigg{)}$
$\displaystyle\quad+\big{[}u(x_{i}+\varepsilon^{2}+2\varepsilon)-u(x_{i}+)-u(x_{i}-\varepsilon^{2}-2\varepsilon)+u(x_{i}-)\big{]}\varphi(x_{i})$
$\displaystyle\quad-\int_{x_{i}-\varepsilon^{2}-2\varepsilon}^{x_{i}+\varepsilon^{2}+2\varepsilon}\varphi
g\mathop{}\\!\mathrm{d}x$

for every $i\in\\{1,\ldots,N-1\\}$. Since we also have
$|\varphi(x)-\varphi(x_{i})|\leq|x-x_{i}|\leq\varepsilon^{2}$ on
$(x_{i}-\varepsilon^{2},x_{i}+\varepsilon^{2})$, we deduce from the bound
$\left\lVert u\right\rVert_{L^{\infty}(a,b)}\leq K$ and the Cauchy–Schwarz
inequality

$\displaystyle\left\lVert\gamma_{\varepsilon}-\gamma\right\rVert_{\text{flat}}$
$\displaystyle\leq 2K(N-1)\varepsilon^{2}$
$\displaystyle\quad+\sum_{i=1}^{N-1}\big{[}|u(x_{i}+\varepsilon^{2}+2\varepsilon)-u(x_{i}+)|+|u(x_{i}-\varepsilon^{2}-2\varepsilon)-u(x_{i}-)|\big{]}$
$\displaystyle\quad+\left\lVert
g\right\rVert_{L^{2}(a,b)}\big{[}2(N-1)(\varepsilon^{2}+2\varepsilon)\big{]}^{\frac{1}{2}}.$

This shows the convergence of $\\{\gamma_{\varepsilon}\\}_{\varepsilon}$ to
$\gamma$ in the flat norm. It only remains to establish the energy estimate.
From the construction of $(u_{\varepsilon},\gamma_{\varepsilon})$, we clearly
have $|u^{\prime}_{\varepsilon}-g_{\varepsilon}|\leq|u^{\prime}-g|$ on
$(a,b)$. Therefore, the elastic energy contribution in
$E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})$ is estimated by

$\int_{a}^{b}|u^{\prime}_{\varepsilon}-g_{\varepsilon}|^{2}\,\textup{d}x\leq\int_{a}^{b}|u^{\prime}-g|^{2}\,\textup{d}x.$

Due to the monotonicity of $f$ and $f(t)\leq c_{0}t$ for all $t\geq 0$, we
estimate the non-local energy term by

$\displaystyle\frac{1}{\varepsilon}\int_{a}^{b}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|g_{\varepsilon}|\,\textup{d}t}\bigg{)}\,\textup{d}x$
$\displaystyle\leq\frac{1}{\varepsilon}\int_{(a,b)\setminus\bigcup_{i=1}^{N-1}(x_{i}-\varepsilon^{2}-\varepsilon,x_{i}+\varepsilon^{2}+\varepsilon)}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}|g|\,\textup{d}t\bigg{)}\,\textup{d}x$
$\displaystyle\quad+\frac{1}{\varepsilon}\int_{\bigcup_{i=1}^{N-1}(x_{i}-\varepsilon^{2}-\varepsilon,x_{i}+\varepsilon^{2}+\varepsilon)}f\bigg{(}\frac{1}{2}\int_{(x_{i}-\varepsilon^{2},x_{i}+\varepsilon^{2})}|u_{\varepsilon}^{\prime}|\,\textup{d}t\bigg{)}\,\textup{d}x$
$\displaystyle\leq c_{0}\int_{a}^{b}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}|g|\,\textup{d}t\,\textup{d}x$
$\displaystyle\quad+\frac{2(\varepsilon^{2}+\varepsilon)}{\varepsilon}\sum_{i=1}^{N-1}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{x_{i}-\varepsilon^{2}-\varepsilon}^{x_{i}+\varepsilon^{2}+\varepsilon}{f\bigg{(}\frac{1}{2}|u(x_{i}+\varepsilon^{2}+2\varepsilon)-u(x_{i}-\varepsilon^{2}-2\varepsilon)|\bigg{)}}\,\textup{d}x.$

With the continuity of $u$ outside of the jump set $J_{u}$ we can pass to the
limit $\varepsilon\to 0$ on the right-hand side. In this way, we finally
arrive at

