# Non-convex functionals penalizing simultaneous oscillations along two
independent directions: structure of the defect measure

M. Goldman111CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris,
91120 Palaiseau, France, email<EMAIL_ADDRESS>B. Merlet222Univ.
Lille, CNRS, UMR 8524, Inria - Laboratoire Paul Painlevé, F-59000 Lille,
email<EMAIL_ADDRESS>
###### Abstract

We continue the analysis of a family of energies penalizing oscillations in
oblique directions: they apply to functions $u(x_{1},x_{2})$ with
$x_{l}\in\mathbb{R}^{n_{l}}$ and vanish when $u(x)$ is of the form
$u_{1}(x_{1})$ or $u_{2}(x_{2})$. We mainly study the rectifiability
properties of the _defect measure_ $\nabla_{1}\nabla_{2}u$ of functions with
finite energy.

The energies depend on a parameter $\theta\in(0,1]$ and the set of functions
with finite energy grows with $\theta$. For $\theta<1$ we prove that the
defect measure is $(n_{1}-1,n_{2}-1)$-tensor rectifiable in
$\Omega_{1}\times\Omega_{2}$. We first get the result for $n_{1}=n_{2}=1$ and
deduce the general case through slicing using White’s rectifiability
criterion.

When $\theta=1$ the situation is less clear as measures of arbitrary
dimensions from zero to $n_{1}+n_{2}-1$ are possible. We show however, in the
case $n_{1}=n_{2}=1$ and for Lipschitz continuous functions, that the defect
measures are $1\,$-rectifiable. This case bears strong analogies with the
study of entropic solutions of the eikonal equation.

##  1 Introduction

As in [GM21], we decompose the euclidean space $\mathbb{X}=\mathbb{R}^{n}$ as,

$\mathbb{R}^{n}=\mathbb{X}_{1}\oplus\mathbb{X}_{2},\quad\text{with}\quad
n_{l}:=\dim\mathbb{X}_{l}\geq 1\ \text{ for }l=1,2.$

The spaces $\mathbb{X}_{1}$ and $\mathbb{X}_{2}$ are assumed to be orthogonal.
We then set

$K:=\mathbb{X}_{1}\cup\mathbb{X}_{2}.$

We consider a domain $\Omega\subset\mathbb{R}^{n}$ which writes as
$\Omega=\Omega_{1}+\Omega_{2}$ where, for $l=1,2$,
$\Omega_{l}\subset\mathbb{X}_{l}$ is a nonempty bounded domain. In this paper,
we push further the analysis of the differential inclusion

$\nabla\times v=0\qquad\text{and }\qquad v\in K\ \text{a.e.\hbox{} missing}$
(1.1)

Here $\nabla\times v=0$ means that $\partial_{i}v_{j}=\partial_{j}v_{i}$ for
$1\leq i,j\leq n$. Writing then $v=\nabla u$ and using the notation
$\nabla_{l}$ for the derivatives with respect to the variable in
$\mathbb{X}_{l}$, (1.1) is equivalent to

$|\nabla_{1}u||\nabla_{2}u|=0\quad\ \text{almost everywhere in }\Omega.$ (1.2)

As noticed in [GM21], even under an $L^{\infty}$ assumption on $v$
(correspondingly in the class of Lipschitz functions $u$), (1.1) is far from
rigid. In particular it is not strong enough to characterize the subset of
functions $u\in\operatorname{Lip}(\Omega)$ of the form
$u(x_{1}+x_{2})=u_{1}(x_{2})$ or $u(x_{1}+x_{2})=u_{2}(x_{2})$ for
$x_{1}\in\Omega_{1}$, $x_{2}\in\Omega_{2}$. This motivates the introduction of
an energy based on a discrete version of (1.2) (see also [GR19] for another
motivation). We fix $\theta_{1},\theta_{2}>0$ and assume

$\theta:=\theta_{1}+\theta_{2}\leq 1.$

We also fix a radial function $\rho\in L^{1}(\mathbb{R}^{n},\mathbb{R}_{+})$
supported in $B_{1}$ and such that $\int\rho=1$. Denoting $L(\Omega)$ the
space of measurable functions over $\Omega$, for $\varepsilon>0$ and $u\in
L(\Omega)$, we define the energy:

$\mathcal{E}_{\varepsilon}(u):=\int_{\mathbb{R}^{n}}\int_{\Omega^{\varepsilon}}\dfrac{|Du(x,z_{1})|^{\theta_{1}}|Du(x,z_{2})|^{\theta_{2}}}{|z|^{2}}\,dx\,\rho_{\varepsilon}(z)\,dz,$
(1.3)

where we use the following conventions.

  1. (i)

$\rho_{\varepsilon}(z):=\varepsilon^{-n}\rho(\varepsilon^{-1}z)$;

  2. (ii)

for $l=1,2$, $z_{l}$ denotes the component in $\mathbb{X}_{l}$ of
$z\in\mathbb{X}_{1}+\mathbb{X}_{2}$;

  3. (iii)

we use the notation

$Du(x,z):=u(x+z)-u(x);$

  4. (iv)

and we integrate over the restricted domain333Notice that
$\Omega^{\varepsilon}$ is empty when $\varepsilon$ is too large.

$\Omega^{\varepsilon}:=\Omega_{1}^{\varepsilon}+\Omega_{2}^{\varepsilon},\qquad\text{where}\qquad\Omega_{l}^{\varepsilon}:=\\{x_{l}\in\Omega_{l}:d(x_{l},\mathbb{X}_{l}\setminus\Omega_{l})>\varepsilon\\}.$

We then send $\varepsilon$ to 0 to obtain the energy:

$\mathcal{E}(u):=\liminf_{\varepsilon\downarrow
0}\mathcal{E}_{\varepsilon}(u).$

######  Remark 1.1.

The definition of [GM21] allows for $\theta>1$ and involves another parameter
$p>0$ which corresponds to the exponent of $|z|$ in the denominator of (1.3).
Here we have fixed $p=2$ which is the relevant value in the case $\theta\leq
1$ as shown in [GM21].

######  Remark 1.2.

As we will see in Proposition 3.1, if $u$ is Lipschitz continuous with
$\mathcal{E}(u)<\infty$ then $v=\nabla u$ satisfies (1.1). However, while
(1.1) requires at least $v\in L^{1}$, i.e. $u\in W^{1,1}$ to make sense,
$\mathcal{E}(u)$ is well defined as soon as $u$ is measurable.

######  Remark 1.3.

Consider some parameters $\theta^{\prime}_{l}\geq\theta_{l}>0$ for $l=1,2$ and
assume that $\theta^{\prime}:=\theta^{\prime}_{1}+\theta^{\prime}_{2}\leq 1$.
For $u\in L^{\infty}(\Omega)$ we have, with obvious notation,

$\mathcal{E}^{(\theta_{1},\theta_{2})}(u)<\infty\ \implies\
\mathcal{E}^{(\theta^{\prime}_{1},\theta^{\prime}_{2})}(u)<\infty.$

In particular, the larger $\theta_{1},\theta_{2}$ are, the less coercive the
energy is. In the limit case $\theta=1$, the set of functions with finite
energy is much larger and we will see that the results are of a different
nature than in the case $0<\theta<1$.

In [GM21, Theorem I], we established the equivalence $\mathcal{E}(u)=0$ if and
only if $u$ depends only on $x_{1}$ or only on $x_{2}$. That is, $u$ lies in
the _non convex_ set

$S(\Omega):=\\{u\in L(\Omega):\exists\,l\in\\{1,2\\},\ \exists\,u_{l}\in
L(\Omega_{l})\text{ such that }u(x)=u_{l}(x_{l})\text{ in }\Omega\\}.$

We also established some quantitative versions of this fact by showing that
$\mathcal{E}(u)$ controls the distance of $u$ to $S(\Omega)$ in a strong
sense, see [GM21, Theorem R $\And$ Theorem S]. In the proofs of these results
a key step is the control of the $\mathbb{X}_{1}\otimes\mathbb{X}_{2}$-valued
distribution

$\mu[u]:=\nabla_{1}\nabla_{2}u.$

We established [GM21, Proposition M(a)] that if $u\in L^{\infty}(\Omega)$ has
finite energy then $\mu[u]$ is a Radon measure with

$|\mu[u]|(\Omega)\lesssim\|u\|_{\infty}^{1-\theta}\mathcal{E}(u).$ (1.4)

Obviously, the functions for which $\mu[u]=0$ are exactly the functions of the
form $u_{1}(x_{1})+u_{2}(x_{2})$, that is the elements of
$\operatorname{span}(S(\Omega))$. The distribution $\mu[u]$ measures how much
the function $u$ deviates from $\operatorname{span}(S(\Omega))$.

The aim of this paper, is to study the regularity and geometric structure of
$\mu[u]$ that we call from now on: the _defect measure_. As a by-product of
our analysis, for $n=2$ and $\theta<1$, we are able to improve the
quantitative results obtained in [GM21, Theorem R].

Before going further let us recall a useful result from [GM21].

######  Proposition 1.4 ([GM21, Lemma 3.4, Remark 3.5]).

If $u\in L^{1}_{loc}(\Omega)$ is such that $\mathcal{E}(u)<\infty$, then up to
a change of variables, there exist:

  1. (i)

sequences $\varepsilon_{k}\geq r_{k}>0$ tending to $0$;

  2. (ii)

orthonormal bases $(e_{1},\cdots,e_{n_{1}})$ of $\mathbb{X}_{1}$ and
$(f_{1},\cdots,f_{n_{2}})$ of $\mathbb{X}_{2}$;

such that444We use the notation $a\lesssim b$ to indicate that there exists a
constant $C>0$ which can depend only on $n_{1}$, $n_{2}$, $\theta_{1}$,
$\theta_{2}$ and $\rho$, such that $a\leq Cb$.

$\mathcal{E}^{\prime}(u):=\limsup_{k\uparrow\infty}\sum_{1\leq i\leq
n_{1},1\leq j\leq
n_{2}}\int_{\Omega^{\varepsilon_{k}}}\dfrac{q(x,r_{k}(e_{i}+f_{j}))}{r_{k}^{2}}\,dx\lesssim\mathcal{E}(u),$
(1.5)

with the notation

$q(x,z):=\left(|Du(x+z_{2},z_{1})|^{\theta_{1}}+|Du(x,z_{1})|^{\theta_{1}}\right)\left(|Du(x+z_{1},z_{2})|^{\theta_{2}}+|Du(x,z_{2})|^{\theta_{2}}\right).$
(1.6)

As a consequence, if $u\in L^{\infty}$,

$\mathcal{E}^{\prime\prime}(u):=\liminf_{k\to\infty}\sum_{1\leq i\leq
n_{1},1\leq j\leq
n_{2}}\int_{\Omega^{\varepsilon_{k}}}\dfrac{|D[Du(\cdot,r_{k}f_{j})](x,r_{k}e_{i})|}{r_{k}^{2}}\,dx\lesssim\|u\|_{\infty}^{1-\theta}\mathcal{E}(u).$
(1.7)

Throughout the article we implicitly assume that the sequences $r_{k}$,
$\varepsilon_{k}$ and the bases $(e_{1},\cdots,e_{n_{1}})$,
$(f_{1},\cdots,f_{n_{2}})$ are those provided by the proposition. There are
however some exceptions where the symbols $\varepsilon_{k}$, $r_{k}$ are used
for other purposes (as in Lemma 2.4, Remark 2.5 or Proposition 3.20) but this
is clear from the context.
When $n_{1}=n_{2}=1$, we can take $(e_{1},f_{1})$ to be the standard basis of
$\mathbb{R}^{2}$ and we denote it by $(e_{1},e_{2})$.

######  Remark 1.5.

In [GM21] we actually derive (1.4) from the stronger estimate,
$|\mu[u]|(\Omega)\lesssim\mathcal{E}^{\prime\prime}(u)$, see [GM21, Lemma
3.6].

###  1.1 The case $\theta<1$

We first consider the case $\theta<1$ with $n_{1}=n_{2}=1$ so that
$\Omega=I_{1}\times I_{2}$ with $I_{l}$ open intervals. As shown in [GM21,
Proposition P], the typical example of a function with $\mathcal{E}(u)<\infty$
in this case is given by the characteristic function of a polyhedron with
sides parallel to the axes (see Figure 1, left). The defect measure is then a
sum of Dirac masses sitting at the vertices of the polyhedron that lie in
$\Omega$.
In the setting of characteristic functions the situation simplifies in the
sense that we only need the following consequence of (1.4).

$\mu[u]=\partial_{1}\partial_{2}u\in\mathcal{M}(\Omega)\ \text{ and }\
u(x)\in\\{0,1\\}\ \text{ almost everywhere in }\Omega.$ (1.8)

$\Omega$$x^{7}$$\bullet$$x^{1}$$\bullet$$x^{2}$$\bullet$$x^{3}$$\bullet$$x^{4}$$\bullet$$x^{5}$$\bullet$$x^{6}$$A$$\bullet$$\Omega$$A^{\prime}$
Figure 1: A polygon $A$ with $\boldsymbol{1}_{A}$ of finite energy and with
$\mu[\boldsymbol{1}_{A}]=\delta_{x^{1}}-\delta_{x^{2}}+\delta_{x^{3}}-\delta_{x^{4}}+2\delta_{x^{5}}-\delta_{x^{6}}-\delta_{x^{7}}$
and a set $A^{\prime}$ with the same defect measure but with infinite
perimeter.

######  Theorem 1.6.

Let $n_{1}=n_{2}=1$, $\Omega=I_{1}\times I_{2}\subset\mathbb{R}^{2}$ be a
nonempty open box and let $A\subset\Omega$ be measurable. If
$u=\boldsymbol{1}_{A}$ satisfies (1.8) then the following properties hold.

  1. (i)

There exist finite sequences $m_{1},\cdots,m_{N}\in\\{\pm 1,\pm 2\\}$ and
$x^{1},\cdots,x^{N}\in\Omega$ with the $x^{i}$’s pairwise distinct such that

$\mu[u]=\sum_{j=1}^{N}m_{j}\delta_{x^{j}}.$

  2. (ii)

The set $A$ is the union of finitely many polygons with sides parallel to the
axes and stripes which are either all vertical or all horizontal. We can thus
find measurable subsets $A_{1}\subset I_{1}$, $A_{2}\subset I_{2}$ with either
$\mathcal{H}^{0}(\partial A_{1})<\infty$ or $\mathcal{H}^{0}(\partial
A_{2})<\infty$ and such that

$u(x)=\boldsymbol{1}_{A_{1}}(x_{1})\pm\boldsymbol{1}_{A_{2}}(x_{2})+w(x)\qquad\text{where}\quad
w(x):=\mu[u]((0,x_{1}]\times(0,x_{2}]).$

Besides $w=0$ (and $\mu[u]=0$) whenever $|\mu[u]|(\Omega)<1$ (see Figure 2 in
Section 2.1 for an example with
$u(x)=\boldsymbol{1}_{A_{1}}(x_{1})-\boldsymbol{1}_{A_{2}}(x_{2})$).

  3. (iii)

As a consequence, there exists $c>0$ such that if $\mathcal{E}(u)<c$ then
$u\in S(\Omega)$, that is, up to a negligible set,

$A=I_{1}\times A^{\prime}\quad\text{ or }\quad A=A^{\prime}\times
I_{2}\quad\text{ for some measurable set }A^{\prime}.$

######  Remark 1.7.

Notice that point (iii) improves [GM21, Theorem R] when $u$ is a
characteristic function. Such result does not hold for functions which can
take values into a set with more than two elements. For instance, if for
$l=1,2$, $A_{l}$ is a finite union of intervals with $\varnothing\neq
A_{l}\varsubsetneq I_{l}$, the function
$u(x):=\boldsymbol{1}_{A_{1}}(x_{1})+\boldsymbol{1}_{A_{2}}(x_{2})$ satisfies
$\mu[u]=\partial_{1}\partial_{2}u\equiv 0$ (and even
$0<\mathcal{E}(u)<\infty$) but $u\notin S(\Omega)$.

######  Remark 1.8.

Let us stress that the theorem does not state that assumption (1.8) implies
that $u$ is the characteristic function of a finite union of polygons. This is
only true up to vertical or horizontal stripes since such stripes do not
contribute to $\mu[u]$ (see Figure 1).

We now turn to the case of generic functions $u$ when $\theta<1$. Unlike the
case of characteristic functions where (1.8) is rigid, we need the stronger
assumption $\mathcal{E}(u)<\infty$.

######  Theorem 1.9.

Let $n_{1}=n_{2}=1$ and let $\Omega=I_{1}\times I_{2}\subset\mathbb{R}^{2}$ be
a nonempty open box. Assume that $\theta<1$ and let $u\in L^{\infty}(\Omega)$
with $\mathcal{E}(u)<\infty$.
Then $\mu[u]=\sum_{j\geq 1}m_{j}\delta_{x^{j}}$ for some $x^{j}\in\Omega$ and
$m_{j}\in\mathbb{R}\backslash\\{0\\}$ and we have the estimate

$\sum_{j\geq 1}|m_{j}|^{\theta}\lesssim\mathcal{E}(u).$ (1.9)

Moreover, if we assume that $u$ is integer-valued then the $m_{j}$’s are
integers.

The proof of Theorem 1.9 is based on the following observation. For every
$r>0$ and almost every $x\in\Omega$ such that
$Q_{x,r}:=x+[0,r)^{2}\subset\Omega$, there holds,

$\displaystyle\mu[u](Q_{x,r})=$ $\displaystyle
u(x+r(e_{1}+e_{2}))-u(x+re_{1})-u(x+re_{2})+u(x)$ $\displaystyle=$
$\displaystyle D[Du(\cdot,re_{2})](x,re_{1}).$

For a rigorous justification, see Lemma 2.1. By Proposition 1.4, this shows
that

$\liminf_{k\uparrow\infty}\int_{\Omega^{\varepsilon_{k}}}\dfrac{|\mu[u](Q_{x,r_{k}})|^{\theta}}{r_{k}^{2}}\,dx\lesssim\mathcal{E}(u).$

Lemma 2.4 then implies that $\mu[u]$ is atomic and that the estimate (1.9)
holds true.

######  Remark 1.10.

We see from Remark 1.7 that $\mu[u]=0$ does not imply $\mathcal{E}(u)=0$. In
particular we cannot expect the energy $\mathcal{E}$ to concentrate on
$\operatorname{supp}\mu[u]$.

Building on the structure of $\mu[u]$ provided by the theorem, we are able to
improve [GM21, Theorem R]. In order to state the result let us introduce the
set $SBV_{\theta}(\Omega)$ as the subset of functions of bounded variation in
$\Omega$ (see [AFP00]) whose distributional derivative has only jump part,
i.e. $\nabla u=(u_{+}-u_{-})\nu_{u}\mathcal{H}^{n-1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,J_{u}$, and such that

$|\nabla
u|_{\theta}:=\int_{J_{u}}|u_{+}-u_{-}|^{\theta}\,d\mathcal{H}^{n-1}<\infty.$

######  Theorem 1.11.

Assume that $n_{1}=n_{2}=1$, that $\Omega=I_{1}\times I_{2}$ is an open box
and that $\theta<1$. Let $u\in L^{\infty}(\Omega)$ with
$\mathcal{E}(u)<\infty$ and $\|u\|_{\infty}\leq 1$.

  1. (i)

There exists $\bar{u}\in S(\Omega)$ such that $u-\bar{u}\in
SBV_{\theta}(\Omega)\cap L^{\infty}(\Omega)$ with the estimate

$\|u-\bar{u}\|_{\infty}+|\nabla[u-\bar{u}]|_{\theta}(\Omega)\lesssim(1+\mathcal{H}^{1}(I_{1})+\mathcal{H}^{1}(I_{2}))\left(\mathcal{E}(u)+\sqrt{\mathcal{E}(u)\,}\right).$
(1.10)

  2. (ii)

If moreover $\mu[u]=0$ and $u\notin S(\Omega)$, then there exist two non-
constant functions $u_{l}\in SBV_{\theta_{l}}(I_{l})$ for $l=1,2$ such that
$u(x)=u_{1}(x_{1})+u_{2}(x_{2})$ and

$|\nabla u_{1}|_{\theta_{1}}(I_{1})\,|\nabla
u_{2}|_{\theta_{2}}(I_{2})\,\lesssim\,\mathcal{E}(u).$ (1.11)

  3. (iii)

As a consequence, if $u\in L^{\infty}(\Omega)\cap BV(\Omega)$ is such that
$\mathcal{E}(u)<\infty$ and $\nabla u$ has no jump part then $u\in S(\Omega)$.

We now turn to the higher dimensional case $n\geq 3$. Our main result is the
$(n-2)$\- rectifiability of $\mu[u]$, more precisely, its
$(n_{1}-1,n_{2}-1)\,$-tensor rectifiability.

######  Theorem 1.12.

Assume $\theta<1$ and $u\in L^{\infty}(\Omega)$ with $\mathcal{E}(u)<\infty$
then the following hold.

  1. (i)

$\mu[u]$ is a $(n-2)\,$-rectifiable measure, i.e. there exist a
$(n-2)\,$-rectifiable set $\Sigma\subset\Omega$ and a Borel mapping
$m:\Omega\to\mathbb{R}$ such that:
($*$) the approximate tangent space at $\mathcal{H}^{n-2}\,$-almost every
$x\in\Sigma$ is of the form $(\nu_{1}(x),\nu_{2}(x))^{\perp}$ where
$\nu_{l}(x)\in\mathbb{X}_{l}\setminus\\{0\\}$ for $l=1,2$,
($*$)

$\mu[u]=m(\nu_{1}\otimes\nu_{2})\,\mathcal{H}^{n-2}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Sigma.$

  2. (ii)

We have the estimate,

$\mathbf{M}_{\theta}(\mu[u]):=\int_{\Sigma}|m|^{\theta}\,d\mathcal{H}^{n-2}\
\lesssim\ \mathcal{E}(u).$

  3. (iii)

We can choose $\Sigma$ such that $\Sigma\subset\Sigma_{1}+\Sigma_{2}$ for some
$(n_{l}-1)\,$-rectifiable subsets $\Sigma_{l}\subset\Omega_{l}$ for $l=1,2$.
We say that $\mu[u]$ is $(n_{1}-1,n_{2}-1)$-tensor rectifiable.

The main two observations in the proof of (i) and (ii) in Theorem 1.12 are the
following. First, we can identify the defect measure $\mu[u]$ with a
$(n-2)\,$-current $T[u]$ (see Proposition 2.6). Moreover, since
$\mu[u]=\nabla_{1}\nabla_{2}u$ is a finite measure, we get that $T[u]$ is a
finite mass cycle, i.e. $\partial(T[u])=0$ and $\mathbf{M}(T[u])<\infty$.
We then argue by slicing (Theorem 2.7) and apply the slicing rectifiability
criterion of White [Whi99b], see also [Jer02]. For this we have to show that
that the $0\,$-slices of $T[u]$ are rectifiable.

