#  Approximation of topological singularities
through free discontinuity functionals:
the critical and super-critical regimes

V. Crismale  Dipartimento di Matematica “Guido Castelnuovo”, Sapienza
Università di Roma, Piazzale Aldo Moro 2, I-00185 Roma, Italy
<EMAIL_ADDRESS>,  L. De Luca  Istituto per le Applicazioni del
Calcolo “M. Picone”, IAC-CNR, 00185 Rome, Italy<EMAIL_ADDRESS>and  R.
Scala  Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche,
Università di Siena, 53100 Siena, Italy<EMAIL_ADDRESS>
###### Abstract.

We further investigate the properties of an approach to topological
singularities through free discontinuity functionals of Mumford-Shah type
proposed in [28]. We prove the variational equivalence between such energies,
Ginzburg-Landau, and Core-Radius for anti-plane screw dislocations energies in
dimension two, in the relevant energetic regimes $|\log\varepsilon|^{a}$,
$a\geq 1$, where $\varepsilon$ denotes the linear size of the process zone
near the defects.

Further, we remove the _a priori_ restrictive assumptions that the
approximating order parameters have compact jump set. This is obtained by
proving a new density result for $\mathbb{S}^{1}$-valued $SBV^{p}$ functions,
approximated through functions with essentially closed jump set, in the strong
$BV$ norm. Keywords: Functions of Bounded Variation; Strict Convergence;
Jacobian determinant; Topological Singularities; $\Gamma$-convergence;
Ginzburg-Landau Model; Core-Radius Approach. AMS subject classifications:
49J45 49Q20 26B30 74B15

###### Contents

  1. 1 Preliminary results
  2. 2 Density results in $SBV^{p}(\Omega;\mathbb{S}^{1})$
  3. 3 Description of the problem
  4. 4 Core radius approach
  5. 5 Proof of Theorem 3.3
  6. 6 Proof of Theorem 3.4

## Introduction

This paper concerns the analysis of topological singularities, which is a
central topic in models arising in Physics and Materials Science. Vortices in
superconductivity and superfluidity and (screw end edge) dislocations in
single crystal plasticity are the main examples of such phenomenon [4, 41, 43,
49, 50, 51].

In the last decades several models have been introduced to describe the
emergency of these objects. Among them, the most celebrated is the Ginzburg-
Landau (GL) model, mainly studied in the context of superconductivity. In such
a model, the order parameter is a function $u\in H^{1}(\Omega;\mathbb{R}^{2})$
and the energy functional (in its simplest form) reads as

(0.1)
$\mathcal{E}^{\mathrm{GL}}_{\varepsilon}(u):=\frac{1}{2}\int_{\Omega}|\nabla
u|^{2}\;\mathrm{d}x+\frac{1}{\varepsilon^{2}}\int_{\Omega}\big{(}1-|u|^{2}\big{)}^{2}\;\mathrm{d}x,$

where the parameter $\varepsilon>0$ is referred to as coherence length. Here
and below $\Omega\subset\mathbb{R}^{2}$ is a bounded open set with Lipschitz
continuous boundary. A topological singularity is nothing but a point around
which $u$ has non-trivial winding number and hence the main object to look at
is the Jacobian determinant (of $u$) $Ju:=\det\nabla u$. Denoting by
$\mathbb{S}^{1}$ the set of unitary vectors in the plane, we notice that close
to a topological singularity, $u$ cannot be $\mathbb{S}^{1}$-valued (a
singularity can be seen somehow as a zero of the order parameter); therefore,
the parameter $\varepsilon$ can be interpreted as the size of the region where
$u$ fails to take values in $\mathbb{S}^{1}$ and hence as the core-radius of
the topological singularity.

The variational analysis of the (GL) functional has been first systematized in
the monography [6] (see also [59] and the references therein for the
asymptotic analysis in terms of $\Gamma$-convergence), where the (GL) model is
compared with (and shown somehow to be “equivalent” to) the so-called core-
radius (CR) approach, in antiplane elasticity. Within this framework, the main
variable is represented by the distribution of topological singularities
$\mu=\sum\alpha_{i}\delta_{\xi_{i}}$ (with integer weights $\alpha_{i}$) but
the energy functional depends both on $\mu$ \- which here plays the role of
the Jacobian in (GL) - and on a map $u\in
H^{1}(\Omega_{\varepsilon}(\mu);\mathbb{S}^{1})$ “compatible with $\mu$”.
Here, $\varepsilon$ is the core-radius,
$\Omega_{\varepsilon}(\mu):=\Omega\setminus\bigcup_{i}\overline{B}_{\varepsilon}(\xi_{i})$
and the notion of compatibility is given by the fact that $\deg(u,\partial
B_{\varepsilon}(\xi_{i}))=\alpha_{i}$ (assuming that the balls
$B_{\varepsilon}(\xi_{i})$ are pairwise disjoint). The energy of the system
thus writes

(0.2)
$\mathcal{E}^{\mathrm{CR}}_{\varepsilon}(\mu,u):=\frac{1}{2}\int_{\Omega_{\varepsilon}(\mu)}|\nabla
u|^{2}\;\mathrm{d}x+|\mu|(\Omega).$

Here, the quantity $|\mu|(\Omega)$ plays the same role of the potential term
in $\mathcal{E}^{\mathrm{CR}}_{\varepsilon}$, namely, avoids that the cores do
not cover the whole domain; in other words, it serves only to guarantee
compactness and does not provide any energy contribution in the asymptotics as
$\varepsilon\to 0$.

The (CR) approach is mostly used to model screw dislocations in semi-discrete
theories. Loosely speaking, in pure (anti-plane) elasticity the bulk energy is
determined by the Hooke’s law, and reads as $\frac{1}{2}\int_{\Omega}|\nabla
w|^{2}\;\mathrm{d}x$, where the displacement $w$ lies in $H^{1}(\Omega)$. In
presence of a finite distribution $\mu=\sum_{i}\alpha_{i}\delta_{x_{i}}$ of
(scalar) defects, the material has a purely plastic behavior in the cores
$B_{\varepsilon}(\xi_{i})$ and, oversimplifying, such a plastic contribution
can be expressed by $|\mu|(\Omega)$. Moreover, along a closed circuit
enclosing the singularity $\xi_{i}$, a displacement $w$ compatible with $\mu$
should have a jump $[w]$ equal to $\alpha_{i}$. Therefore, the displacement
$w$ is only in $SBV^{2}(\Omega_{\varepsilon}(\mu))$ with $[w]\in\mathbb{Z}$
and its elastic energy should be given by
$\frac{1}{2}\int_{\Omega_{\varepsilon}(\mu)}|\nabla w|^{2}\;\mathrm{d}x$,
where $\nabla w$ is the absolutely continuous part of $\mathrm{D}w$. Setting

(0.3) $u=e^{2\pi\imath w},$

one obtains that the total energy associated to the pair $(\mu,u)$ is given by
$\mathcal{E}^{\mathrm{CR}}_{\varepsilon}$.

In this paper, we adopt a different viewpoint, following the approach proposed
in [28]. The main feature is that the order parameter is now an
$\mathbb{S}^{1}$-valued map, as in the (CR) approach, defined on the whole
$\Omega$, as in the (GL) approach. Clearly, in presence of topological
singularities, such a map cannot be in $H^{1}(\Omega;\mathbb{S}^{1})$. But,
instead of removing small disks around the singularities (as in (CR)) or to
weaken the $\mathbb{S}^{1}$-constraint (as in (GL)), the map $u$ is now
allowed to jump. More precisely, $u$ is a special function of bounded
variation with square-integrable approximate gradient (i.e., $u\in
SBV^{2}(\Omega;\mathbb{S}^{1})$). The energy functional we consider is

(0.4) $\mathcal{F}_{\varepsilon}(u):=\int_{\Omega}\frac{1}{2}|\nabla
u|^{2}\;\mathrm{d}x+\frac{1}{\varepsilon}\mathcal{H}^{1}(S_{u}),$

where $\varepsilon>0$ is a small parameter determining the size of the jump
set $S_{u}$ of $u$. Here and throughout the paper $\mathcal{H}^{1}$ denotes
the (one-dimensional) Hausdorff measure.

Formally, the functional $\mathcal{F}_{\varepsilon}$ has the structure of the
Mumford-Shah functional [56], but the $\mathbb{S}^{1}$-constraint makes the
analysis completely different. Indeed, having in mind the identity (0.3) and
the (CR) approach for screw dislocations, jumps of the map $u$ correspond to
non-integer jumps of the displacement $w$ and should pay energy. In other
words, the (amplitude of the) jump $[w]$ of the displacement exhibits a
transition between integers in a little portion of $S_{w}$. The transition is
assumed to have length of the order of $\varepsilon$, and corresponds to the
presence of singularities. In this respect, $\mathcal{H}^{1}(S_{u})$ is the
analogue of the potential term in (GL) and of the plastic term in (CR) and the
parameter $\varepsilon$ can be understood also in this case as the core-radius
of the singularity.

We highlight that for $SBV$ maps the definition of topological degree as well
as that of Jacobian determinant are not so standard so that the notion of
topological singularity is not so clear as in (GL) and in (CR). Nevertheless,
in [28], using the minimal lifting in [45], a notion of Jacobian determinant
is provided also for $SBV$ functions; we recall such a definition in Section 1
(see [53] where in a more restrictive setting this notion was first introduce,
and see also [9] for a different definition of Jacobian determinant in
dimension 2). In a nutshell, given a map $u\in SBV(\Omega;\mathbb{S}^{1})$,
the Jacobian $Ju$ of $u$ is defined as the boundary of the $1$-current $T_{u}$
defined by

(0.5) $\displaystyle T_{u}:=$
$\displaystyle\frac{1}{2}(-u^{1}\partial_{x_{2}}u^{2}+u^{2}\partial_{x_{2}}u^{1};u^{1}\partial_{x_{1}}u^{2}-u^{2}\partial_{x_{1}}u^{1})+\frac{1}{2}(u^{+}\wedge
u^{-})\bm{\tau}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits S_{u}$ $\displaystyle=:$
$\displaystyle\,T_{u}^{D}+T_{u}^{S}\,.$

We stress that dealing with the energy considered in [28], that is

(0.6) $\mathcal{G}_{\varepsilon}(u):=\int_{\Omega}\frac{1}{2}|\nabla
u|^{2}\;\mathrm{d}x+\frac{1}{\varepsilon}\mathcal{H}^{1}(\overline{S}_{u}),$

rules out, for instance, $SBV^{2}$ functions with jump set dense in the whole
$\Omega$; such a fact is crucial to prove compactness, rendering the analysis
more simpler.

In [28, Theorem 3.1] the $\Gamma$-convergence analysis of the functional
$\mathcal{G}_{\varepsilon}$ at the energy regime $|\log\varepsilon|$ has been
developed. As one may expect, such an analysis reveals that the functional
$\mathcal{G}_{\varepsilon}$ shares the same compactness and
$\Gamma$-convergence properties of the functionals
$\mathcal{E}_{\varepsilon}^{\mathrm{CR}}$ and
$\mathcal{E}_{\varepsilon}^{\mathrm{GL}}$. Specifically, as $\varepsilon\to
0$, the Jacobian determinant tends to concentrate around a finite number of
effective singularities and the $\Gamma$-limit of the functionals
$\frac{\mathcal{G}_{\varepsilon}}{|\log\varepsilon|}$ is given (up to
multiplicative constants) by the total variation of the limiting measure of
the Jacobians. Since, in view of the possibile presence of short dipoles, a
uniform bound on the total variation of the dislocations’ distributions is not
available, the natural setting for such an asymptotic analysis is the (strong)
flat convergence for Jacobian determinants, the flat topology being the strong
topology in the dual of Lipschitz continuous functions with compact support in
$\Omega$.

In the present paper we generalize the analysis done in [28] along two
directions. On the one hand, we show that the penalization term can be
“weakened” considering only the length of the jump set instead of its closure,
i.e., working with the functional $\mathcal{F}_{\varepsilon}$ rather than with
$\mathcal{G}_{\varepsilon}$. Therefore it is not needed to assume _a priori_
that the jump set is compact. On the other hand, we show that the functional
$\mathcal{F}_{\varepsilon}$ shares the same asymptotic behavior of the
functionals $\mathcal{E}_{\varepsilon}^{\mathrm{CR}}$ and
$\mathcal{E}_{\varepsilon}^{\mathrm{GL}}$ also in other energy regimes.

The first improvement is obtained by means of a density result in
$SBV^{p}(\Omega;\mathbb{S}^{1})$, $p>1$, with respect to energies
$\mathcal{F}_{\varepsilon}$ for fixed $\varepsilon>0$, through functions in
$SBV^{p}(\Omega;\mathbb{S}^{1})$ with (essentially) closed jump set,
converging in the strong $BV$ norm and such that also the two unilateral
traces of the approximants along the jump set converge; in particular, by
using the characterization of $Ju$ as the boundary of $T_{u}$ in (0.5), the
strong convergence of Jacobian determinants with respect to flat norm follows.

Our result hinges on tools developed in a slightly different setting, that is
when only the symmetric part of the diffuse gradient is controlled in some
$L^{p}$, for $p>1$, rather than the whole diffuse gradient. Mechanically, this
corresponds to consider fracture models for general linearized elasticity
without the anti-plane assumption, described by the Griffith functional [40]
instead of the Mumford-Shah one.

In fact, the main tool for density results developed in the context of
Mumford-Shah functional is an approximated Poincaré-Wirtinger inequality for
$SBV^{p}$ functions with small ($\mathcal{H}^{d-1}$-measure of the) jump set,
due to De Giorgi-Carriero-Leaci ([25]): given $u\in SBV^{p}$ there exists a
truncation in $W^{1,p}$ such that $u$ differs from $w$ on an exceptional set
$\omega$ whose volume is controlled by $(\mathcal{H}^{d-1}(S_{u}))^{1^{*}}$,
$d\geq 2$ being the space dimension and $1^{*}:=d/d{-}1$. In the same paper,
this result has been used to prove that the jump set of Mumford-Shah
minimizers is essentially closed, namely the $\mathcal{H}^{d-1}$ measure of
the jump set equals that of its closure. After short time a generalization for
$SBV^{p}(\Omega;\mathbb{S}^{k-1})$ maps has been proven in [16]; by combining
such a generalization with an argument in [8] (cf. Lemma 5.2 therein), one can
show that the Mumford-Shah functional in $SBV^{p}(\Omega;\mathbb{S}^{1})$ can
be approximated through $SBV^{p}(\Omega;\mathbb{S}^{1})$ functions having
essentially closed jump set and converging pointwise. Unfortunately, pointwise
convergence of a sequence of functions does not guarantee convergence of the
corresponding Jacobians; since we need convergence in the flat norm such an
approach is not satisfactory for our purposes. For functions with finite
Griffith energy (with exponent $p$), that is in the space $GSBD^{p}$ ([23]), a
fundamental tool is the approximated Poincaré-Korn inequality in [10], stating
that for any $u\in GSBD^{p}$ with small jump set there exists an infinitesimal
rigid motion $a$ (i.e. an affine function with null symmetrized gradient) such
that the $L^{\frac{dp}{d-1}}$-norm of $u-a$ is estimated by the $L^{p}$ norm
of $e(u)$, the symmetrized diffuse gradient of $u$, outside an exceptional set
$\omega$ whose volume is controlled by $(\mathcal{H}^{d-1}(J_{u}))^{{1^{*}}}$.
Moreover, a convolution of $u\chi_{\omega^{c}}+a\chi_{\omega}$ at the same
scale of the domain provides a function with $L^{p}$-norm of the symmetrized
diffuse gradient controlled by those of $u$.

This result, on which other contribution in this direction rest (see, e.g.,the
approximation in $GSBD^{p}$ through functions with essentially closed jump set
[19, 11] and the analogue of [25] for the Griffith functional [20, 17, 12]),
has been generalized by [18] and [13]: here, any $u\in GSBD^{p}$ is
approximated, in terms of the Griffith energy, by functions $W^{1,p}$ on a
slightly smaller domain, with essentially closed jump set, which differ from
$u$ on a set $\omega$ whose boundary is controlled by
$\mathcal{H}^{d-1}(J_{u})$; further, in [18] it is shown that in dimension two
it is possible to guarantee that the approximants coincide with $u$ in the
boundary neighborhood where they are not in $W^{1,p}$. In [18] such an
approximation is used to prove an integral representation result, while in
[13] the main result is the approximation of any $u\in GSBD^{p}$, with respect
to the Griffith energy, through functions with essentially closed jump set
differing from $u$ on sets of vanishing perimeter.

Moreover, Friedrich [32] proved a _piecewise Korn inequality_ in dimension
two, showing that up to subtracting piecewise rigid functions (finite sums of
infinitesimal rigid motions multiplied by characteristic functions), any $u$
in $GSBD^{p}$ can be approximated by functions in $SBV^{q}\cap L^{\infty}$,
for $q<p$, in particular the diffuse gradient of the approximants is estimated
on the whole domain by $e(u)$; this is a very powerful tool allowing to
overcome the lack of a Coarea Formula in $GSBD$ and then to show, e.g.,
existence of quasi-static evolutions for Brittle Fracture models (see, for
instance, [34]). In the same spirit, in [33] a similar result has been shown
in the Mumford Shah setting, namely dealing only with full diffuse gradients.

Eventually, we refer to [29] for the two dimensional analogue of [16] for maps
in $SBV^{p(\cdot)}(\Omega;\mathbb{S}^{k-1})$, (with
$\Omega\subset\mathbb{R}^{2}$) whose approximate gradient is integrable with
respect to the variable exponent $p(\cdot)$ over $\Omega$ and whose jump set
has finite $\mathcal{H}^{1}$-measure (see also [48] for the variable exponents
analogue of [25]), obtained under the assumption that the function $p(\cdot)$
is regular enough and takes values in $(1,2)$. This uses the analogue of the
approximation of [18], proven by employing retractions
$\mathcal{P}\colon\mathbb{R}^{k}\setminus\mathcal{X}\to\mathbb{S}^{k-1}$ with
locally $q$-integrable gradient for $q\in[1,2)$, where $\mathcal{X}$ is a
smooth complex of codimension two (cf. e.g. [14]).

We then compare our main density result Corollary 2.2 with [13, Theorem 5.1]:
we are in two dimensions and consider the full diffuse gradient instead of its
symmetrized part, however we keep in the approximation the constraint of being
$\mathbb{S}^{1}$-valued. We notice that also a version with symmetrized
diffuse gradient is readily shown with essentially the same proof, see Theorem
2.6.

Since in our application the case $p=2$ is the relevant one, we cannot follow
a strategy based on retractions. Moreover, the proof of [18], [13], and [33]
is not compatible with a non-convex target space such as $\mathbb{S}^{1}$.

Our approach is based on the existence of a _lifting_ $\varphi\in
SBV^{p}(\Omega)$ (i.e., such that $u=e^{2\pi\imath\varphi}$) with
$\pi\|\varphi\|_{BV}\leq\|u\|_{BV}$ ([24]), for which we provide a suitable
approximation (Theorem 2.1) and then compose the approximants with
$e^{2\pi\imath\cdot}$. We observe that, since in [13, Theorem 5.1] the set on
which the traces of the approximants differ from those of the given function
has only finite $\mathcal{H}^{d-1}$ measure and then could be dense in the
original jump set, after an application of [13, Theorem 5.1] to $\varphi$ and
the composition with $e^{2\pi\imath\cdot}$, one could obtain approximating
functions with jump set dense in the integer jump set of $\varphi$, whose
$\mathcal{H}^{1}$-measure could be of order $\|u\|_{BV}$.

Therefore we need a more refined density result, which allows to approximate
integer jumps with integer jumps as well. The strategy is to work locally near
points with integer jump at a scale for which the jump is almost flat and
assumes a constant integer value. Then locally most of the jump set of
$\varphi$ can be transferred (up to a small error) into a flat segment $S$ on
which the jump has the same value; such $S$ can be chosen in such a way that
the remaining small jump set of $\varphi$, on any square with arbitrarily
small sidelength with a side contained in $S$, is small compared to the
sidelength: this follows by a two dimensional argument drawn from [18].

In this way, the approximation for functions with small jump set of [18] may
be applied at every scale; therefore a Whitney-type argument combined with the
fact that the approximants coincide with the original function on the boundary
of any square ensures that the traces are the same on both sides of $S$, so
the new jump is still integer. This strategy may be replicated for different
target manifolds, provided a lifting with good $BV$ bounds as in [24] exists
(in this respect see e.g. [15]), but limiting to two dimensional domains. We
notice that in the present context several hard issues arise when considering
space dimension $d\geq 3$, such as the lack of the analogue of the Ball
Construction.

The density result described above allows to develop the $\Gamma$-convergence
analysis for the functional $\mathcal{F}_{\varepsilon}$ also in different
energetic regimes (as it applies for fixed $\varepsilon>0$), thus generalizing
the setting of [28].

First, we develop the $\Gamma$-convergence analysis in the so-called critical
regime, that is $|\log\varepsilon|^{2}$. Loosely speaking, since
$|\log\varepsilon|$ is the energy cost of an isolated singularity, the fact
that $\mathcal{F}_{\varepsilon}(u_{\varepsilon})\sim|\log\varepsilon|^{2}$
implies that, for $\varepsilon>0$, the number of singularities of the
Jacobians $Ju_{\varepsilon}$ is of order $|\log\varepsilon|$; therefore, the
Jacobians $Ju_{\varepsilon}$, once rescaled by $|\log\varepsilon|$, should
converge (in the flat norm) to a measure $\mu$ that is not anymore atomic but
diffuse. Furthermore, we prove that such a measure $\mu$ lies also in
$H^{-1}(\Omega)$. Indeed, by standard compactness results in $L^{2}(\Omega)$,
also the fields $T^{\mathrm{D}}_{u_{\varepsilon}}$, once scaled by
$|\log\varepsilon|$, should converge (weakly in $L^{2}(\Omega)$) to a field
$T^{\mathrm{D}}$, whose distributional divergence is shown to be given by
$-\pi\mu$. As one may expect, the $\Gamma$-limit accounts both for the plastic
contribution of $\mu$ as well as for the elastic energy of $T^{\mathrm{D}}$.
That is the reason why the $|\log\varepsilon|^{2}$ regime is called critical,
since in such a case the elastic and plastic effects are of the same order.
The $\Gamma$-convergence analysis for the functional
$\mathcal{F}_{\varepsilon}$ is provided in Theorems 3.3 and 3.4 which are
proved in Section 5. Second, the proofs of the compactness and of the lower
bound are obtained combining the corresponding results for the core-radius
approach together with the refined ball construction machinery introduced in
[28] to analyze the $|\log\varepsilon|$ regime.