$\displaystyle E^{\prime\prime}(u,\gamma)$
$\displaystyle\leq\limsup_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})$
$\displaystyle\leq\int_{a}^{b}|u^{\prime}-g|^{2}\,\textup{d}x+c_{0}\int_{a}^{b}|g|\,\textup{d}x+2\sum_{i=1}^{N-1}f\bigg{(}\frac{1}{2}|[u](x_{i})|\bigg{)}$
$\displaystyle=E(u,\gamma).\qed$

######  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
$u_{\varepsilon}$ a function $v_{\varepsilon}\in W^{1,p}_{0}(a,b)$ such that
$v_{\varepsilon}\to 0$ in $L^{p}(a,b)$ and

$\limsup_{\varepsilon\to
0}\int_{a}^{b}W(u^{\prime}_{\varepsilon}+v^{\prime}_{\varepsilon}-g_{\varepsilon})\,\textup{d}x\leq\int_{a}^{b}W^{**}(u^{\prime}-g)\,\textup{d}x.$

So also in this case we have
$E^{\prime\prime}(u,\gamma)\leq\limsup_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon}+v_{\varepsilon},\gamma_{\varepsilon})\leq
E(u,\gamma)$.

By approximation with $SBV^{2}$-functions, we can now give the proof of the
second part of Theorem 2.1.

######  Theorem 6.3.

For every $(u,\gamma)\in BV(a,b)\times\mathcal{M}(a,b)$ with $\left\lVert
u\right\rVert_{L^{\infty}(a,b)}\leq K$, $\gamma=D^{s}u+g\mathcal{L}^{1}$ and
$u^{\prime}-g\in L^{2}(a,b)$ we have

$E^{\prime\prime}(u,\gamma)\leq E(u,\gamma).$

###### Proof.

We here want to construct a sequence
$\\{(\hat{u}_{h},\hat{\gamma}_{h})\\}_{h}$ in
$SBV^{2}(a,b)\times\mathcal{M}(a,b)$ (with
$\left\lVert\hat{u}_{h}\right\rVert_{L^{\infty}(a,b)}\leq K$,
$\hat{\gamma}_{h}=D^{s}\hat{u}_{h}+\hat{g}_{h}\mathcal{L}^{1}$ and
$\hat{u}_{h}^{\prime}-\hat{g}_{h}\in L^{2}(a,b)$ for every $h>0$) such that
$\hat{u}_{h}\to u$ in $L^{1}(a,b)$, $\hat{\gamma}_{h}\to\gamma$ in the flat
norm and

$\liminf_{h\to 0}E(\hat{u}_{h},\hat{\gamma}_{h})\leq E(u,\gamma).$ (38)

This is indeed sufficient since by lower semi-continuity of the $\Gamma$-upper
limit $E^{\prime\prime}$ and by Proposition 6.1 we then conclude with

$E^{\prime\prime}(u,\gamma)\leq\liminf_{h\to
0}E^{\prime\prime}(\hat{u}_{h},\hat{\gamma}_{h})\leq\liminf_{h\to
0}E(\hat{u}_{h},\hat{\gamma}_{h})\leq E(u,\gamma).$

We first recall that in dimension one every function $u\in BV(a,b)$ can be
represented as $u_{a}+u_{j}+u_{c}$, see (7) where $u_{a}\in W^{1,1}(a,b)$,
$u_{j}$ is a pure jump function and $u_{c}$ is a Cantor function. This allows
us to modify the three parts of $u$ separately. We start with the jump
function $u_{j}$. We define $u_{j,h}$ by

$u_{j,h}(x):=u_{j}(a+)+\sum_{y\in
J_{u_{j}}(h)\cap(a,x]}[u_{j}](y)\quad\text{for }x\in(a,b),$

where $J_{u_{j}}(h)\coloneqq\\{y\in
J_{u_{j}}\colon|[u_{j}](y)|>h\\}=J_{u_{j,h}}$. We observe
$\\#J_{u_{j,h}}<\infty$, $u_{j,h}\to u_{j}$ in $L^{1}(a,b)$ as $h\to 0$, and
for all $h$ the estimate

$\sum_{x\in J_{u_{j,h}}}f\left(\frac{1}{2}|[u_{j,h}](x)|\right)\leq\sum_{x\in
J_{u_{j}}}f\left(\frac{1}{2}|[u_{j}](x)|\right).$ (39)