  1. ($*$)

We first notice that as a consequence of the formula
$\mu[u]=\nabla_{1}\nabla_{2}u$, all the slices of $T[u]$ with respect to
coordinate $(n-2)\,$-spaces orthogonal to a plane of the form
$\operatorname{span}(e_{i_{1}},e_{i_{2}})$ or
$\operatorname{span}(f_{j_{1}},f_{j_{2}})$ vanish.

  2. ($*$)

Next, we consider instead a $2\,$-plane of the form
$\operatorname{span}(e_{i},f_{j})$. On the one hand, slicing and partial
boundary operations commute, see (2.31). On the other hand, by Fubini and
Fatou, the energy $\mathcal{E}^{\prime\prime}(u)$ (recall (1.7)) also behaves
well with respect to slicing. We may thus apply the two dimensional result
Theorem 1.9 to conclude that the $0\,$-slices of $T[u]$ are rectifiable.

Once rectifiability is obtained, (ii) follows from the corresponding bound
(1.9) in the case $n_{1}=n_{2}=1$. The proof of (iii) is considerably more
involved and motivated the development of the theory of tensor-rectifiable
flat chains in [GM22b]. In a nutshell, the idea of the proof of [GM22b,
Theorem 1.3] is to first treat the case $n_{1}=1$, $n_{2}\geq 2$. In this
case, based on the decomposition of $T[u]$ in indecomposable components proven
in [GM22a] we obtain in [GM22b, Proposition 6.2] the stronger result below
(rephrased in the language of the present paper),

######  Proposition 1.13.

Let $n_{1}=1$ and $n_{2}\geq 2$ so that $\Omega=I_{1}+\Omega_{2}$ for some
open interval $I_{1}$. Assume that $\theta<1$ and that $u\in
L^{\infty}(\Omega)$ is such that $\mathcal{E}(u)<\infty$.
Then there exist sequences $y^{j}_{1}\in I_{1}$ and $u_{2}^{j}\in
SBV^{\theta}(\Omega_{2})$ for $j\geq 1$ such that

$\mu[u]=\sum_{j\geq 1}\delta_{x^{j}_{1}}\otimes\nabla_{2}u_{2}^{j}$ (1.12)

and

$\mathbf{M}_{\theta}(\mu[u])=\sum_{j\geq
1}|\nabla_{2}u_{2}^{j}|_{\theta}(\Omega_{2}).$ (1.13)

As a consequence, there exist $u^{0}_{1}\in L^{\infty}(I_{1})$ and
$u^{0}_{2}\in L^{\infty}(\Omega_{2})$ such that

$u(x)=u^{0}_{1}(x_{1})+u^{0}_{2}(x_{2})+\sum_{j\geq
1}\boldsymbol{1}_{(-\infty,x^{j}_{1})}(x_{1})\,u_{2}^{j}(x_{2}).$

In the general case $n_{1},n_{2}\geq 2$, we can formally interpret $T[u]$ as a
$(n_{1}-1)\,$-flat chain over $\Omega_{1}$ with coefficients in the infinite
dimensional space of $(n_{2}-1)\,$-flat chains over $\Omega_{2}$. By (1.12) we
see that if we slice this flat-chain with respect to a hyperplane of
$\mathbb{X}_{1}$, we obtain a $0\,$-rectifiable flat chain in $\Omega_{1}$
(still with coefficients in the space of $(n_{2}-1)\,$-flat chains over
$\Omega_{2}$). Applying White’s rectifiability criterion then concludes the
proof.

######  Remark 1.14.

Let us point out that for $n_{1},n_{2}\geq 2$ we cannot expect a decomposition
analog to (1.12) satisfying also the counterpart of identity (1.13) (see the
counterexample of [GM22b, Proposition 6.5]).

######  Remark 1.15.

The energy $\mathbf{M}_{\theta}$ coincides with a so-called $h\,$-mass studied
for instance in [Whi99a, CDRMS17]. Partly due to its connection with branched
transport models, this type of functionals has received a lot of attention in
the past few years, see e.g. [BW18, CFM19, CDRM21]. It would be interesting to
understand further this connection with the energy $\mathcal{E}(u)$.

######  Remark 1.16.

Let us finally observe that since $\mu=\mu[u]=\nabla_{1}\nabla_{2}u$, it
satisfies the linear PDE constraints

$\nabla_{1}\times\mu=0\qquad\text{and}\qquad\nabla_{2}\times\mu=0,$

where for instance $\nabla_{1}\times\mu=0$ means that for every
$i,k\in[1,n_{1}]$ and every $j\in[1,n_{2}]$,

$\dfrac{\partial\mu_{i,j}}{\partial e_{k}}=\dfrac{\partial\mu_{k,j}}{\partial
e_{i}}.$

Letting $\mathbb{A}$ be the associated symbol, we have for
$\xi=\xi_{1}+\xi_{2}\in\mathbb{R}^{n}$ with $\xi_{i}\neq 0$

$\ker(\mathbb{A}(\xi))=\mathbb{R}\,\xi_{1}\otimes\xi_{2}$

and for $\xi_{1}\neq 0$,
$\ker(\mathbb{A}(\xi_{1}))=\xi_{1}\otimes\mathbb{X}_{2}$ and similarly for
$\ker(\mathbb{A}(\xi_{2}))$. In the language of [ARDPHR19], we thus find
$\Lambda_{\mathcal{A}}^{n-2}=\\{0\\}$ so that we could appeal to [DPR16,
ARDPHR19] and obtain that the most singular part of $\mu$ is
$(n-2)\,$-rectifiable. In comparison with Theorem 1.12 this would however not
exclude the presence of a more diffuse part of the measure.

###  1.2 The case $\theta=1$ (with $n_{1}=n_{2}=1$)

As proven in [GM21, Proposition P], when $\theta=1$ the set of possible
measures $\mu[u]$ is much richer. The study being much more difficult, we
restrict ourselves to the case $n_{1}=n_{2}=1$. Indeed, besides atomic
measures we can also have measures concentrated on lines (or a mixture of
both). The typical example is given by the “roof” function
$u(x_{1},x_{2}):=\min(x_{1},x_{2})$. It has finite energy (when restricted to
cubes) and $\mu[u]=(1/\sqrt{2})\mathcal{H}^{1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,L$ where $L=\\{x_{1}=x_{2}\\}$ is the
diagonal. It turns out that $\mu[u]$ cannot be more diffuse. We establish this
through a far-reaching refinement of the method used for Theorem 1.8 (but now
in the case $\theta=1$).

######  Theorem 1.17.

Let $n_{1}=n_{2}=1$ and $\theta=1$. Let $u\in L^{1}_{loc}(\Omega)$ be such
that $\mathcal{E}(u)<\infty$. Then $\mu[u]=\mu[u]\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,A$ for some Borel subset
$A\subset\Omega$ such that $\mathcal{H}^{1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,A$ is $\sigma\,$-finite.

######  Remark 1.18.

Measures of all dimensions between zero and one are possible. Indeed, fix
$\nu$ a finite measure on $(-1,1)$ and, for $l=1,2$, let
$y_{l}:(-1,1)\to(-1,1)$ be a smooth and increasing function. Setting for
$x=(x_{1},x_{2})\in(-1,1)^{2}$,

$u(x):=\int_{-1}^{1}\boldsymbol{1}_{(y_{1}(t),1)}(x_{1})\boldsymbol{1}_{(1,y_{2}(t))}(x_{2})\,d\nu(t),$

then $u$ has finite energy and $\mu[u]=(y_{1}\otimes y_{2})\,\\!_{\\#}\,\nu$.

From the examples discussed above it seems then natural to restrict ourselves
to the case of Lipschitz functions. Our last main result states that for such
functions, the defect measure is $1\,$-rectifiable.

######  Theorem 1.19.

Let $n_{1}=n_{2}=1$ and $\theta=1$. Let $u\in L^{\infty}(\Omega)$ be such that
$\|\nabla u\|_{\infty}\leq 1$ and $\mu:=\mu[u]$ is a Radon measure. Then $\mu$
is $\mathcal{H}^{1}$-rectifiable, that is

$\mu=m\mathcal{H}^{1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Sigma$

for some $1\,$-rectifiable set $\Sigma\subset\Omega$ and some Borel measurable
function $m:\Sigma\to\mathbb{R}$.
Moreover, $\nabla u$ has strong traces on $\Sigma$ of the form

$(v^{\infty}_{1},0)\qquad\text{and }\qquad(0,v^{\infty}_{2})$ (1.14)

with $v^{\infty}_{1}\neq 0$, $v^{\infty}_{2}\neq 0$ and

$|m|=\dfrac{|v^{\infty}_{1}||v^{\infty}_{2}|}{\sqrt{|v^{\infty}_{1}|^{2}+|v^{\infty}_{2}|^{2}}}.$

For the proof of Theorem 1.19, we first show in Proposition 3.1 that if $u$
satisfies the hypothesis of the theorem then $v=\nabla u$ satisfies the
differential inclusion (1.1) with the additional constraints that $|v|\leq 1$
and that $\mu[v]$ is a measure. We then prove in Theorem 3.2 the analog of
Theorem 1.19 but for bounded vector fields $v$ satisfying (1.1) and
$\mu[v]\in\mathcal{M}(\Omega)$. Writing $v=\nabla u$ for some Lipschitz
function $u$, the idea is to use the layer-cake formula to decompose it on its
superlevel-sets $\omega_{t}:=\\{u>t\\}$. Defining

$\kappa_{t}:=\mu[\boldsymbol{1}_{\omega_{t}}]=\partial_{1}\partial_{2}\boldsymbol{1}_{\omega_{t}},$

we show in Proposition 3.4 that

$\mu[v]=\int\kappa_{t}\,dt.$

In particular, for almost every $t$, $\kappa_{t}$ is a finite measure.
Therefore, $\boldsymbol{1}_{\omega_{t}}$ satisfies the differential inclusion
(1.8). Applying Theorem 1.6 we conclude that $\omega_{t}$ is a finite union of
polygons with sides parallel to the axes and $\kappa_{t}$ is nothing else than
the sum of the Dirac masses located at the corners. Moreover,
$|\mu[v]|\,$-almost every point $\bar{x}$ is given by such a corner
corresponding to a level $\bar{t}$ at which the function
$t\mapsto|\omega_{t}|$ is continuous. As a consequence, up to discarding sets
of small measures, for $t$ close to $\bar{t}$, every $\omega_{t}$ contains
exactly one corner $x(t)$ in a small neighborhood of $\bar{x}$, see Lemma 3.8.
This gives a sort of local parametrization of $\mu[v]$ around $\bar{x}$. The
main point is then to prove that $x$ is differentiable in an appropriate sense
at $t=\bar{t}$. The central insight is that since $u(x(t))=t$, the velocity
$x^{\prime}(t)$ is governed by $\nabla u$. We prove in Lemma 3.16 that, with
the notation (1.14),

$x^{\prime}_{l}(\bar{t}\,)=\dfrac{1}{v_{l}^{\infty}(x(\bar{t}\,))}\qquad\text{
for }l=1,2.$

######  Remark 1.20.

We call entropy any mapping $\Phi:\mathbb{R}^{2}\to\mathbb{R}^{2}$ which
writes as $\Phi(v)=\varphi_{1}(v_{2})e_{1}+\varphi_{2}(v_{1})e_{2}$ for some
pair of smooth functions $\varphi_{1},\varphi_{2}:\mathbb{R}\to\mathbb{R}$. We
say that $v$ is an entropy solution to (1.1) if for every entropy $\Phi$,

$\mu_{\Phi}:=\nabla\cdot[\Phi(v)]\in\mathcal{M}(\Omega).$

It is then not hard to see that if $v$ satisfies (1.1) it is an entropy
solution if and only if $\mu[v]:=\mu_{\Phi^{0}}$ is a measure for the entropy
$\Phi^{0}$ associated with $(\varphi_{1}^{0},\varphi_{2}^{0})=(\text{Id},0)$.
In this light our result can be compared with [ODL03] where a similar question
is addressed for entropic solutions of the eikonal equation. This corresponds
to replacing

$K\cap B_{1}=([-1,1]\times\\{0\\})\cup(\\{0\\}\times[-1,1])$

by $\partial B_{1}$ as the nonlinear constraint. Thanks to the strong
constraint on the level-sets of the stream function $u$ the analysis in our
case turns out to be substantially simpler than for [ODL03]. In particular we
are able to obtain a stronger result which is the rectifiability of the defect
measure $\mu[v]$. This is still a major open problem for the eikonal equation,
see [Mar21, Mar22] for recent results for related models.

Partly motivated by this analogy with the Aviles-Giga functional, we
investigate in Section 3.2.3 the compactness properties of sequences in

$S^{\infty}(\Omega):=\\{v\in
L^{\infty}(\Omega,\mathbb{R}^{2}):\|v\|_{\infty}\leq 1,\ \nabla\times v=0,\
v\in K\text{ a.e.\hbox{} \,and }\,\mu[v]\in\mathcal{M}(\Omega)\\}.$

######  Proposition 1.21.

If $v^{k}\in S^{\infty}(\Omega)$ is such that

$\sup_{k}|\mu[v^{k}]|(\Omega)<\infty,$

  1. (i)

Then, up to extraction, $v^{k}\longrightarrow v$ for some $v\in
S^{\infty}(\Omega)$ in the weak-$*$ topology of $L^{\infty}$.

  2. (ii)

If moreover

$\limsup_{k\to\infty}|\mu[v^{k}]|(\Omega)=0$

then for some $l\in\\{1,2\\}$, $v=v_{l}e_{l}$ with $v_{l}(x)=v_{l}(x_{l})$ and
$v_{\bar{l}}^{k}$ converges strongly to $0$ in $L^{1}$ (here
$\\{l,\bar{l}\\}=\\{1,2\\}$).

The proof follows the same scheme as in [DKMO01] using the div-curl lemma and
Young measures.

For $x\in\mathbb{R}^{2}$ and $r>0$ we set $Q_{r}(x):=x+(-r,r)^{2}$ and simply
write $Q$ for $(-1,1)^{2}$. Returning to the setting (and to the notation) of
Theorem 1.19, we introduce for $x^{*}\in\Omega$ the family of blow-ups of
$v:=\nabla u$ at $x^{*}$ as the functions $v^{x^{*},r}\in S^{\infty}(Q)$
defined for $0<r<d(x^{*},\mathbb{R}^{2}\setminus\Omega)$ by

$v^{x^{*},r}(x):=v(x^{*}+rx)=\nabla u(x^{*}+rx)\qquad\text{for }x\in Q.$

We notice that for every Borel subset $A\subset Q$ we have

$\mu\left[v^{x^{*},r}\right](A)=\dfrac{1}{r}\mu[v]({x^{*}+rA})$

so that the assumption $|\mu|(\Omega)<\infty$ implies by [AFP00, Theorem 2.56]
that for $\mathcal{H}^{1}$-almost every $x^{*}\in\Omega\setminus\Sigma$ there
holds

$\limsup_{r\downarrow 0}|\mu[v^{x^{*},r}]|(Q)=\limsup_{r\downarrow
0}\dfrac{|\mu[v]|(Q_{r}(x^{*}))}{r}=0.$

Applying Proposition 1.21 to the family $v^{x^{*},r}\in S^{\infty}(Q)$ we
deduce that for $\mathcal{H}^{1}$-almost every
$x^{*}\in\Omega\setminus\Sigma$, there exists $l\in\\{1,2\\}$ such that, _up
to extraction_ of a subsequence $r_{k}\downarrow 0$, there hold

$v^{x^{*},r_{k}}_{l}\ \text{ converges weakly-$*$ in }L^{\infty}(Q)\qquad\
\text{ and }\ \qquad v^{x^{*},r_{k}}_{\bar{l}}\to 0\text{ in }L^{1}(Q).$
(1.15)

These points parallel the “VMO points” of [ODL03] in the setting of the
eikonal equation which are conjectured to be $\mathcal{H}^{1}$-almost all
Lebesgue points of $v=\nabla u$ (see [LM23] for recent results in this
direction).
A natural question is whether we can choose the integer $l=1,2$ in (1.15)
independently of the choice of the subsequence. In such a case, $x^{*}$ would
be a Lebesgue point of $v_{\overline{l}}$. The answer is no in general.
Indeed, we construct in Proposition 3.20 a vector field $v\in S^{\infty}(Q)$
satisfying

$\limsup_{r\downarrow 0}\dfrac{|\mu[v]|(Q_{r}(0))}{r}=0$

but such that the integer $l\in\\{1,2\\}$ in (1.15) does depend on the choice
of the subsequence. In particular 0 is neither a Lebesgue point of $v_{1}$ nor
of $v_{2}$.

###  1.3 Conventions and notation

In all the paper, we consider $\theta_{1},\theta_{2}>0$, we note
$\theta=\theta_{1}+\theta_{2}$ their sum and we assume that $\theta\leq 1$.
For $x,z\in\mathbb{R}^{n}$ and $u:\Omega\to\mathbb{R}$ we define

$Du(x,z):=u(x+z)-u(x).$

We denote by $(e_{1},\cdots,e_{n_{1}})$ (respectively
$(f_{1},\cdots,f_{n_{2}})$) an orthonormal basis of $\mathbb{X}_{1}$
(respectively of $\mathbb{X}_{2}$).
If $n_{1}=n_{2}=1$, for $x\in\mathbb{R}^{2}$ and $z\in(0,+\infty)^{2}$, we
write

$Q_{x,z}:=[x_{1},x_{1}+z_{1})\times[x_{2},x_{2}+z_{2}).$

We also use the notation $Q_{x,r}:=x+[0,r)^{n}$ for $x\in\mathbb{R}^{n}$ and
$r>0$.
The open ball in $\mathbb{R}^{n}$ with radius $r>0$ and centered at $x$ is
denoted $B_{r}(x)$ and we write $B_{r}$ when $x=0$. The dimension $n$ is
always clear from the context.
Given $\mathcal{S}\subset\mathbb{X}$ where $\mathbb{X}$ is some vector space,
$\operatorname{span}(\mathcal{S})$ is the space spanned by $\mathcal{S}$. We
denote $\mathcal{H}^{k}(A)$ the $k$-dimensional Hausdorff measure of a set
$A\subset\mathbb{R}^{n}$ and if $A\subset\mathbb{R}^{n}$ is a measurable
subset, we note $|A|$ its volume.
We write $a\lesssim b$ when $a\leq Cb$ for some $C>0$ which may only depend on
$\theta_{1},\theta_{2}$, $n$ or on the kernel $\rho$.
We use standard notation for functional spaces such as $L^{p}(\omega)$,
$W^{1,p}(\omega)$, $BV(\omega)$. $\mathcal{M}(\omega)$ is the space of Radon
measures on $\omega$.
Unless otherwise specified the sequences are indexed from $1$. We often write
$\sup a_{j}$ as a shortcut for $\sup_{j\geq 1}a_{j}$ and similarly for series,
we write $\sum a_{j}$ for $\sum_{j\geq 1}a_{j}$.

###  1.4 Outline of the paper

The paper is organized as follows. In Section 2 we consider the case
$\theta<1$ and start by treating the two dimensional case in Section 2.1. We
first prove Theorem 1.6 on the structure of the defect measures for
characteristic functions. We then consider the case of arbitrary functions,
i.e. Theorem 1.9. Finally, we use it to prove Theorem 1.11. In Section 2.2 we
consider the higher dimensional case and prove Theorem 1.12. In Section 3 we
turn to the case $\theta=1$. We first prove Theorem 1.17 for arbitrary
functions and in Section 3.2, we consider the case of Lipschitz functions. We
derive the differential inclusion satisfied by functions of finite energy in
Section 3.2.1 before proving Theorem 3.2 about the rectifiability of the
defect measure in Section 3.2.2. We finally prove the compactness result,
Proposition 1.21 in Section 3.2.3.

##  2 The case $\theta<1$

###  2.1 The two-dimensional case

In this section we assume that $n_{1}=n_{2}=1$. We first show the following
simple lemma used in several places.

######  Lemma 2.1.

Let $\Omega=\mathring{Q}_{\bar{x},\bar{z}}\subset\mathbb{R}^{2}$ be a nonempty
open box (that is $\bar{x}\in\mathbb{R}^{2}$ and $\bar{z}\in(0,+\infty)^{2}$)
and let $u\in L^{1}_{loc}(\Omega)$ be such that
$\mu[u]=\partial_{1}\partial_{2}u$ is a Radon measure.
Defining $w$ as

$w(x):=\mu((\bar{x}_{1},x_{1}]\times(\bar{x}_{2},x_{2}])\qquad\text{ for
}x\in\Omega,$ (2.1)

we have $w\in BV(\Omega)$ with the estimate

$\|w\|_{\infty}+|\nabla w|(\Omega)\lesssim(1+|\bar{z}|)|\mu[u]|(\Omega),$
(2.2)

and there exist functions $u_{1}(x_{1})$ and $u_{2}(x_{2})$ in $S(\Omega)$
(which are bounded if $u$ is bounded) such that

$u(x)=u_{1}(x_{1})+u_{2}(x_{2})+w(x)\qquad\text{ for almost every
}x\in\Omega.$ (2.3)

As a consequence, for every $z\in(0,+\infty)^{2}$, we have for almost every
$x\in\Omega$ with $Q_{x,z}\subset\Omega$,

$\mu[u](Q_{x,z})=D[Du(\cdot,z_{2})](x,z_{1})\qquad\text{ and
}\qquad|\mu[u]|(\partial Q_{x,z})=0.$ (2.4)

In particular, $|\mu[u](Q_{x,z})|\leq 4\|u\|_{\infty}$.

###### Proof.

Estimate (2.2) is a direct consequence of the definition of $w$. Since
$\partial_{1}\partial_{2}(u-w)=0$, (2.3) is also readily obtained (see for
instance the proof of [GM21, Theorem 3.7]). We thus only need to show (2.4).

Let $I_{l}:=(\bar{x}_{l},\bar{x}_{l}+\bar{z}_{l})$ for $l=1,2$ (so that
$\Omega=I_{1}\times I_{2}$). Since $\mu:=\mu[u]$ is a Radon measure, there
exist $J_{1}\subset I_{1}$ and $J_{2}\subset I_{2}$ with full Lebesgue
measures such that for every $x_{1}\in J_{1}$, $|\mu|(\\{x_{1}\\}\times
I_{2})=0$ and for every $x_{2}\in J_{2}$, $|\mu|(I_{1}\times\\{x_{2}\\})=0$.
Hence, for every box $Q$ with vertices in $J_{1}\times J_{2}$, we have
$|\mu|(\partial Q)=0$. We may assume moreover that all the points of $J_{l}$
are Lebesgue points of $u_{l}$ for $l=1,2$.
Then, using (2.3), for every box $Q_{x,z}$ with vertices in $J_{1}\times
J_{2}$ we have

$\displaystyle D[Du(\cdot,z_{2})](x,z_{1})$
$\displaystyle=D[Dw(\cdot,z_{2})](x,z_{1})$
$\displaystyle=w(x)-w(x_{1},x_{2}+z_{2})-w(x_{1}+z_{1},x_{2})+w(x+z).$

By definition of $w$ this implies (2.4). Eventually, if $z\in(0,+\infty)$ is
fixed then almost every box $Q_{x,z}\subset\Omega$ has vertices in
$J_{1}\times J_{2}$. This concludes the proof. ∎

Let us prove Theorem 1.6 which deals with the case of characteristic
functions.