Finally, adopting the same strategy, in Theorem 3.4 we analyze also the super-
critical regimes $|\log\varepsilon|^{2}\ll
N_{\varepsilon}\ll\frac{1}{\varepsilon}$. In such a case, the interaction
elastic energy is larger and larger than the core energy, so that (unless
scaling differently the two quantities $Ju_{\varepsilon}$ and
$T^{\mathrm{D}}_{u_{\varepsilon}}$) one keeps track of the only
$T^{\mathrm{D}}$ and the Jacobian determinants do not play any role when
computing the effective energy.

We highlight that the $\Gamma$-convergence analysis for the functional
$\mathcal{E}_{\varepsilon}^{\mathrm{GL}}$ in the regime
$|\log\varepsilon|^{2}$ has been developed in [47, 60, 61], where the authors
consider also the case with magnetic field. The analysis for
$\mathcal{E}_{\varepsilon}^{\mathrm{CR}}$ is provided in Section 4 and is
somehow a short self-contained resume of the results above, along the lines of
[2].

However, a similar result in the context of edge dislocations within the (CR)
approach is proven in [35] under the well-separation assumption for the
singularities (see also [55] for such an analysis in the nonlinear elasticity
framework); such an assumption has been removed in [39]. In view of the
asymptotic equivalence result [1] between the Ginzburg-Landau model and the
purely discrete models of $XY$ spin systems and screw dislocations, we have
that the analysis in the (GL) context extends also to such discrete models.

The paper is organized as follows: After recalling some notations and
preliminary results in Section 1, we prove in Section 2 a general density
result for $SBV^{p}(\Omega)$ functions in Theorem 2.1 which implies, as a
consequence, Corollary 2.2. This is the result we employ to obtain energy
density in the $\Gamma$-convergence results of Section 3, actually allowing us
to restrict such analysis to $\mathbb{S}^{1}$-valued functions with
essentially closed jump set. The latter results are Theorems 3.3 and 3.4,
stated in Section 3 after recalling the main features of our model. In order
to prove them we recall in Section 4 the classical core radius approach, which
is the starting point of our analysis. Finally, the proofs of Theorems 3.3 and
3.4 are given, respectively, in Sections 5 and 6.

Acknowledgements: The authors are members of the Gruppo Nazionale per
l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the
Istituto Nazionale di Alta Matematica (INdAM).

VC acknowledges the financial support of PRIN 2022J4FYNJ “Variational methods
for stationary and evolution problems with singularities and interfaces”, PNRR
Italia Domani, funded by the European Union under NextGenerationEU, CUP
B53D23009320006.

LDL acknowledges the financial support of PRIN 2022HKBF5C “Variational
Analysis of complex systems in Materials Science, Physics and Biology”, PNRR
Italia Domani, funded by the European Union via the program NextGenerationEU,
CUP B53D23009290006.

RS also acknowledges the partial financial support of the F-cur project number
2262-2022-SR-CONRICMIUR-PC-FCUR2022${}_{-}002$ of the University of Siena, and
the of the PRIN project 2022PJ9EFL ”Geometric Measure Theory: Structure of
Singular Measures, Regularity Theory and Applications in the Calculus of
Variations”, PNRR Italia Domani, funded by the European Union via the program
NextGenerationEU, CUP B53D23009400006.

##  1\. Preliminary results

In this section we collect some preliminary notions on the flat norm of
measures and currents, as well as some properties of $BV$ functions that will
be used throughout the paper.

#### Flat norm of Radon measures

Let $n\geq 1$ be an integer and let $U\subset\mathbb{R}^{n}$ be a bounded and
open set. We denote by $\mathcal{M}_{b}(U)$ the space of Radon measures on $U$
with finite total variation. If $\mu\in\mathcal{M}_{b}(U)$, we denote by
$|\mu|(U)$ the total variation of $\mu$ . We recall that a sequence
$\mu_{k}\in\mathcal{M}_{b}(U)$ converges tightly to $\mu\in\mathcal{M}_{b}(U)$
if $\mu_{k}$ converges to $\mu$ weakly* as measure, and
$|\mu_{k}|(U)\rightarrow|\mu|(U)$. We also introduce the concept of flat norm
of a measure $\mu$, denoted by $\|\mu\|_{\mathrm{flat}}$ , as

(1.1)
$\displaystyle{\|\mu\|_{\mathrm{flat}}:=\sup_{{\genfrac{}{}{0.0pt}{1}{\varphi\in
C_{\mathrm{c}}^{0,1}(U)}{\|\varphi\|_{C^{0,1}(U)}\leq
1}}}\int_{U}\varphi\;\mathrm{d}\mu\,.}$

Here and below, the Lipschitz norm $\|\varphi\|_{C^{0,1}(U)}$ is defined by

$\|\varphi\|_{C^{0,1}(U)}:=\|\varphi\|_{L^{\infty}(U)}+\sup_{\genfrac{}{}{0.0pt}{1}{x,y\in
U}{x\neq y}}\frac{|\varphi(x)-\varphi(y)|}{|x-y|}\,.$

By a density argument we easily see that the supremum in (1.1) can be
equivalently computed among smooth and compactly supported (in $U$) functions
$\varphi$ with $\|\varphi\|_{C^{0,1}(U)}\leq 1$ .

#### Flat norm of $k$-currents.

Let $n\geq 2$ be an integer and let $U\subset\mathbb{R}^{n}$ be an open set.
For every $k\in\mathbb{N}$ with $0\leq k\leq n$ , we denote by
$\mathcal{D}^{k}(U)$ the topological vector space of smooth and compactly
supported $k$-forms on $U$, and by $\mathcal{D}_{k}(U)$ its dual, i.e., the
space of $k$-currents on $U$.

The mass $|T|$ of a current $T\in\mathcal{D}_{k}(U)$ is defined as

$|T|=\sup\\{\langle
T,\omega\rangle:\;\omega\in\mathcal{D}^{k}(U),\|\omega\|_{L^{\infty}}\leq
1\\}\,.$

As done in (1.1) for measures, we define the flat norm of a current
$T\in\mathcal{D}_{k}(U)$ in $U$ by

(1.2)
$\|T\|_{\mathrm{flat},U}:=\sup_{\genfrac{}{}{0.0pt}{1}{\omega\in\mathcal{D}^{k}(U)}{\|\omega\|_{F,U}\leq
1}}\langle T,\omega\rangle,$

where

$\|\omega\|_{F,U}:=\|\omega\|_{L^{\infty}(U)}+\|\mathrm{d}\omega\|_{L^{\infty}(U)}\,.$

In the special case that $T$ is a $0$-current and has finite mass, then it can
be standardly identified with a measure, and the flat norm of $T$ coincides
with the flat norm of the measure $T$ defined in (1.1).

#### Jacobian for $\mathbb{S}^{1}$-valued Sobolev maps

Let $U\subset\mathbb{R}^{2}$ be a bounded and open set. Given a map $u\in
W^{1,1}(U;\mathbb{S}^{1})$ we recall that the distributional Jacobian
$Ju=\textrm{Det}(\nabla u)$ of $u$ is defined by

(1.3) $\langle
Ju,\varphi\rangle_{U}:=\int_{U}\nabla\varphi\cdot\lambda_{u}\,\;\mathrm{d}x,\qquad\qquad\textrm{for
every }\varphi\in C_{\mathrm{c}}^{\infty}(U),$

where

(1.4) $\displaystyle\lambda_{u}:=\frac{1}{2}\Big{(}-u^{1}\frac{\partial
u^{2}}{\partial x_{2}}+u^{2}\frac{\partial u^{1}}{\partial
x_{2}};u^{1}\frac{\partial u^{2}}{\partial x_{1}}-u^{2}\frac{\partial
u^{1}}{\partial x_{1}}\Big{)}\,;$

notice that $\lambda_{u}\in L^{1}(U;\mathbb{R}^{2})$.

Moreover, denoting by $j(u)\in L^{1}(U;\mathbb{R}^{2})$ the current associated
to $u$ , i.e.,

(1.5) $j(u):=\frac{1}{2}(u^{1}\nabla u^{2}-u^{2}\nabla u^{1})\,,$

one has $j^{\perp}(u)=\lambda_{u}$ and $j(u)=\pi\nabla w$ , where $w$ is a
generic lifting of $u$ , i.e., a map in $SBV^{2}(U)$ satisfying (0.3) and
$\nabla w$ is the approximate gradient of $w$. Furthermore, it is easy to
check that

$Ju=-\mathrm{Div}\lambda_{u}=\mathrm{curl}\,j(u)=\pi\mathrm{curl}(\nabla w),$

holds in the sense of distributions.

In the sequel we will use the fact that a function $u\in
H^{1}(U;\mathbb{S}^{1})$ satisfies $\textrm{Det}(\nabla u)=0$ in the sense of
distributions. Moreover, if $u\in
H^{1}(U\setminus\overline{B};\mathbb{S}^{1})$, where $B\subset U$ is a ball,
then, integrating by parts,

$\int_{U\setminus\overline{B}}\lambda_{u}\cdot\nabla\varphi\;\mathrm{d}x=\int_{\partial
B}\lambda_{u}\cdot{\bm{\nu}}\varphi\;\mathrm{d}\mathcal{H}^{1}=\int_{\partial
B}j(u)\cdot{\bm{\tau}}\varphi\;\mathrm{d}\mathcal{H}^{1},\qquad\textrm{for
every }\varphi\in C_{\mathrm{c}}^{\infty}(U),$

where ${\bm{\nu}}$ is the inner normal vector to $\partial B$,
${\bm{\tau}}=-{\bm{\nu}}^{\perp}$ is the counter-clockwise tangent vector to
$\partial B$. Notice that
$j(u)\cdot{\bm{\tau}}=\frac{1}{2}(u^{1}\frac{\partial
u^{2}}{\partial{\bm{\tau}}}-u^{2}\frac{\partial u^{1}}{\partial{\bm{\tau}}})$
on $\partial B$.

We recall that $\deg(u,\partial B)\in\mathbb{Z}$ is defined as

(1.6) $\displaystyle\deg(u,\partial B):=\frac{1}{\pi}\int_{\partial
B}j(u)\cdot{\bm{\tau}}\;\mathrm{d}\mathcal{H}^{1}=\frac{1}{\pi}\int_{\partial
B}\lambda_{u}\cdot{\bm{\nu}}\;\mathrm{d}\mathcal{H}^{1}\,,$

whenever $u\in H^{\frac{1}{2}}(\partial B;\mathbb{S}^{1})$.

#### Jacobian for $\mathbb{S}^{1}$-valued $SBV$ maps

Let $U\subset\mathbb{R}^{2}$ be a bounded and open set. For any
$p\in[1,+\infty)$ the symbol $SBV^{p}(U;\mathbb{R}^{2})$ denotes the space of
functions $u\in BV(A;\mathbb{R}^{2})$ such that the Cantor part
$\mathrm{D}^{c}u\equiv 0$ , and $\nabla u\in L^{p}(A;\mathbb{R}^{2\times 2})$,
where $\nabla u$ is the density of $\mathrm{D}^{a}u$ , i.e.,
$\mathrm{D}^{a}u:=\nabla u\mathcal{L}^{2}$ . The space
$SBV^{p}(U;\mathbb{S}^{1})$ denotes the set of the functions $u\in
SBV^{p}(U;\mathbb{R}^{2})$ such that $|u|=1$ a.e. in $U$.

The following result, proven in [28, Corollary 2.1], is specialized here to
maps taking values in $\mathbb{R}^{2}$.

######  Proposition 1.1.

Let $u\in SBV(U;\mathbb{R}^{2})\cap L^{\infty}(U;\mathbb{R}^{2})$ ; then there
exists a unique measure $\nu_{u}\in\mathcal{M}_{b}(U;\mathbb{R}^{2\times
2\times 2})$ such that, whenever $\\{v_{k}\\}_{k\in\mathbb{N}}\subset
C^{1}(U;\mathbb{R}^{2})\cap W^{1,1}(U;\mathbb{R}^{2})\cap
L^{\infty}(U;\mathbb{R}^{2})$ satisfies
$\|v_{k}\|_{{L^{\infty}(U;\mathbb{R}^{2})}}\leq C<+\infty$ for all $k\geq 1$
and $v_{k}\overset{\mathrm{strict}}{\rightharpoonup}u$ in
$BV(U;\mathbb{R}^{2})$ , then $v_{k}\otimes\nabla v_{k}\rightarrow\nu_{u}$ ,
where $(\nu_{u})_{j}^{i,h}$ is defined (for all $\varphi\in
C_{\mathrm{c}}(U)$) by

$\displaystyle\int_{U}\varphi(x)\;\mathrm{d}(\nu_{u})^{i,h}_{j}=$
$\displaystyle\int_{U\setminus
S_{u}}\phi(x)u^{h}(x)\partial_{x_{j}}u^{i}(x)\;\mathrm{d}x$
$\displaystyle+\frac{1}{2}\int_{S_{u}}\phi(x)(u^{h,+}(x)+u^{h,-}(x))(u^{i,+}(x)-u^{i,-}(x))\bm{\nu}_{j}(x)\;\mathrm{d}\mathcal{H}^{1}(x)\,,$

for every $i,j,h\in\\{1,2\\}$ . Finally, if
$\\{u_{k}\\}_{k\in\mathbb{N}}\subset SBV(U;\mathbb{R}^{2})\cap
L^{\infty}(U;\mathbb{R}^{2})$ with

(1.7) $\|u_{k}\|_{L^{\infty}(U;\mathbb{R}^{2})}\leq C,$

for some constant $C>0$, and
$u_{k}\overset{\mathrm{strict}}{\rightharpoonup}u$ in $BV(U;\mathbb{R}^{2})$ ,
then

(1.8)
$\nu_{u_{k}}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\nu_{u}\qquad\textrm{in
}\mathcal{M}_{b}(U;\mathbb{R}^{2\times 2\times 2})\,.$

In the following, for every map $u\in SBV(U;\mathbb{R}^{2})\cap
L^{\infty}(U;\mathbb{R}^{2})$ , we set

$[u^{h}\mathrm{D}_{j}u^{i}]:=(\nu_{u})^{i,h}_{j},\;\;\;\;i,j,h\in\\{1,2\\}\,,$

For any map $u\in SBV(U;\mathbb{R}^{2})\cap L^{\infty}(U;\mathbb{R}^{2})$ we
introduce the $1$-current $T_{u}$ defined by

(1.9) $\displaystyle T_{u}:=$
$\displaystyle\frac{1}{2}(-[u^{1}\mathrm{D}_{2}u^{2}]+[u^{2}\mathrm{D}_{2}u^{1}];[u^{1}\mathrm{D}_{1}u^{2}]-[u^{2}\mathrm{D}_{1}u^{1}])$
$\displaystyle=$
$\displaystyle\frac{1}{2}(-u^{1}\partial_{x_{2}}u^{2}+u^{2}\partial_{x_{2}}u^{1};u^{1}\partial_{x_{1}}u^{2}-u^{2}\partial_{x_{1}}u^{1})+\frac{1}{2}(u^{+}\wedge
u^{-})\bm{\tau}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits S_{u}$ $\displaystyle=:$
$\displaystyle\,T_{u}^{D}+T_{u}^{S}\,.$

where we have noted
$\alpha\wedge\beta=-\alpha\cdot\beta^{\perp}=\det(\alpha,\beta).$ Notice that
$T_{u}^{D}\in L^{1}(U;\mathbb{R}^{2})$ and that, if $u\in
W^{1,1}(U;\mathbb{S}^{1})$ , then $T_{u}=T_{u}^{D}=\lambda_{u}$ with
$\lambda_{u}$ defined in (1.4). Finally, we highlight that if $u\in
SBV(U;\mathbb{S}^{1})$ , for any lifting $w\in SBV(U)$ of $u$, i.e.,
satisfying (0.3), it holds that

(1.10) $T^{D}_{u}:=\pi\nabla^{\perp}w;\qquad\qquad
T^{S}_{u}:=\frac{1}{2}\sin(2\pi(w^{-}-w^{+}))\bm{\tau}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits S_{u}\,.$

The distributional Jacobian $Ju\in\mathcal{D}_{0}(U)$ of $u$ is defined as the
boundary of $T_{u}$, namely

(1.11) $\displaystyle Ju:=\partial T_{u}\qquad\qquad\text{in
}\mathcal{D}_{0}(U).$

Essentially by definition, it easily follows that

$\|Ju\|_{\text{{\rm flat}},U}\leq C\|u\|_{BV},$

for all $u\in SBV(\Omega;\mathbb{S}^{1})$, for a universal constant $C>0$.

######  Remark 1.2.

We point out that in general $Ju$ is not a Radon measure. This notion of
Jacobian determinant was first introduced in [53] under some special
hypotheses on $u$. Under these hypotheses it turns out that $Ju$ is also a
Radon measure.

##  2\. Density results in $SBV^{p}(\Omega;\mathbb{S}^{1})$

In this section we prove that any function $u\in
SBV^{p}(\Omega;\mathbb{S}^{1})$ can be approximated - in the strong $BV$ norm
- by (sequence of) functions in $SBV^{p}(\Omega;\mathbb{S}^{1})$ with closed
jump set. As a consequence (see Corollary 2.3), we deduce that the
corresponding currents can be approximated in the flat norm.

In what follows for every function $\varphi\in SBV^{p}(\Omega)$ (with $p\geq
1$) we denote by $S_{\varphi}^{\mathrm{frac}}$ the fractional jump set of
$\varphi$ , i.e., $S^{\mathrm{frac}}_{\varphi}:=\\{x\in
S_{\varphi}\colon[\varphi]\notin\mathbb{Z}\\}$ , and by
$S_{\varphi}^{\mathrm{int}}$ the integer jump set , namely,
$S_{\varphi}^{\mathrm{int}}:=S_{\varphi}\setminus S_{\varphi}^{\mathrm{frac}}$
. The main result of this section is the following.

######  Theorem 2.1.

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded open set with finite perimeter,
$p\in(1,+\infty)$, and $\varepsilon>0$. Then for every $\varphi\in
SBV^{p}(\Omega)$ there exist:

  * •

closed sets $\Gamma^{\mathrm{int}}=\Gamma^{\mathrm{int}}_{\varepsilon}$,
$\Gamma^{\mathrm{frac}}=\Gamma^{\mathrm{frac}}_{\varepsilon}$, finite unions
of disjoint $C^{1}$ curves;

  * •

a set $\widetilde{\omega}=\widetilde{\omega}_{\varepsilon}$, finite union of
cubes;

  * •

a set of finite perimeter $\widehat{\omega}=\widehat{\omega}_{\varepsilon}$;

  * •

a function $\theta=\theta_{\varepsilon}\in SBV^{p}(\Omega)\cap
W^{1,p}(\Omega\setminus({\Gamma^{\mathrm{int}}\cup\Gamma^{\mathrm{frac}}}\cup\overline{\widetilde{\omega}}))$
;

such that

(2.1)
$\\{\nabla\varphi\neq\nabla\theta\\}\subset\widetilde{\omega}\cup\widehat{\omega},\quad\mathcal{L}^{2}(\\{\varphi\neq\theta\\})<\varepsilon,\quad\theta=0\textrm{
in }\widetilde{\omega},$

$[\theta](x)\in\mathbb{Z}$ for $\mathcal{H}^{1}$-a.e.
$x\in\Gamma^{\mathrm{int}}$ ,

(2.2)
$|\mathcal{H}^{1}(S^{\mathrm{int}}_{\varphi})-\mathcal{H}^{1}(\Gamma^{\mathrm{int}})|+\mathcal{H}^{1}(S^{\mathrm{frac}}_{\varphi}\triangle\Gamma^{\mathrm{frac}})+\mathcal{H}^{1}(\partial\widetilde{\omega})+\mathcal{H}^{1}(\partial^{*}\widehat{\omega})\leq\varepsilon,$

and

(2.3)
$\int_{\Omega}|\nabla\theta|^{p}\;\mathrm{d}x\leq(1+\varepsilon)\int_{\Omega}|\nabla\varphi|^{p}\;\mathrm{d}x.$

Moreover,
$\mathcal{H}^{1}(\Gamma^{\mathrm{frac}}\cap\\{\theta^{+}\neq\varphi^{+}\\})+\mathcal{H}^{1}(\Gamma^{\mathrm{frac}}\cap\\{\theta^{-}\neq\varphi^{-}\\})\leq\varepsilon$,
where $\theta^{\pm}$ and $\varphi^{\pm}$ denote the traces of $\theta$ and
$\varphi$ on the two sides of $\Gamma$.

Before providing the proof of Theorem 2.1 we state and prove our desired
approximation results for maps in $SBV^{p}(\Omega;\mathbb{S}^{1})$.

######  Corollary 2.2.

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded open set of finite perimeter,
$p\in(1,+\infty)$, and $\varepsilon>0$. Then for every $u\in
SBV^{p}(\Omega;\mathbb{S}^{1})$ there exist:

  * •

a closed set $\Gamma=\Gamma_{\varepsilon}$, finite union of disjoint $C^{1}$
curves;

  * •

a set $\widetilde{\omega}=\widetilde{\omega}_{\varepsilon}$, finite union of
cubes;

  * •

a set of finite perimeter $\widehat{\omega}=\widehat{\omega}_{\varepsilon}$;

  * •

a function $v=v_{\varepsilon}\in SBV^{p}(\Omega;\mathbb{S}^{1})\cap
W^{1,p}(\Omega\setminus(\Gamma\cup\overline{\widetilde{\omega}});\mathbb{S}^{1})$;

such that

(2.4) $\\{\nabla u\neq\nabla
v\\}\subset\widetilde{\omega}\cup\widehat{\omega},\quad\mathcal{L}^{2}(\\{u\neq
v\\})<\varepsilon,\quad\nabla v=0\textrm{ $\mathcal{L}^{2}$-a.e. in
}\widetilde{\omega},$

and

(2.5)
$\begin{split}\mathcal{H}^{1}(S_{u}\triangle\Gamma)+\mathcal{H}^{1}(\partial\widetilde{\omega})+\mathcal{H}^{1}(\partial^{*}\widehat{\omega})\leq\varepsilon,\quad\int_{{\Omega}}|\nabla
v|^{p}\;\mathrm{d}x\leq(1+\varepsilon)\int_{\Omega}|\nabla
u|^{p}\;\mathrm{d}x.\end{split}$

Moreover, $\mathcal{H}^{1}(\Gamma\cap\\{v^{+}\neq
u^{+}\\})+\mathcal{H}^{1}(\Gamma\cap\\{v^{-}\neq u^{-}\\})\leq\varepsilon$,
where $v^{\pm}$ and $u^{\pm}$ denote the traces of $v$ and $u$ on the two
sides of $\Gamma$.