For the Cantor function $u_{c}$ we use the density of smooth functions in
$BV(a,b)$ with respect to the strict topology. In this way, we find a sequence
$\\{u_{c,h}\\}_{h}$ in $W^{1,1}(a,b)$ with $u_{c,h}\to u_{c}$ in $L^{1}(a,b)$
and

$\displaystyle\lim_{h\to
0}\int_{a}^{b}|u^{\prime}_{c,h}|\,\textup{d}x=|Du_{c}|(a,b).$ (40)

The absolutely continuous part $u_{a}$ is first extended to a
$W^{1,1}({\mathbb{R}})$ function with compact support and we then set
$u_{a,h}\coloneqq u_{a}*\psi_{h}$ for all $h>0$, where $\psi_{h}$ is a
standard $h$-mollifier given by $\psi_{h}(x)\coloneqq h^{-1}\psi(h^{-1}x)$ for
$x\in{\mathbb{R}}$, for a fixed non-negative, symmetric function $\psi\in
C^{\infty}({\mathbb{R}})$ with compact support and normalized to
$\int_{\mathbb{R}}\psi\,\textup{d}x=1$. We then have $u_{a,h}\in
C^{\infty}({\mathbb{R}})$ for all $h>0$, $u_{a,h}\to u_{a}$ in $L^{1}(a,b)$
and $u_{a,h}^{\prime}=u_{a}^{\prime}*\psi_{h}$, see e.g. [20, Theorem 4.2.1].
Then we set

$u_{h}\coloneqq u_{a,h}+u_{j,h}+u_{c,h}\quad\text{for all }h>0.$

We clearly have $\\{u_{h}\\}_{h}$ in $SBV^{2}(a,b)$ for all $h>0$ and
$u_{h}\to u$ in $L^{1}(a,b)$, which implies $Du_{h}\to Du$ in the flat norm.

We next address the modification of $\gamma$. We extend the absolutely
continuous part $g$ outside of $(a,b)$ by $0$ and set $g_{a,h}\coloneqq
g*\psi_{h}$ for all $h>0$. Then we have $g_{a,h}\in C^{\infty}({\mathbb{R}})$
for all $h>0$ and

$\displaystyle g_{a,h}\to g\text{ in }L^{1}(a,b).$ (41)

Because of $u_{a}^{\prime}-g\in L^{2}(a,b)$ and
$u_{a,h}^{\prime}-g_{a,h}=(u_{a}^{\prime}-g)*\psi_{h}$ we further notice

$\displaystyle u_{a,h}^{\prime}-g_{a,h}\to u_{a}^{\prime}-g\text{ in
}L^{2}(a,b).$ (42)

Now, we set

$\gamma_{h}:=Du_{j,h}+g_{h}\mathcal{L}^{1}\quad\text{with }g_{h}\coloneqq
g_{a,h}+u_{c,h}^{\prime}\quad\text{for all }h>0.$

With the convergences $Du_{h}\to Du$ and $g_{a,h}-u_{a,h}^{\prime}\to
g-u_{a}^{\prime}$ in the flat norm (via (42)), we infer $\gamma_{h}\to\gamma$
in the flat norm. Since $u_{c,h}^{\prime}$ is canceled in the first term, we
deduce

$\int_{a}^{b}|u_{h}^{\prime}-g_{h}|^{2}\,\textup{d}x+c_{0}\int_{a}^{b}|g_{h}|\,\textup{d}x+2\sum_{x\in
J_{u_{h}}}f\left(\frac{1}{2}|[u_{h}](x)|\right)\\\
\leq\int_{a}^{b}|u_{a,h}^{\prime}-g_{a,h}|^{2}\,\textup{d}x+c_{0}\int_{a}^{b}\big{(}|g_{a,h}|+|u_{c,h}^{\prime}|\big{)}\,\textup{d}x+2\sum_{x\in
J_{u_{j,h}}}f\left(\frac{1}{2}|u_{j,h}](x)|\right),$

which, via (39), (40), (41) and (42), implies

$\limsup_{h\to\infty}\bigg{[}\int_{a}^{b}|u_{h}^{\prime}-g_{h}|^{2}\,\textup{d}x+c_{0}\int_{a}^{b}|g_{h}|\,\textup{d}x+2\sum_{x\in
J_{u_{h}}}f\left(\frac{1}{2}|[u_{h}](x)|\right)\bigg{]}\leq E(u,\gamma).$ (43)