###### Proof of Theorem 1.6.

Let $u=\boldsymbol{1}_{A}$ be such that
$\mu:=\partial_{1}\partial_{2}u\in\mathcal{M}(\Omega)$ as in the theorem.

(i) We first prove that $\mu$ is atomic. Notice that since $u\in\\{0,1\\}$,
(2.4) implies that for every fixed $z$, Lebesgue almost every rectangle
$Q_{x,z}\subset\Omega$ satisfies

$\mu(Q_{x,z})\in\\{0,\pm 1,\pm 2\\}$ (2.5)

and by approximation this actually holds for every $Q_{x,z}\subset\Omega$.
Therefore, if $m\delta_{x}$ is a non trivial atom of $\mu$, we deduce that
$m=\mu(\\{x\\})=\lim_{k}\mu(Q_{x^{k},z^{k}})\in\pm\\{1,2\\}$, where we take
the limit over a sequence of boxes $Q_{x^{k},z^{k}}$ with $x\in
Q_{x^{k},z^{k}}$ and $|z^{k}|\to 0$. Since $\mu$ is a finite measure on
$\Omega$, its atomic part writes as a finite sum
$\mu^{a}=\sum_{j=1}^{N}m_{j}\delta_{x^{j}}$ with $m_{j}\in\\{\pm 1,\pm 2\\}$,
$x^{j}\in\Omega$ and $N\leq\sum|m_{j}|\leq|\mu|(\Omega)$.
Let us prove that $\mu(Q_{x,z})=0$ for every rectangle
$Q_{x,z}\subset\Omega^{\prime}:=\Omega\backslash\operatorname{supp}\mu^{a}$.
Let us assume by contradiction that $\mu(Q_{x,z})\neq 0$ so that
$\mu(Q_{x,z})\in\\{\pm 1,\pm 2\\}$. We split $Q_{x,z}$ into four equal
disjoint rectangles $Q_{x^{1},z^{1}},\cdots,Q_{x^{4},z^{4}}$. By (2.5), we
have $\mu(Q_{x^{j},z^{j}})\in\pm\\{0,1,2\\}$ for $j=1,\cdots,4$. Since
$\sum\mu(Q_{x^{j},z^{j}})=\mu(Q_{x,z})\neq 0$, we can pick $j$ with
$\mu(Q_{x^{j},z^{j}})\in\\{\pm 1,\pm 2\\}$. We note $Q^{1}:=Q_{x^{j},z^{j}}$
and iterate the construction to produce a sequence of nested boxes $Q^{k}$
with $\mu(Q^{k})\in\\{\pm 1,\pm 2\\}$ for $k\geq 1$ and
$\operatorname{diam}Q^{k}\to 0$. We deduce from the monotone convergence
theorem that $\cap_{k}Q^{k}=\\{x^{*}\\}$ for some $x^{*}\in Q_{x,z}$ with
$\mu(\\{x^{*}\\})\neq 0$. This contradicts
$Q_{x,z}\cap\operatorname{supp}\mu^{a}=\varnothing$ and proves the claim.
Eventually, since the rectangles of the form $Q_{x,z}\subset\Omega^{\prime}$
generate the $\sigma\,$-algebra of Borel sets of $\Omega^{\prime}$, we
conclude that $\mu\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Omega^{\prime}=0$ so that
$\mu=\mu^{a}$ as claimed and (i) is proved.

(ii) We turn to the proof of (ii). Without loss of generality, we may assume
that $I_{l}=(0,\ell_{l})$ for $l=1,2$.
We start with the following simple observation. If $Q_{0,z}\subset\Omega$ is
such that $\mu\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,Q_{0,z}=0$, then $w\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,Q_{0,z}=0$ (recall the definition
(2.1)) and thus by (2.3),

$u(x)=u_{1}(x_{1})+u_{2}(x_{2})\qquad\text{for }x\in Q_{0,z}$

for some functions $u_{1}$, $u_{2}\in S(Q_{0,z})$. However, since $u$ takes
only two values this implies that $u\in S(Q_{0,z})$ (see [DM95] for an
application of this argument in a different context). Notice that since $\mu$
is quantized this implies in particular (iii).
Let $z$ be such that $\mu$ does not contain any Dirac mass in the $L\,$-shaped
domain $\mathcal{L}:=Q_{0,(\ell_{1},z_{2})}\cup Q_{0,(z_{1},\ell_{2})}$, see
Figure 2.

$z$$(0,z_{2})$$(z_{1},0)$$(l_{1},0)$$(0,l_{2})$$0$$\Bigg{\\}}\\!\\!\\!\\!\\!\Bigg{\\}}\,Q_{0,(l_{1},z_{2})}$$Q_{0,(z_{1},l_{2})}$$\overbrace{\phantom{a\,aaaaaa}}$
Figure 2:  The $L\,$-shaped domain $\mathcal{L}=Q_{0,(\ell_{1},z_{2})}\cup
Q_{0,(z_{1},\ell_{2})}$. Here $u=1$ on the gray zones.

By the above observation applied in $Q_{0,(\ell_{1},z_{2})}$ and then in
$Q_{0,z_{1},\ell_{2}}$ and using obvious notation, there exist
$u^{\mathrm{h}}_{1},u^{\mathrm{h}}_{2}\in S(Q_{0,(\ell_{1},z_{2})})$ and
$u^{\mathrm{v}}_{1},u^{\mathrm{v}}_{2}\in S(Q_{0,z_{1},\ell_{2}})$ such
that555Throughout the article, the superscripts $\\_^{\mathrm{h}}$ and
$\\_^{\mathrm{v}}$ stand for “horizontal” and “vertical”.

$\displaystyle u(x)$
$\displaystyle=u_{1}^{\mathrm{h}}(x_{1})+u_{2}^{\mathrm{h}}(x_{2})\qquad\text{for
}x\in Q_{0,(\ell_{1},z_{2})},$ $\displaystyle u(x)$
$\displaystyle=u_{1}^{\mathrm{v}}(x_{1})+u_{2}^{\mathrm{v}}(x_{2})\qquad\text{for
}x\in Q_{0,(z_{1},\ell_{2})}.$

In the intersection $Q_{0,z}$, there holds

$u_{1}^{\mathrm{h}}(x_{1})+u_{2}^{\mathrm{h}}(x_{2})=u(x)=u_{1}^{\mathrm{v}}(x_{1})+u_{2}^{\mathrm{v}}(x_{2}),$

and substituting $(u_{1}^{\mathrm{v}}-c,u_{2}^{\mathrm{v}}+c)$ for
$(u_{1}^{\mathrm{v}},u_{2}^{\mathrm{v}})$ for some $c\in\mathbb{R}$ we may
assume that $u_{l}^{\mathrm{v}}=u_{l}^{\mathrm{h}}$ in $Q_{0,z}$ for $l=1,2$.
Then, setting $u_{1}=u^{\mathrm{h}}_{1}$ and $u_{2}=u^{\mathrm{v}}_{2}$ we
have

$u(x)=u_{1}(x_{1})+u_{2}(x_{2})\quad\text{for
}x\in\mathcal{L}\qquad\text{and}\qquad u\in S(Q_{0,(\ell_{1},z_{2})})\cap
S(Q_{0,(z_{1},\ell_{2})}).$

If $u_{1}$ is not constant in $(0,\ell_{1})$, then $u_{2}$ is constant in
$(0,z_{2})$. Up to the addition of a constant, we may assume without loss of
generality that $u_{2}=0$ in $(0,z_{2})$. This implies that $u=u_{1}$ in
$Q_{0,(\ell_{1},z_{2})}$ and since $u$ is a characteristic function we see
that in any case $u_{1}=\boldsymbol{1}_{A_{1}}$ for some measurable set
$A_{1}\subset(0,\ell_{1})$. The exact same considerations in
$Q_{0,(z_{1},\ell_{2})}$ show that $u_{2}=\pm\boldsymbol{1}_{A_{2}}$ for some
measurable set $A_{2}\subset(0,\ell_{2})$. Eventually, since
$\partial_{1}\partial_{2}(u-w)=0=\partial_{1}\partial_{2}(u_{1}+u_{2})$ in the
rectangle $\Omega$, the identity $u-w=u_{1}+u_{2}$, valid in $\mathcal{L}$,
propagates to $\Omega$. This concludes the proof. ∎

We now establish an elementary measure theoretical lemma used in the proofs of
Theorem 1.9 and Theorem 1.17.

######  Lemma 2.2.

Let $\mathcal{X}$ be a locally compact, complete and separable metric space,
let $\mu$ be a finite Radon measure on $\mathcal{X}$. For $k\geq 1$, let
$\mathcal{Q}^{k}$ be a collection of pairwise disjoint Borel subsets of
$\mathcal{X}$ such that
$d_{k}:=\sup\\{\operatorname{diam}Q\,:\,Q\in\mathcal{Q}^{k}\\}$ goes to $0$ as
$k\uparrow\infty$. Then,

$|\mu|\left(\cap_{k}\cup\mathcal{Q}^{k}\right)\leq\liminf_{k\uparrow\infty}\sum_{Q\in\mathcal{Q}^{k}}|\mu(Q)|.$

######  Remark 2.3.

The lemma is false if we do not assume that the elements of $\mathcal{Q}^{k}$
are disjoint. In the sequel we apply the result with $\mathcal{X}=\Omega$, a
bounded open set of $\mathbb{R}^{n}$ and with $\mathcal{Q}^{k}$ a finite set
of disjoint boxes.

###### Proof of Lemma 2.2.

In the proof we denote ${\mathcal{Y}}^{k}:=\cup\mathcal{Q}^{k}$ and
$\mathcal{Y}:=\cap_{k}{\mathcal{Y}}^{k}$.
Writing $\mu=\zeta|\mu|$ the polar decomposition of $\mu$, the function
$\zeta$ is Borel measurable and takes values in $\\{-1,1\\}$. Let
$\varepsilon>0$, there exists $\zeta_{\varepsilon}\in
C_{c}(\mathcal{X},[-1,1])$ such that

$\|\zeta_{\varepsilon}-\mathbf{1}_{\mathcal{Y}}\zeta\|_{L^{1}(\mathcal{X},|\mu|)}<\varepsilon.$
(2.6)

For $k\geq 1$ and $Q\in\mathcal{Q}^{k}$, we set

$\zeta_{\varepsilon}(Q):=\dfrac{1}{|\mu|(Q)}\int_{Q}\zeta_{\varepsilon}\,d|\mu|\quad\text{
with the convention }\zeta_{\varepsilon}(Q)=0\text{ if }|\mu|(Q)=0.$

By construction, $-1\leq\zeta_{\varepsilon}(Q)\leq 1$ and we deduce

$\sum_{Q\in\mathcal{Q}^{k}}|\mu(Q)|\geq\sum_{Q\in\mathcal{Q}^{k}}\zeta_{\varepsilon}(Q)\mu(Q)=\int_{{\mathcal{Y}}^{k}}\zeta_{\varepsilon}\,d\mu+\sum_{Q\in\mathcal{Q}^{k}}\int_{Q}\left(\zeta_{\varepsilon}(Q)-\zeta_{\varepsilon}\right)\,d\mu.$
(2.7)

The last term is bounded from below by

$-|\mu|(\mathcal{X})\max\\{|\zeta_{\varepsilon}(x)-\zeta_{\varepsilon}(y)|:x,y\in\mathcal{X},\,|x-y|\leq
d_{k}\\}.$

By uniform continuity of $\zeta_{\varepsilon}$ and the assumption
$d_{k}\downarrow 0$, this goes to $0$ as $k\uparrow\infty$.
Taking the infimum limit of (2.7) as $k\uparrow\infty$, we obtain,

$\liminf_{k\uparrow\infty}\sum_{Q\in\mathcal{Q}^{k}}|\mu(Q)|\,\geq\,\liminf_{k\uparrow\infty}\int_{{\mathcal{Y}}^{k}}\zeta_{\varepsilon}\,d\mu\stackrel{{\scriptstyle\eqref{lemma:tech_prf1}}}{{\geq}}\int_{\mathcal{Y}}\zeta\,d\mu-\varepsilon=|\mu|(\mathcal{Y})-\varepsilon.$

Recalling that $\varepsilon>0$ is arbitrary, the lemma is proved. ∎

In the proof of Theorem 1.9 we use the following characterization of discrete
measures. Let us first recall the notation $Q_{x,r}=x+r[0,1)^{n}$ for
$x\in\Omega$ and $r>0$ and let us notice that for $x\in\Omega^{\varepsilon}$
and $0<\sqrt{n}\,r\leq\varepsilon$, we have $Q_{x,r}\subset\Omega$.

######  Lemma 2.4.

Let $n\geq 1$, $\Omega\subset\mathbb{R}^{n}$ be a bounded open set and $\mu$
be a finite Radon measure on $\Omega$ such that there exist $0<\theta<1$ and
sequences $r_{k}$, $\varepsilon_{k}$ with
$0<\sqrt{n}\,r_{k}\leq\varepsilon_{k}\downarrow 0$ for which

$\liminf_{k\uparrow\infty}\int_{\Omega^{\varepsilon_{k}}}\dfrac{|\mu(Q_{x,r_{k}})|^{\theta}}{r_{k}^{n}}\,dx<\infty.$
(2.8)

Then there exist sequences $x^{j}\in\Omega$,
$m_{j}\in\mathbb{R}\backslash\\{0\\}$ such that $\mu=\sum m_{j}\delta_{x^{j}}$
with the estimate

$\sum|m_{j}|^{\theta}\leq\liminf_{k\uparrow\infty}\int_{\Omega^{\varepsilon_{k}}}\dfrac{|\mu(Q_{x,r_{k}})|^{\theta}}{r_{k}^{n}}\,dx.$
(2.9)

######  Remark 2.5.

In the statement of the lemma, the sequences $\varepsilon_{k}$ and $r_{k}$ are
not a priori those of Proposition 1.4.

###### Proof of Lemma 2.4.

For $y\in\Omega$, we define

$M_{k}(y):=\sup\left\\{|\mu|(Q_{x,r_{k}}):x\in\Omega\text{ such that }y\in
Q_{x,r_{k}}\subset\Omega\right\\}.$

and then for $\eta>0$,

$\Sigma_{\eta}:=\left\\{y\in\Omega\,:\,\limsup_{k\uparrow\infty}M_{k}(y)\geq\eta\right\\}.$

For every $y\in\Sigma_{\eta}$ we have $|\mu(\\{y\\})|\geq\eta$ and thus
$\Sigma_{\eta}$ is a finite set. Taking the union of the nested family
$\\{\Sigma_{\eta}\\}_{\eta>0}$, we see that the set

$\Sigma:=\,\\!\\!\bigcup_{\eta\downarrow
0}\Sigma_{\eta}=\left\\{y\in\Omega\,:\,|\mu|(\\{y\\})>0\right\\}$

is at most countable.

We claim that $|\mu|(\Omega\setminus\Sigma)=0$. By definition, for every
$\varepsilon>0$,

$\Omega^{\varepsilon}\setminus\Sigma=\left\\{y\in\Omega^{\varepsilon}\,:\,\lim_{k\uparrow\infty}M_{k}(y)=0\right\\}.$

By Egoroff theorem, there exists a Borel set
$\mathcal{Y}\subset\Omega^{\varepsilon}\setminus\Sigma$ with
$|\mu|(\mathcal{Y})\geq|\mu|(\Omega^{\varepsilon}\setminus\Sigma)/2$ and

$\lim_{k\uparrow\infty}t_{k}=0\qquad\text{ where }\
t_{k}:=\sup_{y\in{\mathcal{Y}}}M_{k}(y).$ (2.10)

For $k\geq 1$ and $x\in Q_{0,r_{k}}$ we consider the set

$\mathcal{Q}^{k}_{x}:=\\{Q\ :\ Q=Q_{x+r_{k}z,r_{k}}\ \text{ for some
}z\in\mathbb{Z}^{n},\ Q\subset\Omega\text{ and
}Q\cap\mathcal{Y}\neq\varnothing\\}.$

Using the definition of $\mathcal{Q}^{k}_{x}$, we have,

$\dfrac{1}{r_{k}^{n}}\int_{Q_{0,r_{k}}}\sum_{Q\in\mathcal{Q}^{k}_{x}}|\mu(Q)|\,dx\,\leq\,\dfrac{t_{k}^{1-\theta}}{r_{k}^{n}}\int_{Q_{0,r_{k}}}\sum_{Q\in\mathcal{Q}^{k}_{x}}|\mu(Q)|^{\theta}\,dx\,\leq\,t_{k}^{1-\theta}\int_{\Omega^{\varepsilon_{k}}}\dfrac{|\mu(Q_{y,r_{k}})|^{\theta}}{r_{k}^{n}}\,dy,$

where we used Fubini for the last identity. Taking the infimum limit as
$k\uparrow\infty$, we see that by assumption (2.8), the right-hand side tends
to zero. In particular, there exists a sequence $(x^{k})_{k\geq 1}$ with
$x^{k}\in Q_{0,r_{k}}$ such that

$\liminf_{k\uparrow\infty}\sum_{Q\in\mathcal{Q}^{k}_{x^{k}}}|\mu(Q)|\,=\,0.$

By (2.10), $\mathcal{Q}^{k}_{x^{k}}$ covers $\mathcal{Y}$ for $k$ large enough
(as soon as $r_{k}<\varepsilon/\sqrt{n}$). Applying Lemma 2.2 to the measure
$\mu$ and to the sequence $\\{\mathcal{Q}^{k}_{x^{k}}\\}$, we obtain

$|\mu|(\Omega^{\varepsilon}\setminus\Sigma)/2\leq|\mu|(\mathcal{Y})\leq\liminf_{k\uparrow\infty}\sum_{Q\in\mathcal{Q}^{k}_{x^{k}}}|\mu(Q)|\,=\,0.$

Since $\varepsilon>0$ is arbitrary we conclude that $\mu=\mu\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Sigma$.

Recalling that $\Sigma$ is at most countable, we write $\Sigma=(x^{j})_{j\geq
1}$ with $x^{j}$ pairwise distinct. We then have $\mu=\sum
m_{j}\delta_{x^{j}}$ for some summable sequence of real numbers $(m_{j})$. For
every $N\in\mathbb{N}$ and every $t>0$, there exists $r(N,t)>0$ such that for
$0<r\leq r(N,t)$ and $x\in\Omega$,

$\displaystyle Q_{x,r}\cap\\{x^{1},\cdots,x^{N}\\}\text{ has at most one
element}$ $\displaystyle\text{and if }x^{j}\in Q_{x,r}\text{ for some }1\leq
j\leq N\text{ then }|\mu(Q_{x,r})|\geq(1-t)|m_{j}|.$

We thus have

$\liminf_{k\uparrow\infty}\int_{\Omega^{\varepsilon_{k}}}\dfrac{|\mu(Q_{x,r_{k}})|^{\theta}}{r_{k}^{n}}\,dx\geq\liminf_{k\uparrow\infty}\sum_{j=1}^{N}\int_{\Omega}\boldsymbol{1}_{Q_{x,r_{k}}}(x^{j})\dfrac{|\mu(Q_{x,r_{k}})|^{\theta}}{r_{k}^{n}}\,dx\geq(1-t)^{\theta}\sum_{j=1}^{N}|m_{j}|^{\theta}.$

Sending $t\downarrow 0$ and $N\uparrow\infty$, we obtain (2.9). ∎

###### Proof of Theorem 1.9.

The proof goes as explained in the introduction. Using Lemma 2.1, the theorem
follows from Proposition 1.4 and Lemma 2.4 together with the triangle
inequality in the form of

$q(x,z)\gtrsim|\mu[u](Q_{x,z})|^{\theta}.$

Finally, if $u$ is integer-valued, we can argue as in the proof of Theorem 1.6
that for every $x\in\operatorname{supp}\mu[u]$ there exists a sequence
$x^{k}\to x$ and $z^{k}\to 0$ such that (2.4) holds. Therefore
$\mu[u](\\{x\\})=\lim_{k\to\infty}\mu[u](Q_{x^{k},z^{k}})=\lim_{k\to\infty}D[Du(\cdot,z_{2}^{k})](x^{k},z^{k}_{1})\in\mathbb{Z}.$
∎

###### Proof of Theorem 1.11.

The proof is a refinement of the proof of [GM21, Theorem 4.1] using the
additional information given by Theorem 1.9. Without loss of generality, we
assume that $I_{l}=(0,\ell_{l})$ for $l=1,2$. Let $u\in L^{\infty}(\Omega)$
with $\|u\|_{\infty}\leq 1$ and $\mathcal{E}(u)<\infty$.