###### Proof.

Let $u\in SBV^{p}(\Omega;\mathbb{S}^{1})$ and let $\varepsilon>0$ . Then, by
[24, Theorem 1.1], there exists $\varphi\in SBV^{p}(\Omega)$ such that
$u=e^{2\pi\imath\varphi}$ with $\pi\|\varphi\|_{BV}\leq\|u\|_{BV}$ . Let
$\Gamma^{\mathrm{int}},\,\Gamma^{\mathrm{frac}},\,\widetilde{\omega},\,\widehat{\omega}$
be the sets and let $\theta$ be the function provided by Theorem 2.1 . We set
$v:=e^{2\pi\imath\theta}$ . Then,
$\\{\nabla\varphi=\nabla\theta\\}\equiv\\{\nabla u=\nabla v\\}$
$\mathcal{L}^{2}$-a.e. and $\\{\varphi=\theta\\}\subset\\{u=v\\}$ , so that,
by (2.1) we immediately deduce (2.4). Furthermore, since
$S_{\varphi}^{\mathrm{frac}}\equiv S_{u}$ , taking
$\Gamma=\Gamma^{\mathrm{frac}}$, by Theorem 2.1 we deduce also the last part
of the claim. ∎

######  Proposition 2.3.

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded open set with finite perimeter,
let $p\in(1,+\infty)$, and let $u\in SBV^{p}(\Omega;\mathbb{S}^{1})$. Then
there exists $\\{u_{n}\\}_{n\in\mathbb{N}}\subset
SBV^{p}(\Omega;\mathbb{S}^{1})$ with
$\mathcal{H}^{1}(\overline{S}_{u_{n}}\setminus S_{u_{n}})=0$ for all $n$ such
that

(2.6) $\displaystyle\|u_{n}-u\|_{BV(\Omega;\mathbb{R}^{2})}\to 0,$
$\displaystyle\|\nabla u_{n}\|_{L^{p}(\Omega;\mathbb{R}^{2\times
2})}\rightarrow\|\nabla u\|_{L^{p}(\Omega;\mathbb{R}^{2\times 2})},$
$\displaystyle\mathcal{H}^{1}(S_{u_{n}})\rightarrow\mathcal{H}^{1}(S_{u}).$

Furthermore

(2.7) $|T_{u_{n}}-T_{u}|(\Omega)\to 0,$

where $T_{u}$ and $T_{u_{n}}$ are the measures provided by (1.9); in
particular

$\|Ju-Ju_{n}\|_{\mathrm{flat},\Omega}\to 0,$

with $\|\cdot\|_{\mathrm{flat},\Omega}$ is the norm defined in (1.2).

###### Proof.

Let $u\in SBV^{p}(\Omega;\mathbb{S}^{1})$ be fixed. For every $n\in\mathbb{N}$
let $u_{n}$ be the function provided by Corollary 2.2 for
$\varepsilon=\varepsilon_{n}=\frac{1}{n}$ . By (2.4), we have that, for all
$s\in[1,+\infty)$,

(2.8) $\mathcal{L}^{2}(\\{u\neq
u_{n}\\})\leq\frac{1}{n},\qquad\|u-u_{n}\|_{L^{s}(\Omega;\mathbb{R}^{2})}\leq\frac{1}{n^{\frac{1}{s}}},$

and

(2.9) $\displaystyle\|\nabla u-\nabla
u_{n}\|_{L^{1}(\Omega;\mathbb{R}^{2\times 2})}\leq$
$\displaystyle\,\big{(}\|\nabla u\|_{L^{p}(\Omega;\mathbb{R}^{2\times
2})}+\|\nabla u_{n}\|_{L^{p}(\Omega;\mathbb{R}^{2\times
2})}\big{)}\big{(}|\widetilde{\omega}_{n}|+|\widehat{\omega}_{n}|\big{)}^{\frac{1}{p^{\prime}}}$
$\displaystyle\leq$ $\displaystyle\,\frac{C}{n^{2}}\|\nabla
u\|_{L^{p}(\Omega;\mathbb{R}^{2\times 2})}\,.$

Furthermore, since

$\mathrm{D}^{\mathrm{S}}u=(u^{+}-u^{-})\otimes\bm{\nu}_{u}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
S_{u}\;\mathrm{d}\mathcal{H}^{1},\qquad\qquad\mathrm{D}^{\mathrm{S}}u_{n}=(u_{n}^{+}-u_{n}^{-})\otimes\bm{\nu}_{u_{n}}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
S_{u_{n}}\;\mathrm{d}\mathcal{H}^{1}\,,$

by triangle inequality, using that $\bm{\nu}_{u}=\bm{\nu}_{u_{n}}$ on
$S_{u}\cap S_{u_{n}}$, we get

(2.10)
$\displaystyle|\mathrm{D}^{\mathrm{S}}u-\mathrm{D}^{\mathrm{S}}u_{n}|(\Omega)\leq$
$\displaystyle\,\big{(}|u^{+}-u^{+}_{n}|+|u^{-}-u^{-}_{n}|\big{)}\mathcal{H}^{1}(S_{u}\cap
S_{u_{n}})$ $\displaystyle\,+|u^{+}-u^{-}|\mathcal{H}^{1}(S_{u}\setminus
S_{u_{n}})+|u_{n}^{+}-u_{n}^{-}|\mathcal{H}^{1}(S_{u_{n}}\setminus S_{u})$
$\displaystyle\leq$
$\displaystyle\,4\mathcal{H}^{1}(\Gamma_{n}\cap\\{u_{n}^{+}\neq
u^{+}\\})+4\mathcal{H}^{1}(\Gamma_{n}\cap\\{u_{n}^{-}\neq
u^{-}\\})+2\mathcal{H}^{1}(S_{u}\triangle S_{u_{n}})$ $\displaystyle\leq$
$\displaystyle\,\frac{C}{n},$

where in the last inequality we have used (2.5) and the fact that
$S_{u_{n}}\subset\Gamma_{n}\cup\partial\widetilde{\omega}_{n}$, to deduce that

$\displaystyle\mathcal{H}^{1}(S_{u}\triangle S_{u_{n}})\leq$
$\displaystyle\,\mathcal{H}^{1}(\Gamma_{n}\cap(S_{u}\setminus
S_{u_{n}}))+\mathcal{H}^{1}(S_{u}\triangle\Gamma_{n})+\mathcal{H}^{1}(\partial\widetilde{\omega}_{n})$
$\displaystyle\leq$
$\displaystyle\,\mathcal{H}^{1}(\Gamma_{n}\cap\\{u_{n}^{+}\neq
u^{+}\\})+\mathcal{H}^{1}(\Gamma_{n}\cap\\{u_{n}^{-}\neq
u^{-}\\})+\mathcal{H}^{1}(S_{u}\triangle\Gamma_{n})+\mathcal{H}^{1}(\partial\widetilde{\omega}_{n})\leq\frac{2}{n}.$

By (2.8), (2.9), (2.10), we immediately deduce the first condition in (2.6),
whereas the other two easily follow from (2.5). Notice that (2.6) together
with the fact that $u_{n},u\in SBV^{p}(\Omega;\mathbb{S}^{1})$ implies that

(2.11) $\int_{S_{u}\cap
S_{u_{n}}}|[u]-[u_{n}]|\otimes\bm{\nu}_{u}\;\mathrm{d}\mathcal{H}^{1}\to
0\quad\textrm{and}\quad\mathcal{H}^{1}(S_{u}\triangle S_{u_{n}})\to
0\qquad\textrm{as }n\to+\infty.$

Furthermore, using triangle inequality, (2.8) and (2.9), it is easy to check
that

$|T^{D}_{u}-T^{D}_{u_{n}}|(\Omega)\leq 4\|\nabla u\|_{L^{1}(\\{u\neq
u_{n}\\};\mathbb{R}^{2\times
2})}\|u-u_{n}\|_{L^{\infty}(\Omega;\mathbb{R}^{2})}+4\|u_{n}\|_{L^{\infty}(\Omega;\mathbb{R}^{2})}\|\nabla
u-\nabla u_{n}\|_{L^{1}(\Omega;\mathbb{R}^{2\times 2})}\to 0$

and that

$\displaystyle|T^{S}_{u}-T^{S}_{u_{n}}|(\Omega)\leq$
$\displaystyle\frac{1}{2}\int_{S_{u}\cap S_{u_{n}}}|u^{+}_{n}\wedge
u^{-}_{n}-u^{+}\wedge u^{-}|\;\mathrm{d}\mathcal{H}^{1}$
$\displaystyle\,+\frac{1}{2}\int_{S_{u}\setminus S_{u_{n}}}|u^{+}\wedge
u^{-}|\;\mathrm{d}\mathcal{H}^{1}+\frac{1}{2}\int_{S_{u_{n}}\setminus
S_{u}}|u_{n}^{+}\wedge
u_{n}^{-}|\;\mathrm{d}\mathcal{H}^{1}\leq\mathcal{H}^{1}(\\{[u]\neq[u_{n}]\\})\leq\frac{3}{n},$

since
$S_{u_{n}}\subset\Gamma_{n}\cup\partial\widetilde{\omega}_{n}\cup\partial^{*}\widehat{\omega}_{n}$
and $\\{[u]\neq[u_{n}]\\}=(S_{u}\triangle S_{u_{n}})\cup(S_{u}\cap
S_{u_{n}}\cap\\{[u]\neq[u_{n}]\\})$. ∎

We now turn to the proof of Theorem 2.1.

###### Proof of Theorem 2.1.

Let $\varrho$, $\alpha_{1}$, $\alpha_{2}$ be small positive constants to be
determined later. We divide the proof into three steps.

Step 1: Covering the jump set. Since the sets

$\widehat{S}_{u}^{0}:=S_{u}^{\mathrm{frac}}\qquad\text{and}\qquad\widehat{S}_{u}^{z}:=\\{x\in
S_{u}\colon[u](x)=z\\}$

for all $z\in\mathbb{Z}^{*}:=\mathbb{Z}\setminus\\{0\\}$ are countably
$(\mathcal{H}^{1},1)$ rectifiable with finite $\mathcal{H}^{1}$ measure, by
[31, Theorem, 3.2.29] for every $z\in\mathbb{Z}$ there exists a countable
family $\\{M_{k}^{z}\\}_{k\in\mathbb{N}}$ such that

(2.12)
$\mathcal{H}^{1}\Big{(}\widehat{S}_{u}^{z}\setminus\bigcup_{k=1}^{\infty}M_{k}^{z}\Big{)}=0$

and, by [3, Theorem 2.76] we may assume that for each $z\in\mathbb{Z}$ and
$k\in\mathbb{N}$ the manifold $M_{k}^{z}$ is a graph of a $C^{1}$ and
Lipschitz function with Lipschitz constant less than $\alpha_{1}$ . Let us fix
$z\in\mathbb{Z}$ such that $\mathcal{H}^{1}(\widehat{S}_{u}^{z})>0$ (in
particular, for the application of the present theorem in this paper, this
holds for $z=0$). Then, since for $\mathcal{H}^{1}$-a.e.
$x\in\widehat{S}_{u}^{z}\cap M_{k}^{z}$, $x$ has $\mathcal{H}^{1}$-density 1
both for $\widehat{S}_{u}^{z}$ and $M_{k}^{z}$, for every $k\in\mathbb{N}$ and
every such $x\in\widehat{S}_{u}^{z}\cap M_{k}^{z}$ there exists
$\eta(\alpha_{2},x)\in(0,\varrho)$ such that

(2.13)
$\displaystyle|\mathcal{H}^{1}(\overline{Q}_{r}(x)\cap\widehat{S}_{u}^{z})-2r|\leq
2\alpha_{2}r,\quad|\mathcal{H}^{1}(\overline{Q}_{r}(x)\cap M_{k}^{z})-2r|\leq
2\alpha_{2}r,$
$\displaystyle|\mathcal{H}^{1}(\overline{Q}_{r}(x)\cap(\widehat{S}_{u}^{z}\cap
M_{k}^{z}))-2r|\leq 2\alpha_{2}r.$

for every $r\leq\eta(\alpha_{2},x)$, and moreover

(2.14) $\mathcal{H}^{1}(\overline{Q}_{r}(x)\cap(\widehat{S}_{u}^{z}\triangle
M_{k}^{z}))\leq\alpha_{2}\mathcal{H}^{1}(\overline{Q}_{r}(x)\cap\widehat{S}_{u}^{z}),$

for every $r\leq\eta(\alpha_{2},x)$; here we recall that $Q_{r}(x)$ denotes
the (open) cube $Q^{\nu(x)}_{r}(x)$, centered at $x$, with sidelength $2r$ and
with a side normal to $\nu(x)$, the approximate normal to $S_{u}$ (or
$\widehat{S}_{u}^{z}$) at $x$. We notice that (2.13) holds also for $S_{u}$ in
place of $\widehat{S}_{u}^{z}$ or $\widehat{S}_{u}^{z}\cap M_{k}^{z}$, that
is, for $\mathcal{H}^{1}$-a.e. $x\in S_{u}$, we may also assume

(2.15) $|\mathcal{H}^{1}(\overline{Q}_{r}(x)\cap S_{u})-2r|\leq 2\alpha_{2}r$

for every $r\leq\eta(\alpha_{2},x)$. Now we introduce

$M:=S_{u}\cap\bigcup_{z\in\mathbb{Z},\,k\in\mathbb{N}}M_{k}^{z}.$

We also denote by $\widetilde{M}\subset M$, the set of points $x$ satisfying
(2.13), (2.14), and (2.14). From what observed,
$\mathcal{H}^{1}(M\setminus\widetilde{M})=0$; so, since the family
$\\{\overline{Q}_{r}(x)\colon x\in M,\,r\leq\eta(\alpha_{2},x)\\}$ is a fine
cover of $\widetilde{M}$, Vitali-Besicovitch’s Covering Theorem (see [30,
Theorem 1.10] for its version for cubes) ensures the existence of a disjoint
subfamily $\\{\overline{Q}_{r(\alpha_{2},x)}(x)\colon x\in M^{\prime}\\}$, for
a countable set
$M^{\prime}=\\{x_{j}\\}_{j\in\mathbb{N}}\subset\widetilde{M}\subset M$ and
$r(\alpha_{2},x)\leq\eta(\alpha_{2},x)$, such that

$\mathcal{H}^{1}\Big{(}S_{u}\setminus\bigcup_{j\in\mathbb{N}}\overline{Q}_{j}\Big{)}=0,$

where we have noted $Q_{j}:=Q_{r_{j}}(x_{j})$ and $r_{j}:=r(\alpha_{2},x_{j})$
for every $j\in\mathbb{N}$. Then there exists $J=J(\alpha_{2})\in\mathbb{N}$
such that

(2.16)
$\mathcal{H}^{1}\Big{(}S_{u}\setminus\bigcup_{j=1}^{J}\overline{Q}_{j}\Big{)}<\alpha_{2}.$

For every $j\in\\{1,\dots,J\\}$, let $k_{j}\in\mathbb{N}$,
$z_{j}\in\mathbb{Z}$ be the indeces such that,
$x_{j}\in\widehat{S}_{u}^{z_{j}}\cap M_{k_{j}}^{z_{j}}$, and (2.13), (2.14),
(2.15) hold (for all $j$, such indeces are unique). Then we set

$\Gamma_{j}:=\overline{Q}_{j}\cap M_{k_{j}}^{z_{j}}\;.$

We now see that for every $j\in\\{1,\dots,J\\}$, there hold

(a) $\Gamma_{j}$ is the graph of a $C^{1}$ and Lipschitz function with
Lipschitz constant less than $\alpha_{1}$; (b)
$|\mathcal{H}^{1}(\overline{Q}_{r}(x_{j})\cap\Gamma_{j})-2r|\leq 2\alpha_{2}r$
for all $0<r\leq r_{j}$; (c)
$\mathcal{H}^{1}(\overline{Q}_{j}\cap(\widehat{S}^{z_{j}}_{u}\triangle\Gamma_{j}))+\mathcal{H}^{1}(\overline{Q}_{j}\cap(S_{u}\setminus{\widehat{S}}^{z_{j}}_{u}))\leq
8\alpha_{2}r_{j}$; (d)
$\mathcal{H}^{1}\Big{(}S_{u}\triangle\bigcup_{j=1}^{J}\Gamma_{j}\Big{)}\leq\alpha_{2}(1+5\mathcal{H}^{1}(S_{u}))$;
(e) $\mathcal{L}^{2}\Big{(}\bigcup_{j=1}^{J}\overline{Q}_{j}\Big{)}\leq
3\varrho\,\mathcal{H}^{1}(S_{u})$.

Property (a) follows by definition, since $M^{z}_{k}$ are graphs of Lipschitz
maps with constant less than $\alpha_{1}$. Property (b) follows immediately
from (2.13). As for the proof of (c), by (2.14) and (2.13), choosing

(2.17) $\displaystyle\alpha_{2}\leq 1\;,$

we have that

(2.18)
$\mathcal{H}^{1}(\overline{Q}_{j}\cap(\widehat{S}^{z_{j}}_{u}\triangle\Gamma_{j}))\leq\alpha_{2}\mathcal{H}^{1}(\overline{Q}_{j}\cap\widehat{S}^{z_{j}}_{u})\leq
2\alpha_{2}(1+\alpha_{2})r_{j}\leq 4\alpha_{2}r_{j}\,;$

moreover, by (2.15) and (2.13) we have that

$\displaystyle\mathcal{H}^{1}(\overline{Q}_{j}\cap(S_{u}\setminus\widehat{S}^{z_{j}}_{u}))=$
$\displaystyle\mathcal{H}^{1}(\overline{Q}_{j}\cap
S_{u})-\mathcal{H}^{1}(\overline{Q}_{j}\cap\widehat{S}_{u}^{z_{j}})$
$\displaystyle\leq$
$\displaystyle\,2r_{j}(1+\alpha_{2})-2r_{j}(1-\alpha_{2})=4\alpha_{2}r_{j}\,,$

which, combined with (2.18), yields property (c). Property (d) follows from
the decomposition

$S_{u}\setminus\bigcup_{j=1}^{J}\Gamma_{j}=\left(S_{u}\setminus\bigcup_{j=1}^{J}\overline{Q}_{j}\right)\cup\bigcup_{j=1}^{J}\big{(}\overline{Q}_{j}\cap(S_{u}\setminus\Gamma_{j})\big{)},$

combined with (2.16) and the estimate

$\displaystyle\mathcal{H}^{1}\big{(}\overline{Q}_{j}\cap(S_{u}\triangle\Gamma_{j})\big{)}=$
$\displaystyle\,\mathcal{H}^{1}\big{(}\overline{Q}_{j}\cap(\widehat{S}_{u}^{z_{j}}\setminus\Gamma_{j})\big{)}+\mathcal{H}^{1}\big{(}\overline{Q}_{j}\cap((S_{u}\setminus\widehat{S}_{u}^{z_{j}})\setminus\Gamma_{j})\big{)}+\mathcal{H}^{1}\big{(}\overline{Q}_{j}\cap(\Gamma_{j}\setminus
S_{u})\big{)}$ $\displaystyle\leq$
$\displaystyle\,\mathcal{H}^{1}\big{(}\overline{Q}_{j}\cap(\widehat{S}_{u}^{z_{j}}\triangle\Gamma_{j})\big{)}+\mathcal{H}^{1}\big{(}\overline{Q}_{j}\cap(S_{u}\setminus\widehat{S}_{u}^{z_{j}})\big{)}$
$\displaystyle\leq$ $\displaystyle\,8\alpha_{2}r_{j}\leq
4\alpha_{2}\left(\mathcal{H}^{1}(\overline{Q}_{j}\cap
S_{u})+2\alpha_{2}r_{j})\right)$ $\displaystyle\leq$
$\displaystyle\,5\alpha_{2}\mathcal{H}^{1}(\overline{Q}_{j}\cap S_{u})$

recalling that the cubes $\overline{Q}_{j}$ are pairwise disjoint. Here the
first inequality follows from the fact that $\widehat{S}_{u}^{z_{j}}\subset
S_{u}$ , the second one from (c), the third one from (2.15), the fourth again
from (2.15) choosing

(2.19) $\displaystyle\alpha_{2}<\frac{1}{5}\;.$

Eventually, recalling that $r_{j}\leq\varrho$ for every $j$, by (2.15), since
$\alpha_{2}<\frac{1}{5}<\frac{1}{3}$, and using again that the cubes $Q_{j}$
are pairwise disjoint, we have

$\mathcal{L}^{2}\big{(}\bigcup_{j=1}^{J}\overline{Q}_{j}\big{)}=\sum_{j=1}^{J}4r_{j}^{2}\leq
2\varrho\sum_{j=1}^{J}\left(\mathcal{H}^{1}(\overline{Q}_{j}\cap
S_{u})+2\alpha_{2}r_{j}\right)\leq
3\varrho\sum_{j=1}^{J}\mathcal{H}^{1}(\overline{Q}_{j}\cap S_{u})\leq
3\varrho\mathcal{H}^{1}(S_{u}),$

from which (e) follows. Moreover, using (2.16) with (c), and arguing as done
to prove (d), we obtain that

$\widehat{\Gamma}^{\mathrm{int}}:=\bigcup_{j\colon z_{j}\neq
0}\Gamma_{j},\qquad\quad\widehat{\Gamma}^{\mathrm{frac}}:=\bigcup_{j\colon
z_{j}=0}\Gamma_{j},\quad\qquad\widehat{\Gamma}:=\widehat{\Gamma}^{\mathrm{int}}\cup\widehat{\Gamma}^{\mathrm{frac}}=\bigcup_{j=1}^{J}\Gamma_{j}$

are finite unions of disjoint $C^{1}$ curves and

(2.20)
$\mathcal{H}^{1}(S^{\mathrm{int}}_{u}\triangle\widehat{\Gamma}^{\mathrm{int}})+\mathcal{H}^{1}(S^{\mathrm{frac}}_{u}\triangle\widehat{\Gamma}^{\mathrm{frac}})\leq\alpha_{2}(1+5\mathcal{H}^{1}(S_{u})).$

Step 2: Approximation in the cubes $Q_{j}$. We perform two different
approximations depending on whether the cube $Q_{j}=Q_{r_{j}}(x_{j})$ is such
that $z_{j}=0$ or not. To shorten the notation, we set $\nu_{j}:=\nu(x_{j})$,
where $\nu(x_{j})$ is the approximate normal to $S_{u}$ at $x_{j}$.