This does not yet show (38), since $\left\lVert
u_{h}\right\rVert_{L^{\infty}(a,b)}\leq K$ might not be satisfied for all
$h>0$. We resolve this problem in two steps. With $\left\lVert
u\right\rVert_{L^{\infty}(a,b)}\leq K$ and $u_{h}\to u$ in $L^{1}(a,b)$, we
can fix a sequence $\\{\eta_{h}\\}_{h}$ in ${\mathbb{R}}^{+}$ with
$\eta_{h}\to 0^{+}$ as $h\to 0$ and

$|\\{x\in(a,b)\colon|u_{h}(x)|\geq K+\eta_{h}\\}|\to 0\quad\text{as }\,h\to
0.$ (44)

We next define the truncated versions

$\tilde{u}_{h}(x)\coloneqq\min\\{\max\\{u_{h}(x),-K-\eta_{h}\\},K+\eta_{h}\\}\quad\text{for
all }h>0.$

We then have $\tilde{u}_{h}\to u$ in $L^{1}(a,b)$, $D\tilde{u}_{h}\to Du$ in
the flat norm and, in addition, also
$\left\lVert\tilde{u}_{h}\right\rVert_{L^{\infty}(a,b)}\leq K+\eta_{h}$ for
all $h>0$. Correspondingly, we set

$\tilde{\gamma}_{h}\coloneqq
D^{j}\tilde{u}_{h}+g_{h}\mathbbm{1}_{\\{\tilde{u}_{h}=u_{h}\\}}\mathcal{L}^{1}\quad\text{for
all }h>0.$

By using (9) and by applying subsequently the Cauchy–Schwarz inequality, we
get

$\displaystyle\left\lVert\tilde{\gamma}_{h}-\gamma_{h}\right\rVert_{\textnormal{flat}}$
$\displaystyle\leq\left\lVert(\tilde{\gamma}_{h}-D\tilde{u}_{h})-(\gamma_{h}-Du_{h})\right\rVert_{\textnormal{flat}}+\left\lVert
D\tilde{u}_{h}-Du_{h}\right\rVert_{\textnormal{flat}}$
$\displaystyle=\left\lVert(g_{h}-u_{h}^{\prime})\mathbbm{1}_{\\{\tilde{u}_{h}\neq
u_{h}\\}}\mathcal{L}^{1}\right\rVert_{\textnormal{flat}}+\left\lVert
D\tilde{u}_{h}-Du_{h}\right\rVert_{\textnormal{flat}}$
$\displaystyle=\left\lVert(g_{a,h}-u_{a,h}^{\prime})\mathbbm{1}_{\\{\tilde{u}_{h}\neq
u_{h}\\}}\mathcal{L}^{1}\right\rVert_{\textnormal{flat}}+\left\lVert
D\tilde{u}_{h}-Du_{h}\right\rVert_{\textnormal{flat}}$
$\displaystyle\leq\left\lVert(g_{a,h}-u_{a,h}^{\prime})\mathbbm{1}_{\\{\tilde{u}_{h}\neq
u_{h}\\}}\right\rVert_{L^{1}(a,b)}+\left\lVert\tilde{u}_{h}-u_{h}\right\rVert_{L^{1}(a,b)}$
$\displaystyle\leq\left\lVert
g_{a,h}-u_{a,h}^{\prime}\right\rVert_{L^{2}(a,b)}|\\{x\in(a,b)\colon\tilde{u}_{h}\neq
u_{h}\\}|^{\frac{1}{2}}+\left\lVert\tilde{u}_{h}-u_{h}\right\rVert_{L^{1}(a,b)}.$

If we pass to the limit $h\to 0$ on the right-hand side, the first term
vanishes because of the uniform boundedness of
$u_{h}^{\prime}-g_{h}=u_{a,h}^{\prime}-g_{a,h}$ in $L^{2}(a,b)$ due to (42)
combined with the convergence