Proof of (i), Step 1. Decomposition of $u$ and control of
$\|w\|_{\infty}+|\nabla w|_{\theta}(\Omega)$.
Let us first recall that by (2.3), there exist $u_{1}\in L^{\infty}(I_{1})$,
$u_{2}\in L^{\infty}(I_{2})$ such that

$u(x)=u_{1}(x_{1})+u_{2}(x_{2})+w(x)\qquad\text{a.e.\hbox{} in }\Omega.\vskip
3.0pt plus 1.0pt minus 1.0pt$ (2.11)

By Theorem 1.9, $\mu:=\mu[u]=\partial_{1}\partial_{2}u$ is a finite Radon
measure which writes as

$\mu=\sum m_{j}\delta_{x^{j}}\qquad\text{ with
}\qquad\sum|m_{j}|^{\theta}\lesssim\mathcal{E}(u).$ (2.12)

By direct computation, the definition (2.1) of $w$ yields the identities

$\partial_{1}w=\sum m_{j}\mathcal{H}^{1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,I_{1}^{j},\qquad\qquad\partial_{2}w=\sum
m_{j}\mathcal{H}^{1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,I_{2}^{j},$

where, for $j\geq 1$, we denote,
$I_{1}^{j}:=\\{x_{1}^{j}\\}\times(0,x_{2}^{j})$ and
$I_{2}^{j}:=(0,x_{1}^{j})\times\\{x_{2}^{j}\\}$. We deduce that $w\in
SBV_{\theta}(\Omega)$ with the estimate

$|\nabla
w|_{\theta}(\Omega)=\sum(|x_{1}^{j}|+|x_{2}^{j}|)|m_{j}|^{\theta}\leq(\ell_{1}+\ell_{2})\sum|m_{j}|^{\theta}\stackrel{{\scriptstyle\eqref{encoresommempowteta}}}{{\lesssim}}(\ell_{1}+\ell_{2})\mathcal{E}(u).$
(2.13)

Besides, by definition of $w$ we have $w\in L^{\infty}(\Omega)$ with

$\|w\|_{\infty}\leq|\mu|(\Omega)\lesssim\mathcal{E}(u).$ (2.14)

Eventually, let us notice for later use that by Lemma 2.1 there holds, for
every $j\geq 1$,

$|m_{j}|\leq\sup_{Q\subset\Omega}|\mu(Q)|\leq 4\|u\|_{\infty}\leq 4.$ (2.15)

Proof of (i), Step 2. Control of the oscillations of $u_{1}$ or of $u_{2}$.
We proceed as in the proof of [GM21, Theorem 4.1]. For every fixed
$z\in\mathbb{R}^{2}$, almost every $x\in\Omega$ with $Q_{x,z}\subset\Omega$
satisfies $Du(x,z_{l})=Du_{l}(x_{l},z_{l})+Dw(x,z_{l})$ for $l=1,2$. Since,
$Dw(x,z_{1})=\mu((x_{1},x_{1}+z_{1}]\times(0,x_{2}])$ and
$Dw(x,z_{2})=\mu((0,x_{1}]\times(0,x_{2}+z_{2}])$, we deduce from the triangle
inequality,

$\displaystyle\psi_{1}(x_{1},z_{1})$
$\displaystyle:=\Big{[}|Du_{1}(x_{1},z_{1})|-|\mu|((x_{1},x_{1}+z_{1}]\times
I_{2})\Big{]}_{+}\leq|Du(x,z_{1})|,$ (2.16)
$\displaystyle\psi_{2}(x_{2},z_{2})$
$\displaystyle:=\Big{[}|Du_{2}(x_{2},z_{2})|-|\mu|(I_{1}\times(x_{2},x_{2}+z_{2}])\Big{]}_{+}\leq|Du(x,z_{2})|,$
(2.17)

where, as usual, $a_{+}:=\max(a,0)$ denotes the positive part of $a$. Raising
inequality (2.16) to the power $\theta_{1}$ and (2.17) to the power
$\theta_{2}$, taking the product and integrating, we have by Proposition 1.4,

$\limsup_{k\uparrow\infty}\left(\int_{I_{1}^{\varepsilon_{k}}}\dfrac{[\psi_{1}(x_{1},z_{1}^{k})]^{\theta_{1}}}{r_{k}}\,dx_{1}\right)\left(\int_{I_{2}^{\varepsilon_{k}}}\dfrac{[\psi_{2}(x_{2},z_{2}^{k})]^{\theta_{2}}}{r_{k}}\,dx_{2}\right)\,\lesssim\,\mathcal{E}(u).$

We obtain that either for $l=1$ or for $l=2$ and up to extraction,

$\limsup_{k\uparrow\infty}\int_{I_{l}^{\varepsilon_{k}}}\dfrac{\left[\psi_{l}(x_{l},z_{l}^{k})\right]^{\theta_{l}}}{r_{k}}\,dx_{l}\,\lesssim\,\mathcal{E}(u)^{1/2}.$
(2.18)

Let us assume for instance that (2.18) holds with $l=1$ and let
$\theta_{*}\in[\theta,1]$. Since $\theta_{*}\leq 1$, the function
$s\in\mathbb{R}_{+}\mapsto s^{\theta_{*}}$ is subaddititive and we have for
every $k\geq 1$,

$\int_{I_{1}^{\varepsilon_{k}}}|Du_{1}(x_{1},r_{k})|^{\theta_{*}}\,dx_{1}\\\
\leq\int_{I_{1}^{\varepsilon_{k}}}\left[\psi_{1}(x_{1},r_{k}e_{1})\right]^{\theta_{*}}\,dx_{1}+\int_{I_{1}^{\varepsilon_{k}}}\left[|\mu|((x_{1},x_{1}+r_{k}]\times
I_{2})\right]^{{\theta_{*}}}\,dx_{1}.$ (2.19)

To estimate the second term in the right-hand side, we use the atomic
decomposition of $\mu$ and the subadditivity of $s\in\mathbb{R}_{+}\mapsto
s^{\theta_{*}}$. We have,

$\int_{I_{1}^{\varepsilon_{k}}}\left[|\mu|((x_{1},x_{1}+r_{k}]\times(0,\ell_{2}))\right]^{{\theta_{*}}}\,dx_{1}=\int_{I_{1}^{\varepsilon_{k}}}\left(\sum_{\\{0<x_{1}^{j}-x_{1}\leq
r_{k}\\}}|m_{j}|\right)^{\theta_{*}}\,dx_{1}\\\
\leq\int_{I_{1}^{\varepsilon_{k}}}\sum_{\\{0<x_{1}^{j}-x_{1}\leq
r_{k}\\}}|m_{j}|^{\theta_{*}}\,dx_{1}\leq(\sup|m_{j}|)^{\theta_{*}-\theta}\int_{I_{1}^{\varepsilon_{k}}}\sum_{\\{j\,:\,0<x_{1}^{j}-x_{1}\leq
r_{k}\\}}|m_{j}|^{\theta}\,dx_{1}\\\
\leq(\sup|m_{j}|)^{\theta_{*}-\theta}\sum\left[|m_{j}|^{\theta}\mathcal{H}^{1}((x_{1}^{j}-r_{k},x_{1}^{j}])\right]\stackrel{{\scriptstyle\eqref{encoresommempowteta}\eqref{supml}}}{{\lesssim}}r_{k}\mathcal{E}(u).$
(2.20)

To recover the conclusion of [GM21, Theorem 4.1], we first pick
$\theta_{*}=1$. Dividing (2.19) by $r_{k}$, taking the superior limit as
$k\uparrow\infty$ and using (2.18) and (2.20), we obtain

$\limsup_{k\uparrow\infty}\int_{I_{1}^{\varepsilon_{k}}}\dfrac{|D_{1}u_{1}(x_{1},r_{k})|}{r_{k}}\,dx_{1}\lesssim\mathcal{E}(u)^{1/2}+\mathcal{E}(u).$

This readily yields $u_{1}\in BV(I_{1})$ and up to the subtraction of a
constant, we have

$\|u_{1}\|_{\infty}+|\nabla_{1}u_{1}|(I_{1})\lesssim\mathcal{E}(u)^{1/2}+\mathcal{E}(u).$
(2.21)

Next, we use $\theta_{*}=\theta$ to get

$\limsup_{k\uparrow\infty}\int_{I_{1}^{\varepsilon_{k}}}\dfrac{|D_{1}u_{1}(x_{1},r_{k})|^{\theta}}{r_{k}}\,dx_{1}\lesssim\mathcal{E}(u)^{1/2}+\mathcal{E}(u).$

We apply Lemma 2.4 with $n=1$ to the measure $\mu_{1}:=\partial_{e_{1}}u_{1}$,
we can write

$\mu_{1}=\sum\widetilde{m}_{j}\delta_{\widetilde{x}_{1}^{j}}$

for some sequences $(\widetilde{x}_{1}^{j})\subset I_{1}$ and
$\widetilde{m}_{j}\in\mathbb{R}$ with
$\sum|\widetilde{m}_{j}|^{\theta}<\infty$. We deduce that $u_{1}\in
SBV_{\theta}(I_{1})$, with

$|\nabla_{1}u_{1}|_{\theta}(I_{1})\lesssim\mathcal{E}(u)^{1/2}+\mathcal{E}(u).$
(2.22)

Eventually, we set $\bar{u}(x)=u_{2}(x_{2})$, we have $\bar{u}\in S(\Omega)$
and $u(x)-\bar{u}(x)=u_{1}(x_{1})+w(x)$. Collecting (2.13), (2.14), (2.21) &
(2.22), we conclude that $u-\bar{u}\in SBV_{\theta}(\Omega)$ with the estimate
(1.10).

Proof of (ii).
Let us now turn to the proof of the second part of the theorem and assume that
$\mu[u]=0$ as well as $u\notin S(\Omega)$. This implies that $w=0$ in (2.11).
By Proposition 1.4, we have

$\limsup_{k\uparrow\infty}\left(\int_{I_{1}^{\varepsilon_{k}}}\dfrac{|Du_{1}(x_{1},z_{1})|^{\theta_{1}}}{r_{k}}\,dx_{1}\right)\left(\int_{I_{2}^{\varepsilon_{k}}}\dfrac{|Du_{2}(x_{2},z_{2})|^{\theta_{2}}}{r_{k}}\,dx_{2}\right)\,\lesssim\,\mathcal{E}(u).$

Therefore, there exist constants $\lambda_{1},\lambda_{2}\in[0,\infty]$ such
that $\lambda_{1}\lambda_{2}\lesssim\mathcal{E}(u)$ and for $l=1,2$,

$\liminf_{k\uparrow\infty}\int_{I_{l}^{\varepsilon_{k}}}\dfrac{|Du_{l}(x_{l},z_{l})|^{\theta_{l}}}{r_{k}}\,dx_{l}\,\leq\lambda_{l}.$

Since both $u_{1}$ and $u_{2}$ are not constant, this implies that
$\lambda_{l}>0$ for $l=1,2$ and thus also $\lambda_{1},\lambda_{2}<\infty$.
Arguing exactly as in (i) we get that $u_{l}\in SBV_{\theta_{l}}(I_{l})$ for
$l=1,2$ with $|\nabla u_{l}|_{\theta_{l}}(I_{l})\leq\lambda_{l}$ with the
estimate (1.11).

Proof of (iii).
We finally assume that $u\in L^{\infty}(\Omega)\cap BV(\Omega)$ has finite
energy and that moreover666we note $\nabla^{\mathrm{j}}v$ the jump part of the
distributional gradient of a $BV$ function $v$. $\nabla^{\mathrm{j}}u=0$. We
have $\nabla^{\mathrm{j}}w=\nabla w$, so that
$0=\nabla^{\mathrm{j}}_{1}u=\partial_{1}w+\nabla^{\mathrm{j}}_{1}u_{1}$.
Taking the derivative with respect to $x_{2}$, we obtain
$0=\partial_{2}\partial_{1}w=\mu[u]$. By (ii), $u\in S(\Omega)$ since
otherwise we could write it as $u=u_{1}+u_{2}$ where $u_{l}\in
SBV_{\theta_{l}}(I_{l})$ are non constant functions for $l=1,2$ and would
contradict the hypothesis $\nabla^{j}u=0$. ∎

###  2.2 The higher dimensional case

We now turn to the case $n=n_{1}+n_{2}>2$. We recall that we have fixed
orthonormal bases $(e_{1},\cdots,e_{n_{1}})$ and $(f_{1},\cdots,f_{n_{2}})$ of
$\mathbb{X}_{1}$ and $\mathbb{X}_{2}$ and have set

$\mu[u]=\nabla_{1}\nabla_{2}u=\sum_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
1\leq j\leq n_{2}\end{subarray}}\dfrac{\partial^{2}u}{\partial
x^{1}_{i}\partial x^{2}_{j}}\,e_{i}\otimes
f_{j}=\sum_{\begin{subarray}{c}1\leq i\leq n_{1}\\\ 1\leq j\leq
n_{2}\end{subarray}}\mu_{i,j}\,e_{i}\otimes f_{j}.$ (2.23)

As outlined in the introduction we will identify $\mu[u]$ with a
$(n-2)\,$-dimensional current. To this aim we recall some notation from
Geometric Measure Theory, see [Fed69, KP08]. We note $\mathcal{D}^{k}(\Omega)$
the space of smooth and compactly supported $k\,$-differential forms on
$\Omega$ on which acts the differential operator
$d:\mathcal{D}^{k}(\Omega)\to\mathcal{D}^{k+1}(\Omega)$. Its dual space is the
space of $k\,$-currents $\mathcal{D}_{k}(\Omega)$ on which acts the dual
operator $\partial:\mathcal{D}_{k}(\Omega)\to\mathcal{D}_{k-1}(\Omega)$. We
use the standard notation $\wedge$ for the exterior product. We recall that a
$k\,$-current $T$ is rectifiable if there exists a $k\,$-rectifiable set
$\Sigma\subset\Omega$ oriented by a unitary simple $k\,$-vector field $\xi$
and a Borel measurable multiplicity function $m:\Sigma\to\mathbb{R}$ such that
for $\omega\in\mathcal{D}^{k}(\Omega)$,

$\left<T,\omega\right>=\int_{\Sigma}m\,\left<\omega,\xi\right>d\mathcal{H}^{k}.$
(2.24)

We introduce the “partial” differentials $d_{1}$, $d_{2}$ as follows:

$\begin{array}[]{rclcrcl}d_{1}:\mathcal{D}^{k_{1}}(\Omega_{1})\wedge\mathcal{D}^{k_{2}}(\Omega_{2})&\to&\mathcal{D}^{k_{1}+k_{2}+1}(\Omega)&&d_{2}:\mathcal{D}^{k_{1}}(\Omega_{1})\wedge\mathcal{D}^{k_{2}}(\Omega_{2})&\to&\mathcal{D}^{k_{1}+k_{2}+1}(\Omega)\\\
\omega_{1}\wedge\omega_{2}&\mapsto&d\omega_{1}\wedge\omega_{2},&&\omega_{1}\wedge\omega_{2}&\mapsto&\omega_{1}\wedge
d\omega_{2}.\end{array}$

For $0\leq k\leq n$, we then extend by linearity and density the operators
$d_{1}$, $d_{2}$ on

$\mathcal{D}^{k}(\Omega)=\overline{\operatorname{span}\left(\oplus_{l=0}^{k}\mathcal{D}^{l}(\Omega_{1})\wedge\mathcal{D}^{k-l}(\Omega_{2})\right)}.$

By duality, this defines continuous partial boundary operators on the space of
currents:

$\left<\partial_{1}T,\omega\right>:=\left<T,d_{1}\omega\right>,\quad\left<\partial_{2}T,\omega\right>:=\left<T,d_{2}\omega\right>,\quad\text{for
}T\in\mathcal{D}_{k}(\Omega),\ \omega\in\mathcal{D}^{k-1}(\Omega).$

For $u\in L^{1}_{loc}(\Omega)$, we define the current

$[[u]]:=u\,e_{1}\wedge\cdots\wedge e_{n_{1}}\wedge f_{1}\wedge\cdots\wedge
f_{n_{2}}\ \in\mathcal{D}_{n}(\Omega),$

from which, we derive the $(n-2)\,$-dimensional current

$T[u]:=\partial_{1}\partial_{2}[[u]].$

For every $(n-2)\,$-current $T$ we define the $\theta\,$-mass of $T$ by

$\mathbf{M}_{\theta}(T):=\begin{cases}\displaystyle\int_{\Sigma}|m|^{\theta}\,d\mathcal{H}^{n-2}&\text{if
}T\text{ is rectifiable,}\\\\[8.0pt]
\qquad+\infty&\text{otherwise}.\end{cases}$

We say that a rectifiable $(n-2)\,$-current is tensor-rectifiable if we can
choose the set $\Sigma$ from (2.24) such that
$\Sigma\subset\Sigma^{1}+\Sigma^{2}$ where $\Sigma^{l}\subset\Omega_{l}$ is
$(n_{l}-1)\,$-rectifiable. Notice that in this case, the $k\,$-vector field
$\xi$ tangent to $\Sigma$ must be of the form $\xi=\xi^{1}\wedge\xi^{2}$ with
$\xi^{l}$ tangent to $\Sigma^{l}$. For $l=1,2$ we set

$dx^{l}=dx_{1}^{l}\wedge\cdots\wedge dx_{n_{l}}^{l}$

and introduce the following simple multi-covectors for $1\leq i\leq n_{l}$,

$dx^{l}_{\bar{\imath}}:=dx^{l}_{1}\wedge\cdots\wedge dx^{l}_{i-1}\wedge
dx^{l}_{i+1}\wedge\cdots\wedge dx^{l}_{n_{l}}.$

Similarly, we define the simple multi-vectors

$e_{\bar{\imath}}:=e_{1}\wedge\cdots\wedge e_{i-1}\wedge
e_{i+1}\wedge\cdots\wedge e_{n_{1}},\qquad
f_{\bar{\jmath}}:=f_{1}\wedge\cdots\wedge f_{j-1}\wedge
f_{j+1}\wedge\cdots\wedge f_{n_{2}}.$

We define the Hodge star operator on $\mathbb{X}_{1}$, first on the simple
$(n_{1}-1)\,$-vectors $e_{\bar{\imath}}$ by $\star
e_{\bar{\imath}}=(-1)^{i}e_{i}$, and then extend it by linearity. Similarly we
set $\star f_{\bar{\jmath}}=(-1)^{j}f_{j}$.
Every $\omega\in\mathcal{D}^{n-2}(\Omega)$ decomposes as

$\omega=\left[\sum_{\begin{subarray}{c}1\leq i\leq n_{1}\\\ 1\leq j\leq
n_{2}\end{subarray}}\omega_{i,j}\,dx^{1}_{\bar{\imath}}\wedge
dx^{2}_{\bar{\jmath}}\right]+\omega_{1}\,\wedge
dx^{2}\,+\,dx^{1}\wedge\omega_{2}.$ (2.25)

######  Proposition 2.6.

For every $u\in L^{1}_{loc}(\Omega)$ we have with the notation (2.23),

$T[u]=\sum_{\begin{subarray}{c}1\leq i\leq n_{1}\\\ 1\leq j\leq
n_{2}\end{subarray}}(-1)^{i+j}\mu_{i,j}\,e_{\bar{\imath}}\wedge
f_{\bar{\jmath}}.$ (2.26)

As a consequence, $\mu[u]$ is rectifiable (respectively tensor-rectifiable) if
and only if $T[u]$ is rectifiable (respectively tensor-rectifiable). Moreover
$\partial T[u]=0$ (we say that $T[u]$ is a cycle),

$\mathbf{M}(T[u])\lesssim|\mu[u]|(\Omega)\qquad\text{and}\qquad\mathbf{M}_{\theta}(T[u])=\mathbf{M}_{\theta}(\mu[u]).$
(2.27)

###### Proof.

We first establish (2.26). Fix $\omega\in\mathcal{D}^{n-2}(\Omega)$ and
decompose it as in (2.25). Since

$d_{2}(\omega_{1}\wedge dx^{2})=\omega_{1}\wedge
d(dx^{2})=0=d_{1}(dx^{1}\wedge\omega_{2}),$ (2.28)

we have

$\left<T[u],\omega\right>=\left<[[u]],d_{2}d_{1}\omega\right>=\sum_{i,j}\left<[[u]],d_{2}d_{1}\left(\omega_{i,j}\,dx^{1}_{\bar{\imath}}\wedge
dx^{2}_{\bar{\jmath}}\right)\right>\\\
=\sum_{i,j}(-1)^{i+j}\int_{\Omega}\dfrac{\partial^{2}\omega_{i,j}}{\partial
x^{1}_{i}\partial
x^{2}_{j}}u\,dx=\sum_{i,j}(-1)^{i+j}\int_{\Omega}\omega_{i,j}\,d\mu_{i,j}.$

This proves (2.26).
Assume now that $T[u]$ is rectifiable. Let $(\Sigma,\xi,m)$ be as in (2.24).
First, by (2.26) we can write $\xi=\xi^{1}\wedge\xi^{2}$ for some simple
$(n_{l}-1)\,$-vectors $\xi^{l}$ of $X_{l}$. We let $\nu^{l}=\star\xi^{l}\in
X_{l}$ be the normals to $\Sigma$. If we decompose $\xi^{1}$ as

$\xi^{1}=\sum_{i}\xi^{1}_{\bar{\imath}}e_{\bar{\imath}}$

we have by definition of the Hodge star operator
$\xi^{1}_{\bar{\imath}}=(-1)^{i}\nu^{1}_{i}$. Similarly
$\xi^{2}_{\bar{\jmath}}=(-1)^{j}\nu^{2}_{j}$. Using (2.26) we find

$\mu_{i,j}=m(-1)^{i+j}\xi^{1}_{\bar{\imath}}\xi^{2}_{\bar{\jmath}}\,\mathcal{H}^{n-2}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Sigma=m\nu^{1}_{i}\nu^{2}_{j}\,\mathcal{H}^{n-2}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Sigma.$

This proves that $\mu[u]=m\nu^{1}\otimes\nu^{2}\,\mathcal{H}^{n-2}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Sigma$ and thus that $\mu[u]$ is
rectifiable. If moreover $T[u]$ is tensor rectifiable we see that also
$\mu[u]$ is tensor rectifiable. Since we can revert the argument, this also
proves that if $\mu[u]$ is rectifiable (respectively tensor rectifiable), then
$T[u]$ is rectifiable (respectively tensor rectifiable).
Since on
$\mathcal{D}^{k_{1}}(\Omega_{1})\wedge\mathcal{D}^{k_{2}}(\Omega_{2})$,
$d=d_{1}+(-1)^{k_{1}}d_{2}$ we have $d_{2}d_{1}d=0$ and thus $\partial
T[u]=0$. We finally prove (2.27). For the first identify we write using the
decomposition (2.25) and (2.26),

$\mathbf{M}(T[u])=\sup_{|\omega|\leq
1}\left<T[u],\omega\right>=\sup_{|\omega|\leq
1}\sum_{i,j}(-1)^{i+j}\int_{\Omega}\omega_{i,j}\,d\mu_{i,j}\lesssim|\mu|(\Omega).$

By the previous discussion, the equality
$\mathbf{M}_{\theta}(T[u])=\mathbf{M}_{\theta}(\mu[u])$ is immediate (notice
that both terms are finite only if $T[u]$ is rectifiable). ∎

Thanks to Proposition 2.6, we can reduce the proof of Theorem 1.12 to the
analog result for $T[u]$.

######  Theorem 2.7.

Let $u\in L^{\infty}(\Omega)$ be such that

$\mathcal{E}^{\prime\prime}(u):=\liminf_{k\uparrow\infty}\sum_{1\leq i\leq
n_{1},1\leq j\leq
n_{2}}\int_{\Omega^{\varepsilon_{k}}}\dfrac{|D[Du(\cdot,r_{k}f_{j})](x,r_{k}e_{i})|^{\theta}}{r_{k}^{2}}\,dx<\infty.$

Then $T[u]$ is a tensor rectifiable cycle with

$\mathbf{M}(T[u])\lesssim\|u\|_{\infty}^{1-\theta}\mathcal{E}^{\prime\prime}(u)\qquad\text{and
}\qquad\mathbf{M}_{\theta}(T[u])\lesssim\mathcal{E}^{\prime\prime}(u).$ (2.29)

Moreover, if $u$ is integer-valued then $T[u]$ is integer rectifiable.

###### Proof.

Step 1. Preliminary observations. We first notice that by Remark 1.5, we have
$|\mu[u]|(\Omega)\lesssim\|u\|_{\infty}^{1-\theta}\mathcal{E}^{\prime\prime}(u)$
and thus by Proposition 2.6, $T[u]$ is a normal current and the first
inequality in (2.29) holds. We thus need to show that $T[u]$ is tensor
rectifiable and the second inequality in (2.29).