Step 2.1: The case $z_{j}\neq 0$. This implies that
$[u](x_{j})=z_{j}\in\mathbb{Z}^{*}$ . In this case we first show that there
exists a “big” set of segments (parallel to $\nu_{j}^{\perp}$) in the cube
$\overline{Q}_{j}$ that do not intersect the jump set $S_{u}$ of $u$ and such
that (small) stripes centered at each of this segment contain a “few” portion
of $S_{u}\setminus\Gamma_{j}$ . To this end, denoting by
$x_{j}+\nu_{j}^{\perp}$ the straight line orthogonal to $\nu_{j}$ and passing
through $x_{j}$ , we define the (signed) distance from such a line as
$\mathrm{dist}(x,x_{j}+\nu_{j}^{\perp}):=(x-x_{j})\cdot\nu_{j}$ . Moreover,
for every $\gamma\in(-r_{j}/2,r_{j}/2)$ we define

$T_{j}^{\gamma}:=\overline{Q}_{j}\cap\\{\mathrm{dist}(\cdot,x_{j}+\nu_{j}^{\perp})=\gamma\\}$

and, for every $k\in\mathbb{N}$ , we set

$C^{\gamma,k}_{j}:=\overline{Q}_{j}\cap\\{\mathrm{dist}(\cdot,x_{j}+\nu_{j}^{\perp})\in[\gamma-2^{-k}r_{j},\gamma+2^{-k}r_{j}]\\}.$

Let

(2.21) $\tilde{\eta}:=\frac{1}{160}\eta,$

where $\eta$ is the constant from Proposition 2.5 below. We set
$S_{j}:=\overline{Q}_{j}\cap(S_{u}\setminus\Gamma_{j})$ . We claim that there
exists a set $I^{\tilde{\eta}}_{j}\subset(-r_{j}/2,r_{j}/2)$ with

(2.22)
$\mathcal{L}^{1}(I^{\tilde{\eta}}_{j})\leq\frac{160\,\alpha_{2}}{\tilde{\eta}}r_{j}$

such that, for every $\gamma\in(-r_{j}/2,r_{j}/2)\setminus
I^{\tilde{\eta}}_{j}$, it holds

(2.23) $\mathcal{H}^{1}(S_{j}\cap
C^{\gamma,k}_{j})<\tilde{\eta}2^{-(k+1)}r_{j}\quad\text{for all
}k\in\mathbb{N}$

and

(2.24) $\mathcal{H}^{1}(T^{\gamma}_{j}\cap S_{u})=0.$

Indeed, for $\delta_{j}^{k}:=2^{-k}r_{j}$, we argue as in the proof of [18,
Theorem 2.1], considering the family

(2.25)
$\mathscr{I}^{\tilde{\eta}}_{j}:=\left\\{[\gamma-\delta^{k}_{j}\color[rgb]{0,0,0},\gamma+\delta^{k}_{j}\color[rgb]{0,0,0}]\quad\colon\quad\mathcal{H}^{1}(S_{j}\cap
C^{\gamma,k}_{j})\geq\frac{\tilde{\eta}}{2}\delta^{k}_{j}\,,\quad
k\in\mathbb{N}\,,\quad\gamma\in(-r_{j}/2,r_{j}/2)\right\\}$

and $I^{\tilde{\eta}}_{j}:=\bigcup_{I\in\mathscr{I}^{\tilde{\eta}}_{j}}I$ . By
Vitali’s covering theorem, there exists a countable set
$\\{(\gamma^{l},k^{l})\\}_{l\in\mathbb{N}}$ such that the intervals
$[\gamma^{l}-\delta^{k^{l}}_{j}\color[rgb]{0,0,0},\gamma^{l}+\delta^{k^{l}}_{j}\color[rgb]{0,0,0}]$
in $\mathscr{I}^{\tilde{\eta}}_{j}$ are pairwise disjoint and

$I^{\tilde{\eta}}_{j}\subset\bigcup_{l\in\mathbb{N}}[\gamma^{l}-5\delta^{k^{l}}_{j}\color[rgb]{0,0,0},\gamma^{l}+5\delta^{k^{l}}_{j}\color[rgb]{0,0,0}]\,.$

By property (c) of $\Gamma_{j}$ we have

$8\alpha_{2}r_{j}\geq\mathcal{H}^{1}(S_{j}\cap
Q_{j})\geq\sum_{l\in\mathbb{N}}\mathcal{H}^{1}(S_{j}\cap
C_{j}^{\gamma^{l},k^{l}})\geq\sum_{l\in\mathbb{N}}\frac{\tilde{\eta}}{2}\delta_{j}^{k^{l}}\geq\frac{\tilde{\eta}}{20}\mathcal{L}^{1}({I}_{j}^{\tilde{\eta}}).$

Then (2.22) follows. By definition of $I^{\tilde{\eta}}_{j}$, every
$\gamma\in(-r_{j}/2,r_{j}/2)\setminus I^{\tilde{\eta}}_{j}$ does not belong to
any interval of the family (2.25) and then satisfies (2.23). Since
$\mathcal{H}^{1}(T_{j}^{\gamma}\cap S_{u})=0$ for every
$\gamma\in(-r_{j}/2,r_{j}/2)$ except at most countable many, we may enforce
also (2.24). Then the claim is confirmed.

Let us choose $\gamma_{j}\in(-r_{j}/2,r_{j}/2)\setminus I^{\tilde{\eta}}_{j}$
satisfying (2.23), (2.24) with

(2.26) $\gamma_{j}\in\left(0,\frac{161\alpha_{2}}{\tilde{\eta}}r_{j}\right).$

We consider the function

(2.27) $\widehat{u}_{j}:=u+z_{j}\chi_{H_{j}}\quad\text{in }\overline{Q}_{j},$

where $H_{j}\subset Q_{j}$ is the closed region delimited by $\Gamma_{j}$,
$T_{j}^{\gamma_{j}}$, and the two segments $\Sigma_{j}^{1}$,
$\Sigma_{j}^{2}\subset\partial Q_{j}$ joining the two couples of intersection
points of $\Gamma_{j}$ and $T_{j}^{\gamma_{j}}$ with the two boundary segments

$B_{j}^{\pm}:=\\{x_{j}\pm
r_{j}\tau_{j}+t\nu_{j}\colon|t|<r_{j}\\}\subset\partial Q_{j},$

being the ‘tangent vector’ $\tau_{j}$ the unit vector forming an angle of
$\pi/2$ with $\nu_{j}$. By property (c) we deduce that

(2.28) $\begin{split}\mathcal{H}^{1}(Q_{j}&\cap(S_{\widehat{u}_{j}}\setminus
T_{j}^{\gamma_{j}}))\color[rgb]{0,0,0}\leq\color[rgb]{0,0,0}\mathcal{H}^{1}(Q_{j}\cap(S_{u}\setminus\color[rgb]{0,0,0}\Gamma_{j}\color[rgb]{0,0,0}))+\mathcal{H}^{1}(\Gamma_{j}\setminus\widehat{S}_{u}^{z_{j}})\leq
8\alpha_{2}r_{j};\end{split}$

by (2.24) it holds that

(2.29)
$\mathcal{H}^{1}(T_{j}^{\gamma_{j}}\cap\\{[\widehat{u}_{j}]\neq\color[rgb]{0,0,0}z_{j}\color[rgb]{0,0,0}\\})=0.$

Further, for every $s\in(0,r_{j})$, denoting $Q_{j,s}:=x_{j}+Q_{s}$ and
$\Sigma_{j,s}^{1}$, $\Sigma_{j,s}^{2}\subset\partial Q_{j,s}$ the two segments
joining the two couples of intersection points of $\Gamma_{i}$ and
$T_{j}^{\gamma_{j}}$ with the boundary segments

$B_{j,s}^{\pm}:=\\{x_{j}\pm
s\tau_{j}+t\nu_{j}\colon|t|<r_{j}\\}\subset\partial Q_{j,s},$

in view of (a) and (2.26)

(2.30)
$\mathcal{H}^{1}(\Sigma_{j,s}^{1}\color[rgb]{0,0,0}\cup\color[rgb]{0,0,0}\Sigma_{j,s}^{2})\leq
2\left(\alpha_{1}s\color[rgb]{0,0,0}+\frac{161\alpha_{2}}{\tilde{\eta}}r_{j}\right).\color[rgb]{0,0,0}$

Arguing as done before to ensure (2.23), (2.24) for $\gamma$ outside a small
set, it is possible to find
$\widehat{r}_{j}\in((1-\sqrt{\alpha_{2}})r_{j},r_{j})$ (for $\alpha_{2}$ small
enough) such that, denoting

$\displaystyle Q^{+}_{j,\widehat{r}_{j}}:=\\{x\in
Q_{j}\colon(x-x_{j})\cdot\nu_{j}\in(\gamma_{j},\gamma_{j}+\widehat{r}_{j}),\,(x-x_{j})\cdot\tau_{j}\in(-\widehat{r}_{j},\widehat{r}_{j})\\},$
(2.31) $\displaystyle Q^{-}_{j,\widehat{r}_{j}}:=\\{x\in
Q_{j}\colon(x-x_{j})\cdot\nu_{j}\in(\gamma_{j}-\widehat{r}_{j},\gamma_{j}),\,(x-x_{j})\cdot\tau_{j}\in(-\widehat{r}_{j},\widehat{r}_{j})\\},$

it holds that

(2.32) $\begin{split}&\mathcal{H}^{1}(S_{\widehat{u}_{j}}\cap\partial
Q^{\pm}_{j,\widehat{r}_{j}})=0,\\\
&\mathcal{H}^{1}\Big{(}S_{\widehat{u}_{j}}\cap
Q^{\pm}_{j,\widehat{r}_{j}}\cap\big{(}\partial
Q^{\pm}_{j,\widehat{r}_{j}}+B_{2^{-k}r_{j}}(0)\big{)}\Big{)}<\tilde{\eta}2^{-(k+1)}r_{j}\text{
for every }k\in\mathbb{N}.\end{split}$

In fact, the same argument as above shows that there exist sets
$I_{\mathrm{hor}}^{\pm}$, $I_{\mathrm{ver}}^{\pm}\subset(-r_{j},r_{j})$ with
$\mathcal{L}^{1}(I_{\mathrm{hor}}^{\pm})\leq\frac{320\alpha_{2}}{\tilde{\eta}}r_{j}$,
$\mathcal{L}^{1}(I_{\mathrm{ver}}^{\pm})\leq\frac{320\alpha_{2}}{\tilde{\eta}}r_{j}$
such that, for

$\begin{split}&C_{\mathrm{hor},-}^{\widetilde{\gamma},k}:=\\{x\in
Q_{j}\colon(x-x_{j})\cdot\nu_{j}\in[\widetilde{\gamma},\widetilde{\gamma}+2^{-k}r_{j}]\\},\quad
C_{\mathrm{hor},+}^{\widetilde{\gamma},k}:=\\{x\in
Q_{j}\colon(x-x_{j})\cdot\nu_{j}\in[\widetilde{\gamma}-2^{-k}r_{j},\widetilde{\gamma}]\\},\\\
&C_{\mathrm{ver},-}^{\widetilde{\gamma},k}:=\\{x\in
Q_{j}\colon(x-x_{j})\cdot\tau_{j}\in[\widetilde{\gamma},\widetilde{\gamma}+2^{-k}r_{j}]\\},\quad
C_{\mathrm{ver},+}^{\widetilde{\gamma},k}:=\\{x\in
Q_{j}\colon(x-x_{j})\cdot\tau_{j}\in[\widetilde{\gamma}-2^{-k}r_{j},\widetilde{\gamma}]\\},\end{split}$

it holds that for every $\widetilde{\gamma}\notin I_{\mathrm{hor}}^{\pm}$,
$I_{\mathrm{ver}}^{\pm}$

$\mathcal{H}^{1}(S_{j}\cap
C_{\mathrm{hor},\pm}^{\widetilde{\gamma},k})\leq\tilde{\eta}2^{-(k+3)}r_{j},\quad\mathcal{H}^{1}(S_{j}\cap
C_{\mathrm{ver},\pm}^{\widetilde{\gamma},k})\leq\tilde{\eta}2^{-(k+3)}r_{j}\quad\text{
for every }k\in\mathbb{N}.$

Therefore, for $\alpha_{2}$ small enough it is possible to find

$\widehat{r}_{j}\in((1-\sqrt{\alpha_{2}})r_{j},r_{j})\quad\text{ such that
}\gamma_{j}\pm\widehat{r}_{j}\notin
I_{\mathrm{hor}}^{\pm},\,\pm\widehat{r}_{j}\notin I_{\mathrm{ver}}^{\pm},$

so the second condition in (2.32) is satisfied. Then (2.32) follows, since the
first condition holds true for every $\widehat{r}_{j}$ except at most
countably many.

Let111The intersection can be assumed to be nonempty, up to choosing
$\alpha_{2}$ smaller if necessary.

(2.33)
$\widehat{\delta}_{j}\in\Big{(}8\frac{\alpha_{2}}{\tilde{\eta}},16\frac{\alpha_{2}}{\tilde{\eta}}\Big{)}\cap\frac{\widehat{r}_{j}}{r_{j}\mathbb{N}},$

so that the rectangles $Q^{\pm}_{i,\widehat{r}_{j}}$ are partitioned into
cubes of sidelength $\widehat{\delta}_{j}r_{j}$. Moreover, let
$\widehat{k}_{j}\in\mathbb{N}$ be such that

(2.34)
$\widehat{\delta}_{j}\in[2^{-(\widehat{k}_{j}+1)},2^{-\widehat{k}_{j}}).$

By property (a), we have that

(2.35) $\mathcal{H}^{1}((Q_{j}\setminus
Q_{j,\widehat{r}_{j}})\cap\Gamma_{j})\leq 2\alpha_{1}\sqrt{\alpha_{2}}r_{j}.$

We now subdivide $Q^{\pm}_{j,\widehat{r}_{j}}$ into cubes whose sidelength
vanishes in a dyadic way towards the boundaries such that in any of them there
is a small amount of jump of $\widehat{u}_{j}$ compared to the sidelength, in
the sense of Proposition 2.5.

Let us assume, for simplicity of notation, that $x_{j}=0$ and
$\nu(x_{j})=e_{2}$. We introduce two sets ${\mathcal{Q}}_{i}^{\pm}$ of dyadic
squares of sidelength
$\widetilde{\delta}_{k}:=2^{-k}\widehat{\delta}_{j}\color[rgb]{0,0,0}\,r_{j}$,
$k\in\mathbb{N}$, which refine towards $\partial Q^{\pm}_{j,\widehat{r}_{j}}$,
as follows: let ${\mathcal{Q}}^{\pm}_{j,0}$ be the family of squares
$q\in\\{z+(0,\widehat{\delta}_{j}\color[rgb]{0,0,0}r_{j}]^{2}\colon
z\in\widehat{\delta}_{j}\color[rgb]{0,0,0}r_{j}\mathbb{Z}^{2}\\}$, $q\subset
Q^{\pm}_{j,\widehat{r}_{j}}$ such that $\mathrm{dist}(q,\partial
Q^{\pm}_{j,\widehat{r}_{j}})>\widehat{\delta}_{j}\color[rgb]{0,0,0}r_{j}$;
recursively, for $k\geq 1$, let ${\mathcal{Q}}^{\pm}_{j,k}$ be the family of
squares $q\in\\{z+(0,\widetilde{\delta}_{k}]^{2}\colon
x\in\widetilde{\delta}_{k}\mathbb{Z}^{2}\\}$, $q\subset
Q^{\pm}_{j,\widehat{r}_{j}}$ such that $\mathrm{dist}(q,\partial
Q^{\pm}_{j,\widehat{r}_{j}})>\widetilde{\delta}_{k}$ and $q$ does not
intersect any cube in ${\mathcal{Q}}_{j,l}^{\pm}$, for $l<k$; we define

${\mathcal{Q}}^{\pm}_{j}:=\bigcup_{k=0}^{\infty}{\mathcal{Q}}^{\pm}_{j,k}.$

For each $q\in{\mathcal{Q}}_{j}^{\pm}$ let $q^{\prime}$ and $q^{\prime\prime}$
denote squares concentric with $q$ with sidelength 10% and 20% longer,
respectively, so that $l(q^{\prime})=\frac{11}{12}l(q^{\prime\prime})$ and
$l(q^{\prime\prime})=\frac{6}{5}l(q)$; here and below, $l(\tilde{q})$ denotes
the sidelength of a cube $\tilde{q}$. By (2.21), (2.28), and (2.33), for any
$q\in{\mathcal{Q}}_{j,0}^{\pm}$, we get that

(2.36) $\mathcal{H}^{1}(q^{\prime\prime}\cap S_{\widehat{u}_{j}})\leq
8\alpha_{2}r_{j}\leq\tilde{\eta}\widehat{\delta}_{j}\color[rgb]{0,0,0}r_{j}<\eta\frac{1}{40}l(q)=\eta\frac{1}{10}\frac{5}{6}\frac{l(q^{\prime\prime})}{4}=\eta\Big{(}1-\frac{11}{12}\Big{)}\frac{l(q^{\prime\prime})}{4}\,,$

so that all the squares $q^{\prime\prime}$ “coming from” squares
$q\in{\mathcal{Q}}_{j,0}^{\pm}$ satisfy the hypotheses of Proposition 2.5 for
$s=\frac{11}{12}$.

Moreover, let $k\in\mathbb{N}$ . By (2.21), (2.32), (2.33), and (2.34), for
any $q\in{\mathcal{Q}}_{j,k}^{\pm}$ we have (since
$q^{\prime\prime}\subset\partial
Q^{\pm}_{j,\widehat{r}_{j}}+B_{2^{-(k+\widehat{k}_{j}-2)}r_{j}}(0)$)

(2.37) $\mathcal{H}^{1}(q^{\prime\prime}\cap
S_{\widehat{u}_{j}})\leq\tilde{\eta}2^{-(k+\widehat{k}_{j}\color[rgb]{0,0,0}-1)}r_{j}<\frac{\eta}{40}\widetilde{\delta}_{k}\color[rgb]{0,0,0}=\frac{\eta}{10}\frac{5}{6}\frac{l(q^{\prime\prime})}{4}=\eta\Big{(}1-\frac{11}{12}\Big{)}\frac{l(q^{\prime\prime})}{4}\,,$

so that all the squares $q^{\prime\prime}$ “coming from” squares
$q\in{\mathcal{Q}}_{j,k}^{\pm}$ satisfy the hypotheses of Proposition 2.5 for
$s=\frac{11}{12}$ . By (2.36) and (2.37) we thus deduce that all the squares
$q^{\prime\prime}$ “coming from” squares $q\in{\mathcal{Q}}_{j}^{\pm}$ satisfy
the hypotheses of Proposition 2.5 for $s=\frac{11}{12}$.

Therefore, by Proposition 2.5 applied to $\widehat{u}_{j}\in
SBV^{p}(q^{\prime\prime})$, for each $q\in{\mathcal{Q}}^{\pm}_{j}$ there is a
set of finite perimeter $\omega_{q}\subset q^{\prime\prime}$,
$\omega_{q}=\cup_{\mathcal{F}_{q}}B$ such that

(2.38) $\begin{split}w_{q}\in W^{1,p}(q^{\prime}),&\quad
w_{q}=\widehat{u}_{j}\text{ in }q^{\prime\prime}\setminus\omega_{q}\\\
\mathcal{H}^{1}(\partial^{*}\omega_{q})&\leq
c/\eta\mathcal{H}^{1}(S_{\widehat{u}_{j}}\cap q^{\prime\prime}),\\\
\int_{\omega_{q}}|\nabla w_{q}|^{p}\;\mathrm{d}x&\leq
c\int_{\omega_{q}}|\nabla\widehat{u}_{j}|^{p}\;\mathrm{d}x.\end{split}$

We define

$\omega_{j}^{\pm}:=\bigcup_{q\in{\mathcal{Q}}_{j}^{\pm}}\omega_{q}.$

Since the cubes $q^{\prime\prime}$ overlap at most 8 times, by the second and
third property in (2.38) we deduce that

(2.39) $\begin{split}\mathcal{H}^{1}(\partial^{*}\omega_{j}^{\pm})&\leq
8c/\eta\,\mathcal{H}^{1}(S_{\widehat{u}_{j}}\cap
Q_{j,\widehat{r}_{j}}^{\pm}),\\\ \int_{\omega_{j}^{\pm}}|\nabla
w_{q}|^{p}\;\mathrm{d}x&\leq
8c\int_{\omega_{j}^{\pm}}|\nabla\widehat{u}_{j}|^{p}\;\mathrm{d}x\color[rgb]{0,0,0}\end{split}$

Following [13, proof of Theorem 4.1], we construct regularized functions
$v_{j}^{\pm}$ on $Q_{j,\widehat{r}_{j}}^{\pm}$ which are convex combinations
of the functions $w_{q}$. We notice that in our setting all the cubes are
“good”, according to the definition in [13], that is the jump inside has small
$\mathcal{H}^{1}$-measure compared to the sidelength.

We set

(2.40) $v_{j}^{\pm}:=\sum_{q\in{\mathcal{Q}}_{j}^{\pm}}w_{q}\varphi_{q},$

where

$\begin{split}\varphi_{q}:=&\frac{\psi_{q}}{\sum_{\widehat{q}\in{\mathcal{Q}}_{j}^{\pm}}\psi_{\widehat{q}}},\quad\psi_{q}(x):=\psi\Big{(}\frac{x-c_{q}}{l(q)}\Big{)}\text{
for }q=c_{q}+\Big{(}-\frac{l(q)}{2},\frac{l(q)}{2}\Big{)}^{2},\\\ &\psi\in
C_{c}^{\infty}\Big{(}(-11/20,11/20)^{2};[0,1]\Big{)},\,\psi=1\text{ on
}[-1/2,1/2]^{2}.\end{split}$

By construction, $\psi_{q}\in C_{\mathrm{c}}^{\infty}(q^{\prime};[0,1])$ and
$\psi_{q}\equiv 1$ in $q$, for any $q\in{\mathcal{Q}}_{j}^{\pm}$. Since, by
(2.38), $w_{q}\in W^{1,p}(q^{\prime})$ for every
$q\in{\mathcal{Q}}_{j}^{\pm}$, we deduce that

(2.41) $v_{j}^{\pm}\in
W^{1,p}\Big{(}\bigcup_{q\in{\mathcal{Q}}_{j}^{\pm}}q^{\prime}\Big{)}.$

Eventually, we define

(2.42)
$v_{j}:=v_{j}^{+}\chi_{Q^{+}_{j,\widehat{r}_{j}}}+v_{j}^{-}\chi_{Q^{-}_{j,\widehat{r}_{j}}},\quad\omega_{j}:=\omega^{+}_{j}\cup\omega^{-}_{j}.$

By (2.28), (2.38), (2.39), (2.40), (2.42) it follows that

(2.43) $\begin{split}\int_{\omega_{j}}&|\nabla v_{j}|^{p}\;\mathrm{d}x\leq
8c\color[rgb]{0,0,0}\int_{\omega_{j}}|\nabla u|^{p}\;\mathrm{d}x,\quad
v_{j}=\widehat{u}_{j}\text{ in }Q_{j,\widehat{r}_{j}}\setminus\omega_{j},\\\
&\mathcal{H}^{1}(\partial^{*}\omega_{j})\leq
8\,c/\eta\mathcal{H}^{1}\big{(}Q_{j,\widehat{r}_{j}}\cap(S_{\widehat{u}_{j}}\setminus\widehat{S}_{j}^{\gamma_{j}})\big{)}\leq
64\,c/\eta\,\,\alpha_{2}r_{j}.\end{split}$

We observe that the first estimate above is obtained arguing as in [13, Step
3.3 in Theorem 5.1] with the full gradient in place of the symmetrized
gradient.