$\displaystyle\\{x\in(a,b)\colon\tilde{u}_{h}(x)\neq
u_{h}\\}=\\{x\in(a,b)\colon|u_{h}(x)|>K+\eta_{h}\\}\to 0\quad\text{as }h\to 0$

as a consequence from (44). Since with $u_{h}\to u$ and $\tilde{u}_{h}\to u$
in $L^{1}(a,b)$ also the second term vanishes, we conclude that
$\tilde{\gamma}_{h}-\gamma_{h}\to 0$ in the flat norm. Consequently, we have
established $\tilde{\gamma}_{h}\to\gamma$ in the flat norm. For $h>0$ we
finally define

$\displaystyle\hat{u}_{h}(x)\coloneqq\frac{K}{K+\eta_{h}}\tilde{u}_{h}\quad\text{and}\quad\hat{\gamma}_{h}(x)\coloneqq\frac{K}{K+\eta_{h}}\tilde{\gamma}_{h}(x)\quad\text{for
}x\in(a,b).$

We clearly have $\left\lVert\hat{u}_{h}\right\rVert_{L^{\infty}(a,b)}\leq K$,
$J_{u_{h}}\subset J_{\hat{u}_{h}}$ and $|[\hat{u}_{h}]|\leq|[u_{h}]|$ for all
$h>0$. In view of $K/(K+\eta_{h})\to 1$, we also have $\hat{u}_{h}\to u$ in
$L^{1}(a,b)$ and $\hat{\gamma}_{h}\to\gamma$ in the flat norm. Moreover, if we
denote the density of $\hat{\gamma}_{h}$ with respect to $\mathcal{L}^{1}$ by
$\hat{g}_{h}$, we observe
$|\hat{u}_{h}^{\prime}-\hat{g}_{h}|\leq|u_{h}^{\prime}-g_{h}|$ and
$|\hat{g}_{h}|\leq|g_{h}|$ on $(a,b)$ for all $h>0$. This shows, that the
energy $E(\hat{u}_{h},\hat{\gamma}_{h})$ is finite for all $h>0$, with

$\displaystyle E(\hat{u}_{h},\hat{\gamma}_{h})$
$\displaystyle=\int_{a}^{b}|\hat{u}_{h}^{\prime}-\hat{g}_{h}|^{2}\,\textup{d}x+c_{0}\int_{a}^{b}|\hat{g}_{h}|\,\textup{d}x+2\sum_{x\in
J_{\hat{u}_{h}}}f\left(\frac{1}{2}|[\hat{u}_{h}](x)|\right)$
$\displaystyle\leq\int_{a}^{b}|u_{h}^{\prime}-g_{h}|^{2}\,\textup{d}x+c_{0}\int_{a}^{b}|g_{h}|\,\textup{d}x+2\sum_{x\in
J_{u_{h}}}f\left(\frac{1}{2}|[u_{h}](x)|\right).$

By taking into account (43), we then obtain the claim (38) (even for the
$\limsup$), which ends the proof. ∎

######  Remark 6.4.

The function $\hat{u}_{h}$ can even be chosen such that $\hat{u}_{h}(x)=u(a+)$
on $(a,\delta_{h})$ and $\hat{u}_{h}(x)=u(b-)$ on $(b-\delta_{h},b)$ for a
sequence $\delta_{h}\searrow 0$. To see this, note that from $\hat{u}_{h}\to
u$ a.e. and $\lim_{x\searrow a}u(x)=u(a+)$, $\lim_{x\nearrow b}u(x)=u(b-)$,
one finds $\delta_{h}\searrow 0$ such that $a+\delta_{h}$ and $b-\delta_{h}$
are not contained in $J_{u_{h}}$ and
$\lim_{h\to\infty}u_{h}(a+\delta_{h})=u(a+)$,
$\lim_{h\to\infty}u_{h}(b-\delta_{h})=u(b-)$. Now consider
$\hat{\hat{u}}_{h}\in SBV^{2}(a,b)$ defined by

$\hat{\hat{u}}_{h}(x)=\begin{cases}u(a+)&\text{for }x\in(a,a+\delta_{h}],\\\
\hat{u}_{h}(x)&\text{for }x\in(a+\delta_{h},b-\delta_{h}),\\\ u(b-)&\text{for
}x\in[b-\delta_{h},b)\\\ \end{cases}$

and

$\hat{\hat{\gamma}}_{h}=\hat{\gamma}_{h}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(a+\delta_{h},b-\delta_{h})+\big{(}u_{h}(a+\delta_{h})-u(a+)\big{)}\delta_{a+\delta_{h}}+\big{(}u(b-)-u_{h}(b-\delta_{h})\big{)}\delta_{b-\delta_{h}}$

The claim follows from observing that

$E(\hat{\hat{u}}_{h},\hat{\hat{\gamma}}_{h})\leq
E(\hat{u}_{h},\hat{\gamma}_{h})+2c_{0}|u_{h}(a+\delta_{h})-u(a+)|+2c_{0}|u(b-)-u_{h}(b-\delta_{h})|$

where the last two terms on the right-hand side vanish as $h\to\infty$ by
construction.