Step 2. The case where $\Omega$ is a cube.
In this step we prove the claim assuming that $\Omega$ is a $n$-cube. By
scaling we assume without loss of generality that $\Omega=Q^{n}:=(0,1)^{n}$.
We start by recalling the definition of slicing of currents by coordinate
$(n-2)\,$-planes (see [GMS98, Fed69, Whi99b, Jer02] for more details). For
this we introduce some notation. For $\alpha^{1}\subset\\{1,\cdots,n_{1}\\}$
and $\alpha^{2}\subset\\{1,\cdots,n_{2}\\}$ we set

$\alpha:=\alpha_{1}\cup(n_{1}+\alpha_{2})\qquad\text{and}\qquad\bar{\alpha}:=\\{1,\cdots,n\\}\setminus\alpha.$

We then define
$\mathbb{X}_{\alpha^{1}}=\operatorname{span}\\{e_{i}\\}_{i\in\alpha^{1}}$,
$\mathbb{X}_{\alpha^{2}}=\operatorname{span}\\{f_{j}\\}_{j\in\alpha^{2}}$ and
set $\mathbb{X}_{\alpha}=\mathbb{X}_{\alpha^{1}}\oplus\mathbb{X}_{\alpha^{2}}$
so that $\mathbb{R}^{n}=\mathbb{X}_{\alpha}\oplus\mathbb{X}_{\bar{\alpha}}$.
We decompose correspondingly every $x\in\mathbb{R}^{n}$ as
$x=x^{\alpha}+x^{\bar{\alpha}}$. With a slight abuse of notation we set
$Q^{\alpha}=Q^{n}\cap\mathbb{X}_{\alpha}$. For a function $u\in L^{1}(Q^{n})$
and $x\in Q^{n}$ we set $u_{x^{\bar{\alpha}}}(x^{\alpha}):=u(x)$ so that by
Fubini $u_{x^{\bar{\alpha}}}\in L^{1}(Q^{\alpha})$ for almost every
$x^{\bar{\alpha}}$. For every $\alpha$ with $|\alpha|=2$ and every
$T\in\mathcal{D}_{n-2}(Q^{n})$, we define the slices
$\operatorname{Sl}_{\bar{\alpha}}^{x^{\bar{\alpha}}}T\in\mathcal{D}^{0}(Q^{\alpha})$
by the requirement (see [Fed69, Theorem 4.3.2] or [GMS98, Section 2.5]) that
for every $\omega\in\mathcal{D}^{0}(Q^{n})$,

$\left<T,\omega\,dx^{\bar{\alpha}}\right>=\int_{Q^{\bar{\alpha}}}\left<\operatorname{Sl}_{\bar{\alpha}}^{x^{\bar{\alpha}}}T,\omega_{x^{\bar{\alpha}}}\right>\,dx^{\bar{\alpha}}.$
(2.30)

Here $dx^{\bar{\alpha}}$ denotes the canonical $(n-2)\,$-form.

Step 2.1. We claim that for every $\alpha$ with $|\alpha|=2$ and almost every
$x^{\bar{\alpha}}$, the slices
$\operatorname{Sl}_{\bar{\alpha}}^{x^{\bar{\alpha}}}T[u]$ are
$0\,$-rectifiable. Using White’s rectifiability criterion [Whi99b, Jer02] this
would prove that $T[u]$ is $(n-2)\,$-rectifiable.
By (2.26), if $\alpha_{1}=\emptyset$ or $\alpha_{2}=\emptyset$, then
$\operatorname{Sl}_{\bar{\alpha}}^{x^{\bar{\alpha}}}T[u]=0$ and there is
nothing to prove. Indeed, if for instance $\alpha_{2}=\emptyset$ then
$\\{n_{1}+1,\dots,n\\}\subset\bar{\alpha}$ so that for every
$\omega\in\mathcal{D}^{0}(Q^{n})$ we have
$\omega\,dx^{\bar{\alpha}}=\omega_{1}\wedge dx^{\bar{\alpha}}$ in the
decomposition (2.25). Hence, using (2.28) as in the proof of (2.26), we
compute

$\left<T[u],\omega\,dx^{\bar{\alpha}}\right>=\left<[[u]],d_{2}d_{1}[\omega
dx^{\bar{\alpha}}]\right>=\left<[[u]],d_{2}0\right>=0.$

We conclude by the definition (2.30) that
$\operatorname{Sl}_{\bar{\alpha}}^{x^{\bar{\alpha}}}T[u]=0$. The case
$\alpha_{1}=\emptyset$ is similar using $d_{2}d_{1}=-d_{1}d_{2}$.
We may thus assume that $\alpha_{1}=\\{i\\}$ and $\alpha_{2}=\\{j\\}$ so that
$dx^{\bar{\alpha}}=dx^{1}_{\bar{\imath}}\wedge dx^{2}_{\bar{\jmath}}$. Let us
first prove that the operator $\partial_{1}\partial_{2}$ commutes with slicing
in the sense that

$\operatorname{Sl}_{\bar{\alpha}}^{x^{\bar{\alpha}}}T[u]=(-1)^{i+j}\partial_{1}\partial_{2}[[u_{x^{\bar{\alpha}}}]]\qquad\text{
for a.e.\hbox{} missing}\,x^{\bar{\alpha}}\in Q^{\bar{\alpha}}.$ (2.31)

Indeed, for $\omega\in\mathcal{D}^{0}(Q^{n})$, we compute using (2.26) and
(2.23)

$\left<T[u],\omega dx^{1}_{\bar{\imath}}\wedge
dx^{2}_{\bar{\jmath}}\right>=(-1)^{i+j}\int_{Q^{n}}\omega\,d\mu_{i,j}=(-1)^{i+j}\int_{Q^{n}}\dfrac{\partial^{2}\omega}{\partial
x^{1}_{i}\partial x^{2}_{j}}\,u\,dx.$

Then by Fubini,

$\displaystyle\left<T[u],\omega dx^{1}_{\bar{\imath}}\wedge
dx^{2}_{\bar{\jmath}}\right>$
$\displaystyle=(-1)^{i+j}\int_{Q^{\bar{\alpha}}}\left[\int_{Q^{\alpha}}\dfrac{\partial^{2}\omega_{x^{\bar{\alpha}}}}{\partial
x^{1}_{i}\partial
x^{2}_{j}}\,u_{x^{\bar{\alpha}}}\,dx^{\alpha}\right]\,dx^{\bar{\alpha}}$
$\displaystyle=(-1)^{i+j}\int_{Q^{\bar{\alpha}}}\left<\partial_{1}\partial_{2}[[u_{x^{\bar{\alpha}}}]],\omega_{x^{\bar{\alpha}}}\right>\,dx^{\bar{\alpha}}.$

By definition (2.30) of
$\operatorname{Sl}_{\bar{\alpha}}^{x^{\bar{\alpha}}}T[u]$, this concludes the
proof of (2.31).
We now prove that for almost every $x^{\bar{\alpha}}\in Q^{\bar{\alpha}}$,
$T[u_{x^{\bar{\alpha}}}]=\partial_{1}\partial_{2}[[u_{x^{\bar{\alpha}}}]]$ is
$0\,$-rectifiable. To simplify notation we write $Q^{n}_{\varepsilon}$ for
$(Q^{n})^{\varepsilon}$. Let us prove that

$\mathcal{E}^{\prime\prime}(u)\geq\int_{Q^{\bar{\alpha}}}\mathcal{E}^{\prime\prime}(u_{x^{\bar{\alpha}}})\,dx^{\bar{\alpha}}.$
(2.32)

For this we use Fubini and Fatou to obtain

$\displaystyle\mathcal{E}^{\prime\prime}(u)$
$\displaystyle\geq\liminf_{k\uparrow\infty}\int_{Q^{n}_{\varepsilon_{k}}}\dfrac{|D[Du(\cdot,r_{k}f_{j})](x,r_{k}e_{i})|^{\theta}}{r_{k}^{2}}\,dx$
$\displaystyle=\liminf_{k\uparrow\infty}\int_{Q^{\bar{\alpha}}}\boldsymbol{1}_{Q^{\bar{\alpha}}_{\varepsilon_{k}}}(x^{\bar{\alpha}})\left[\int_{Q^{\alpha}_{\varepsilon_{k}}}\dfrac{|D[Du_{x^{\bar{\alpha}}}(\cdot,r_{k}f_{j})](x^{\alpha},r_{k}e_{i})|^{\theta}}{r_{k}^{2}}\,dx^{\alpha}\right]\,dx^{\bar{\alpha}}$
$\displaystyle\geq\int_{Q^{\bar{\alpha}}}\liminf_{k\uparrow\infty}\boldsymbol{1}_{Q^{\bar{\alpha}}_{\varepsilon_{k}}}(x^{\bar{\alpha}})\left[\int_{Q^{\alpha}_{\varepsilon_{k}}}\dfrac{|D[Du_{x^{\bar{\alpha}}}(\cdot,r_{k}f_{j})](x^{\alpha},r_{k}e_{i})|^{\theta}}{r_{k}^{2}}\,dx^{\alpha}\right]\,dx^{\bar{\alpha}}$
$\displaystyle=\int_{Q^{\bar{\alpha}}}\liminf_{k\uparrow\infty}\left[\int_{Q^{\alpha}_{\varepsilon_{k}}}\dfrac{|D[Du_{x^{\bar{\alpha}}}(\cdot,r_{k}f_{j})](x^{\alpha},r_{k}e_{i})|^{\theta}}{r_{k}^{2}}\,dx^{\alpha}\right]\,dx^{\bar{\alpha}}$
$\displaystyle=\int_{Q^{\bar{\alpha}}}\mathcal{E}^{\prime\prime}(u_{x^{\bar{\alpha}}})\,dx^{\bar{\alpha}}.$

This proves (2.32). Therefore, for almost every $x^{\bar{\alpha}}\in
Q^{\bar{\alpha}}$, $\mathcal{E}^{\prime\prime}(u_{x^{\bar{\alpha}}})<\infty$.
By Theorem 1.9, for every such $x^{\bar{\alpha}}$, there exist countable
sequences $x^{l}\in Q^{\alpha}$ and $m_{l}\in\mathbb{R}\setminus\\{0\\}$ with
$T[u_{x^{\bar{\alpha}}}]=\sum_{l}m_{l}\delta_{x^{l}}$ and such that

$\sum_{l}|m_{l}|\lesssim\|u\|_{\infty}^{1-\theta}\mathcal{E}^{\prime\prime}(u_{x^{\bar{\alpha}}})\qquad\text{
and
}\qquad\mathbf{M}_{\theta}(T[u_{x^{\bar{\alpha}}}])=\sum_{l}|m_{l}|^{\theta}\lesssim\mathcal{E}^{\prime\prime}(u_{x^{\bar{\alpha}}}).$
(2.33)

In particular, $T[u_{x^{\bar{\alpha}}}]$ is $0\,$-rectifiable. Moreover, if
$u$ is integer-valued then by Theorem 1.9, so is $T[u_{x^{\bar{\alpha}}}]$.

Step 2.2. We now prove the second inequality in (2.29). For this we notice
that if $(\Sigma,\xi,m)$ are as in (2.24) for $T=T[u]$, then as in the proof
of Proposition 2.6, we can write $\xi=\xi^{1}\wedge\xi^{2}$ for some simple
(and unitary) $(n_{l}-1)\,$-vectors $\xi^{l}\in\mathbb{X}_{l}$. We write

$\xi^{1}=\sum_{i}\xi^{1}_{i}e_{\bar{\imath}}\qquad\text{and
}\qquad\xi^{2}=\sum\xi^{2}_{j}f_{\bar{\jmath}}$

so that by triangle inequality

$\mathbf{M}_{\theta}(T[u])=\int_{\Sigma}|m|^{\theta}\,d\mathcal{H}^{n-2}=\int_{\Sigma}|m|^{\theta}|\xi^{1}\wedge\xi^{2}|\,d\mathcal{H}^{n-2}\leq\sum_{i,j}\int_{\Sigma}|m|^{\theta}|\xi^{1}_{i}\xi^{2}_{j}|\,d\mathcal{H}^{n-2}.$
(2.34)

Fix $i\in\\{1,\cdots,n_{1}\\}$, $j\in\\{1,\cdots,n_{2}\\}$ and let
$\alpha_{1}=\\{i\\}$, $\alpha_{2}=\\{j\\}$. By the co-area formula for
rectifiable sets (see [Fed69, Theorem 3.2.22] or [KP08, Theorem 5.4.9]),

$\int_{\Sigma}|m|^{\theta}|\xi^{1}_{i}\xi^{2}_{j}|\,d\mathcal{H}^{n-2}=\int_{Q^{\bar{\alpha}}}\sum_{x^{\alpha}\in\Sigma\cap(Q^{\alpha}+x^{\bar{\alpha}})}|m(x)|^{\theta}dx^{\bar{\alpha}}=\int_{Q^{\bar{\alpha}}}\mathbf{M}_{\theta}(\operatorname{Sl}_{\bar{\alpha}}^{x^{\bar{\alpha}}}T[u])dx^{\bar{\alpha}}.$

Using (2.31) and (2.33) we find

$\int_{\Sigma}|m|^{\theta}|\xi^{1}_{i}\xi^{2}_{j}|\,d\mathcal{H}^{n-2}\lesssim\int_{Q^{\bar{\alpha}}}\mathcal{E}^{\prime\prime}(u_{x^{\bar{\alpha}}})dx^{\bar{\alpha}}\stackrel{{\scriptstyle\eqref{fatouG}}}{{\leq}}\mathcal{E}^{\prime\prime}(u).$

Plugging this in (2.34) proves

$\mathbf{M}_{\theta}(T[u])\lesssim\mathcal{E}^{\prime\prime}(u).$

Step 2.3. We finally show that $T[u]$ is tensor-rectifiable. Since it is a
local statement it is enough to prove that around each point $\bar{x}\in
Q^{n}$ there is a ball $B_{r}(\bar{x})\subset Q^{n}$ such that $T[u]\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,B_{r}(\bar{x})$ is tensor rectifiable.
As $T[u]$ is normal and rectifiable, $T_{\bar{x},r}=T[u]\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,B_{r}(\bar{x})$ is also a normal and
rectifiable current for every $\bar{x}\in Q^{n}$ and almost every $r>0$
(depending on $\bar{x}$) such that $B_{r}(\bar{x})\subset Q^{n}$. In
particular $T_{\bar{x},r}$ is a rectifiable flat chain (notice however that in
general $T_{\bar{x},r}$ is not a cycle). Moreover, the tangent
$(n-2)\,$-vector $\xi$ to $\Sigma$ can be written as $\xi^{1}\wedge\xi^{2}$
with $\xi^{l}\in\mathbb{X}^{l}$ so that for $\mathcal{H}^{n-2}$-every
$x\in\Sigma\cap B_{r}(\bar{x})$ the approximate tangent $(n-2)$-plane
$T_{x}\Sigma$ is of the form $L^{1}(x)\times L^{2}(x)$ for some the
hyperplanes $L^{l}(x)=(\xi^{l})^{\perp}\cap\mathbb{X}^{l}$ for $l=1,2$.
Therefore we may appeal to [GM22b, Theorem 1.3] which yields that
$T_{\bar{x},r}$ is tensor rectifiable.

Step 3. The general case.
In this final step we use a covering argument to prove the claim in a more
general domain $\Omega$. For this we need to introduce localized versions of
the energy. For any open set $A=A_{1}+A_{2}\subset\Omega$, we set

$\mathcal{E}^{\prime\prime}(u,A):=\liminf_{k\uparrow\infty}\sum_{1\leq i\leq
n_{1},1\leq j\leq
n_{2}}\int_{A^{\varepsilon_{k}}}\dfrac{|D[Du(\cdot,r_{k}f_{j})](x,r_{k}e_{i})|^{\theta}}{r_{k}^{2}}\,dx.$

We then set for $u\in L^{\infty}(\Omega)$,
$u_{A}:=u\boldsymbol{1}_{A}\in\mathcal{D}_{n}(A)$ so that as elements of
$\mathcal{D}_{n-2}(A)$ (this is of course not true in
$\mathcal{D}_{n-2}(\Omega)$),

$T[u_{A}]=T[u]\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,A.$

Let $\mathcal{Q}^{\prime}$ be a Whitney partition of $\Omega$, see e.g.
[Gra14, Appendix J]. By definition, there exists $\lambda=\lambda(d)>0$ such
that

$\mathcal{Q}=\\{\lambda Q\ :\ Q\in\mathcal{Q^{\prime}}\\}$

is a cover of $\Omega$ with finite overlap, i.e.

$\boldsymbol{1}_{\Omega}\leq\sum_{Q\in\mathcal{Q}}\boldsymbol{1}_{Q}\lesssim\boldsymbol{1}_{\Omega}.$
(2.35)

Therefore, letting

$F_{k}(x)=\sum_{1\leq i\leq n_{1},1\leq j\leq
n_{2}}\dfrac{|D[Du(\cdot,r_{k}f_{j})](x,r_{k}e_{i})|^{\theta}}{r_{k}^{2}},$

we have

$\mathcal{E}^{\prime\prime}(u)=\liminf_{k\uparrow\infty}\int_{\Omega^{\varepsilon_{k}}}\boldsymbol{1}_{\Omega}(x)F_{k}(x)\,dx\gtrsim\liminf_{k\uparrow\infty}\sum_{\mathcal{Q}}\int_{\Omega^{\varepsilon_{k}}}\boldsymbol{1}_{Q}(x)F_{k}(x)\,dx\\\
\geq\liminf_{k\uparrow\infty}\sum_{\mathcal{Q}}\int_{Q^{\varepsilon_{k}}}F_{k}(x)\,dx\geq\sum_{\mathcal{Q}}\liminf_{k\uparrow\infty}\int_{Q^{\varepsilon_{k}}}F_{k}(x)\,dx=\sum_{\mathcal{Q}}\mathcal{E}^{\prime\prime}(u,Q).$

As a consequence, for every $Q\in\mathcal{Q}$ we have
$\mathcal{E}^{\prime\prime}(u_{Q})=\mathcal{E}^{\prime\prime}(u,Q)<\infty$ so
that by Step 2, $T[u_{Q}]$ is tensor rectifiable with

$\mathbf{M}_{\theta}(T[u_{Q}])\lesssim\mathcal{E}^{\prime\prime}(u,Q).$

On the one hand this yields that $T[u]$ is also tensor rectifiable. On the
other hand, by (2.35),

$\mathbf{M}_{\theta}(T[u])\leq\sum_{\mathcal{Q}}\mathbf{M}_{\theta}(T[u]\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,Q)=\sum_{\mathcal{Q}}\mathbf{M}_{\theta}(T[u_{Q}])\lesssim\sum_{\mathcal{Q}}\mathcal{E}^{\prime\prime}(u,Q)\lesssim\mathcal{E}^{\prime\prime}(u).$

This concludes the proof. ∎

##  3 The case $\theta=1$

When $\theta=1$, we only consider the case $n_{1}=n_{2}=1$. In this last
section we assume without loss of generality that $\Omega=(-1,1)^{2}$.

###  3.1 The general case

In the case $\theta=1$ (and $n_{1}=n_{2}=1)$ the defect measure does not in
general concentrate on a set of Hausdorff dimension $n_{1}+n_{2}-2=0$. The
examples of Remark 1.18 show that it may concentrate on a set with Hausdorff
dimension $s$ for any $s\in[0,1]$. We establish that 1 is the largest possible
dimension provided $u$ is integrable and $\mathcal{E}(u)<\infty$.

###### Proof of Theorem 1.17.

Let $u\in L^{1}_{loc}(\Omega)$ be such that $\mathcal{E}(u)<\infty$. By (1.4),
we know that $\mu=\partial_{1}\partial_{2}u$ is a measure with
$|\mu|(\Omega)\lesssim\mathcal{E}(u)$. By Lemma 2.1 we have for $k\geq 1$ and
almost every $x\in\Omega^{\varepsilon_{k}}$,

$\mu(Q_{x,r_{k}})=Du(x+r_{k}e_{2},r_{k}e_{1})-Du(x,r_{k}e_{1})\qquad\text{and
}\qquad|\mu|(\partial Q_{x,r_{k}})=0.$ (3.1)

Recalling the definition of $q(x,z)$ in (1.6) and the subadditivity of
$s\mapsto s^{\theta_{l}}$, we deduce that

$|\mu(Q_{x,r_{k}})|^{\theta_{1}}(|Du(x,z^{k}_{2})|+|Du(x+z^{k}_{1},z^{k}_{2})|)^{\theta_{2}}\leq
q(x,z^{k}).$

Here we used the notation $z^{k}_{1}:=r_{k}e_{1}$, $z^{k}_{2}:=r_{k}e_{2}$
where $(e_{1},e_{2})$ is the standard basis of $\mathbb{R}^{2}$. Denoting
$z^{k}:=z^{k}_{1}+z^{k}_{2}=(r_{k},r_{k})$, the inequality (1.5) then leads to

$\limsup_{k\uparrow\infty}\int_{\Omega^{\varepsilon_{k}}}\dfrac{|\mu(Q_{x,r_{k}})|^{\theta_{1}}(|Du(x,z^{k}_{2})|+|Du(x+z^{k}_{1},z^{k}_{2})|)^{\theta_{2}}}{r_{k}^{2}}\,dx\lesssim\mathcal{E}(u).$
(3.2)

The rest of the proof is of the same flavor as the proof of Lemma 2.4 but the
construction of the partitions of $\Omega$ into rectangles in Steps 2.1—2.3
below is more involved.
Step 1. For $y\in\Omega$, we define

$N_{k}(y):=\sup\left\\{\dfrac{|\mu|(Q_{x,r_{k}})}{r_{k}}:x\in\Omega^{\varepsilon_{k}}\text{
such that }y\in Q_{x,r_{k}}\right\\}.$

Then for $\eta>0$, we consider the set

$\Sigma_{\eta}:=\left\\{y\in\Omega\,:\,\limsup_{k\uparrow\infty}N_{k}(y)\geq\eta\right\\}.$

By Besicovitch covering theorem [AFP00, Theorem 2.17], we have
$\mathcal{H}^{1}(\Sigma_{\eta})\lesssim|\mu|(\Omega)/\eta$. Next, the sets
$\\{\Sigma_{\eta}\\}_{\eta>0}$ form a decreasing family of Borel sets and
their union is

$\Sigma:=\left\\{y\in\Omega\,:\,\limsup_{k\uparrow\infty}N_{k}(y)>0\right\\}.$

We deduce that $\Sigma$ is a Borel subset of $\Omega$ and that the measure
$\mathcal{H}^{1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Sigma$ is $\sigma\,$-finite.

Step 2. Let us show that $|\mu|(\Omega\backslash\Sigma)=0$. Let
$\varepsilon>0$. By definition, a point $y$ of $\Omega^{\varepsilon}$ belongs
to $\Omega\backslash\Sigma$ if and only if

$\lim_{k\uparrow\infty}\left[\sup\left\\{\dfrac{|\mu|(Q_{x,r_{k}})}{r_{k}}\,:\,Q_{x,r_{k}}\subset\Omega\text{
with }y\in Q_{x,r_{k}}\right\\}\right]=0.$ (3.3)

By Egoroff theorem, there exists a measurable set
$\mathcal{Y}\subset\Omega^{\varepsilon}\setminus\Sigma$ with

$|\mu|(\mathcal{Y})\geq|\mu|(\Omega^{\varepsilon}\setminus\Sigma)/2$

such that (3.3) holds uniformly in $\mathcal{Y}$. In particular, there exists
a sequence $t_{k}>0$ with $t_{k}\downarrow 0$ and

$|\mu|(Q_{x,r_{k}})\leq t_{k}r_{k}\quad\text{for every }x\text{ such that
}Q_{x,r_{k}}\cap\mathcal{Y}\neq\varnothing\text{ and every }k\geq 1.$ (3.4)

Substituting $\max(t_{k},r_{k})$ for $t_{k}$ we assume without loss of
generality that $t_{k}\geq r_{k}$.