Furthermore, by construction we have that

(2.44) $v_{j}=\widehat{u}_{j}\text{ on }\partial
Q_{j,\widehat{r}_{j}},\quad[v_{j}]=[\widehat{u}_{j}]\text{ on
}T_{j}^{\gamma_{j}}.$

We observe that the latter property above follows from the fact that we
employed a Whitney-type approximation towards $T_{j}^{\gamma_{j}}$. In view of
(2.30),

(2.45) $\mathcal{H}^{1}(\partial Q_{j,\widehat{r}_{j}}\cap\\{v_{j}\neq
u\\})=\mathcal{H}^{1}(\Sigma_{j,\widehat{r}_{j}}^{1}\cap\Sigma_{j,\widehat{r}_{j}}^{2})\leq
2\alpha_{1}\widehat{r}_{j}+\frac{322\alpha_{2}}{\tilde{\eta}}r_{j}\color[rgb]{0,0,0},$

and, by property (b) and the definition of $T_{j}^{\gamma_{j}}$,

(2.46)
$|\mathcal{H}^{1}(Q_{j,\widehat{r}_{j}}\cap\Gamma_{j})-\mathcal{H}^{1}(Q_{j,\widehat{r}_{j}}\cap
T_{j}^{\gamma_{j}})|\leq 2\alpha_{2}\widehat{r}_{j}.$

Step 2.2: the case $z_{j}=0$. In this case, $x_{j}\in S^{\mathrm{frac}}_{u}$,
and as done before, we find radii
$\widehat{r}_{j}\in((1-\sqrt{\alpha_{2}})r_{j},r_{j})$ such that, denoting
$Q_{j,s}:=x_{j}+Q_{s}$, it holds that

(2.47) $\begin{split}&\mathcal{H}^{1}(S_{\widehat{u}_{j}}\cap\partial
Q_{j,\widehat{r}_{j}})=0,\\\ &\mathcal{H}^{1}\Big{(}S_{\widehat{u}_{j}}\cap
Q_{j,\widehat{r}_{j}}\cap\big{(}Q_{j,\widehat{r}_{j}}+B_{2^{-k}r_{j}}(0)\big{)}\Big{)}<\tilde{\eta}2^{-(k+1)}r_{j}\text{
for every }k\in\mathbb{N}.\end{split}$

By this choice, we can slightly amend the construction in [13, Theorem 4.1],
to find in both

$Q^{\pm}_{j,\widehat{r}_{j}}\quad\text{connected components of
}Q_{j,\widehat{r}_{j}}\setminus\Gamma_{j}$

two sets of finite perimeter $\omega_{j}^{\pm}$ and functions $v_{j}^{\pm}\in
W^{1,p}(Q^{\pm}_{j,\widehat{r}_{j}})$ such that, for suitable
$c_{j}^{\pm}=c_{j}^{\pm}(p)>0$,

(2.48) $\begin{split}&v_{j}^{\pm}=u\text{ in
}Q^{\pm}_{j,\widehat{r}_{j}}\setminus\omega_{j}^{\pm},\quad\int_{\omega_{j}^{\pm}}|\nabla
v_{j}^{\pm}|^{p}\;\mathrm{d}x\leq c_{j}^{\pm}\int_{\omega_{j}^{\pm}}|\nabla
u|^{p}\;\mathrm{d}x,\\\ &\mathcal{H}^{1}(\partial^{*}\omega_{j}^{\pm})\leq
c_{j}^{\pm}\mathcal{H}^{1}(S_{u}\cap
Q^{\pm}_{j,\widehat{r}_{j}}),\quad\mathcal{H}^{1}(\partial
Q_{j,\widehat{r}_{j}}\cap(\omega_{j}^{+}\cup\omega_{j}^{-}))=0.\end{split}$

We notice that the last condition is new with respect to [13, Theorem 4.1]: it
comes from the Whitney-type construction as in the previous substep, in turn
allowed by the choice of $\widehat{r}_{j}$ for which (2.32) holds, which is
possible in 2d. Moreover, as in [13, proof of Theorem 5.1, Step 2.2], one
proves that the constant

(2.49) $\tilde{c}:=\max\\{c_{j}^{\pm}\colon j\text{ s.t.\ }x_{j}\in
S_{u}^{\mathrm{frac}}\\}$

is bounded uniformly with respect to $\alpha_{2}$ (in particular, even if the
sidelenghts of cubes decrease and the number of cubes increases; notice that
increasing the number of cubes one may assume that the Lipschitz constant
corresponding to $\Gamma_{j}$ decreases). As above, we set

$v_{j}:=v_{j}^{+}\chi_{Q^{+}_{j,\widehat{r}_{j}}}+v_{j}^{-}\chi_{Q^{-}_{j,\widehat{r}_{j}}},\quad\omega_{j}:=\omega^{+}_{j}\cup\omega^{-}_{j}.$

Step 3: conclusion. Following the lines of [13, proof of Theorem 5.1, Step 3],
let us consider $\delta\in(0,0.4\,\sqrt{2}\alpha_{2}\min_{j=1,\dots,J}r_{j})$
and the families:

$\begin{split}&{\mathscr{Q}}_{1}:=\bigg{\\{}q_{z,\delta}=\delta
z+[0,\delta]^{2}\colon
z\in\mathbb{Z}^{2},\,q_{z,\delta}\cap\Big{(}\mathbb{R}^{2}\setminus\bigcup_{j=1}^{J}Q_{j,\widehat{r}_{j}}\Big{)}\neq\emptyset\bigg{\\}},\\\
&{\mathscr{Q}}_{2}:=\bigg{\\{}q_{z,\delta}=\delta z+[0,\delta]^{2}\colon
z\in\mathbb{Z}^{2},\,q_{z,\delta}\notin{\mathscr{Q}}_{1}\text{ and intersects
some cubes in }{\mathscr{Q}}_{1}\bigg{\\}},\\\
&{\mathscr{Q}}:={\mathscr{Q}}_{1}\cup{\mathscr{Q}}_{2}.\end{split}$

For each $q\in{\mathscr{Q}}$, let $q^{\prime}$ and $q^{\prime\prime}$ be the
(closed) cubes concentric with $q$ and having side length
$l(q^{\prime})=\frac{9}{8}\delta$ and
$l(q^{\prime\prime})=\frac{10}{8}\delta=\frac{10}{9}l(q^{\prime})$,
respectively. Let

(2.50) $\widehat{v}(x):=\begin{dcases}v_{j}(x),&\quad x\in
Q_{j,\widehat{r}_{j}},\\\ u(x),&\quad
x\in\Omega\setminus\bigcup_{j=1,\dots,J}Q_{i,\widehat{r}_{j}},\end{dcases}$

and, recalling the definition of $c$, $\eta$ from Proposition 2.5, set

${\mathscr{Q}}_{g}:=\Big{\\{}q\in{\mathscr{Q}}\,\colon\,\mathcal{H}^{1}(S_{\widehat{v}}\cap
q^{\prime\prime})\leq{\frac{1}{32}}\eta\delta\Big{\\}}{=\Big{\\{}q\in{\mathscr{Q}}\,\colon\,\mathcal{H}^{1}(S_{\widehat{v}}\cap
q^{\prime\prime})\leq\eta\Big{(}1-\frac{9}{10}\Big{)}\frac{l(q^{\prime\prime})}{4}\Big{\\}}},\quad{\mathscr{Q}}_{b}:={\mathscr{Q}}\setminus{\mathscr{Q}}_{g}.$

For every $q\in{\mathscr{Q}}_{g}$, by Proposition 2.5 applied to
$\widehat{v}\in SBV^{p}(q^{\prime\prime})$ (in correspondence to $s=0.9$)
there exist $w_{q}\in SBV^{p}(q^{\prime\prime})$ and $\omega_{q}\subset
q^{\prime\prime}$, $\omega_{q}=\cup_{\mathcal{F}_{q}}B$ such that

(2.51) $\begin{split}w_{q}\in W^{1,p}(q^{\prime}),&\quad
w_{q}=\widehat{v}\text{ in }q^{\prime\prime}\setminus\omega_{q}\\\
\mathcal{H}^{1}(\partial^{*}\omega_{q})&\leq
c/\eta\mathcal{H}^{1}(S_{\widehat{v}}\cap q^{\prime\prime}),\\\
\int_{\omega_{q}}|\nabla w_{q}|^{p}\;\mathrm{d}x&\leq
c\int_{\omega_{q}}|\nabla{\widehat{v}}|^{p}\;\mathrm{d}x.\end{split}$

Up to reducing the threshold in the definition of ${\mathscr{Q}}_{g}$, it
holds that if $q^{\prime}\cap\Gamma_{j}\neq\emptyset$ for some $j=1,\dots,J$,
then $q\notin{\mathscr{Q}}_{g}$, so that if $q\in{\mathscr{Q}}_{g}$ is such
that $q^{\prime}\subset Q_{j,\widehat{r}_{j}}$ it holds that
$q^{\prime}\subset Q^{\pm}_{j,\widehat{r}_{j}}$ and then $w_{q}=\widehat{v}$
(and $\omega_{q}\cap q^{\prime}=\emptyset$), since $\widehat{v}=v_{j}\in
W^{1,p}(Q^{\pm}_{j,\widehat{r}_{j}})$.

We set, recalling (2.42) (and the analogue for $j$ s.t. $x_{j}\in
S_{u}^{\mathrm{frac}}$)

$G:=\bigcup_{q\in{\mathscr{Q}}_{g}}q,\quad\widetilde{\omega}:=\bigcup_{q\in{\mathscr{Q}}_{b}}q,$
$\Gamma^{\mathrm{int}}:=\bigcup_{j\colon x_{j}\notin
S_{u}^{\mathrm{frac}}}(Q_{j,\widehat{r}_{j}}\cap
T_{j}^{\gamma_{j}}),\quad\Gamma^{\mathrm{frac}}:=\bigcup_{j\colon x_{j}\in
S_{u}^{\mathrm{frac}}}(Q_{j,\widehat{r}_{j}}\cap\Gamma_{j}),\qquad\widehat{\omega}:=\bigcup_{q\in{\mathscr{Q}}_{g}}\omega_{q}\cup\bigcup_{j=1,\dots,J}\omega_{j},$

and

(2.52)
$v:=\begin{dcases}\sum_{q\in{\mathscr{Q}}_{g}}w_{q}\varphi_{q},&\quad\text{in
}G,\\\ 0,&\quad\text{in }\widetilde{\omega},\\\ \widehat{v},&\quad\text{in
}\Omega\setminus(G\cup\widetilde{\omega}),\end{dcases}$

where

$\begin{split}\varphi_{q}:=&\frac{\psi_{q}}{\sum_{\widehat{q}\in{\mathcal{Q}}_{g}}\psi_{\widehat{q}}},\quad\psi_{q}(x):=\psi\Big{(}\frac{x-c_{q}}{l(q)}\Big{)}\text{
for }q=c_{q}+\Big{(}-\frac{l(q)}{2},\frac{l(q)}{2}\Big{)}^{2},\\\ &\psi\in
C_{\mathrm{c}}^{\infty}\Big{(}(-9/16,9/16)^{2};[0,1]\Big{)},\,\psi=1\text{ on
}[-1/2,1/2]^{2}.\end{split}$

By triangle inequality, (d), (b), (2.15) using that $0\leq
r_{j}-\widehat{r}_{j}\leq\sqrt{\alpha_{2}}r_{j}$ and that the cubes $Q_{j}$
are pairwise disjoint, we obtain

(2.53)
$\displaystyle|\mathcal{H}^{1}(S_{u}^{\mathrm{int}})-\mathcal{H}^{1}(\Gamma^{\mathrm{int}})|\leq$
$\displaystyle\,\mathcal{H}^{1}(S_{u}^{\mathrm{int}}\setminus\widehat{\Gamma}^{\mathrm{int}})+|\mathcal{H}^{1}(S_{u}^{\mathrm{int}}\cap\widehat{\Gamma}^{\mathrm{int}})-\mathcal{H}^{1}(\Gamma^{\mathrm{int}})|$
$\displaystyle\leq$
$\displaystyle\,\mathcal{H}^{1}(S_{u}^{\mathrm{int}}\triangle\widehat{\Gamma}^{\mathrm{int}})+\sum_{j:x_{j}\notin
S_{u}^{\mathrm{frac}}}|\mathcal{H}^{1}(S_{u}^{\mathrm{int}}\cap\Gamma_{j})-\mathcal{H}^{1}(T_{j}^{\gamma_{j}}\cap
Q_{j,\widehat{r}_{j}})|$ $\displaystyle\leq$
$\displaystyle\,\mathcal{H}^{1}(S_{u}^{\mathrm{int}}\triangle\widehat{\Gamma}^{\mathrm{int}})+\sum_{j:x_{j}\notin
S_{u}^{\mathrm{frac}}}\mathcal{H}^{1}(\Gamma_{j}\setminus
S_{u}^{\mathrm{int}})$
$\displaystyle\,\phantom{\mathcal{H}^{1}(S_{u}^{\mathrm{int}}\triangle\widehat{\Gamma}^{\mathrm{int}})}+\sum_{j:x_{j}\notin
S_{u}^{\mathrm{frac}}}|\mathcal{H}^{1}(\Gamma_{j})-2\widehat{r}_{j}|$
$\displaystyle\leq$
$\displaystyle\,\mathcal{H}^{1}(S_{u}^{\mathrm{int}}\triangle\widehat{\Gamma}^{\mathrm{int}})+\alpha_{2}(1+5\mathcal{H}^{1}(S_{u}))+(\alpha_{2}+\sqrt{\alpha_{2}})\mathcal{H}^{1}(S_{u}).$

Furthermore, by construction,

$\displaystyle S_{u}^{\mathrm{frac}}\triangle\Gamma^{\mathrm{frac}}=$
$\displaystyle\,\Big{(}S_{u}^{\mathrm{frac}}\setminus\bigcup_{j:x_{j}\in
S_{u}^{\mathrm{frac}}}\overline{Q}_{j}\Big{)}\cup\bigcup_{j:x_{j}\in
S_{u}^{\mathrm{frac}}}\big{(}\overline{Q}_{j}\cap(S_{u}^{\mathrm{frac}}\triangle(\Gamma_{j}\cap
Q_{j,\widehat{r}_{j}}))\big{)}$ $\displaystyle\subseteq$
$\displaystyle\,\Big{(}S_{u}^{\mathrm{frac}}\setminus\bigcup_{j:x_{j}\in
S_{u}^{\mathrm{frac}}}\overline{Q}_{j}\Big{)}\cup\bigcup_{j:x_{j}\in
S_{u}^{\mathrm{frac}}}\big{(}\overline{Q}_{j}\cap(S_{u}^{\mathrm{frac}}\triangle\Gamma_{j})\big{)}$
$\displaystyle\,\cup\bigcup_{j=1}^{J}\big{(}S_{u}\cap(\overline{Q}_{j}\setminus
Q_{j,\widehat{r}_{j}})\big{)}$ $\displaystyle=$
$\displaystyle\,\big{(}S_{u}^{\mathrm{frac}}\triangle\widehat{\Gamma}^{\mathrm{frac}}\big{)}\cup\bigcup_{j=1}^{J}\big{(}S_{u}\cap(\overline{Q}_{j}\setminus
Q_{j,\widehat{r}_{j}})\big{)}\,,$

whence, using (2.35) and property (d), we deduce that

(2.54)
$\displaystyle\mathcal{H}^{1}(S_{u}^{\mathrm{frac}}\triangle\Gamma^{\mathrm{frac}})\leq$
$\displaystyle\,\mathcal{H}^{1}(S_{u}^{\mathrm{frac}}\triangle\widehat{\Gamma}^{\mathrm{frac}})+\sum_{j=1}^{J}\mathcal{H}^{1}(S_{u}\cap(Q_{j}\setminus
Q_{j,\widehat{r}_{j}}))$ $\displaystyle\leq$
$\displaystyle\,\mathcal{H}^{1}(S_{u}^{\mathrm{frac}}\triangle\widehat{\Gamma}^{\mathrm{frac}})+\sum_{j=1}^{J}\mathcal{H}^{1}(\Gamma_{j}\cap(Q_{j}\setminus
Q_{j,\widehat{r}_{j}}))+\mathcal{H}^{1}\Big{(}S_{u}\triangle\bigcup_{j=1}^{J}\Gamma_{j}\Big{)}$
$\displaystyle\leq\,\mathcal{H}^{1}(S_{u}^{\mathrm{frac}}\triangle\widehat{\Gamma}^{\mathrm{frac}})+2\alpha_{1}\sqrt{\alpha_{2}}\mathcal{H}^{1}(S_{u})+\alpha_{2}(1+5\mathcal{H}^{1}(S_{u})).$

By summing (2.53) and (2.54), using (2.20), we obtain

$|\mathcal{H}^{1}(S_{u}^{\mathrm{int}})-\mathcal{H}^{1}(\Gamma^{\mathrm{int}})|+\mathcal{H}^{1}(S_{u}^{\mathrm{frac}}\triangle\Gamma^{\mathrm{frac}})\leq\alpha_{2}(3+16\mathcal{H}^{1}(S_{u}))+\sqrt{\alpha_{2}}(1+2\alpha_{1})\mathcal{H}^{1}(S_{u}).$

By (2.29) and (2.44) it follows that $[v](x)\in\mathbb{Z}$ for
$\mathcal{H}^{1}$-a.e. $x\in\Gamma^{\mathrm{int}}$.

By definition and (2.41), (2.48), (2.51) it is immediate that $v\in
SBV^{p}(\Omega)\cap
W^{1,p}(\Omega\setminus(\Gamma\cup\overline{\widetilde{\omega}}))$, that
$\\{\nabla u\neq\nabla v\\}\subset\widetilde{\omega}\cup\widehat{\omega}$
(since $\nabla(\widehat{u}_{j}-u)=0$ in $Q_{j}$, see (2.27)), that $\\{u\neq
v\\}\subset\\{\nabla u\neq\nabla v\\}\cup\bigcup_{j\colon z_{j}\neq 0}H_{j}$
(whose $\mathcal{L}^{2}$-measure vanishes with $\alpha_{2}$ and $\varrho$ from
property (e)), and that $v=0$ in $\widetilde{\omega}$.

Summing up (2.51) over $q\in{\mathscr{Q}}_{g}$ we obtain (since the cubes
$q^{\prime\prime}$ may overlap at most 8 times)

(2.55)
$\mathcal{H}^{1}(\partial^{*}\bigcup_{q\in{\mathscr{Q}}_{g}}\omega_{q})\leq
8(c\vee\tilde{c})/\eta\,\mathcal{H}^{1}\Big{(}S_{\widehat{v}}\setminus\bigcup_{j=1}^{J}Q_{j,\widehat{r}_{j}}\Big{)}\leq
C(\alpha_{1},\alpha_{2},p)$

($\tilde{c}$ is the constant in (2.49)) with $C(\alpha_{1},\alpha_{2},p)$
vanishes with $\alpha_{2}$ (for $\alpha_{1}\leq 1/4$), since

$S_{\widehat{v}}\setminus\bigcup_{j=1,\dots,J}Q_{j,\widehat{r}_{j}}\subset\Big{(}S_{u}\setminus\bigcup_{j=1,\dots,J}\overline{Q}_{j}\Big{)}\cup\bigcup_{j\colon
x_{j}\notin
S_{u}^{\mathrm{frac}}}(\Sigma_{j,\widehat{r}_{j}}^{1}\cup\Sigma_{j,\widehat{r}_{j}}^{2})\cup\bigcup_{j\colon
x_{j}\notin S_{u}^{\mathrm{frac}}}((Q_{j}\setminus
Q_{j,\widehat{r}_{j}})\cap(\Gamma_{j}\cup T_{j}^{\gamma_{j}})$

and from the properties of $\Gamma_{j}$, (2.30), (2.35), (2.45), (2.46).
Therefore, adding the estimates of the $\mathcal{H}^{1}$-measures of
$\partial^{*}\omega_{j}$ in (2.43) over $j$ such that $x_{j}\notin
S_{u}^{\mathrm{frac}}$ plus $\partial^{*}\omega_{j}^{\pm}$ in (2.48) over $j$
such that $x_{j}\in S_{u}^{\mathrm{frac}}$ together with (2.55), we conclude
that $\mathcal{H}^{1}(\partial^{*}\widehat{\omega})$ vanishes with
$\alpha_{2}$. In view of the definition of $\widetilde{\omega}$ (in particular
of ${\mathscr{Q}}_{b}$) we get

$\mathcal{H}^{1}(\partial\widetilde{\omega})\leq\frac{40}{9}\eta\,\mathcal{H}^{1}\Big{(}S_{\widehat{v}}\setminus\bigcup_{j=1,\dots,J}Q_{j,\widehat{r}_{j}}\Big{)},$

where above a factor 8 accounts for the overlapping of squares
$q^{\prime\prime}$; as well, $\mathcal{H}^{1}(\partial\widetilde{\omega})$
vanishes with $\alpha_{2}$ by (2.55).

Eventually, arguing as in [13, Step 3.1 in Theorem 5.1] for the cubes $Q_{j}$
such that $x_{j}\in S_{u}^{\mathrm{frac}}$ one proves that
$\mathcal{H}^{1}(\Gamma^{\mathrm{frac}}\cap\\{v^{\pm}\neq u^{\pm}\\})$
vanishes with $\alpha_{2}$, while (again following [13, Step 3.3 in Theorem
5.1] with the full gradient in place of the symmetrized gradient) one deduces
from the last estimate in (2.51) that

$\int_{\bigcup_{q\in{\mathscr{Q}}_{g}}\omega_{q}}|\nabla
v|^{p}\;\mathrm{d}x\leq
8c\color[rgb]{0,0,0}\int_{\bigcup_{q\in{\mathscr{Q}}_{g}}\omega_{q}}|\nabla\widehat{v}|^{p}\;\mathrm{d}x,$

which together with the estimates on the gradients in (2.43) and (2.48) gives
that

$\begin{split}\int_{\widehat{\omega}}|\nabla v|^{p}\;\mathrm{d}x\leq
C^{\prime}(\alpha_{2},p)\int_{\widehat{\omega}}|\nabla
u|^{p}\;\mathrm{d}x,\end{split}$

for $C^{\prime}(\alpha_{2},p)$ a positive constant vanishing with
$\alpha_{2}$. Being $\\{\nabla u\neq\nabla
v\\}\subset\widetilde{\omega}\cup\widehat{\omega}$, $v=0$ in
$\widetilde{\omega}$, and since the measure of $\widehat{\omega}$ vanishes
with $\alpha_{2}$, we obtain

$\begin{split}\int_{\Omega}|\nabla
v|^{p}\;\mathrm{d}x\leq(1+C^{\prime\prime}(\alpha_{2},\varrho,p))\int_{\Omega}|\nabla
u|^{p}\;\mathrm{d}x,\end{split}$

where $C^{\prime\prime}(\alpha_{2},p)>0$ vanishes with $\alpha_{2}$.