Again, the case $K=\infty$ is a direct consequence.

######  Corollary 6.5.

For every $(u,\gamma)\in
L^{0}((a,b);\overline{{\mathbb{R}}})\times\mathcal{M}(a,b)$ there is a
sequence $\\{(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$
converging to $(u,\gamma)$ in
$L^{0}((a,b);\overline{{\mathbb{R}}})\times\mathcal{M}(a,b)$ such that

$\limsup_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\leq E(u,\gamma).$

###### Proof.

Without loss of generality we assume $u\in BV_{\infty,\mathcal{P}}(a,b)$. Let
$a=x_{0}<\ldots<x_{m}=b$ such that
$\\{x_{1},\ldots,x_{m}\\}=\\{x\in(a,b):|[u](x)|=\infty\\}$. Note
$\|u\|_{L^{\infty}(\mathcal{F}(u))}<\infty$. With the help of Theorem 6.3 we
choose $\tilde{u}_{\varepsilon},\tilde{g}_{\varepsilon}:(a,b)\to{\mathbb{R}}$
such that
$\big{(}\tilde{u}_{\varepsilon}|_{(x_{i-1},x_{i})},\tilde{g}_{\varepsilon}\mathcal{L}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(x_{i-1},x_{i})\big{)}$ is a
recovery sequence for $\big{(}u|_{(x_{i-1},x_{i})},\gamma\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(x_{i-1},x_{i})\big{)}$ whenever
$(x_{i-1},x_{i})\subset\mathcal{F}(u)$ and
$\tilde{u}_{\varepsilon}\equiv\pm\varepsilon^{-1}$,
$\tilde{g}_{\varepsilon}\equiv 0$ on $(x_{i-1},x_{i})$ in case
$u\equiv\pm\infty$ on $(x_{i-1},x_{i})$.

As in the proof of Proposition 6.1 now define $u_{\varepsilon}\in
W^{1,1}(a,b)$ by linear interpolation near the $x_{i}$ as

$\begin{array}[]{lll}&u_{\varepsilon}\coloneqq\tilde{u}_{\varepsilon}&\text{on
}(a,b)\setminus\bigcup_{i=1}^{m-1}(x_{i}-\varepsilon^{2}-2\varepsilon,x_{i}+\varepsilon^{2}+2\varepsilon),\\\
&u^{\prime}_{\varepsilon}\coloneqq\frac{\tilde{u}_{\varepsilon}(x_{i}+\varepsilon^{2}+2\varepsilon)-\tilde{u}_{\varepsilon}(x_{i}-\varepsilon^{2}-2\varepsilon)}{2\varepsilon^{2}}&\text{on
}(x_{i}-\varepsilon^{2},x_{i}+\varepsilon^{2})\text{ for
}i\in\\{1,\ldots,m-1\\},\\\\[2.84544pt] &u^{\prime}_{\varepsilon}\coloneqq
0&\text{otherwise}.\end{array}$

Clearly, $u_{\varepsilon}\to u$ a.e. on $(a,b)$. Accordingly we define
$\gamma_{\varepsilon}\in\mathcal{M}(a,b)$ as
$\gamma_{\varepsilon}=g_{\varepsilon}\mathcal{L}^{1}$ with

$g_{\varepsilon}:=\begin{cases}\tilde{g}_{\varepsilon}&\text{on
}(a,b)\setminus\bigcup_{i=1}^{m-1}(x_{i}-\varepsilon^{2}-2\varepsilon,x_{i}+\varepsilon^{2}+2\varepsilon),\\\
u_{\varepsilon}^{\prime}&\text{on
}(x_{i}-\varepsilon^{2},x_{i}+\varepsilon^{2})\text{ for
}i\in\\{1,\ldots,m-1\\},\\\ 0&\text{otherwise}.\end{cases}$