Step 2.1. Covering of $\mathcal{Y}$. Let us fix $k\geq 1$ and $x\in
Q_{0,r_{k}}$ and let us consider the sets

$\displaystyle{\mathcal{P}}^{k}_{x}$ $\displaystyle:=\\{Q\ :\
Q=Q_{x+r_{k}z,r_{k}}\ \text{ for some }z\in\mathbb{Z}^{2}\text{ such that
}x+r_{k}z\in\Omega^{\varepsilon_{k}}\\},$ $\displaystyle\mathcal{Q}^{k}_{x}$
$\displaystyle:=\\{Q\in{\mathcal{P}}^{k}_{x}:Q\cap\mathcal{Y}\neq\varnothing\\}.$

We then define $\mathcal{Y}^{k}_{x}:=\cup\mathcal{Q}^{k}_{x}$. Notice that by
(3.4),

$\text{for }k\text{ large enough, }\mathcal{Y}\subset\mathcal{Y}^{k}_{x}\text{
for every }x\in Q_{0,r_{k}}.$ (3.5)

Our task is now to build a covering of some $\mathcal{Y}^{k}_{x}$ (with $x$
depending on $k$) by a collection of boxes $Q$ such that $\sum|\mu(Q)|$ tends
to $0$ as $k$ goes to $+\infty$. For this, we introduce a large number
$\Lambda\geq 1$ and we cover $\mathcal{Q}^{k}_{x}$ with a disjoint union,
$\mathcal{Q}^{k}_{x}\subset{\mathcal{G}}^{k}_{x}\cup\mathcal{B}^{k}_{x}$
defined as follows. For $Q$ of the form $Q_{y,r_{k}}$, let us note $y=x_{Q}$
its bottom left corner. We set,

$\displaystyle{\mathcal{G}}^{k}_{x}$ $\displaystyle:=$
$\displaystyle\left\\{Q\in\mathcal{Q}^{k}_{x}\,:\,|Du(x_{Q},z^{k}_{2})|+|Du(x_{Q}+z^{k}_{1},z^{k}_{2})|\geq\Lambda
t_{k}r_{k}\right\\},$ (3.6) $\displaystyle\mathcal{B}^{k}_{x}$
$\displaystyle:=$
$\displaystyle\left\\{Q\in\mathcal{Q}^{k}_{x}\,:\,|Du(x_{Q},z^{k}_{2})|+|Du(x_{Q}+z^{k}_{1},z^{k}_{2})|<\Lambda
t_{k}r_{k}\right\\}.$ (3.7)

Step 2.2. Estimation of $|\mu|$ on the good set ${\mathcal{G}}^{k}_{x}$ and
selection of $x=x^{k}$.
We first bound the average over $x$ of the sums
$\sum_{{\mathcal{G}}_{x}^{k}}|\mu(Q)|$. By (3.6) and using
$\theta=\theta_{1}+\theta_{2}=1$ we get,

$\dfrac{1}{r_{k}^{2}}\int_{Q_{0,r_{k}}}\sum_{Q\in{\mathcal{G}}_{x}^{k}}|\mu(Q)|\,dx\\\
\leq\dfrac{1}{r_{k}^{2}}\int_{Q_{0,r_{k}}}\sum_{Q\in{\mathcal{G}}_{x}^{k}}|\mu(Q)|^{\theta_{1}}|\mu(Q)|^{\theta_{2}}\dfrac{\left(|Du(x_{Q},z^{k}_{2})|+|Du(x_{Q}+z^{k}_{1},z^{k}_{2})|\right)^{\theta_{2}}}{(\Lambda
t_{k}r_{k})^{\theta_{2}}}\,dx.$

By (3.5) we can use (3.4) for $k$ large enough. We obtain,

$\displaystyle\dfrac{1}{r_{k}^{2}}\int_{Q_{0,r_{k}}}\sum_{Q\in{\mathcal{G}}_{x}^{k}}$
$\displaystyle|\mu(Q)|\,dx$
$\displaystyle\leq\,\dfrac{1}{\Lambda^{\theta_{2}}}\dfrac{1}{r_{k}^{2}}\int_{Q_{0,r_{k}}}\sum_{Q\in{\mathcal{P}}_{x}^{k}}|\mu(Q)|^{\theta_{1}}(|Du(x_{Q},z^{k}_{2})|+|Du(x_{Q}+z^{k}_{1},z^{k}_{2})|)^{\theta_{2}}\,dx$
$\displaystyle=\dfrac{1}{\Lambda^{\theta_{2}}}\int_{\Omega^{\varepsilon_{k}}}\dfrac{|\mu(Q_{y,r_{k}})|^{\theta_{1}}(|Du(y,z_{2}^{k})|+|Du(y+z_{1}^{k},z_{2}^{k})|)^{\theta_{2}}}{r_{k}^{2}}\,dy,$

where we used Fubini for the last identity. By (3.2) the last integral is of
the order of $O(\mathcal{E}(u)/\Lambda^{\theta_{2}})$ as $k\uparrow\infty$.
Hence, there exists a sequence $x^{k}\in Q_{0,r_{k}}$ such that

$\limsup_{k\uparrow\infty}\sum_{Q\in{\mathcal{G}}_{x^{k}}^{k}}|\mu(Q)|\,\lesssim\,\dfrac{\mathcal{E}(u)}{\Lambda^{\theta_{2}}}.$
(3.8)

Moreover, we may assume that (3.1) holds for every square $Q\subset\Omega$ of
the form $Q_{x^{k}+r_{k}z,r_{k}}$ for some $z\in\mathbb{Z}^{2}$.

From now on, we select $x=x^{k}$ and we drop the subscripts $x^{k}$: we write
${\mathcal{P}}^{k}$ for ${\mathcal{P}}_{x^{k}}^{k}$, $\mathcal{Q}^{k}$ for
$\mathcal{Q}_{x^{k}}^{k}$, ${\mathcal{G}}^{k}$ for ${\mathcal{G}}_{x^{k}}^{k}$
and $\mathcal{B}^{k}$ for $\mathcal{B}_{x^{k}}^{k}$.

Step 2.3. Covering of $\cup\mathcal{B}^{k}$.
Let $z_{2}\in\mathbb{Z}$, we denote ${\mathcal{P}}^{k}_{z_{2}}$ the row of
squares $Q\in{\mathcal{P}}^{k}$ such that $x_{Q}=x^{k}+r_{k}(z_{1},z_{2})$ for
some $z_{1}\in\mathbb{Z}$. The set ${\mathcal{P}}^{k}_{z_{2}}$ is totally
ordered by the relation “$<$” defined by

$Q_{y_{k}+r_{k}(z_{1},z_{2})}<Q_{y_{k}+r_{k}(z_{1}^{\prime},z_{2})}\qquad\text{
whenever }\,z_{1}<z_{1}^{\prime}.$

$R_{z_{2},1}$$R_{z_{2},2}$$Q_{1}^{-}$$Q_{1}^{+}$$Q_{2}^{-}$$Q_{2}^{+}$$\mathcal{B}_{z_{2},1}$$\mathcal{B}_{z_{2},2}$
Figure 3: The covering $\mathcal{B}$ of the bad set. The gray squares are the
elements of $\mathcal{B}^{k}$.

Let us define

$\mathcal{B}^{k}_{z_{2}}:=\mathcal{B}^{k}\cap{\mathcal{P}}^{k}_{z_{2}}.$

It is easy to see that $\mathcal{B}^{k}_{z_{2}}$ admits a partition into a
sequence $\mathcal{B}^{k}_{z_{2},1},\cdots,\mathcal{B}^{k}_{z_{2},s_{z_{2}}}$
with the following properties.

  1. (a)

$s_{z_{2}}\lesssim 1/(\Lambda^{2}t_{k})$;

  2. (b)

every subset $\mathcal{B}^{k}_{z_{2},s}$ satisfies
$\operatorname{diam}\left(\bigcup_{\mathcal{B}^{k}_{z_{2},s}}Q\right)\lesssim\Lambda^{2}t_{k}$;

  3. (c)

the sets $\mathcal{B}^{k}_{z_{2},1},\cdots,\mathcal{B}^{k}_{z_{2},s_{z_{2}}}$
are ordered as follows: if $1\leq s_{1}<s_{2}\leq s_{z_{2}}$ then
$\max\mathcal{B}^{k}_{z_{2},s_{1}}<\min\mathcal{B}^{k}_{z_{2},s_{2}}$.

Let $s\in\\{1,\cdots,s_{z_{2}}\\}$ we denote
$Q^{-}_{s}=\min\mathcal{B}^{k}_{z_{2},s}$,
$Q^{+}_{s}=\max\mathcal{B}^{k}_{z_{2},s}$ and we define the box $R_{z_{2},s}$
by gluing together the elements of $\mathcal{Q}^{k}_{z_{2}}$ between
$Q^{-}_{s}$ and $Q^{+}_{s}$, namely (see Figure 3),

$R_{z_{2},s}:=\bigcup\left\\{Q\in\mathcal{Q}^{k}_{z_{2}}:Q^{-}_{s}\leq Q\leq
Q^{+}_{s}\right\\}.$

By construction the sets $R_{z_{2},s}$ are disjoint, satisfy
$\operatorname{diam}(R_{z_{2},s})\lesssim\Lambda^{2}t_{k}$, their number, for
$z_{2}$ fixed is $s_{z_{2}}\lesssim 1(\Lambda^{2}t_{k})$ (so that their total
number is estimated by $1/(r_{k}\Lambda^{2}t_{k})$) and their union covers
$\cup\mathcal{B}^{k}_{z_{2}}$. Moreover, for $s$ fixed, denoting
$x^{-}_{s}=x_{Q^{-}_{s}}$, $x^{+}_{s}=x_{Q^{+}_{s}}$, we have

$|\mu(R_{z_{2},s})|\,\stackrel{{\scriptstyle\eqref{id4pts}}}{{=}}\,|u(x^{-}_{s})-u(x^{+}_{s}+z_{1}^{k})+u(x^{+}_{s}+z^{k})-u(x^{-}_{s}+z_{2}^{k})|\\\
=\,\left|Du(x^{+}_{s}+z_{1}^{k},z_{2}^{k})-Du(x^{-}_{s},z_{2}^{k})\right|\,\stackrel{{\scriptstyle\eqref{defBk}}}{{\leq}}\,2\Lambda
t_{k}r_{k},$

by triangle inequality and because ${Q^{-}_{s}}$ and ${Q^{+}_{s}}$ belong to
$\mathcal{B}^{k}$. Denoting $\widetilde{\mathcal{B}}^{k}$ the collection of
the sets $R_{z_{2},s}$ for $z_{2}\in\mathbb{Z}$ and $1\leq s\leq s_{z_{2}}$,
we deduce from the above discussion that

$\sum_{R\in\widetilde{\mathcal{B}}^{k}}|\mu(R)|\,\lesssim\,\dfrac{1}{r_{k}}\,\dfrac{1}{\Lambda^{2}t_{k}}\,\Lambda
t_{k}r_{k}\,\lesssim\,\dfrac{1}{\Lambda},\qquad\text{and}\qquad\max\limits_{R\in\widetilde{\mathcal{B}}^{k}}\operatorname{diam}(R)\,\lesssim\,\Lambda^{2}t_{k}.$
(3.9)

Moreover, by construction
$\bigcup\mathcal{B}^{k}\,\subset\,\bigcup\widetilde{\mathcal{B}}^{k}$.
Eventually we define $\widetilde{{\mathcal{G}}}^{k}$ as the set of elements
$Q\in{\mathcal{G}}^{k}$ such that
$Q\not\subset\bigcup\widetilde{\mathcal{B}}^{k}$ and set
$\widetilde{\mathcal{Q}}^{k}:=\widetilde{{\mathcal{G}}}^{k}\cup\widetilde{\mathcal{B}}^{k}$.

Step 2.4. Sending $k\uparrow\infty$. By construction the elements of
$\widetilde{\mathcal{Q}}^{k}$ are disjoint rectangles in $\Omega$ and
recalling (3.5), (3.8),(3.9), we have

$\mathcal{Y}\,\subset\,\bigcup\widetilde{\mathcal{Q}}^{k},\qquad\sum_{Q\in\widetilde{\mathcal{Q}}^{k}}|\mu(Q)|\lesssim\dfrac{\mathcal{E}(u)}{\Lambda^{\theta_{2}}}+\dfrac{1}{\Lambda}\quad\text{and}\quad\max\limits_{Q\in\widetilde{\mathcal{Q}}^{k}}\operatorname{diam}(Q)\,\stackrel{{\scriptstyle
k\uparrow\infty}}{{\longrightarrow}}\,0.$

Applying Lemma 2.2 to the measure $\mu$ and to the family
$\\{\widetilde{\mathcal{Q}}^{k}\\}$, we obtain

$|\mu|(\mathcal{Y})\,\leq\,\liminf_{k\uparrow\infty}\sum_{Q\in\widetilde{\mathcal{Q}}^{k}}|\mu(Q)|\,\,\lesssim\,\dfrac{\mathcal{E}(u)}{\Lambda^{\theta_{2}}}+\dfrac{1}{\Lambda}.$

Since $\Lambda\geq 1$ is arbitrary, we obtain that
$|\mu|(\Omega^{\varepsilon}\setminus\Sigma)\leq 2|\mu|(\mathcal{Y})=0$ and
sending $\varepsilon$ to 0 we conclude that $\mu=\mu\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Sigma$. Recalling Step 1,
$\mathcal{H}^{1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Sigma$ is $\sigma\,$-finite and the
theorem is proved. ∎

###  3.2 The case of Lipschitz continuous functions

From now on we assume that $u$ is Lipschitz continuous with $\|\nabla
u\|_{\infty}\leq 1$. The results of this section are stated for $v=\nabla u$
instead of $u$ (we always assume that $u$ and $v$ are related by the identity
$\nabla u=v$).

Let us recall that $\Omega=(-1,1)^{2}$ and letting
$K=(\mathbb{R}\times\\{0\\})\cup(\\{0\\}\times\mathbb{R})$, we are interested
in the mappings $v\in L^{\infty}(\Omega,\mathbb{R}^{2})$ such that

$\displaystyle\nabla\times v=\partial_{1}v_{2}-\partial_{2}v_{1}=0,$ (3.10)
$\displaystyle v(x)\in K\text{ for almost every }x\in\Omega.$ (3.11)

Setting $\mu[v]=\partial_{1}v_{2}$ (which coincides with $\mu[u]$) we consider
the set777This set was already defined in the introduction.

$S^{\infty}(\Omega):=\\{v\in
L^{\infty}(\Omega,\mathbb{R}^{2}):\|v\|_{\infty}\leq 1,\
\text{~{}\eqref{curl0main},\eqref{inKmain} hold and
}\mu[v]\in\mathcal{M}(\Omega)\\}.$

####  3.2.1 Derivation of the differential inclusion

Our first result states that if $u$ is Lipschitz continuous with $\|\nabla
u\|_{\infty}\leq 1$ and $\mathcal{E}(u)<\infty$ then $\nabla u\in
S^{\infty}(\Omega)$. We actually prove a stronger statement.

######  Proposition 3.1.

Let $u_{\varepsilon}$ be a sequence of $1\,$-Lipschitz functions such that

$\mathcal{E}_{0}:=\liminf_{\varepsilon\to
0}\mathcal{E}_{\varepsilon}(u_{\varepsilon})<\infty.$

Then, up to extraction and subtraction of constants, $u_{\varepsilon}$
converges uniformly to a $1\,$-Lipschitz function $u$. Moreover $\nabla u\in
S^{\infty}(\Omega)$ with $|\mu[\nabla u]|(\Omega)\lesssim\mathcal{E}_{0}$.

###### Proof.

Since the functions $u_{\varepsilon}$ are $1\,$-Lipschitz, up to extraction
and subtraction of constants, they converge uniformly to a $1\,$-Lipschitz
function $u$. Let $v:=\nabla u$. Using Fatou and the rescaling $z=\varepsilon
z^{\prime}$ we obtain

$\int_{\mathbb{R}^{2}}\dfrac{\rho(z)}{|z|^{2}}\left[\liminf_{\varepsilon\to
0}\int_{\Omega^{\varepsilon}}\dfrac{|Du_{\varepsilon}(x,\varepsilon
z_{1})|^{\theta_{1}}|Du_{\varepsilon}(x,\varepsilon
z_{2})|^{\theta_{2}}}{\varepsilon^{2}}\,dx\right]dz\leq\mathcal{E}_{0}.$

Arguing as in [GM21, Proposition M], we find the existence of
$\sigma_{1}\in\mathbb{X}_{1}\setminus\\{0\\}$ and
$\sigma_{2}\in\mathbb{X}_{2}\setminus\\{0\\}$ such that

$\displaystyle\liminf_{\varepsilon\to
0}\int_{\Omega^{\varepsilon}}\dfrac{|Du_{\varepsilon}(x,\varepsilon\sigma_{1})|^{\theta_{1}}|Du_{\varepsilon}(x,\varepsilon\sigma_{2})|^{\theta_{2}}}{\varepsilon^{2}}\,dx$
$\displaystyle\lesssim\mathcal{E}_{0},$ (3.12)
$\displaystyle\liminf_{\varepsilon\to
0}\int_{\Omega^{\varepsilon}}\dfrac{|D[Du_{\varepsilon}(\cdot,\varepsilon\sigma_{1})](x,\varepsilon\sigma_{2})|}{\varepsilon^{2}}\,dx$
$\displaystyle\lesssim\mathcal{E}_{0}.$ (3.13)

Step 1. $\mu[v]$ is a measure.
We now prove that $\mu[v]\in\mathcal{M}(\Omega)$ (with
$|\mu[v]|(\Omega)\lesssim\mathcal{E}_{0}$). Fix $\varphi\in
C^{\infty}_{c}(\Omega)$, set
$z^{\varepsilon}:=-\varepsilon(\sigma_{1}+\sigma_{2})$ and consider

$\eta_{\varepsilon}:=\dfrac{1}{|\sigma_{1}||\sigma_{2}|\varepsilon^{2}}\boldsymbol{1}_{Q_{0,-z^{\varepsilon}}}.$

Then $u_{\varepsilon}*\eta_{\varepsilon}$ still converges to $u$ uniformly and
we have

$\displaystyle\left<\mu[v],\varphi\right>=\int_{\Omega}u\,\partial_{1}\partial_{2}\varphi$
$\displaystyle=\lim_{\varepsilon\to
0}\int_{\Omega^{\varepsilon}}(u_{\varepsilon}*\eta_{\varepsilon})\partial_{1}\partial_{2}\varphi$
$\displaystyle=\lim_{\varepsilon\to
0}\int_{\Omega^{\varepsilon}}\dfrac{1}{|\sigma_{1}||\sigma_{2}|\varepsilon^{2}}\left[\int_{\Omega^{\varepsilon}}\boldsymbol{1}_{Q_{y,z^{\varepsilon}}}(x)u_{\varepsilon}(y)\,dy\right]\partial_{1}\partial_{2}\varphi(x)\,dx$
$\displaystyle=\lim_{\varepsilon\to
0}\int_{\Omega^{\varepsilon}}\dfrac{1}{|\sigma_{1}||\sigma_{2}|\varepsilon^{2}}\left[\int_{Q_{y,z^{\varepsilon}}}\partial_{1}\partial_{2}\varphi(x)\,dx\right]u_{\varepsilon}(y)\,dy$
$\displaystyle=\lim_{\varepsilon\to
0}\int_{\Omega^{\varepsilon}}\dfrac{1}{|\sigma_{1}||\sigma_{2}|\varepsilon^{2}}D[D\varphi(\cdot,-\varepsilon\sigma_{1})](y,-\varepsilon\sigma_{2})\,u_{\varepsilon}(y)\,dy$
$\displaystyle=\lim_{\varepsilon\to
0}\int_{\Omega^{\varepsilon}}\dfrac{1}{|\sigma_{1}||\sigma_{2}|\varepsilon^{2}}D[Du_{\varepsilon}(\cdot,\varepsilon\sigma_{1})](x,\varepsilon\sigma_{2})\,\varphi(x)\,dx.$

In combination with (3.13) we obtain that indeed
$\mu[v]\in\mathcal{M}(\Omega)$ with $|\mu[v]|(\Omega)\lesssim\mathcal{E}_{0}$.

Step 2. Proof that $v(x)\in K$ almost everywhere.
Let us denote here $\omega:=Q^{2}=(0,1)^{2}$. We claim the following.

######  Claim.

Let $\alpha\in(0,1)$, there exist $\varepsilon_{0},c,\delta>0$ satisfying the
following property.
Let $p=(p_{1},p_{2})\in\mathbb{R}^{2}$ with $|p_{1}|,|p_{2}|\geq\alpha$ and
let $u:\omega\to\mathbb{R}$ be a $1$-Lipschitz function. If

$|u(x)-p\cdot x|\leq\delta\qquad\text{for every }x\in\omega,$ (3.14)

then there holds, for $0<\varepsilon<\varepsilon_{0}$,

$\int_{\omega^{\varepsilon|\sigma|}}\dfrac{|Du(x,\varepsilon\sigma_{1})|^{\theta_{1}}|Du(x,\varepsilon\sigma_{2})|^{\theta_{2}}}{\varepsilon^{2}}\,dx+\int_{\omega^{\varepsilon|\sigma|}}\dfrac{|D[Du(\cdot,\varepsilon\sigma_{1})](x,\varepsilon\sigma_{2})|}{\varepsilon^{2}}\,dx\geq
c.$ (3.15)

Step 2.1. Proof that $v(x)\in K$ almost everywhere, assuming the claim.
Let us first derive from the claim the conclusion $v\in K$ almost everywhere.
For this we argue by contradiction and assume this is not the case. Recalling
that $v\notin K$ means that $v_{1}v_{2}\neq 0$ we see that there exists
$0<\alpha<1$ such that the set $M=\\{|v_{1}|>\alpha,|v_{2}|>\alpha\\}$ has
positive Lebesgue measure. For $x\in Q^{2}$ such that $\bar{x}+rx\in\Omega$ we
set

$u^{r,\bar{x}}(x)=\dfrac{u(\bar{x}+rx)-u(\bar{x})}{r}\qquad\text{and
similarly}\qquad
u_{\varepsilon}^{r,\bar{x}}(x)=\dfrac{u_{\varepsilon}(\bar{x}+rx)-u_{\varepsilon}(\bar{x})}{r}.$

Then, at every point of differentiability $\bar{x}$ of $u$ we have

$\lim_{r\to 0}|u^{r,\bar{x}}(x)-v(\bar{x})\cdot x|=0\qquad\text{uniformly with
respect to }x.$

By Egoroff we may further assume that this limit is uniform in $M$. Let
$\delta>0$ be given by the claim. We deduce that for $r$ small enough, there
holds for every $\bar{x}\in M$ and every $x\in Q^{2}$ such that
$\bar{x}+rx\in\Omega$,

$|u^{r,\bar{x}}(x)-v(\bar{x})\cdot x|\leq\delta/2.$

Let us fix such $r$. Since $\|u_{\varepsilon}-u\|_{\infty}\to 0$ we deduce
that there exists $\varepsilon_{1}>0$ depending on $\delta$ and $r$ such that
for $0<\varepsilon<\varepsilon_{1}$, we have

$|u_{\varepsilon}^{r,\bar{x}}(x)-v(\bar{x})\cdot x|\leq\delta$

for $\bar{x}\in M$ and $x\in Q^{2}$ such that $\bar{x}+rx\in\Omega$.
Therefore, $u_{\varepsilon}^{r,\bar{x}}$ satisfies (3.14) with $p=v(\bar{x})$
and moreover $\min(|p_{1}|,|p_{2}|)\geq\alpha$ as in the statement of the
claim. Furthermore, since $u$ is $1\,$-Lipschitz, the functions
$u^{r,\bar{x}}$ are also $1\,$-Lipschitz.