We conclude since $\alpha_{1}$, $\alpha_{2}$, $\varrho$ may be fixed
arbitrarily small. ∎

We recall a fundamental technical tool, [18, Proposition 3.2]. In [18] the
result is stated for balls, and it holds for cubes as well. Moreover, it holds
true also for $GSBD^{p}$ functions, in place of $SBD^{p}$ (see e.g. [17, Proof
of Proposition 3.1]). As usual, $Q_{\rho}:=(-\rho,\rho)^{2}$.

######  Proposition 2.4.

For every $p\in(1,\infty)$ there exist $c>0$ and $\eta\in(0,1)$ such that if
$u\in GSBD^{p}(Q_{2\varrho})$, $\varrho>0$, satisfies

$\mathcal{H}^{1}(S_{u}\cap Q_{2\varrho})<\eta(1-s)\varrho$

for some $s\in(0,1)$, then there is a countable family $\mathscr{F}=\\{B\\}$
of closed balls of radius $r_{B}<2(1-s)\varrho$ and center
$x_{B}\in\overline{Q}_{2s{\varrho}}$ such that their union is compactly
contained in $B_{2\varrho}$, and a field $w\in SBD^{p}(Q_{2\varrho})$ such
that

  * (i)

$\varrho^{-1}\sum_{B\in\mathscr{F}}\mathcal{L}^{2}(B)+\sum_{B\in\mathscr{F}}\mathcal{H}^{1}(\partial
B)\leq c/\eta\,\,\mathcal{H}^{1}(S_{u}\cap Q_{2\varrho})$;

  * (ii)

$\mathcal{H}^{1}(S_{u}\cap\cup_{B\in\mathscr{F}}\partial
B)=\mathcal{H}^{1}\big{(}(S_{u}\cap
Q_{2s\varrho})\setminus\cup_{B\in\mathscr{F}}B\big{)}=0$;

  * (iii)

$w=u$ $\mathcal{L}^{2}$-a.e. on
$Q_{2\varrho}\setminus\cup_{B\in\mathscr{F}}B$;

  * (iv)

$w\in W^{1,p}(Q_{2s\varrho};\mathbb{R}^{2})$ and
$\mathcal{H}^{1}(S_{w}\setminus S_{u})=0$;

  * (v)

(2.56) $\int_{\cup_{B\in\mathscr{F}}B}|e(w)|^{p}\;\mathrm{d}x\leq
c\int_{\cup_{B\in\mathscr{F}}B}|e(u)|^{p}\;\mathrm{d}x.$

The previous result may be directly modified to obtain a $SBV^{p}$ version.

######  Proposition 2.5.

For every $p\in(1,\infty)$ there exist $c>0$ and $\eta\in(0,1)$ such that if
$u\in SBV^{p}(Q_{2\varrho})$, $\varrho>0$, satisfies

$\mathcal{H}^{1}(S_{u}\cap Q_{2\varrho})<\eta(1-s)\varrho$

for some $s\in(0,1)$, then there is a countable family $\mathscr{F}=\\{B\\}$
of closed balls of radius $r_{B}<2(1-s)\varrho$ and center
$x_{B}\in\overline{Q}_{2s{\varrho}}$ such that their union is compactly
contained in $Q_{2\varrho}$, and a field $w\in SBV^{p}(Q_{2\varrho})$ such
that

  * (i)

$\varrho^{-1}\sum_{B\in\mathscr{F}}\mathcal{L}^{2}(B)+\sum_{B\in\mathscr{F}}\mathcal{H}^{1}(\partial
B)\leq c/\eta\,\,\mathcal{H}^{1}(S_{u}\cap Q_{2\varrho})$;

  * (ii)

$\mathcal{H}^{1}(S_{u}\cap\cup_{B\in\mathscr{F}}\partial
B)=\mathcal{H}^{1}\big{(}(S_{u}\cap
Q_{2s\varrho})\setminus\cup_{\mathscr{F}}B\big{)}=0$;

  * (iii)

$w=u$ $\mathcal{L}^{2}$-a.e. on
$Q_{2\varrho}\setminus\cup_{B\in\mathscr{F}}B$;

  * (iv)

$w\in W^{1,p}(Q_{2s\varrho})$ and $\mathcal{H}^{1}(S_{w}\setminus S_{u})=0$;

  * (v)

(2.57) $\int_{\cup_{B\in\mathscr{F}}B}|\nabla w|^{p}\;\mathrm{d}x\leq
c\int_{\cup_{B\in\mathscr{F}}B}|\nabla u|^{p}\;\mathrm{d}x.$

###### Proof.

We notice that it is enough to follow the proof of [18, Theorem 2.1], from
which [18, Proposition 3.2] follows, and use the fact that, if $u\in SBV^{p}$,
one can control the components of the constant matrix $\nabla\phi(u)$ in place
of those of $e(\phi(u))$ (see (2.12) in [18] and its consequences) by

$\nabla\phi(u)\cdot(x-y)=\phi(u)(x)-\phi(u)(y)=\int_{S_{x,y}}(u^{\nu}_{z})^{\prime}(t)\,\mathrm{d}t,$

where $u^{\nu}_{z}(t):=u(z+t\nu)$, for $\nu:=\frac{x-y}{|x-y|}$,
$z:=(\mathrm{Id}-\nu\otimes\nu)x$. Moreover, a constant in place of an
infinitesimal rigid motion appears in the Poincaré’s inequality for $u$ on
$Q_{\overline{x},\overline{y}}$. ∎

By arguing as in the proof of Theorem 2.1, using Proposition 2.4 in place of
Proposition 2.5, one can show that also the following result holds true.

######  Theorem 2.6.

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded open set of finite perimeter,
$p\in(1,+\infty)$, $u\in GSBD^{p}(\Omega)$, and $\varepsilon>0$. Then there
exist:

  * •

closed sets $\Gamma^{\mathrm{int}}$, $\Gamma^{\mathrm{frac}}$, finite unions
of disjoint $C^{1}$ curves;

  * •

a set $\tilde{\omega}$, finite union of cubes;

  * •

a set of finite perimeter $\widehat{\omega}$;

  * •

a function $v\in GSBD^{p}(\Omega)\cap
W^{1,p}(\Omega\setminus(\Gamma\cup\overline{\tilde{\omega}});\mathbb{R}^{2})$,
where $\Gamma:=\Gamma^{i}\cup\Gamma^{\mathrm{frac}}$;

such that $\\{\nabla u\neq\nabla
v\\}\subset\tilde{\omega}\cup\widehat{\omega}$, $\mathcal{L}^{2}(\\{u\neq
v\\})<\varepsilon$, $v=0$ in $\tilde{\omega}$, $[v](x)\in\mathbb{Z}^{2}$ for
$\mathcal{H}^{1}$-a.e. $x\in\Gamma^{\mathrm{int}}$, and

$\begin{split}|\mathcal{H}^{1}(S^{\mathrm{int}}_{u})-\mathcal{H}^{1}(\Gamma^{\mathrm{int}})|+\mathcal{H}^{1}(S^{\mathrm{frac}}_{u}\triangle\Gamma^{\mathrm{frac}})+\mathcal{H}^{1}(\partial\tilde{\omega})+\mathcal{H}^{1}(\partial^{*}\widehat{\omega})\leq\varepsilon,\quad\int_{\Omega\setminus\tilde{\omega}}|e(v)|^{p}\;\mathrm{d}x\leq(1+\varepsilon)\int_{\Omega}|e(u)|^{p}\;\mathrm{d}x,\end{split}$

where $S^{\mathrm{frac}}_{u}:=\\{x\in S_{u}\colon[u]\notin\mathbb{Z}^{2}\\}$.
Moreover, $\mathcal{H}^{1}(\Gamma\cap\\{v^{\pm}\neq
u^{\pm}\\})\leq\varepsilon$, where $v^{\pm}$ and $u^{\pm}$ denote the traces
of $v$ and $u$ on the two sides of $\Gamma$, and, if $u\in SBD^{p}(\Omega)$,
then also $v\in SBD^{p}(\Omega)$.

##  3\. Description of the problem

Let $\Omega$ be a bounded and open subset of $\mathbb{R}^{2}$ with Lipschitz
continuous boundary and let $\Omega^{\prime}\subset\subset\Omega$ be an open
set. We introduce

(3.1) $\mathcal{AD}(\Omega,\Omega^{\prime}):=\\{u\in
SBV^{2}(\Omega;\mathbb{S}^{1})\,:\,\overline{S}_{u}\subset\overline{\Omega^{\prime}}\\},$

where $S_{u}$ denotes the jump set of $u$. For every $\varepsilon>0$ , let
$\mathcal{G}_{\varepsilon}:{SBV^{2}(\Omega;\mathbb{S}^{1})}\to[0,\infty]$ be
the functional defined by

(3.2)
$\mathcal{G}_{\varepsilon}(u):=\left\\{\begin{array}[]{ll}\displaystyle\int_{\Omega}\frac{1}{2}|\nabla
u|^{2}\;\mathrm{d}x+\frac{1}{\varepsilon}\mathcal{H}^{1}(\overline{S}_{u})&\textrm{if
}u\in\mathcal{AD}(\Omega,\Omega^{\prime})\\\ +\infty&\textrm{elsewhere in
}{SBV^{2}(\Omega;\mathbb{S}^{1})}\,.\end{array}\right.$

In what follows, we will adopt also localized versions of the functional
$\mathcal{G}_{\varepsilon}$; more precisely, for any
$u\in\mathcal{AD}(\Omega,\Omega^{\prime})$ and for any open set $A$ with
$\Omega^{\prime}\subset\subset A\subset\subset\Omega$ , we will denote by
$\mathcal{G}_{\varepsilon}(u;A)$ the functional in (3.2) with $\Omega$
replaced by $A$ .

Notice that, since $u\in
H^{1}(\Omega\setminus\overline{\Omega}^{\prime};\mathbb{S}^{1})$, it follows
that

(3.3)
$\displaystyle\mathrm{supp}\,{Ju}\subseteq\overline{\Omega}^{\prime}\qquad\textrm{for
every }u\in\mathcal{AD}(\Omega,\Omega^{\prime}).$

Indeed, let $\varphi\in
C_{\mathrm{c}}^{\infty}(\Omega\setminus\overline{\Omega}^{\prime})$, and write

$\displaystyle\langle Ju,\varphi\rangle_{\Omega}$
$\displaystyle=\frac{1}{2}\int_{\Omega}\frac{\partial\varphi}{\partial
x_{2}}\;\mathrm{d}([u^{1}\mathrm{D}_{1}u^{2}]-[u^{2}\mathrm{D}_{1}u^{1}])-\frac{1}{2}\int_{\Omega}\frac{\partial\varphi}{\partial
x_{1}}\;\mathrm{d}([u^{1}\mathrm{D}_{2}u^{2}]-[u^{2}\mathrm{D}_{2}u^{1}])$
$\displaystyle=\frac{1}{2}\int_{\Omega\setminus\overline{\Omega}^{\prime}}\frac{\partial\varphi}{\partial
x_{2}}\Big{(}u^{1}\frac{\partial u^{2}}{\partial x_{1}}-u^{2}\frac{\partial
u^{1}}{\partial
x_{1}}\Big{)}\;\mathrm{d}x-\frac{1}{2}\int_{\Omega\setminus\overline{\Omega}^{\prime}}\frac{\partial\varphi}{\partial
x_{1}}\Big{(}u^{1}\frac{\partial u^{2}}{\partial x_{2}}-u^{2}\frac{\partial
u^{1}}{\partial x_{2}}\Big{)}\;\mathrm{d}x$ $\displaystyle=\langle
Ju,\varphi\rangle_{\Omega\setminus\overline{\Omega}^{\prime}}=0,$

where the last equality follows since $u\in
H^{1}(\Omega\setminus\overline{\Omega}^{\prime};\mathbb{S}^{1})$ has null
distributional Jacobian determinant in
$\Omega\setminus\overline{\Omega}^{\prime}$.

###  3.1. $\Gamma$-convergence in the subcritical regime

We introduce the class of atomic measures, namely

$X(\Omega):=\Big{\\{}\mu\in\mathcal{M}(\Omega)\,:\,\mu=\sum_{n=1}^{N}z^{n}\delta_{x^{n}},\;x^{n}\in\Omega,\;z^{n}\in\mathbb{Z}\setminus\\{0\\},\;N\in\mathbb{N}\Big{\\}}.$

In [28, Theorem 3.1], the authors show that the rescaled functional
$|\log\varepsilon|^{-1}\mathcal{G}_{\varepsilon}$ $\Gamma$-converges to the
functional $\mathcal{F}:X(\Omega)\rightarrow\mathbb{R}^{+}$ defined as
$\mathcal{F}(\mu)=\pi|\mu|(\Omega)$. Using a density argument, and in
particular Proposition 2.3, this result can be easily extended to the
following setting, where the energy functional does not take into account of
the closure of the jump set: We introduce

(3.4)
$\mathcal{F}_{\varepsilon}(u):=\left\\{\begin{array}[]{ll}\displaystyle\int_{\Omega}\frac{1}{2}|\nabla
u|^{2}\;\mathrm{d}x+\frac{1}{\varepsilon}\mathcal{H}^{1}({S}_{u})&\textrm{if
}u\in\mathcal{AD}(\Omega,\Omega^{\prime})\\\ +\infty&\textrm{elsewhere in
}{SBV^{2}(\Omega;\mathbb{S}^{1})}\,.\end{array}\right.$

Then the following $\Gamma$-convergence result holds:

######  Theorem 3.1.

Let $\Omega$ and $\Omega^{\prime}$ be as above; then there hold

  * (i)

(Compactness) Let
$\\{u_{\varepsilon}\\}_{\varepsilon}\subset{SBV^{2}(\Omega;\mathbb{S}^{1})}$
be such that

(3.5)
$\sup_{\varepsilon>0}\frac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})}{|\log\varepsilon|}\leq
C,$

for some $C>0$ . Then there exists $\mu\in X(\Omega)$ with
$\mathrm{supp}\,\mu\subseteq\overline{\Omega}^{\prime}$ such that, up to a
subsequence, $\|Ju_{\varepsilon}-\pi\mu\|_{\mathrm{flat},\Omega}\to 0$ (as
$\varepsilon\to 0$).

  * (ii)

($\Gamma$-liminf inequality) For every $\mu\in X(\Omega)$ with
$\mathrm{supp}\,\mu\subseteq\overline{\Omega}^{\prime}$ and for every
$\\{u_{\varepsilon}\\}_{\varepsilon}\subset{SBV^{2}(\Omega;\mathbb{S}^{1})}$
such that $\|Ju_{\varepsilon}-\pi\mu\|_{\mathrm{flat},\Omega}\to 0$ (as
$\varepsilon\to 0$) , it holds

(3.6) $\pi|\mu|(\Omega)\leq\liminf_{\varepsilon\to
0}\frac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})}{|\log\varepsilon|}\,.$

  * (iii)

($\Gamma$-limsup inequality) For every $\mu\in X(\Omega)$ with
$\mathrm{supp}\,\mu\subseteq\overline{\Omega}^{\prime}$, there exists
$\\{u_{\varepsilon}\\}_{\varepsilon}\subset{SBV^{2}(\Omega;\mathbb{S}^{1})}$
with $\|Ju_{\varepsilon}-\pi\mu\|_{\mathrm{flat},\Omega}\to 0$ (as
$\varepsilon\to 0$) , such that

(3.7) $\pi|\mu|(\Omega)\geq\limsup_{\varepsilon\to
0}\frac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})}{|\log\varepsilon|}\,.$

Actually, by arguing as above and going through the proof of [28, Theorem
3.1], one can prove the following more general result.

######  Theorem 3.2.

Let $\Omega$ and $\Omega^{\prime}$ be as above; and let
$\\{E_{\varepsilon}\\}_{\varepsilon}\subset(0,+\infty)$ with
$c|\log\varepsilon|\leq E_{\varepsilon}\ll|\log\varepsilon|^{2}$ for some
constant $c>0$ (independent of $\varepsilon$). Then the following
$\Gamma$-convergence result holds true.

  * (i)

(Compactness) Let
$\\{u_{\varepsilon}\\}_{\varepsilon}\subset{SBV^{2}(\Omega;\mathbb{S}^{1})}$
be such that

$\sup_{\varepsilon>0}\frac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})}{E_{\varepsilon}}\leq
C,$

for some $C>0$. Then there exists $\mu\in X(\Omega)$ with
$\mathrm{supp}\,\mu\subseteq\overline{\Omega}^{\prime}$ such that, up to a
subsequence,
$\|\frac{|\log\varepsilon|}{E_{\varepsilon}}Ju_{\varepsilon}-\pi\mu\|_{\mathrm{flat},\Omega}\to
0$ (as $\varepsilon\to 0$).

  * (ii)

($\Gamma$-liminf inequality) For every $\mu\in X(\Omega)$ with
$\mathrm{supp}\,\mu\subseteq\overline{\Omega}^{\prime}$ and for every
$\\{u_{\varepsilon}\\}_{\varepsilon}\subset{SBV^{2}(\Omega;\mathbb{S}^{1})}$
such that
$\|\frac{|\log\varepsilon|}{E_{\varepsilon}}Ju_{\varepsilon}-\pi\mu\|_{\mathrm{flat},\Omega}\to
0$ (as $\varepsilon\to 0$) , it holds

(3.8) $\pi|\mu|(\Omega)\leq\liminf_{\varepsilon\to
0}\frac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})}{E_{\varepsilon}}\,.$

  * (iii)

($\Gamma$-limsup inequality) For every $\mu\in X(\Omega)$ with
$\mathrm{supp}\,\mu\subseteq\overline{\Omega}^{\prime}$, there exists
$\\{u_{\varepsilon}\\}_{\varepsilon}\subset{SBV^{2}(\Omega;\mathbb{S}^{1})}$
with
$\|\frac{|\log\varepsilon|}{E_{\varepsilon}}Ju_{\varepsilon}-\pi\mu\|_{\mathrm{flat},\Omega}\to
0$ (as $\varepsilon\to 0$) , such that

(3.9) $\pi|\mu|(\Omega)\geq\limsup_{\varepsilon\to
0}\frac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})}{E_{\varepsilon}}\,.$

###### Proof of Theorem 3.1.

Although the argument is standard, we briefly discuss how to prove points (i)
and (ii), (iii) being identical to the case of [28]. Assume (3.5); by
Proposition 2.3 (applied to the domain $\Omega^{\prime}$), for all
$\varepsilon>0$ we choose $\widehat{u}_{\varepsilon}$ such that

$\displaystyle\int_{\Omega}\frac{1}{2}|\nabla\widehat{u}_{\varepsilon}|^{2}dx\leq\int_{\Omega}\frac{1}{2}|\nabla
u_{\varepsilon}|^{2}dx+\varepsilon,$
$\displaystyle\mathcal{H}^{1}(\overline{S}_{\widehat{u}_{\varepsilon}})=\mathcal{H}^{1}(S_{\widehat{u}_{\varepsilon}})\leq\mathcal{H}^{1}(S_{u_{\varepsilon}})+\varepsilon,$
(3.10) $\displaystyle\|J\widehat{u}_{\varepsilon}-Ju_{\varepsilon}\|_{{\rm
flat},\Omega}\leq\varepsilon,$

so that it follows

$\sup_{\varepsilon>0}\frac{\mathcal{G}_{\varepsilon}(\widehat{u}_{\varepsilon})}{|\log\varepsilon|}\leq
C+1.$

The compactness result in [28, Theorem 3.1 (i)] and the third condition in
(3.1) imply (i). In a similar way also (ii) is a consequence of [28, Theorem
3.1 (ii)] and of the same density result. ∎

We do not discuss the proof of Theorem 3.2 since it follows from the same
result with $\mathcal{G}_{\varepsilon}$ in place of
$\mathcal{F}_{\varepsilon}$, which in turn has the same proof of [28, Theorem
3.1].

###  3.2. $\Gamma$-convergence in the critical and supercritical regimes

Our main results are the following.

######  Theorem 3.3.

The following $\Gamma$-convergence result holds true.

  * (i)

(Compactness) Let
$\\{u_{\varepsilon}\\}_{\varepsilon}\subset{SBV^{2}(\Omega;\mathbb{S}^{1})}$
be such that

(3.11)
$\sup_{\varepsilon>0}\frac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})}{|\log\varepsilon|^{2}}\leq
C,$

for some $C>0$ . Then there exist a measure $\mu\in\mathcal{M}(\Omega)\cap
H^{-1}(\Omega)$ with $\mathrm{supp}\,\mu\subseteq\overline{\Omega}^{\prime}$
and a map $T^{D}\in L^{2}(\Omega;\mathbb{R}^{2})$ with
$-\mathrm{Div}\,T^{D}=\pi\mu$ such that, up to a subsequence,

(FJ)
$\displaystyle\big{\|}\frac{Ju_{\varepsilon}}{\pi|\log\varepsilon|}-\mu\big{\|}_{\mathrm{flat},\Omega}\to
0$ (ACJ)
$\displaystyle\frac{T^{D}_{u_{\varepsilon}}}{|\log\varepsilon|}\rightharpoonup
T^{D}\textrm{ in }L^{2}(\Omega;\mathbb{R}^{2})\,.$

  * (ii)

($\Gamma$-liminf inequality) For every
$(\mu,T^{D})\in\big{(}\mathcal{M}(\Omega)\cap H^{-1}(\Omega)\big{)}\times
L^{2}(\Omega;\mathbb{R}^{2})$ as in (i) and for every
$\\{u_{\varepsilon}\\}_{\varepsilon}\subset{SBV^{2}(\Omega;\mathbb{S}^{1})}$
satisfying (FJ) and (ACJ), it holds

(3.12)
$\pi|\mu|(\Omega)+2\int_{\Omega}|T^{D}|^{2}\;\mathrm{d}x\leq\liminf_{\varepsilon\to
0}\frac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})}{|\log\varepsilon|^{2}}\,.$

  * (iii)

($\Gamma$-limsup inequality) For every $(\mu,T^{D})\in(\mathcal{M}(\Omega)\cap
H^{-1}(\Omega))\times L^{2}(\Omega;\mathbb{R}^{2})$ as in (i) there exists
$\\{u_{\varepsilon}\\}_{\varepsilon}\subset{SBV^{2}(\Omega;\mathbb{S}^{1})}$
satisfying (FJ) and (ACJ), such that

(3.13)
$\pi|\mu|(\Omega)+2\int_{\Omega}|T^{D}|^{2}\;\mathrm{d}x\geq\limsup_{\varepsilon\to
0}\frac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})}{|\log\varepsilon|^{2}}\,.$

######  Theorem 3.4.

Let $\\{N_{\varepsilon}\\}_{\varepsilon>0}$ be such that $|\log\varepsilon|\ll
N_{\varepsilon}\ll\varepsilon^{-1}$. The following $\Gamma$-convergence result
holds true.