So still $\gamma_{\varepsilon}\to\gamma$ locally in the flat norm on each
$(x_{i-1},x_{i})\subset\mathcal{F}(u)$. Estimating the energy as

$\displaystyle E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})$
$\displaystyle\leq\int_{\mathcal{F}(u)}|\tilde{u}_{\varepsilon}^{\prime}-\tilde{g}_{\varepsilon}|^{2}\,\textup{d}x+\frac{1}{\varepsilon}\int_{\mathcal{F}(u)}f\bigg{(}\varepsilon\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I_{\varepsilon}(x)\cap(a,b)}{|\tilde{g}_{\varepsilon}|\,\textup{d}t}\bigg{)}\,\textup{d}x$
$\displaystyle\quad+\frac{1}{\varepsilon}\int_{\bigcup_{i=1}^{m-1}(x_{i}-\varepsilon^{2}-\varepsilon,x_{i}+\varepsilon^{2}+\varepsilon)}f\bigg{(}\frac{1}{2}\int_{(x_{i}-\varepsilon^{2},x_{i}+\varepsilon^{2})}|u_{\varepsilon}^{\prime}|\,\textup{d}t\bigg{)}\,\textup{d}x,$

where the last term on the right-hand side is bounded by
$2(1+\varepsilon)(m-1)c_{0}=2(1+\varepsilon)c_{0}\\#J_{u}\setminus\mathcal{F}(u)$,
we find that indeed

$\limsup_{\varepsilon\to\infty}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\leq
E(u,\gamma).\qed$

##  7 $\Gamma$-convergence for the minimal energies with respect to $\gamma$

We finally prove the $\Gamma$-convergence result in Corollary 2.8 for the
minimal energies with respect to the second variable $\gamma$, i.e., we
consider the energies $\tilde{E}_{\varepsilon}$ and $\tilde{E}$ from (5) and
(6), respectively. We only treat the case $K<\infty$, the necessary
modifications for $K=\infty$ are straightforward. Notice that, as a direct
consequence of the fact that the function $g^{\ast}$ from (3) solves the
optimization problem in (2), for every $u\in BV(a,b)$ there holds

$\tilde{E}(u)=E(u,D^{s}u+g^{\ast}\mathcal{L}^{1})=E(u,\gamma^{\textup{opt}}).$
(45)

For completeness we state also the corresponding compactness result.

######  Corollary 7.1 (Compactness of the minimal energies with respect to
$\gamma$).

Let $\\{u_{\varepsilon}\\}_{\varepsilon}$ be a sequence in ${L^{1}}(a,b)$ with

$\tilde{E}_{\varepsilon}(u_{\varepsilon})\leq C_{0}$

for a positive constant $C_{0}$ and all $\varepsilon>0$. There exists a
function $u\in BV(a,b)$ with $\left\lVert
u\right\rVert_{{L^{\infty}}(a,b)}\leq K$ such that, up to a subsequence,
$\\{u_{\varepsilon}\\}_{\varepsilon}$ converges to $u$ in ${L^{1}}(a,b)$.

###### Proof.

We choose a low energy sequence $\\{\gamma_{\varepsilon}\\}_{\varepsilon}$ in
$\mathcal{M}(a,b)$ with
$E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\leq\tilde{E}_{\varepsilon}(u_{\varepsilon})+1$
for all $\varepsilon$. Since there holds
$E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\leq C_{0}+1$ for all
$\varepsilon$, according to Theorem 4.2 there exists a function $u\in BV(a,b)$
with $\left\lVert u\right\rVert_{{L^{\infty}}(a,b)}\leq K$ such that
$u_{\varepsilon}\to u$ in $L^{1}(a,b)$. ∎

###### Proof of Corollary 2.8.

It is again sufficient to establish the $\Gamma$-$\liminf$ inequality and the
${\Gamma\text{-}\limsup}$-inequality only for $u\in BV(a,b)$ with $\left\lVert
u\right\rVert_{L^{\infty}(a,b)}\leq K$ since the estimates are trivial
otherwise.