Let us now show that we can find $\bar{x}\in M$ such that (3.15) is violated
for $u_{\varepsilon}^{r,\bar{x}}$ and $\varepsilon$ small enough. Thanks to
the claim, this would provide the desired contradiction. From (3.12), (3.13),
Markov inequality, Fubini and Fatou, we find $\bar{x}\in M$ such that

$\liminf_{\varepsilon\to
0}\int_{Q_{\bar{x},r}}\dfrac{|Du_{\varepsilon}(x,\varepsilon\sigma_{1})|^{\theta_{1}}|Du_{\varepsilon}(x,\varepsilon\sigma_{2})|^{\theta_{2}}}{\varepsilon^{2}}\,dx+\int_{Q_{\bar{x},r}}\dfrac{|D[Du_{\varepsilon}(\cdot,\varepsilon\sigma_{1})](x,\varepsilon\sigma_{2})|}{\varepsilon^{2}}\,dx\lesssim\mathcal{E}_{0}r^{2}.$

Notice that using the change of variable $x=ry$ and denoting
$\varepsilon^{\prime}=\varepsilon/r$, we have

$\dfrac{1}{r}\int_{Q_{\bar{x},r}}\dfrac{|Du_{\varepsilon}(x,\varepsilon\sigma_{1})|^{\theta_{1}}|Du_{\varepsilon}(x,\varepsilon\sigma_{2})|^{\theta_{2}}}{\varepsilon^{2}}\,dx=\int_{Q^{2}}\dfrac{|Du^{r,\bar{x}}_{r\varepsilon^{\prime}}(y,\varepsilon^{\prime}\sigma_{1})|^{\theta_{1}}|Du^{r,\bar{x}}_{r\varepsilon^{\prime}}(y,\varepsilon^{\prime}\sigma_{2})|^{\theta_{2}}}{(\varepsilon^{\prime})^{2}}\,dy.$

Therefore

$\liminf_{\varepsilon\to
0}\int_{Q^{2}}\dfrac{|Du^{r,\bar{x}}_{r\varepsilon}(x,\varepsilon\sigma_{1})|^{\theta_{1}}|Du^{r,\bar{x}}_{r\varepsilon}(x,\varepsilon\sigma_{2})|^{\theta_{2}}}{\varepsilon^{2}}\,dx\lesssim\mathcal{E}_{0}r.$

Arguing similarly for the second term and choosing $r>0$ and then
$\varepsilon>0$ small enough, we get that $u^{r,\bar{x}}_{\varepsilon r}$
contradicts the claim. Hence $v\in K$ almost everywhere in $\Omega$. This
establishes the proposition assuming that the claim stated at the beginning of
Step 2 holds true.

Step 2.2. Proof of the claim.
As a preliminary remark, notice that if the claim holds for some constants
$\alpha,\varepsilon_{0},c,\delta,$ it also holds for
$\alpha,\varepsilon_{0},c^{\prime},\delta^{\prime},$ for $0<c^{\prime}\leq c$
and $0<\delta^{\prime}\leq\delta$.
We argue by contradiction. Let $\alpha\in(0,1)$, $c>0$ and $\delta>0$. Let
$p=(p_{1},p_{2})$ be such that $|p_{1}|,|p_{2}|\geq\alpha$. Assume that there
exists $u$ such that (3.14) holds true but not (3.15) for some sequence
$\varepsilon=\varepsilon_{k}$ with $\varepsilon_{k}\downarrow 0$. We show
below that this leads to a contradiction.

Taking into account the preliminary remark we decrease $\delta>0$ or $c>0$ or
both if necessary to ensure

$\delta<\dfrac{\alpha}{4}\qquad\text{ and }\qquad
c<\dfrac{\alpha^{3}}{2^{6}}.$ (3.16)

Moreover, to simplify a bit the notation we denote $\varepsilon$ for
$\varepsilon_{k}$ and we assume from now on that $\sigma_{1}=e_{1}$ and
$\sigma_{2}=e_{2}$.

Recalling that assumption (3.15) does not hold and using the following
identity valid for every integrable function $f$,

$\int_{\omega}f=\int_{Q_{0,\varepsilon}}\left(\sum_{z\in(\varepsilon\mathbb{Z})^{2}\cap(\omega-x)}f(x+z)\right)\,dx,$

we see that there exists $x^{\varepsilon}\in Q_{0,\varepsilon}$ such that with
the notation $x^{(z)}=x^{\varepsilon}+z$, we have

$\sum_{z\in(\varepsilon\mathbb{Z})^{2}\cap[\omega-x^{\varepsilon}]^{\varepsilon}}\left[|Du(x^{(z)},\varepsilon
e_{1})|^{\theta_{1}}|Du(x^{(z)},\varepsilon
e_{2})|^{\theta_{2}}+|D[Du(\cdot,\varepsilon e_{1})](x^{(z)},\varepsilon
e_{2})|\right]<c.$ (3.17)

Let us assume without loss of generality that $x^{\varepsilon}=0$. Analogously
to Lemma 2.1, we set

$w(x)=u(x)+u(0)-u(x_{1},0)-u(0,x_{2}).$ (3.18)

We thus have

$u(x)=u_{1}(x_{1})+u_{2}(x_{2})+w(x)$

with $u_{1}(x_{1})=u(x_{1},0)-u(0)$ and $u_{2}(x_{2})=u(0,x_{2})$. We check by
direct computation that for $x\in\omega^{\varepsilon}$ and $s,t\in\mathbb{R}$
with $|s|,|t|\leq\varepsilon$ we have the identity

$D[Du(\cdot,se_{1})](x,te_{2})=D[Dw(\cdot,se_{1})](x,te_{2}).$

Let us now establish the estimate

$\sum_{z\in(\varepsilon\mathbb{Z})^{2}\cap\omega^{\varepsilon}}\varepsilon[|Dw(x^{(z)},\varepsilon
e_{1})|+|Dw(x^{(z)},\varepsilon e_{2})|]<2c.$ (3.19)

We focus for definiteness on the first term. Using (3.18), we compute

$Dw(x^{(z)},\varepsilon e_{1})=Du(x^{(z)},\varepsilon
e_{1})-Du_{1}(x^{(z)}_{1},\varepsilon)=Du(x^{(z)},\varepsilon
e_{1})-Du(x^{(z)}_{1},\varepsilon e_{1}).$

This leads to

$\displaystyle\sum_{z\in(\varepsilon\mathbb{Z})^{2}\cap\omega^{\varepsilon}}\varepsilon|Dw(x^{(z)},\varepsilon
e_{1})|$
$\displaystyle=\sum_{z\in(\varepsilon\mathbb{Z})^{2}\cap\omega^{\varepsilon}}\varepsilon|Du(x^{(z)},\varepsilon
e_{1})-Du(x^{(z)}_{1},\varepsilon e_{1})|$
$\displaystyle\leq\sum_{z\in(\varepsilon\mathbb{Z})^{2}\cap\omega^{\varepsilon}}\varepsilon\sum_{j\in(\varepsilon\mathbb{Z})\cap(0,x^{(z)}_{2}-\varepsilon)}|D[Du(\cdot,\varepsilon
e_{1})](x^{(z)}_{1}+je_{2},\varepsilon e_{2})|$
$\displaystyle\leq\sum_{z\in(\varepsilon\mathbb{Z})^{2}\cap\omega^{\varepsilon}}|D[Du(\cdot,\varepsilon
e_{1})](x^{(z)},\varepsilon
e_{2})|\stackrel{{\scriptstyle\eqref{prf_of_claim_prop:differincl_0}}}{{<}}c.$

To get the first inequality of the last line, we used Fubini (for sums) and
the fact that for $z_{1}\in\varepsilon\mathbb{Z}$ fixed, the cardinal of the
set $\\{z_{2}\in\varepsilon\mathbb{Z}:(z_{1},z_{2})\in\omega^{\varepsilon}\\}$
is bounded by $1/\varepsilon$.
Let us now define

$F:=\left\\{z\in(\varepsilon\mathbb{Z})^{2}\cap\omega^{\varepsilon}\,:\,|Dw(x^{(z)},\varepsilon
e_{1})|+|Dw(x^{(z)},\varepsilon
e_{2})|\geq\dfrac{2^{4}c\varepsilon}{\alpha^{2}}\right\\}$

so that from (3.19),

$\mathcal{H}^{0}(F)<\dfrac{\alpha^{2}}{8\varepsilon^{2}}.$ (3.20)

Next, let

$E_{1}:=\left\\{z_{1}\in(\varepsilon\mathbb{Z})\cap Q^{1}\ :\
|Du_{1}(z_{1},\varepsilon)|\geq\dfrac{1}{2}|p_{1}|\varepsilon\right\\}$

and define similarly $E_{2}$. Let us estimate from below
$\mathcal{H}^{0}(E_{1})$. Recall that $u_{1}(x_{1})=u(x_{1},0)-u(0)$ and thus
(3.14) implies

$\|u_{1}-p_{1}x_{1}\|_{L^{\infty}(Q^{1})}\leq\delta$

with $|p_{1}|\geq\alpha$. Moreover since $u_{1}$ is $1\,$-Lipschitz, there
holds $|Du_{1}(x_{1},\varepsilon)|\leq\varepsilon$. With these observations,
we compute

$|p_{1}|-\delta\leq|u_{1}(1)-u_{1}(0)|=\Big{|}\sum_{z_{1}\in(\varepsilon\mathbb{Z})\cap(0,1-\varepsilon)}Du_{1}(z_{1},\varepsilon)\Big{|}\\\
\leq\sum_{E_{1}}|Du_{1}(z_{1},\varepsilon)|+\sum_{E_{1}^{c}}|Du_{1}(z_{1},\varepsilon)|\leq\mathcal{H}^{0}(E_{1})\varepsilon+\dfrac{|p_{1}|}{2}\mathcal{H}^{0}(E_{1}^{c})\varepsilon.$

Now since
$\mathcal{H}^{0}(E_{1}^{c})=\varepsilon^{-1}-\mathcal{H}^{0}(E_{1})$, we
deduce that

$\mathcal{H}^{0}(E_{1})\geq\left(\dfrac{\dfrac{|p_{1}|}{2}-\delta}{1-\dfrac{|p_{1}|}{2}}\right)\dfrac{1}{\varepsilon}\,\stackrel{{\scriptstyle\eqref{delta_c}}}{{>}}\dfrac{\alpha}{4\varepsilon}.$
(3.21)

A similar estimate holds also for $\mathcal{H}^{0}(E_{2})$. Then by (3.20) and
(3.21),

$\mathcal{H}^{0}((E_{1}\times E_{2})\cap
F^{c})\geq\mathcal{H}^{0}(E_{1})\mathcal{H}^{0}(E_{2})-\mathcal{H}^{0}(F)>\dfrac{\alpha^{2}}{4\varepsilon^{2}}-\dfrac{\alpha^{2}}{8\varepsilon^{2}}=\dfrac{\alpha^{2}}{8\varepsilon^{2}}.$
(3.22)

Eventually, let us estimate $|Du(x^{(z)},\varepsilon e_{l})|$ for $l=1,2$ and
$z\in(E_{1}\times E_{2})\cap F^{c}$. Using the the triangle inequality and the
definitions of $E_{1}$, $E_{2}$ and $F$, we compute

$|Du(x^{(z)},\varepsilon
e_{l})|=|Du_{l}(x^{(z)}_{l},\varepsilon)-Dw(x^{(z)},\varepsilon
e_{l})|\geq|Du_{l}(x^{(z)}_{l},\varepsilon)|-|Dw(x^{(z)},\varepsilon
e_{l})|\\\
\geq\dfrac{|p_{l}|}{2}\varepsilon-\dfrac{2^{4}c}{\alpha^{2}}\varepsilon\geq\left(\dfrac{\alpha}{2}-\dfrac{2^{4}c}{\alpha^{2}}\right)\varepsilon\stackrel{{\scriptstyle\eqref{delta_c}}}{{>}}\dfrac{\alpha\varepsilon}{4}.$
(3.23)

We conclude with (3.17) that

$c>\sum_{z\in(E_{1}\times E_{2})\cap F^{c}}|Du(x^{(z)},\varepsilon
e_{1})|^{\theta_{1}}|Du(x^{(z)},\varepsilon
e_{2})|^{\theta_{2}}\,\stackrel{{\scriptstyle\eqref{goodestim:diffincl}}}{{\geq}}\,\mathcal{H}^{0}\left((E_{1}\times
E_{2})\cap
F^{c}\right)\dfrac{\alpha\varepsilon}{4}\,\stackrel{{\scriptstyle\eqref{volgood}}}{{>}}\,\dfrac{\alpha^{3}}{2^{5}\varepsilon}$

which gives a contradiction for $\varepsilon=\varepsilon_{k}$ small enough.
This concludes the proof of the claim and therefore of the proposition. ∎

####  3.2.2 Rectifiability of the defect measure

Let us state the main result of this section which is a detailed version of
Theorem 1.19.

######  Theorem 3.2.

Let $v\in S^{\infty}(\Omega)$ and assume that
$\mu=\mu[v]\in\mathcal{M}(\Omega)$. Then, for $|\mu|\,$-a.e. $\bar{x}$, there
exists
$v^{\infty}=(v_{1}^{\infty},v_{2}^{\infty})\in(\mathbb{R}\setminus\\{0\\})^{2}$
such that letting

$L:=\operatorname{span}\left(v_{2}^{\infty},v_{1}^{\infty}\right)\qquad\text{
and }\qquad
c(\bar{x}):=\dfrac{|v_{1}^{\infty}||v_{2}^{\infty}|}{|v^{\infty}|},$

we have for every $\varphi\in C_{c}(\mathbb{R}^{2})$,

$\lim_{r\to
0}\dfrac{1}{r}\int_{\mathbb{R}^{2}}\varphi\left(\dfrac{x-\bar{x}}{r}\right)d|\mu|(x)=c(\bar{x})\int_{L}\varphi\,d\mathcal{H}^{1}.$
(3.24)

As a consequence $\mu=m\mathcal{H}^{1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Sigma$ for some $1$-rectifiable set
$\Sigma$ and some Borel function $m$ with $|m|=c$. Moreover, denoting

$\nu:=\dfrac{\operatorname{sign}m}{|v^{\infty}|}(-v_{1}^{\infty},v_{2}^{\infty}),$

a unit normal to $\Sigma$ at $\bar{x}$ and then

$V^{\infty}(y):=\begin{cases}(v^{\infty}_{1},0)&\text{if }y\cdot\nu>0,\\\
(0,v_{2}^{\infty})&\text{if }y\cdot\nu<0,\end{cases}$

we have

$\lim_{r\to
0}\frac{1}{r^{2}}\int_{B_{r}}\left|v(\bar{x}+y)-V^{\infty}(y)\right|\,dy=0.$
(3.25)

Therefore $v$ has traces on $\Sigma$.

######  Remark 3.3.

Let us point out that the sign of $m(\bar{x})$ cannot be directly computed
from the values of $v^{\infty}$. Indeed, for both
$u(x_{1},x_{2})=\min(x_{1},x_{2})$ and
$\tilde{u}(x_{1},x_{2})=\max(x_{1},x_{2})$ we have $v^{\infty}=(1,1)$ on
$\Sigma=\\{x_{1}=x_{2}\\}$ but
$\mu[u]=-\mu[\tilde{u}]=(1/\sqrt{2})\mathcal{H}^{1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Sigma$.

We start with some preliminaries. Let $v\in S^{\infty}(\Omega)$ such that
$\mu[v]\in\mathcal{M}(\Omega)$ and $u$ be such that $v=\nabla u$. For
$t\in\mathbb{R}$, we let $\Gamma_{t}=\\{u=t\\}$ and $\omega_{t}=\\{u>t\\}$.
The function $t\mapsto|\omega_{t}|$ is bounded and decreasing thus have
bounded variation888Since $|\omega_{t}|=|\Omega|>0$ for $t<\inf u$, the
function $t\in\mathbb{R}\mapsto|\omega_{t}|$ is not integrable over
$\mathbb{R}$, so, strictly speaking, it is only locally $BV$.. By the co-area
formula (see for instance [ABC13]), we have that for almost every $t$,
$\omega_{t}$ is a set of finite perimeter with, up to a
$\mathcal{H}^{1}$-negligible set, $\partial\omega_{t}\cap\Omega=\Gamma_{t}$.
We have moreover

$\int_{\mathbb{R}}\mathcal{H}^{1}(\Gamma_{t})\,dt=\int_{\Omega}|\nabla
u|\,\leq|\Omega|.$ (3.26)

Our first goal is to establish that the measure $\mu$ decomposes naturally on
the level sets of $u$. For this purpose we introduce a measure $\tilde{\mu}$
on $\Omega\times\mathbb{R}$ defined by the property:

$\int_{\Omega\times\mathbb{R}}\varphi(x,t)\,d\tilde{\mu}(x,t)=\int_{\Omega}\varphi(x,u(x))\,d\mu(x)\qquad\text{for
}\varphi\in C_{c}(\Omega\times\mathbb{R}).$ (3.27)

Notice that the definition makes sense since $u\in\mathrm{Lip}(\Omega)\subset
C(\Omega)$. The next result characterizes $\tilde{\mu}$ it in terms of the
family of measures

$\kappa_{t}:=\partial_{1}\partial_{2}\boldsymbol{1}_{\omega_{t}}\quad\qquad\text{for
}t\in\mathbb{R}.$ (3.28)

######  Proposition 3.4.

For almost every $t\in\mathbb{R}$ there holds
$\kappa_{t}\in\mathcal{M}(\Omega)$ and we have the identity

$\tilde{\mu}=\kappa_{t}\otimes dt.$ (3.29)

As a consequence, for every Borel set $A\subset\Omega$,

$|\mu|(A)=\int_{\mathbb{R}}|\kappa_{t}|(A)\,dt.$ (3.30)

###### Proof.

Since $\Omega$ is bounded and $u$ is Lipschitz continuous, after possibly
adding a constant, we assume that $\inf u=0$ and we set $T:=\sup u\geq 0$.

Step 1. Let us establish that for every $\varphi\in C^{\infty}_{c}(\Omega)$
and every $\psi\in C^{1}(\mathbb{R})$,

$\int_{\Omega}\varphi(x)\psi(u(x))\,d\mu(x)=\int_{0}^{T}\psi(t)\left<\kappa_{t},\varphi\right>\,dt.$
(3.31)

For this we first prove that, denoting $\Psi(t):=\int_{0}^{t}\psi(s)\,ds$, we
have the identity

$\int_{\Omega}\varphi(x)\psi(u(x))\,d\mu(x)=\int_{\Omega}\Psi(u)\partial_{1}\partial_{2}\varphi\,dx.$
(3.32)

Since $\varphi\in C^{\infty}_{c}(\Omega)$, there exists an open set
$\Omega^{\prime}$ compactly supported in $\Omega$ and such that $\varphi\in
C^{\infty}_{c}(\Omega^{\prime})$. Up to replacing $u$ by $\chi u$ where
$\chi\in C^{\infty}_{c}(\Omega)$ with $\chi=1$ on $\Omega^{\prime}$ we may
assume that $u$ has compact support in $\Omega$. We then let
$u_{\varepsilon}=u*\rho_{\varepsilon}$ so that
$\partial_{1}\partial_{2}u_{\varepsilon}\rightharpoonup\mu$ as measures,
$u_{\varepsilon}\to u$ in $C(\Omega)$ and $\nabla u_{\varepsilon}\to\nabla u$
almost everywhere. Therefore

$\displaystyle\int_{\Omega}\varphi(x)\psi(u(x))\,d\mu(x)$
$\displaystyle=\lim_{\varepsilon\to
0}\int_{\Omega}\varphi(x)\psi(u_{\varepsilon}(x))\partial_{1}\partial_{2}u_{\varepsilon}\,dx$
$\displaystyle=-\lim_{\varepsilon\to
0}\left[\int_{\Omega}\partial_{1}\varphi(x)\psi(u_{\varepsilon}(x))\partial_{2}u_{\varepsilon}\,dx+\int_{\Omega}\varphi(x)\psi^{\prime}(u_{\varepsilon}(x))\partial_{1}u_{\varepsilon}\partial_{2}u_{\varepsilon}\,dx\right].$

On the one hand we have

$\lim_{\varepsilon\to
0}\int_{\Omega}\partial_{1}\varphi(x)\psi(u_{\varepsilon}(x))\partial_{2}u_{\varepsilon}\,dx=\int_{\Omega}\partial_{1}\varphi(x)\psi(u)\partial_{2}u\,dx.$

On the other hand, by dominated convergence theorem, we get

$\lim_{\varepsilon\to
0}\int_{\Omega}\varphi(x)\psi^{\prime}(u_{\varepsilon}(x))\partial_{1}u_{\varepsilon}\partial_{2}u_{\varepsilon}\,dx=\int_{\Omega}\varphi(x)\psi^{\prime}(u(x))\underbrace{\partial_{1}u\,\partial_{2}u}_{=0\text{
a.e.\hbox{} missing}}\,dx=0.$

Thus

$\int_{\Omega}\varphi(x)\psi(u(x))\,d\mu(x)=-\int_{\Omega}\partial_{1}\varphi(x)\psi(u)\partial_{2}u\,dx=-\int_{\Omega}\partial_{1}\varphi(x)\partial_{2}\left[\Psi(u)\right]\,dx.$

Integrating by parts once again we obtain (3.32). Finally, using the layer-
cake formula, see [LL01, Theorem 1.13], we deduce

$\int_{\Omega}\Psi(u)\partial_{1}\partial_{2}\varphi\,dx=\int_{0}^{T}\left[\int_{\omega_{t}}\partial_{1}\partial_{2}\varphi\,dx\right]\psi(t)\,dt.$

By definition of $\kappa_{t}$ this concludes the proof of (3.31).