  * (i)

(Compactness) Let
$\\{u_{\varepsilon}\\}_{\varepsilon}\subset{SBV^{2}(\Omega;\mathbb{S}^{1})}$
be such that

(3.14)
$\sup_{\varepsilon>0}\frac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})}{N^{2}_{\varepsilon}}\leq
C,$

for some $C>0$ . Then there exist a field $T^{D}\in
L^{2}(\Omega;\mathbb{R}^{2})$ such that, up to a subsequence,
$\frac{T^{D}_{u_{\varepsilon}}}{N_{\varepsilon}}\rightharpoonup T^{D}$ in
$L^{2}(\Omega;\mathbb{R}^{2})$ .

  * (ii)

($\Gamma$-liminf inequality) For every $T^{D}\in L^{2}(\Omega;\mathbb{R}^{2})$
and for every
$\\{u_{\varepsilon}\\}_{\varepsilon}\subset{SBV^{2}(\Omega;\mathbb{S}^{1})}$
with $\frac{T^{D}_{u_{\varepsilon}}}{N_{\varepsilon}}\rightharpoonup T^{D}$ in
$L^{2}(\Omega;\mathbb{R}^{2})$ , it holds

(3.15) $2\int_{\Omega}|T^{D}|^{2}\;\mathrm{d}x\leq\liminf_{\varepsilon\to
0}\frac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})}{N^{2}_{\varepsilon}}\,.$

  * (iii)

($\Gamma$-limsup inequality) For every $T^{D}\in L^{2}(\Omega;\mathbb{R}^{2})$
there exists
$\\{u_{\varepsilon}\\}_{\varepsilon}\subset{SBV^{2}(\Omega;\mathbb{S}^{1})}$
with $\frac{T^{D}_{u_{\varepsilon}}}{N_{\varepsilon}}\rightharpoonup T^{D}$ in
$L^{2}(\Omega;\mathbb{R}^{2})$ such that

(3.16) $2\int_{\Omega}|T^{D}|^{2}\;\mathrm{d}x\geq\limsup_{\varepsilon\to
0}\frac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})}{N_{\varepsilon}^{2}}\,.$

By using the density result in Proposition 2.3, one can show that Theorems 3.3
and 3.4 hold true also when replacing $\mathcal{F}_{\varepsilon}$ with
$\mathcal{G}_{\varepsilon}$.

In order to prove Theorems 3.3 and 3.4, we will make use of the corresponding
core radius approach results that for the sake of completeness we state and
prove in Section 4 below.

##  4\. Core radius approach

We first introduce some notation. Let $V\subset\mathbb{R}^{2}$ be a bounded
and open set with Lipschitz continuous boundary. For every finite family of
pairwise (essentially222That is, whose closures are mutually disjoint.)
disjoint open balls $\mathscr{B}:=\\{B^{n}\\}_{n=1,\ldots,N}$ (with
$N\in\mathbb{N}$) we set

$V(\mathscr{B}):=V\setminus\bigcup_{n=1}^{N}\overline{B}^{n},$

and we denote by $\mathcal{R}ad(\mathscr{B})$ the sum of the radii of the
balls $B^{n}$, namely

$\mathcal{R}ad(\mathscr{B}):=\sum_{n=1}^{N}r(B^{n})\,,$

where $r(B)$ denotes the radius of the ball $B$ . Moreover, for every $\mu\in
X(V)$ with $\mu\neq 0$ of the form

(4.1) $\displaystyle\mu:=\sum_{n=1}^{N}z^{n}\delta_{x(B^{n})}\qquad\text{ with
}z^{n}\in{\mathbb{Z}}\setminus\\{0\\}\,,$

we set

(4.2) $\mathscr{A}(\mathscr{B},\mu,V):=\\{u\in
H^{1}(V(\mathscr{B});\mathbb{S}^{1})\,:\,\deg(u,\partial B^{n})=z^{n}\textrm{
for every }n=1,\ldots,N\\}\,.$

Here and below, $x(B)$ denotes the center of the ball $B$ .

######  Definition 4.1 (Merging procedure).

Given a finite family $\mathscr{B}=\\{B_{r^{i}}(x^{i})\\}_{i=1,\ldots,I}$
($I\in\mathbb{N}$) of balls in $\mathbb{R}^{2}$, we define a new family
$\widehat{\mathscr{B}}$ as follows. If the closures of two balls in
$\mathscr{B}$ are not disjoint, then we replace the two balls with a unique
ball which contains both of them and has radius less than or equal to the sum
of the radii of the original balls. After this, we repeat this replacement
recursively, until as all the balls in the family are mutually essentially
disjoint. The final family is $\widehat{\mathscr{B}}$. The procedure of
passing from $\mathscr{B}$ to $\widehat{\mathscr{B}}$ is called merging
procedure applied to $\mathscr{B}$. Notice that a merging procedure does not
increase the sum of all the radii of the balls in the family.

The following result is proven in [27, Proposition 2.2].

######  Proposition 4.2.

Let $V\subset\mathbb{R}^{2}$ be a bounded and open set with Lipschitz
continuous boundary, let $\mathscr{B}$ be a finite family of pairwise
essentially disjoint balls in $\mathbb{R}^{2}$, let $\mu\in X(V)$ be of the
form (4.1), and let $u\in\mathscr{A}(\mathscr{B},\mu,V)$ . Then, there exists
a one-parameter family of open balls $\mathscr{B}(t)$ with $t\geq 0$ such
that, setting $U(t):=\bigcup_{B\in\mathscr{B}(t)}B$, the following properties
hold true:

  1. (1)

$\mathscr{B}(0)=\mathscr{B}\,$;

  2. (2)

$U(t_{1})\subset U(t_{2})$ for any $0\leq t_{1}<t_{2}\,$;

  3. (3)

the balls in $\mathscr{B}(t)$ are pairwise (essentially) disjoint for every
$t>0$;

  4. (4)

for any $0\leq t_{1}<t_{2}$ and for any open set $A\subseteq V$ ,

$\frac{1}{2}\int_{(U(t_{2})\setminus\overline{U}(t_{1}))\cap A}|\nabla
u|^{2}\;\mathrm{d}x\geq\pi\sum_{\genfrac{}{}{0.0pt}{1}{B\in\mathscr{B}(t_{2})}{B\subseteq
A}}|\mu(B)|\log\frac{1+t_{2}}{1+t_{1}}\,;$

  5. (5)

for every $t>0$:
$\displaystyle\sum_{B\in\mathscr{B}(t)}r(B)\leq(1+t)\sum_{B\in\mathscr{B}}r(B)$,
where $r(B)$ denotes the radius of $B$ .

For every $\mathscr{B}$ and $\mu$ as in Proposition 4.2, for every $t>0$, we
set ${\mathscr{C}}(t):=\\{B\in\mathscr{B}(t)\,:\,\overline{B}\subset V\\}$ and
we define

(4.3) $\widetilde{\mu}:=\sum_{B\in{\mathscr{C}}(1)}\mu(B)\delta_{x(B)}\,.$

We can now state the crucial result which will be the starting point of the
proof of Theorem 3.3.

######  Theorem 4.3.

Let $V$ be a bounded open set with Lipschitz boundary. For every
$\varepsilon>0$ let
$\mathscr{B}_{\varepsilon}:=\\{B_{\varepsilon}^{n}\\}_{n=1,\ldots,N_{\varepsilon}}$
(with $N_{\varepsilon}\in\mathbb{N}$) be a (finite) family of pairwise
(essentially) disjoint open balls with
$\mathcal{R}ad(\mathscr{B}_{\varepsilon})\to 0$ as $\varepsilon\to 0$ ,
$\mu_{\varepsilon}:=\sum_{n=1}^{N_{\varepsilon}}z_{\varepsilon}^{n}\delta_{x(B_{\varepsilon}^{n})}$
with $z_{\varepsilon}^{n}\in\mathbb{Z}\setminus\\{0\\}$ for every
$n=1,\ldots,N_{\varepsilon}$ . Let moreover
$\\{u_{\varepsilon}\\}_{\varepsilon}$ be such that
$u_{\varepsilon}\in\mathscr{A}(\mathscr{B}_{\varepsilon},\mu_{\varepsilon},V)$
. Assume that

(4.4)
$\sup_{\varepsilon>0}\frac{1}{2|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|^{2}}\int_{V(\mathscr{B}_{\varepsilon})}|\nabla
u_{\varepsilon}|^{2}\;\mathrm{d}x\leq C\,,$

for some constant $C>0$ independent of $\varepsilon$ . Then, the following
facts hold true.

  * (i)

Let $\widetilde{\mu}_{\varepsilon}$ be the measures defined in (4.3) with
${\mathscr{C}}(1)={\mathscr{C}}_{\varepsilon}(1)=\\{B\in\mathscr{B}_{\varepsilon}(1)\,:\,\overline{B}\subset
V\\}$; then $|\widetilde{\mu}_{\varepsilon}|(V)\leq
C|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|^{2}$ for all $\varepsilon>0$
with a constant $C>0$ independent of $\varepsilon$, and there exist a measure
$\mu\in\mathcal{M}(V)$ and a function $\lambda\in L^{2}(V;\mathbb{R}^{2})$
such that, up to a subsequence, as $\varepsilon\to 0$

(4.5) $\displaystyle\frac{\widetilde{\mu}_{\varepsilon}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
V}{|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|}\stackrel{{\scriptstyle\mathrm{flat}}}{{\rightarrow}}\mu,$
(4.6)
$\displaystyle\frac{\lambda_{u_{\varepsilon}}\chi_{V(\mathscr{B}_{\varepsilon})}}{|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|}\rightharpoonup\lambda\qquad\textrm{
weakly in }L^{2}(V;\mathbb{R}^{2})\,;$

  * (ii)

$\pi|\mu|(V)+2\int_{V}|\lambda|^{2}\;\mathrm{d}x\leq\liminf_{\varepsilon\to
0}\frac{1}{2|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|^{2}}\int_{V(\mathscr{B}_{\varepsilon})}|\nabla
u_{\varepsilon}|^{2}\;\mathrm{d}x$ .

Notice that, as
$u_{\varepsilon}\in\mathscr{A}(\mathscr{B}_{\varepsilon},\mu_{\varepsilon},V)$,
we have $\lambda_{u_{\varepsilon}}\in
L^{2}(V(\mathscr{B}_{\varepsilon});\mathbb{R}^{2})$ (see (1.4)). In formula
(4.6), symbol $\lambda_{u_{\varepsilon}}\chi_{V(\mathscr{B}_{\varepsilon})}$
denotes the extension of $\lambda_{u_{\varepsilon}}$ to the constant $(0,0)$
in $V\setminus\overline{V(\mathscr{B}_{\varepsilon})}$.

###### Proof.

We start by proving (i). Our proof closely resembles that of [2, Theorem 3.2]
where the compactness result is proven in the energy regime
$|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|$ .

For every $0<p<1$ and for every $\varepsilon>0$ we set

(4.7)
$t^{p}_{\varepsilon}:=\frac{1}{\mathcal{R}ad^{1-p}(\mathscr{B}_{\varepsilon})}-1\,,\qquad\qquad\nu^{p}_{\varepsilon}:=\nu[t^{p}_{\varepsilon}]\,,$

where we have set, for $t\geq 0$,

$\nu[t]:=\sum_{B\in{\mathscr{C}}_{\varepsilon}(t)}\mu_{\varepsilon}(B)\delta_{x(B)}\,.$

Fix $0<p<1$ . Then, by applying Proposition 4.2(4) (with $t_{1}=0$ and
$t_{2}=1$ and $t_{2}=t^{p}_{\varepsilon}$) and by the energy bound (4.4), we
have that

(4.8) $|\widetilde{\mu}_{\varepsilon}|(V)\leq
C|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|^{2}\,,\qquad\qquad|\nu^{p}_{\varepsilon}|(V)\leq{C(1-p)^{-1}}|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|\,,$

whence we deduce the first statement in claim (i) and the existence of a
measure $\mu^{p}\in\mathcal{M}_{b}(V)$ such that (up to a not-relabeled
subsequence)

(4.9)
$\frac{\nu^{p}_{\varepsilon}}{|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu^{p}\qquad\textrm{as
$\varepsilon\to 0$}\,.$

Now we prove that

(4.10)
$\frac{1}{|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|}\big{(}\widetilde{\mu}_{\varepsilon}-\nu^{p}_{\varepsilon}\big{)}\stackrel{{\scriptstyle\mathrm{flat}}}{{\rightarrow}}0\qquad\textrm{for
every }0<p<1\,,$

from which we deduce also that $\mu^{p}\equiv\mu$ for any $0<p<1$ . To this
purpose, we first observe that
$\widetilde{\mu}_{\varepsilon}(B)=\nu^{p}_{\varepsilon}(B)$ for any
$B\in{\mathscr{C}}_{\varepsilon}(t^{p}_{\varepsilon})=\\{B\in\mathscr{B}_{\varepsilon}(t^{p}_{\varepsilon}):\overline{B}\subset
V\\}$ ; therefore, using (4.8) and Proposition 4.2(5) together with the very
definition of $t^{p}_{\varepsilon}$ , for every sequence
$\\{\varphi_{\varepsilon}\\}_{\varepsilon}\subset C_{\mathrm{c}}^{0,1}(V)$
with $\|\varphi_{\varepsilon}\|_{C^{0,1}}\leq 1$ , we have

$\displaystyle\frac{1}{|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|}\left|\big{\langle}\widetilde{\mu}_{\varepsilon}-\nu^{p}_{\varepsilon},\varphi_{\varepsilon}\big{\rangle}\right|\leq$
$\displaystyle\,\frac{1}{|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|}\left|\sum_{B\in{\mathscr{C}}_{\varepsilon}(t^{p}_{\varepsilon})}\int_{B}\Big{(}\varphi_{\varepsilon}-\fint_{B}\varphi_{\varepsilon}\;\mathrm{d}x\Big{)}\;\mathrm{d}(\widetilde{\mu}_{\varepsilon}-\nu^{p}_{\varepsilon})\right|$
$\displaystyle\,+\frac{1}{|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|}\left|\sum_{B\in\mathscr{B}_{\varepsilon}(t^{p}_{\varepsilon})\setminus{\mathscr{C}}_{\varepsilon}(t^{p}_{\varepsilon})}\int_{B\cap
V}\varphi_{\varepsilon}\;\mathrm{d}(\widetilde{\mu}_{\varepsilon}-\nu^{p}_{\varepsilon})\right|$
$\displaystyle\leq$
$\displaystyle\,\frac{1}{|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|}\sum_{B\in\mathscr{B}_{\varepsilon}(t^{p}_{\varepsilon})}\int_{B}\big{(}\max_{B}\varphi_{\varepsilon}-\min_{B}\varphi_{\varepsilon}\big{)}\big{(}|\widetilde{\mu}_{\varepsilon}|(B)+|\nu^{p}_{\varepsilon}|(B)\big{)}$
$\displaystyle\leq$
$\displaystyle\,\frac{1}{|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|}\sum_{B\in\mathscr{B}_{\varepsilon}(t^{p}_{\varepsilon})}\mathrm{diam}(B)\big{(}|\widetilde{\mu}_{\varepsilon}|(V)+|\nu^{p}_{\varepsilon}|(V)\big{)}$
$\displaystyle\leq$
$\displaystyle\,C\mathcal{R}ad^{p}(\mathscr{B}_{\varepsilon})|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|\,,$

whence (4.10) follows.

Moreover, by the very definition of $\lambda_{u_{\varepsilon}}$ in (1.4) and
by the energy bound (4.4), we immediately have that

(4.11)
$\frac{1}{2}\int_{V(\mathscr{B}_{\varepsilon})}|2\lambda_{u_{\varepsilon}}|^{2}\;\mathrm{d}x=\frac{1}{2}\int_{V(\mathscr{B}_{\varepsilon})}|\nabla
u_{\varepsilon}|^{2}\;\mathrm{d}x\leq
C|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|^{2}\,,$

thus, up to extracting a further subsequence, there exists $\lambda\in
L^{2}(V;\mathbb{R}^{2})$ such that (4.6) holds. Notice that, for $p\in(0,1)$
fixed, since $|U(t^{p}_{\varepsilon})|\rightarrow 0$ as
$\varepsilon\rightarrow 0$, we also deduce

(4.12)
$\frac{\lambda_{u_{\varepsilon}}}{|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|}\chi_{V(\mathscr{B}_{\varepsilon}(t^{p}_{\varepsilon}))}\rightharpoonup\lambda\qquad\text{
weakly in }L^{2}(V;\mathbb{R}^{2}).$

Now we prove (ii). To this end, let $p\in(0,1)$ be fixed; by (4.9) and by
Proposition 4.2(4), we get

(4.13) $\displaystyle\liminf_{\varepsilon\to
0}\frac{1}{2|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|^{2}}\int_{(U(t_{\varepsilon}^{p})\setminus
U(0))\cap V}|\nabla u_{\varepsilon}|^{2}\;\mathrm{d}x\geq$
$\displaystyle\,\pi{(1-p)}\liminf_{\varepsilon\to
0}\frac{|\nu_{\varepsilon}^{p}|(V)}{|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|}$
$\displaystyle\geq$ $\displaystyle\,\pi{(1-p)}|\mu|(V)\,.$

Furthermore, by (4.12), we have that

$\displaystyle\liminf_{\varepsilon\to
0}\frac{1}{2|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|^{2}}\int_{V(\mathscr{B}_{\varepsilon}(t_{\varepsilon}^{p}))}|\nabla
u_{\varepsilon}|^{2}\;\mathrm{d}x=$ $\displaystyle\liminf_{\varepsilon\to
0}\frac{1}{2}\int_{V}\Big{|}\frac{2\lambda_{u_{\varepsilon}}}{|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|}\Big{|}^{2}\chi_{V(\mathscr{B}_{\varepsilon}(t_{\varepsilon}^{p}))}\;\mathrm{d}x$
$\displaystyle\geq$ $\displaystyle\,2\int_{V}|\lambda|^{2}\;\mathrm{d}x\,,$

which, together with (4.13) (letting $p\to 0$), yields (ii). ∎

##  5\. Proof of Theorem 3.3

This section is devoted to the proof of Theorem 3.3. Using the density
argument as in the proof of Theorem 3.1, we can prove Theorem 3.3 with
$\mathcal{F}_{\varepsilon}$ replaced by $\mathcal{G}_{\varepsilon}$ in (3.2).

The proofs of the compactness and of the lower bound are addressed in
Subsection 5.1 and closely resemble those in the $|\log\varepsilon|$ regime
treated in [28], whereas the construction of the recovery sequence is provided
in Subsection 5.2.

###  5.1. Proof of compactness and lower bound

By the energy bound (3.11), together with Corollary 2.2, we have that

(5.1) $\mathcal{H}^{1}(\overline{S}_{u_{\varepsilon}})\leq
C\varepsilon|\log\varepsilon|^{2}\,,$

for every $\varepsilon>0$ . By the very definition of Hausdorff measure, since
$\overline{S}_{u_{\varepsilon}}$ is compact, there exists a finite family
$\mathscr{B}_{\varepsilon}$ of open balls (in $\mathbb{R}^{2}$) such that
$\overline{S}_{u_{\varepsilon}}\subset\bigcup_{B\in\mathscr{B}_{\varepsilon}}B$
and $\mathcal{R}ad(\mathscr{B}_{\varepsilon})\leq
C\varepsilon|\log\varepsilon|^{2}$ . Notice that we can always assume (just by
enlarging an arbitrarily chosen ball in $\mathscr{B}_{\varepsilon}$) that
$\mathcal{R}ad(\mathscr{B}_{\varepsilon})\geq\varepsilon$ , so that, from now
on

(5.2) $\varepsilon\leq\mathcal{R}ad(\mathscr{B}_{\varepsilon})\leq
C\varepsilon|\log\varepsilon|^{2}\,,$

for some $C>0$ . Moreover, by construction,

(5.3) $u_{\varepsilon}\in
H^{1}(\Omega(\mathscr{B}_{\varepsilon});\mathbb{S}^{1})\,,$

where we recall that
$\Omega(\mathscr{B}_{\varepsilon}):=\Omega\setminus\bigcup_{B\in\mathscr{B}_{\varepsilon}}\overline{B}$
. By (5.2) and recalling that
$\overline{S}_{u_{\varepsilon}}\subset\overline{\Omega^{\prime}}$ , we can
assume that, for $\varepsilon$ small enough,

(5.4) $\bigcup_{B\in\mathscr{B}_{\varepsilon}}\overline{B}\subset\Omega\,.$

Up to applying a merging procedure (as described in Definition 4.1) to the
balls in $\mathscr{B}_{\varepsilon}$ , we can assume without loss of
generality that these balls are mutually (essentially) disjoint, and still
satisfy (5.2) and (5.4). For $\varepsilon>0$ small enough we set

(5.5)
$\mu_{\varepsilon}:=\sum_{B\in\mathscr{B}_{\varepsilon}}\deg(u_{\varepsilon},\partial
B)\delta_{x(B)}\,.$

By (3.11), (5.2), and (5.3), for $\varepsilon$ small enough it holds

(5.6) $\frac{1}{2}\int_{\Omega(\mathscr{B}_{\varepsilon})}|\nabla
u_{\varepsilon}|^{2}\,\;\mathrm{d}x\leq\mathcal{G}_{\varepsilon}(u_{\varepsilon})\leq
C|\log\varepsilon|^{2}\leq
C|\log\mathcal{R}ad(\mathscr{B}_{\varepsilon})|^{2}\,.$

Therefore we can apply Theorem 4.3 to the family
$\\{(\mathscr{B}_{\varepsilon};\mu_{\varepsilon})\\}_{\varepsilon}$ . Notice
that, in view of the very definition of $\mathcal{G}_{\varepsilon}$ , we have
that also the family $\mathscr{B}_{\varepsilon}(1)$ satisfies (5.4) (for
$\varepsilon$ small enough), so that
${\mathscr{C}}_{\varepsilon}(1)\equiv\mathscr{B}_{\varepsilon}(1)$ . Setting

$\widetilde{\mu}_{\varepsilon}:=\sum_{B\in\mathscr{B}_{\varepsilon}(1)}\mu_{\varepsilon}(B)\delta_{x(B)}\,,$

by Theorem 4.3(i) (more precisely, by (4.5)), using (5.2), we have that, up to
a subsequence,

(5.7)
$\frac{\widetilde{\mu}_{\varepsilon}}{|\log\varepsilon|}\stackrel{{\scriptstyle\mathrm{flat}}}{{\rightarrow}}\mu\,,$

for some $\mu\in\mathcal{M}(\Omega)$ . By construction,
$\mathrm{supp}\,\mu\subseteq\overline{\Omega^{\prime}}$ . Moreover, by arguing
verbatim as in the proof of [28, formula (3.17)] one can prove that

$\frac{1}{|\log\varepsilon|}\|{Ju_{\varepsilon}}-\pi\widetilde{\mu}_{\varepsilon}\|_{\mathrm{flat},\Omega}\to
0\,,$

which, combined with (5.7), yields (FJ). Furthermore, by (1.10) and by (3.11),
we have that

$\frac{1}{|\log\varepsilon|^{2}}\int_{\Omega}|2T^{D}_{u_{\varepsilon}}|^{2}\;\mathrm{d}x=\frac{1}{|\log\varepsilon|^{2}}\int_{\Omega}|\nabla
u_{\varepsilon}|^{2}\;\mathrm{d}x\leq C\,,$

so that, up to a further subsequence,

(5.8) $\frac{T^{D}_{u_{\varepsilon}}}{|\log\varepsilon|}\rightharpoonup
T^{D}\qquad\textrm{in }L^{2}(\Omega;\mathbb{R}^{2})\,,$

for some field $T^{D}\in L^{2}(\Omega;\mathbb{R}^{2})$ . This proves (ACJ).