We first show the $\Gamma$-$\liminf$ inequality. We consider an arbitrary
sequence $\\{u_{\varepsilon}\\}_{\varepsilon}$ in $L^{1}(a,b)$ with
$u_{\varepsilon}\to u$ in $L^{1}(a,b)$, for which we may assume
$\tilde{E}_{\varepsilon}(u_{\varepsilon})\leq C_{0}$ for some positive
constant $C_{0}$ and all $\varepsilon$. We then select a low energy sequence
$\\{\gamma_{\varepsilon}\\}_{\varepsilon}$ in $\mathcal{M}(a,b)$ with

$E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\leq\tilde{E}_{\varepsilon}(u_{\varepsilon})+\varepsilon\quad\text{
for every }\varepsilon>0.$

By passing to a subsequence if necessary, we may assume that

$\liminf_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})=\lim_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon}).$

At this stage we employ the compactness result of Theorem 4.2: since
$u_{\varepsilon}\to u$ in ${L^{1}}(a,b)$, there exists a function $g\in
L^{1}(a,b)$ with $u^{\prime}-g\in L^{2}(a,b)$ such that, up to a subsequence,
$\gamma_{\varepsilon}\to D^{s}u+g\mathcal{L}^{1}$ in the flat norm. Since by
Theorem 2.1 we have $\Gamma$-convergence of
$\\{E_{\varepsilon}\\}_{\varepsilon>0}$ to $E$ in
$L^{1}(a,b)\times\mathcal{M}(a,b)$ also for every subsequence, we obtain

$\displaystyle\lim_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})=\liminf_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\geq
E(u,D^{s}u+g\mathcal{L}^{1}),$

which, by the choice of the sequence
$\\{\gamma_{\varepsilon}\\}_{\varepsilon}$, shows

$\liminf_{\varepsilon\to
0}\tilde{E}_{\varepsilon}(u_{\varepsilon})\geq\tilde{E}(u).$

We next show the $\Gamma$-$\limsup$ inequality. Via Theorem 2.1 (ii) we find a
recovery sequence $\\{(u_{\varepsilon},\gamma_{\varepsilon})\\}_{\varepsilon}$
of $(u,\gamma^{\textup{opt}})$ in $L^{1}(a,b)\times\mathcal{M}(a,b)$. In
combination with (45) this yields the claim

$\limsup_{\varepsilon\to
0}\tilde{E}_{\varepsilon}(u_{\varepsilon})\leq\limsup_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon},\gamma_{\varepsilon})\leq
E(u,\gamma^{\textup{opt}})=\tilde{E}(u).\qed$

## Acknowledgments

V. A.-V. gratefully acknowledges financial support of the Konrad-Adenauer-
Stiftung. All authors are grateful to the referee for valuable suggestions
that have led to substantial improvements of the results.

## References

  * [1] R. Alessi, J.-J. Marigo, and S. Vidoli.  Gradient damage models coupled with plasticity and nucleation of cohesive cracks.  Arch. Ration. Mech. Anal., 214(2):575–615, 2014.
  * [2] L. Ambrosio, N. Fusco, and D. Pallara.  Functions of bounded variation and free discontinuity problems.  Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.
  * [3] L. Ambrosio and V. M. Tortorelli.  Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence.  Comm. Pure Appl. Math., 43(8):999–1036, 1990.
  * [4] L. Ambrosio and V. M. Tortorelli.  On the approximation of free discontinuity problems.  Boll. Un. Mat. Ital. B (7), 6(1):105–123, 1992.
  * [5] V. Auer-Volkmann, L. Beck, and B. Schmidt.  Eigendamage: an eigendeformation model for the variational approximation of cohesive fracture – antiplane shear deformations.  In preparation, 2021.
  * [6] G. Bouchitté, A. Braides, and G. Buttazzo.  Relaxation results for some free discontinuity problems.  J. Reine Angew. Math., 458:1–18, 1995.
  * [7] B. Bourdin, G. A. Francfort, and J.-J. Marigo.  The variational approach to fracture.  J. Elasticity, 91(1-3):5–148, 2008.
  * [8] A. Braides.  Approximation of free-discontinuity problems, volume 1694 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 1998.
  * [9] A. Braides.  $\Gamma$-convergence for beginners, volume 22 of Oxford Lecture Series in Mathematics and its Applications.  Oxford University Press, Oxford, 2002.
  * [10] A. Braides and G. Dal Maso.  Non-local approximation of the Mumford-Shah functional.  Calc. Var. Partial Differential Equations, 5(4):293–322, 1997.