Step 2. We now prove (3.29). In light of the definition (3.27) of
$\tilde{\mu}$ and of the identity (3.31), we have to check that $\kappa_{t}$
is a measure for almost every $t$ and that the function
$t\mapsto|\kappa_{t}|(\Omega)$ is integrable with respect to the Lebesgue
measure on $\mathbb{R}$.
Let us disintegrate $\tilde{\mu}$ along the level sets $\Gamma_{t}$. We obtain
$\tilde{\mu}=\mu_{t}\otimes\lambda$ where $\lambda$ is a finite positive
measure supported in $\overline{u(\Omega)}$ and $|\mu_{t}|(\Omega)=1$ for
$\lambda\,$-almost every $t$. For $\varphi\in C^{\infty}_{c}(\Omega)$ and
$\psi\in C^{1}(\mathbb{R})$, we have

$\int_{0}^{T}\psi(t)\left(\int_{\Omega}\varphi(x)\,d\mu_{t}(x)\right)\,d\lambda(t)=\int_{\Omega}\varphi(x)\psi(u(x))\,d\mu(x)\stackrel{{\scriptstyle\eqref{decompmu:firsteq}}}{{=}}\int_{0}^{T}\psi(t)\left<\kappa_{t},\varphi\right>\,dt.$

Therefore, for fixed $\varphi$, we have, as measures,

$\left(\int_{\Omega}\varphi(x)\,d\mu_{t}(x)\right)\,d\lambda(t)=\left<\kappa_{t},\varphi\right>\,dt.$
(3.33)

Decomposing $\lambda$ into absolutely continuous and singular parts with
respect to the Lebesgue measure, we write $\lambda=f(t)\,dt+\lambda^{s}$.
Putting this in (3.33) and identifying, we get the identities

$\displaystyle f(t)\left<\mu_{t},\varphi\right>$
$\displaystyle=\left<\kappa_{t},\varphi\right>$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\text{for almost every }t,$
$\displaystyle\left<\mu_{t},\varphi\right>\,d\lambda^{s}(t)$ $\displaystyle=0$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\text{as measure}.$

We deduce that $\kappa_{t}$ is a measure for almost every $t$ with
$\kappa_{t}=f(t)\mu_{t}$ so that $t\mapsto|\kappa_{t}|(\Omega)$ is integrable.
Moreover the contribution of $\mu_{t}\otimes\lambda^{s}$ in the disintegration
vanishes, hence $\lambda^{s}=0$. This concludes the proof of identity (3.29).
∎

We deduce from Theorem 1.6, that for almost every level $t$ the set
$\omega_{t}$ is a polygon.

######  Proposition 3.5.

There exists $\mathcal{I}_{\mathrm{poly}}\subset\mathbb{R}$ of full measure
such that for a.e. $t\in\mathcal{I}_{\mathrm{poly}}$, the set $\omega_{t}$ is
a finite disjoint union of open polygons with sides parallel to the coordinate
axes. Moreover, for such $t$, $N(t):=|\kappa_{t}|(\Omega)$ is the number of
vertices of $\Gamma_{t}$ in $\Omega$. More precisely,

$\kappa_{t}=\sum_{j=1}^{N(t)}\kappa_{t}(x_{t}^{j})\delta_{x^{j}_{t}},$ (3.34)

where the $x^{j}_{t}$’s are the vertices in $\Omega$ of the polygons forming
$\omega_{t}$ and $\kappa_{t}(x_{t}^{j})=\pm 1$.
In the following these inner vertices are called corners and we denote

$\mathcal{C}(\omega_{t}):=\\{x_{t}^{j}:1\leq j\leq N(t)\\}.$

######  Remark 3.6.

(i) Let us point out that Proposition 3.5 implicitly states that
$\overline{\omega}_{t}$ is also made of a finite disjoint union of polygons.
In particular, this excludes polygons intersecting at a corner as in Figure 1.
In other words, we excluded the multiplicities $\pm 2$ from Theorem 1.6.
(ii) For shortness, in the proof of Proposition 3.5 below we deduce this
simplification from the general result999We could also give a more elementary
proof in our context based on the following observation. Since $u$ is
1-Lipschitz continuous, the erosion rate of $\omega_{t}$ as $t$ increases is
bounded from below. Namely,
$\left\\{\begin{array}[]{rcll}\omega_{s}\subset&\left\\{x\in\omega_{t}:d(x,(\omega_{t})^{c})\geq
s-t\right\\}&\subset\subset\omega_{t}&\text{ for }0\leq t<s\leq T,\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\
(\omega_{s})^{c}\subset&\left\\{x\in(\omega_{t})^{c}:d(x,\omega_{t})\geq
t-s\right\\}&\subset\subset(\omega_{t})^{c}&\text{ for }0\leq s<t\leq
T.\end{array}\right.$ [ABC13, Theorem 2.5].

###### Proof of Proposition 3.5.

By Theorem 1.6 applied with $u=\boldsymbol{1}_{\omega_{t}}$ we have for a.e.
$t$,

$\omega_{t}=\omega_{t}^{\text{poly}}\cup S_{t}$

where:

  1. (i)

$\omega_{t}^{\text{poly}}$ is a finite union of polygons with sides parallel
to the coordinate axes and which may intersect only at the corners,

  2. (ii)

$S_{t}$ is a union of stripes, either all vertical or all horizontal.

Moreover, by [ABC13, Theorem 2.5], for almost every $t$, the closure of the
connected components of $\omega_{t}^{\text{poly}}$ are actually disjoint.
Besides, by (3.26) $S_{t}$ is a finite union of stripes for almost every $t$
and since $\omega_{t}=\\{u>t\\}$ is open we conclude that this set is a
disjoint finite union of open polygons with sides parallel to the coordinate
axes.
Eventually, recalling (3.28) and Proposition 3.4, we have
$\kappa_{t}=\partial_{1}\partial_{2}\boldsymbol{1}_{\omega_{t}}$ which yields
identity (3.34) and the related properties. ∎

Thanks to Proposition 3.4, studying the measure $\mu=\mu[\nabla u]$ reduces to
studying the mapping $s\mapsto\kappa_{s}$. The next lemma allows us to focus
on levels $t$ near which $s\mapsto N(s)$ is (almost) constant and such that we
can follow individually the trajectories $s\mapsto x_{s}^{j}$. Later,
establishing the approximate differentiability of these trajectories is an
important step in the proof of the 1-rectifiability of $\mu$.
Let us introduce some further notation.

######  Notation 3.7.

From now on for $x\in\mathbb{R}^{2}$ and $\lambda,r>0$, we denote

$Q_{r}:=(-r,r)^{2},\qquad\qquad Q_{r}(x):=x+Q_{r}\qquad\text{and}\qquad\lambda
Q_{r}(x):=x+\lambda Q_{r}.$

Besides, for a measurable set $J\subset\mathbb{R}$, we define the set of
points of density of $J$ as

$\operatorname{dens}(J):=\left\\{t\in J:\lim_{\eta\downarrow
0}\dfrac{\mathcal{H}^{1}(J\cap(t-\eta,t+\eta))}{2\eta}=1\right\\}.$

Notice that, as opposed to the standard definition, we enforce
$\operatorname{dens}(J)\subset J$. However, with the present definition, the
set $J\setminus\operatorname{dens}(J)$ is still Lebesgue negligible.

######  Lemma 3.8.

Let $t\in\mathcal{I}_{\mathrm{poly}}$ be such that $|\Gamma_{t}|=0$ (_i.e._ a
point of continuity of $t\mapsto|\omega_{t}|$). Then, there exist
$\overline{r}=\overline{r}(t)$ such that the following statements hold true.

  * (i)

For every $0<r\leq\overline{r}$ there exists $\varepsilon=\varepsilon(t,r)>0$
such that

$1=|\kappa_{t}(x)|\leq\sum_{y\in Q_{r}(x)}|\kappa_{s}(y)|\qquad\ \text{for
}s\in(t-\varepsilon,t+\varepsilon)\cap\mathcal{I}_{\mathrm{poly}}\ \text{ and
}\ x\in\mathcal{C}(\omega_{t}).$

  * (ii)

As a consequence, for every such $s$, $N(s)\geq N(t)$.

  * (iii)

If moreover $t$ is a Lebesgue point of $N$ then, denoting, for
$0<r\leq\overline{r}$,

$\mathcal{J}_{t}(r):=\left\\{s\in\mathcal{I}_{\mathrm{poly}}:N(s)=N(t)\text{
and }|\kappa_{s}|(Q_{r}(x))=1\text{ for every
}x\in\mathcal{C}(\omega_{t})\right\\},$

we have $t\in\operatorname{dens}(\mathcal{J}_{t}(r))$.

###### Proof.

We start with the proof of $(i)$ and $(ii)$. Let
$t\in\mathcal{I}_{\mathrm{poly}}$ such that $|\Gamma_{t}|=0$. We then have
$|\omega_{t}\Delta\omega_{s}|\to 0$ as $s\to t$, that is
$\boldsymbol{1}_{\omega_{s}}\to\boldsymbol{1}_{\omega_{t}}$ in
$L^{1}(\Omega)$. Since $\Gamma_{t}$ is a polygon, there exists
$\overline{r}>0$ such that if $x\in\mathcal{C}(\omega_{t})$, there is no other
corner of $\omega_{t}$ in $Q_{\overline{r}}(x)$ and the squares $Q_{\bar{x}}$
for $x\in\mathcal{C}(\omega_{t})$ are pairwise disjoint.
Now let $0<r\leq\overline{r}$. Recalling that
$\kappa_{s}=\partial_{1}\partial_{2}\boldsymbol{1}_{\omega_{s}}$ and that
$\boldsymbol{1}_{\omega_{s}}\to\boldsymbol{1}_{\omega_{t}}$ in $L^{1}$ as
$s\to t$ we get that
$s\in\mathbb{R}\mapsto\kappa_{s}\in\mathcal{D}^{\prime}(\Omega)$ is continuous
at $t$. Therefore, for every $x\in\mathcal{C}(\omega_{t})$,

$\displaystyle|\kappa_{t}(x)|$
$\displaystyle=\sup_{\stackrel{{\scriptstyle\zeta\in
C^{\infty}_{c}(Q_{r},[-1,1])}}{{|\zeta|\leq
1}}}\int\zeta(y-x)\,d\kappa_{t}(y)$
$\displaystyle=\sup_{\stackrel{{\scriptstyle\zeta\in
C^{\infty}_{c}(Q_{r},[-1,1])}}{{|\zeta|\leq 1}}}\lim_{s\to
t}\int\zeta(y-x)\,d\kappa_{s}(y)\,\leq\,\liminf_{s\to
t}|\kappa_{s}|(Q_{r}(x)).$

Since the quantities $|\kappa_{s}|(Q_{r}(x))$ are integers we see that there
exists $\varepsilon>0$ such that for
$s\in\mathcal{I}_{\mathrm{poly}}\cap(t-\varepsilon,t+\varepsilon)$ and every
corner $x\in\mathcal{C}(\omega_{t})$ there holds,

$|\kappa_{t}(x)|\leq|\kappa_{s}|(Q_{r}(x)).$ (3.35)

This proves $(i)$. Eventually, summing (3.35) over
$x\in\mathcal{C}(\omega_{t})$ and recalling that the squares $Q_{r}(x)$ are
disjoint, we get

$N(t)=\sum_{x\in\mathcal{C}(\omega_{t})}|\kappa_{t}|(x)\leq\sum_{x\in\mathcal{C}(\omega_{t})}|\kappa_{s}|(Q_{r}(x))\leq\sum_{y\in\mathcal{C}(\omega_{s})}|\kappa_{s}|(y)=N(s),$
(3.36)

which establishes $(ii)$.

Finally, we prove $(iii)$. If $N(s)=N(t)$ the chain of inequalities in (3.36)
are identities and we get that (3.35) is also an identity. Thus, assuming that
$t\in\mathcal{I}_{\mathrm{poly}}$ is a Lebesgue point of $N$ (notice that by
(3.30), $N\in L^{1}(\mathbb{R})$), we get that
$t\in\operatorname{dens}(\mathcal{J}_{t}(r))$ as claimed. ∎

Even after the preceding lemma, we still need to select further the levels
sets. Indeed the continuity of the trajectories $s\mapsto x_{s}^{i}$ does not
imply the continuity of the shape of the polygons forming $\omega_{s}$ and
unfortunately shape changes are associated with non-differentiability of the
corner trajectories. The following example illustrates this fact

######  Example 3.9.

Let us consider the function defined on $\Omega=(-1,1)^{2}$, by

$u(x_{1},x_{2})=\begin{cases}\quad-x_{1}&\text{for }x_{1}\leq 0,\\\
\min(x_{1},x_{2})&\text{for }x_{1}>0,\,x_{2}\geq 0,\\\
\max(-x_{1},x_{2})&\text{for }x_{1}>0,\,x_{2}<0.\end{cases}$

For this function and $t\in(-1,1)$, the level set $\omega_{t}$ is given by,
see Figure 4,

$y\in\omega_{t}\iff\begin{cases}\quad y_{1}<x_{1}(t)\text{ or
}y_{2}>x_{2}(t)&\text{if }t<0,\\\ x_{1}<-t\text{ or
}\left[y_{1}>x_{1}(t)\text{ and }y_{2}>x_{2}(t)\right]&\text{if }t\geq
0.\end{cases}$

Let us highlight the following facts.

  1. (1)

$\omega_{t}$ is connected for $t<0$, but has two connected components for
$t\geq 0$.

  2. (2)

The set $\omega_{t}$ switches from locally concave to locally convex in the
neighborhood of the vertex $x(t)$.

  3. (3)

In any case, $\mathcal{C}(\omega_{t})$ has a single element $x(t)=(|t|,t)$
with $\kappa_{t}(x(t))=1$ but the derivative of $t\mapsto x(t)$ jumps at
$t=0$.

  4. (4)

The function $s\mapsto\mathcal{H}^{1}(\Gamma_{s})$ is discontinuous at $t=0$.

$\Omega$$\bullet$$x(t_{-})$$\omega_{t_{-}}$$\bullet$$x(t_{+})$$\omega_{t_{+}}$
Figure 4: Left: the vector field $\nabla u$. Middle and right: some sets
$\omega_{t_{\pm}}$, $t_{-}<0<t_{+}$.

The third fact described in the example is the situation we wish to avoid. To
this aim it turns out that the last fact is the most useful. Indeed, the co-
area formula (3.26) provides some control on the quantities
$\mathcal{H}^{1}(\Gamma_{s})$. The idea is then to exclude the points of
discontinuity of $s\mapsto\mathcal{H}^{1}(\Gamma_{s})$. More precisely, we
consider a sequence of local versions of this constraint: we impose that $t$
is a Lebesgue point of $s\mapsto\mathcal{H}^{1}(\Gamma_{s}\cap Q)$ for a dense
countable collection of squares $Q$. However, this is still not sufficient to
ensure the differentiability of the trajectories of the corners at $t$. For
this, we also need to enforce the differentiability of the functions
$s\mapsto|\omega_{s}\cap Q|$. These observations lead to the following
definition.

######  Definition 3.10.

We set

$\mathcal{Q}:=\left\\{Q_{r}(y)\subset\Omega:\text{such that }y_{1},y_{2}\text{
and }r\text{ are rational numbers}\right\\}.$

Moreover, for $Q\in\mathcal{Q}$ we define the function
$\operatorname{Vol}_{Q}:\mathbb{R}\to\mathbb{R}_{+}$ by

$\operatorname{Vol}_{Q}(s):=|\omega_{s}\cap Q|.$

The functions $\operatorname{Vol}_{Q}$ are bounded and nonincreasing, hence
$BV$. We decompose their distributional derivatives
$D\operatorname{Vol}_{Q}(s)$ in their absolutely continuous and singular parts
with respect to the Lebesgue measure:

$D\operatorname{Vol}_{Q}=\operatorname{Vol}_{Q}^{\prime}\,dt+D^{s}\operatorname{Vol}_{Q}.$

Recall that by [AFP00, Theorem 3.28], $\operatorname{Vol}_{Q}$ is
differentiable almost everywhere and its derivative coincides with
$\operatorname{Vol}_{Q}^{\prime}$. We are now ready to introduce the subset of
levels of $t\mapsto|\omega_{t}|$ meeting the constraints we have just
outlined.

######  Definition 3.11.

$\mathcal{I}_{\mathcal{Q}}:=\big{\\{}t\in\mathbb{R}:\text{ for every
}Q\in\mathcal{Q},\ \operatorname{Vol}_{Q}\text{ is differentiable at }t,\\\
\text{ moreover }t\text{ is a Lebesgue point of
}\operatorname{Vol}_{Q}^{\prime}\text{ and of
}s\mapsto\mathcal{H}^{1}(\Gamma_{s}\cap Q)\Big{\\}}.$

######  Lemma 3.12.

The set $\mathcal{I}_{\mathcal{Q}}$ is of full measure in $\mathbb{R}$.

###### Proof.

As a consequence of the co-area formula (i.e. (3.26) with $Q$ in place of
$\Omega$) the function $s\mapsto\mathcal{H}^{1}(\Gamma_{s}\cap Q)$ lies in
$L^{1}(\mathbb{R})$ and the set of its Lebesgue points is of full measure.
Recalling that $\mathcal{Q}$ is countable we get the result. ∎

We can then define the set of good levels $t$.

######  Lemma 3.13.

Let $\mathcal{I}$ be the set of $t$ such that

  * (i)

$t\in\mathcal{I}_{\mathrm{poly}}$ and $|\Gamma_{t}|=0$,

  * (ii)

$t$ is a Lebesgue point of $s\mapsto N(s)$ (consequently
$t\in\operatorname{dens}(\mathcal{J}_{t}(r))$ for $r\leq\overline{r}(t)$),

  * (iii)

$t\in\mathcal{I}_{\mathcal{Q}}$,

  * (iv)

$\mathcal{H}^{1}$-almost every point of $\Gamma_{t}$ is a Lebesgue point of
$\nabla u$.

Then $\mathcal{I}$ is of full measure in $\mathbb{R}$. As a consequence (see
(3.29)), for $|\mu|$-a.e. $x$ there exists $t\in\mathcal{I}$ such that
$x\in\mathcal{C}(\omega_{t})$.

###### Proof.

Points $(i)$ and $(ii)$ follow from Lemma 3.8. Point $(iii)$ from Lemma 3.12.
The last point is the consequence of $\nabla u\in L^{1}(\Omega)$ and of the
co-area formula $\boldsymbol{1}_{\Omega}|\nabla
u|\,dx=(\boldsymbol{1}_{\Gamma_{t}}\mathcal{H}^{1})\otimes dt$. ∎

We may now embark on the proof of Theorem 3.2. By Lemma 3.13, for
$|\mu|\,$-a.e. $\bar{x}$ there exists $\bar{t}\in\mathcal{I}$ such that
$\bar{x}\in\mathcal{C}(\omega_{\bar{t}})$ and
$\kappa_{\bar{t}}(\\{\bar{x}\\})=1$. Up to a translation we assume without
loss of generality that $\bar{t}=0$ and $\bar{x}=0$. Moreover, possibly
replacing $u$ by $\pm u\circ R$ where $R$ is a rotation of angle $k\pi/2$ for
some $k\in\\{0,1,2,3\\}$ we assume that for some ${\overline{r}}>0$ there
holds

$\omega_{0}\cap Q_{\overline{r}}=\\{y\in Q_{\overline{r}}:y_{1}>0,y_{2}>0\\}.$
(3.37)

From now on, we denote (recall the definition of $\mathcal{J}_{0}(r)$ from
point (iii) of Lemma 3.8)

$\mathcal{J}_{0}:=\mathcal{J}_{0}(\overline{r}/4)\cap\mathcal{I}=\left\\{t\in\mathcal{I}:\text{there
exists }x\in Q_{\overline{r}/4}\text{ such that }\mathcal{C}(\omega_{t})\cap
Q_{\overline{r}}=\\{x\\}\right\\},$

and for $t\in\mathcal{J}_{0}$, we denote $x(t)=(x_{1}(t),x_{2}(t))$ the unique
element of $\mathcal{C}(\omega_{t})\cap Q_{\overline{r}}$ so that

$|\kappa_{t}|\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,Q_{\overline{r}}=\delta_{x(t)}\qquad\text{
for }t\in\mathcal{J}_{0}.$ (3.38)

By Lemmas 3.8 & 3.13, there holds

$0\in\operatorname{dens}(\mathcal{J}_{0}).$ (3.39)

Our first goal is to identify the trajectory $t\in\mathcal{J}_{0}\mapsto x(t)$
as the restriction of a mapping $t\in(-\bar{t},\bar{t})\mapsto h(t)$
differentiable at $0$. For instance, for the second coordinate $x_{2}(t)$ we
could use the square
$Q^{\mathrm{h}}_{0}:=(\overline{r}/2,\overline{r})\times(-\overline{r}/4,\overline{r}/4)$
and the identity

$x_{2}(t)=\dfrac{\operatorname{Vol}_{Q^{\mathrm{h}}_{0}}(t)-\operatorname{Vol}_{Q^{\mathrm{h}}_{0}}(0)}{\overline{r}/2}=:h_{2}^{0}(t),$

which is valid for $t\in\mathcal{J}$. Unfortunately, we have no guarantee on
the differentiability of the function $t\mapsto\operatorname{Vol}_{Q}(t)$ at
$0$ for a general square $Q$ unless $Q\in\mathcal{Q}$. To overcome this, we
substitute for $Q^{\mathrm{h}}_{0}$, a square
$Q^{\mathrm{h}}_{+}\in\mathcal{Q}$ such that (see Figure 5),

$\\{\overline{r}/2\\}\times[-\overline{r}/4,\overline{r}/4]\ \subset\
Q^{\mathrm{h}}_{+}\ \subset\
({\overline{r}}/4,{\overline{r}})\times(-{\overline{r}},{\overline{r}}).$
(3.40)

Such square exists by density of $\mathcal{Q}$ in the set of squares inside
$\Omega$. Symmetrically, to control $x_{1}(t)$ we pick
$Q^{\mathrm{v}}_{+}\in\mathcal{Q}$ with

${[-\overline{r}/4,\overline{r}/4]}\times\\{\overline{r}/2\\}\ \subset\
Q^{\mathrm{v}}_{+}\ \subset\
(-{\overline{r}},{\overline{r}})\times({\overline{r}}/4,{\overline{r}}).$
(3.41)

$\bullet$$0$$Q_{\overline{r}}$$\omega_{0}$$Q^{\mathrm{v}}_{+}$$Q^{\mathrm{v}}_{-}$$Q^{\mathrm{h}}_{+}$$Q^{\mathrm{h}}_{-}$
Figure 5:  The set $\omega_{0}\cap Q_{\overline{r}}$ and the four control