It remains to show that $-\mathrm{Div}\,T^{D}=\pi\mu$ , which will imply also
that $\mu\in H^{-1}(\Omega)$ . To this end, let $\varphi\in
C_{\mathrm{c}}^{\infty}(\Omega)$ ; then, by (1.10),

(5.9) $\langle Ju_{\varepsilon},\varphi\rangle=\langle
T_{u_{\varepsilon}},\nabla\varphi\rangle=\langle
T^{D}_{u_{\varepsilon}},\nabla\varphi\rangle+\langle
T^{S}_{u_{\varepsilon}},\nabla\varphi\rangle\,,$

for every $\varepsilon>0$ . By (ACJ), we have that

(5.10) $\frac{1}{|\log\varepsilon|}\langle
T^{D}_{u_{\varepsilon}},\nabla\varphi\rangle\to\langle
T^{D},\nabla\varphi\rangle\qquad\textrm{as }\varepsilon\to 0\,;$

moreover, by (1.10) and (3.11), we have that

(5.11) $\big{|}\langle T^{S}_{u_{\varepsilon}},\nabla\varphi\rangle\big{|}\leq
C\varepsilon\|\nabla\varphi\|_{L^{\infty}}|\log\varepsilon|^{2}\,.$

By, (FJ), (5.9), (5.10), and (5.11),

$\langle\pi\mu,\varphi\rangle=\lim_{\varepsilon\to
0}\frac{1}{|\log\varepsilon|}\langle
Ju_{\varepsilon},\varphi\rangle=\lim_{\varepsilon\to
0}\frac{1}{|\log\varepsilon|}\langle
T^{D}_{u_{\varepsilon}},\nabla\varphi\rangle=\langle
T^{D},\nabla\varphi\rangle=\langle-\mathrm{Div}T^{D},\phi\rangle\,,$

thus concluding the proof of (i).

Now we prove the lower bound (ii). We can assume without loss of generality
that (3.11) holds true. By the first inequality in (5.6) and by Theorem
4.3(ii) we have immediately that

$\liminf_{\varepsilon\to
0}\frac{\mathcal{G}_{\varepsilon}(u_{\varepsilon})}{|\log\varepsilon|^{2}}\geq\pi|\mu|(\Omega)+2\int_{\Omega}|T^{D}|^{2}\;\mathrm{d}x\,,$

where we have used also that the function $T^{D}$ coincides with the field
$\lambda$ in Theorem 4.3. The claim thus follows.

###  5.2. Proof of the upper bound

In order to construct the recovery sequence, we first introduce some notation.
Let $r>0$ be fixed . For every finite sum of Dirac deltas
$\mu:=\sum_{n=1}^{N}\delta_{x^{n}}$ with $|x^{n_{1}}-x^{n_{2}}|\geq 2r$ for
$n_{1}\neq n_{2}$ and for every $0<\rho<r$ we set

(5.12)
$\widehat{\mu}^{\rho}:=\frac{1}{2\pi\rho}\sum_{n=1}^{N}\mathcal{H}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\partial
B_{\rho}(x^{n})\,,\qquad\widetilde{f}^{\rho}:=\frac{1}{\pi\rho^{2}}\sum_{n=1}^{N}\chi_{B_{\rho}(x^{n})}\,,\qquad\textrm{and}\qquad\widetilde{\mu}^{\rho}:=\widetilde{f}^{\rho}\;\mathrm{d}x\,.$

For every $r>0$ and for every $z\in\mathbb{R}^{2}$ , we recall that $Q_{r}(z)$
denotes the (open) square centered at $z$ with sides parallel to the cartesian
axes and side-length equal to $2r$ , i.e., $Q_{r}(z):=z+Q_{r}(0)$ , with
$Q_{r}(0):=(-r,r)^{2}$ .

######  Lemma 5.1.

Let $\mu:=\sum_{l=1}^{L}m^{l}\chi_{\omega^{l}}\mathrm{d}x$, where
$L\in\mathbb{N}$, $m^{l}\in\mathbb{R}$ and $\\{\omega^{l}\\}_{l=1,\ldots,L}$
is a partition of $\Omega$ into sets with Lipschitz continuous boundary. Let
$N_{\varepsilon}\to+\infty$ as $\varepsilon\rightarrow 0$ . For every
$\varepsilon>0$ and for every $l=1,\ldots,L$ with $m^{l}\neq 0$, set

(5.13) $r^{l}_{\varepsilon}:=\frac{1}{2\sqrt{N_{\varepsilon}|m^{l}|}}.$

For every $l=1,\ldots,L$ with $m^{l}\neq 0$, let
$\mathcal{Z}_{\varepsilon}^{l}:=\\{z\in
2r^{l}_{\varepsilon}\mathbb{Z}^{2}\,:\,{{\overline{Q}}_{r^{l}_{\varepsilon}}(z)}\subset\omega^{l}\\}$
and $N_{\varepsilon}^{l}:=\sharp\mathcal{Z}_{\varepsilon}^{l}$ . Then, for
every $l=1,\ldots,L$ with $m^{l}\neq 0$,

(5.14)
$\frac{N^{l}_{\varepsilon}}{N_{\varepsilon}}\to|m^{l}||\omega^{l}|\qquad\textrm{as
}\varepsilon\to 0\,.$

Moreover, setting
$\mu_{\varepsilon}^{l}:=\sum_{n=1}^{N_{\varepsilon}^{l}}\delta_{x_{\varepsilon}^{l,n}}$
(where $\\{x_{\varepsilon}^{l,n}\\}_{n=1}^{N_{\varepsilon}^{l}}$ is the set of
points in $\mathcal{Z}_{\varepsilon}^{l}$) for every $l=1,\ldots,L$ with
$m^{l}\neq 0$, $\mu_{\varepsilon}^{l}\equiv 0$ whenever $m^{l}=0$, and
$\mu_{\varepsilon}:=\sum_{l=1}^{L}\mu_{\varepsilon}^{l}$, we have that the
sequence $\\{\mu_{\varepsilon}\\}_{\varepsilon}\subset\mathcal{M}(\Omega)$
satisfies

  * (a)

for every $\varepsilon>0$ and for every $l=1,\ldots,L$ with $m^{l}\neq 0$ and
$n=1,\ldots,N_{\varepsilon}^{l}$ , the balls
$B_{r^{l}_{\varepsilon}}(x^{l,n}_{\varepsilon})$ are pairwise disjoint and
contained in $\omega^{l}$ ;

  * (b)

$\frac{\mu_{\varepsilon}}{N_{\varepsilon}}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu$
in $\mathcal{M}(\Omega)$ as $\varepsilon\to 0$ ;

  * (c)

$\big{\|}\frac{\widetilde{\mu}_{\varepsilon}}{N_{\varepsilon}}-\mu\big{\|}_{H^{-1}(\Omega)}\leq{C}{{N_{\varepsilon}^{-\frac{1}{4}}}}$
(for $\varepsilon$ small enough),

where
$\widetilde{\mu}_{\varepsilon}:=\sum_{l=1}^{L}\widetilde{\mu}_{\varepsilon}^{l,r^{l}_{\varepsilon}}$,
with $\widetilde{\mu}_{\varepsilon}^{l,r^{l}_{\varepsilon}}$ defined as in
(5.12) starting from $\mu_{\varepsilon}^{l}$ .

###### Proof.

For every $l=1,\ldots,L$ with $m^{l}\neq 0$ we set
$\overline{\omega}^{l}_{\varepsilon}:=\bigcup_{z\in\mathcal{Z}_{\varepsilon}^{l}}\overline{Q}_{r^{l}_{\varepsilon}}(z)$
and we denote by $\omega_{\varepsilon}^{l}:=\text{{\rm
int}}({\overline{\omega}^{l}_{\varepsilon}})$ the set of interior points of
$\overline{\omega}_{\varepsilon}^{l}$ . We set
$r_{\varepsilon}:=\min\\{r_{\varepsilon}^{l}:l=1,\dots,L\\}$ and
$R_{\varepsilon}:=\max\\{r_{\varepsilon}^{l}:l=1,\dots,L,\;m^{l}\neq 0\\}$,
and we notice that $r_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow
0$.

Property (5.14) is straightforward. Indeed, let $l\in\\{1,\ldots,L\\}$ with
$m^{l}\neq 0$; setting
$(\partial\omega^{l})_{2r^{l}_{\varepsilon}}:=\\{x\in\omega^{l}:\mathrm{dist}(x,\partial\omega^{l})<2r^{l}_{\varepsilon}\\}$
, it is sufficient to observe that
$\left|(\omega^{l}\setminus\overline{\omega}^{l}_{\varepsilon})\setminus(\partial\omega^{l})_{2r^{l}_{\varepsilon}}\right|=0$
, and hence, by the Lipschitz continuity of $\partial\omega^{l}$ ,

(5.15)
$|\omega^{l}\setminus\overline{\omega}^{l}_{\varepsilon}|=\mathrm{O}(r^{l}_{\varepsilon})=\mathrm{O}(R_{\varepsilon})\,,$

where $\limsup_{\varepsilon\rightarrow
0}\mathrm{O}(R_{\varepsilon})R_{\varepsilon}^{-1}<C<+\infty$. On the other
hand,
$|\overline{\omega}^{l}_{\varepsilon}||m^{l}|=4(r_{\varepsilon}^{l})^{2}N_{\varepsilon}^{l}|m^{l}|=\frac{N_{\varepsilon}^{l}}{N_{\varepsilon}}$,
which, together with (5.15), yields
$|\omega^{l}|=|\omega^{l}\setminus\overline{\omega}^{l}_{\varepsilon}|+|\overline{\omega}^{l}_{\varepsilon}|=\frac{\mathrm{O}(R_{\varepsilon})}{|m^{l}|}+\frac{N_{\varepsilon}^{l}}{|m^{l}|N_{\varepsilon}}$
, and hence (5.14).

Property (a) follows by construction. Now we prove property (b). Let $\phi\in
C_{\mathrm{c}}(\Omega)$ , and let $m_{\phi}$ be the modulus of continuity of
$\phi$ . Then, setting

(5.16) $M:=\max\\{|m^{l}|:l=1,\dots,L\\},$

we have that, as $\varepsilon\to 0$,

$\displaystyle\Big{|}\Big{\langle}\frac{\mu_{\varepsilon}}{N_{\varepsilon}}-\mu,\phi\Big{\rangle}_{\Omega}\Big{|}\leq$
$\displaystyle\,\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{m^{l}\neq
0}}^{L}m^{l}\sum_{n=1}^{N^{l}_{\varepsilon}}\int_{Q_{r^{l}_{\varepsilon}}(x_{\varepsilon}^{l,n})}|\phi(x_{\varepsilon}^{l,n})-\phi(x)|\,\;\mathrm{d}x+\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{m^{l}\neq
0}}^{L}m^{l}\int_{\omega^{l}\setminus\overline{\omega}^{l}_{\varepsilon}}|\phi(x)|\,\;\mathrm{d}x$
$\displaystyle\leq$
$\displaystyle\,4M\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{m^{l}\neq
0}}^{L}N^{l}_{\varepsilon}(r_{\varepsilon}^{l})^{2}\max_{t\in[0,2\sqrt{2}r^{l}_{\varepsilon}]}m_{\phi}(t)+M\|\phi\|_{L^{\infty}}\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{m^{l}\neq
0}}^{L}|\omega^{l}\setminus\overline{\omega}^{l}_{\varepsilon}|\to 0,$

where we have used (5.13), (5.14), and (5.15); this proves (b).

We are left with the proof of (c). We set

$\displaystyle\omega:=\bigcup_{\genfrac{}{}{0.0pt}{1}{l=1}{m^{l}\neq
0}}^{L}\omega^{l},\qquad\qquad\omega_{\varepsilon}:=\bigcup_{\genfrac{}{}{0.0pt}{1}{l=1}{m^{l}\neq
0}}^{L}\omega_{\varepsilon}^{l},\qquad\qquad\eta_{\varepsilon}:=(\frac{\widetilde{\mu}_{\varepsilon}^{r_{\varepsilon}}}{N_{\varepsilon}}-\mu)\chi_{\omega_{\varepsilon}},$

so that

(5.17)
$\frac{\widetilde{\mu}_{\varepsilon}^{r_{\varepsilon}}}{N_{\varepsilon}}-\mu=\eta_{\varepsilon}-\mu\chi_{\omega\setminus\omega_{\varepsilon}}\,.$

Let $\bar{v}\in H^{1}(Q_{1}(0))$ be a solution to

$\left\\{\begin{array}[]{ll}\Delta v=\frac{4}{\pi}\chi_{B_{1}}-1&\textrm{ in
}Q_{1}(0)\\\ \partial_{\nu}v=0&\textrm{ on }\partial
Q_{1}(0)\,.\end{array}\right.$

We then define for every $\varepsilon>0$ and for every $l=1,\ldots,L,$ and
$n=1,\ldots,N_{\varepsilon}^{l}$ ,

$\displaystyle
v_{\varepsilon}^{l,n}(x):={\frac{1}{2N_{\varepsilon}}}\bar{v}(\frac{x-x_{\varepsilon}^{l,n}}{r^{l}_{\varepsilon}});$

notice that $\nabla
v_{\varepsilon}^{l,n}(x)=\sqrt{\frac{m^{l}}{N_{\varepsilon}}}\nabla\overline{v}(\frac{x-x_{\varepsilon}^{l,n}}{r^{l}_{\varepsilon}})$,
so

(5.18) $\|\nabla
v_{\varepsilon}^{l,n}\|^{2}_{L^{2}(Q_{r^{l}_{\varepsilon}}(x^{l,n}_{\varepsilon});\mathbb{R}^{2})}=\frac{1}{4N_{\varepsilon}^{2}}\|\nabla\bar{v}\|^{2}_{L^{2}(Q_{1}(0);\mathbb{R}^{2})}$

and

$-\Delta
v^{l,n}_{\varepsilon}=\eta_{\varepsilon}\qquad\text{in}\;\;\;Q_{r^{l}_{\varepsilon}}(x^{l,n}_{\varepsilon})\,,\qquad\qquad\partial_{\nu}v_{\varepsilon}^{l,n}=0\;\;\text{
on }\;\;\partial Q_{r^{l}_{\varepsilon}}(x^{l,n}_{\varepsilon}).$

Integrating by parts, using Hölder inequality, (5.18), and Young inequality,
it follows that

(5.19) $\displaystyle\|\eta_{\varepsilon}\|_{H^{-1}(\Omega)}=$
$\displaystyle\,\sup_{\genfrac{}{}{0.0pt}{1}{\phi\in
H^{1}_{0}(\Omega)}{\|\phi\|_{H^{1}(\Omega)}\leq
1}}\int_{\Omega}\phi\;\mathrm{d}\eta_{\varepsilon}=\sup_{\genfrac{}{}{0.0pt}{1}{\phi\in
H^{1}_{0}(\Omega)}{\|\phi\|_{H^{1}(\Omega)}\leq
1}}\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{m^{l}\neq
0}}^{L}\sum_{n=1}^{N^{l}_{\varepsilon}}\int_{Q_{r^{l}_{\varepsilon}}(x_{\varepsilon}^{l,n})}\phi\Delta
v^{l,n}_{\varepsilon}\;\mathrm{d}x$ $\displaystyle=$
$\displaystyle\,\sup_{\genfrac{}{}{0.0pt}{1}{\phi\in
H^{1}_{0}(\Omega)}{\|\phi\|_{H^{1}(\Omega)}\leq
1}}\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{m^{l}\neq
0}}^{L}\left(\sum_{n=1}^{N^{l}_{\varepsilon}}\int_{Q_{r^{l}_{\varepsilon}}(x_{\varepsilon}^{l,n})}\nabla\phi\cdot\nabla
v^{l,n}_{\varepsilon}\;\mathrm{d}x\right)$ $\displaystyle\leq$
$\displaystyle\,\sup_{\genfrac{}{}{0.0pt}{1}{\phi\in
H^{1}_{0}(\Omega)}{\|\phi\|_{H^{1}(\Omega)}\leq
1}}\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{m^{l}\neq
0}}^{L}\sum_{n=1}^{N^{l}_{\varepsilon}}\|\nabla\phi\|_{L^{2}(Q_{r^{l}_{\varepsilon}}(x_{\varepsilon}^{l,n});\mathbb{R}^{2})}\|\nabla
v^{l,n}_{\varepsilon}\|_{L^{2}(Q_{r^{l}_{\varepsilon}}(x_{\varepsilon}^{l,n});\mathbb{R}^{2})}$
$\displaystyle=$ $\displaystyle\,\sup_{\genfrac{}{}{0.0pt}{1}{\phi\in
H^{1}_{0}(\Omega)}{\|\phi\|_{H^{1}(\Omega)}\leq
1}}\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{m^{l}\neq
0}}^{L}\frac{1}{2N_{\varepsilon}}\sum_{n=1}^{N^{l}_{\varepsilon}}\|\nabla\phi\|_{L^{2}(Q_{r^{l}_{\varepsilon}}(x_{\varepsilon}^{l,n});\mathbb{R}^{2})}\|\nabla\bar{v}\|_{L^{2}(Q_{1}(0);\mathbb{R}^{2})}$
$\displaystyle\leq$ $\displaystyle\,\sup_{\genfrac{}{}{0.0pt}{1}{\phi\in
H^{1}_{0}(\Omega)}{\|\phi\|_{H^{1}(\Omega)}\leq
1}}\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{m^{l}\neq
0}}^{L}\frac{1}{4N_{\varepsilon}}\sum_{n=1}^{N_{\varepsilon}^{l}}\Big{(}\frac{1}{N_{\varepsilon}^{\frac{1}{2}}}\|\nabla\bar{v}\|^{2}_{L^{2}(Q_{1}(0);\mathbb{R}^{2})}+N_{\varepsilon}^{\frac{1}{2}}\|\nabla\phi\|^{2}_{L^{2}(Q_{r^{l}_{\varepsilon}}(x_{\varepsilon}^{l,n});\mathbb{R}^{2})}\Big{)}$
$\displaystyle\leq$
$\displaystyle\,\frac{1}{2}\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{m^{l}\neq
0}}^{L}\frac{N_{\varepsilon}^{l}}{N_{\varepsilon}^{\frac{3}{2}}}\|\nabla\bar{v}\|^{2}_{L^{2}(Q_{1}(0);\mathbb{R}^{2})}+\frac{1}{4N_{\varepsilon}^{\frac{1}{2}}}\leq\frac{C}{N_{\varepsilon}^{\frac{1}{2}}}\,,$

where the last inequality follows from (5.14) and (5.13). Finally, by Hölder
inequality, (5.13) and (5.15), we obtain

(5.20)
$\|\mu\chi_{\omega\setminus\omega_{\varepsilon}}\|_{H^{-1}(\Omega)}=\sup_{\genfrac{}{}{0.0pt}{1}{\phi\in
H^{1}_{0}(\Omega)}{\|\nabla\phi\|_{L^{2}(\Omega;\mathbb{R}^{2})}\leq
1}}\int_{\omega\setminus\omega_{\varepsilon}}\phi\;\mathrm{d}\mu\leq\sup_{\genfrac{}{}{0.0pt}{1}{\phi\in
H^{1}_{0}(\Omega)}{\|\nabla\phi\|_{L^{2}(\Omega;\mathbb{R}^{2})}\leq
1}}M\|\phi\|_{L^{2}(\Omega)}|\omega\setminus\omega_{\varepsilon}|^{\frac{1}{2}}\leq\frac{C}{N_{\varepsilon}^{\frac{1}{4}}}\,,$

with $M$ defined in (5.16); this, combined with (5.17) and (5.19), yields (c).

∎

###### Proof of Theorem 3.3(iii).

We divide the proof into two cases.

Case 1: $\mu=\sum_{l=1}^{L}m^{l}\chi_{\omega^{l}}\mathrm{d}x$ , where
$L\in\mathbb{N}$ , $m^{l}\in\mathbb{R}$ , and
$\\{\omega^{l}\\}_{l=1,\ldots,L}$ is a partition of $\Omega$ into sets with
Lipschitz continuous boundary.

We divide the proof into two steps. In the first one we construct the recovery
sequence
$\\{(\mu_{\varepsilon};\overline{\beta}_{\varepsilon})\\}_{\varepsilon}$ for
the core-radius problem; in the second step, we exploit the structure of
$\\{(\mu_{\varepsilon};\overline{\beta}_{\varepsilon})\\}_{\varepsilon}$ to
build up the recovery sequence $\\{u_{\varepsilon}\\}_{\varepsilon}$ for the
functional $\mathcal{G}_{\varepsilon}$ . First step: Construction of the
discrete measure $\mu_{\varepsilon}$ and of the core-radius field
$\overline{\beta}_{\varepsilon}$ . For every $\varepsilon>0$ , we set

(5.21) $N_{\varepsilon}:=\lfloor|\log\varepsilon|\rfloor$

and let

(5.22)
$\mu_{\varepsilon}:=\sum_{l=1}^{L}\mu^{l}_{\varepsilon}=\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{m^{l}\neq
0}}^{L}\sum_{n=1}^{N_{\varepsilon}^{l}}\delta_{x_{\varepsilon}^{l,n}}$

be the measure provided by Lemma 5.1. Set

(5.23) $S:=\\{(0;x_{2})\,:\,x_{2}<0\\}$

and let $\vartheta\in C^{\infty}(\mathbb{R}^{2}\setminus\overline{S})$ be the
function defined by

(5.24)
$\vartheta(x):=\left\\{\begin{array}[]{ll}\arctan\frac{x_{2}}{x_{1}}&\textrm{if
}x_{1}>0\\\ \frac{\pi}{2}&\textrm{if }x_{1}=0\textrm{ and }x_{2}>0\\\
\pi+\arctan\frac{x_{2}}{x_{1}}&\textrm{if }x_{1}<0\\\