# Homogenization of line tension energies

M. Fortuna1 A. Garroni2
Dipartimento di Matematica “Guido Castelnuovo”
Sapienza Università di Roma P.le A. Moro 5 00185 Roma Italy

Abstract: We prove an homogenization result, in terms of $\Gamma$-convergence,
for energies concentrated on rectifiable lines in $\mathbb{R}^{3}$ without
boundary. The main application of our result is in the context of dislocation
lines in dimension $3$. The result presented here shows that the line tension
energy of unions of single line defects converge to the energy associated to
macroscopic densities of dislocations carrying plastic deformation. As a
byproduct of our construction for the upper bound for the $\Gamma$-Limit, we
obtain an alternative proof of the density of rectifiable $1$-currents without
boundary in the space of divergence free fields.

## Keywords

Gamma convergence, dislocations, divergence free fields, homogenization.

##  1 Introduction

11footnotetext<EMAIL_ADDRESS><EMAIL_ADDRESS>
In this paper we prove an homogenization result for energies of the form

$\int_{\gamma}\psi(b,t)d\mathcal{H}^{1},$ (1.1)

where $\mu=b\otimes t\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\gamma$ is a divergence free
matrix valued measure, $\gamma\subset\mathbb{R}^{3}$ is a $1$-rectifiable set,
and $t$ its tangent. The vector $b$ is a multiplicity which belongs to a
discrete lattice $\mathcal{B}$ in $\mathbb{R}^{N}$, with $N\geq 2$, and will
also be called the Burgers vector of $\mu$. Here
$\mathcal{M}_{df}^{1}(\Omega;\mathcal{B}\times\mathbb{S}^{2})$ denotes the set
of such measures, where $\Omega\subset\mathbb{R}^{3}$ is an open bounded and
regular set.

We consider the following scaled version of the energy in (1.1)

$E_{\sigma}(\mu):=\begin{cases}\displaystyle{\int_{\gamma\cap\Omega}\sigma\psi\left(\dfrac{b}{\sigma},t\right)d\mathcal{H}^{1}}\quad&\hbox{if
}\ \mu=b\otimes t\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\gamma\in\mathcal{M}_{df}^{1}(\Omega;\sigma\mathcal{B}\times\mathbb{S}^{2}),\\\
+\infty&\hbox{otherwise}.\end{cases}$ (1.2)

Under some mild assumptions on the density $\psi$ we study the asymptotics of
$E_{\sigma}$ in terms of $\Gamma$-convergence with respect to the weak$*$
topology of measures. The main result of the paper is that the limiting energy
takes the form

$E_{0}(\mu)=\begin{cases}\displaystyle\int_{\Omega}g\left(\dfrac{d\mu(x)}{d\|\mu\|}\right)d\|\mu\|\quad&\hbox{if
}\mu\in[\mathcal{M}(\Omega)]^{N\times 3},\ \operatorname*{div}\mu=0,\\\
{+\infty}&\hbox{otherwise},\end{cases}$ (1.3)

where $g:\mathbb{R}^{N\times 3}\longrightarrow[0,+\infty)$ is a convex
$1$-homogeneous function defined in terms of the density $\psi$ (see Theorem
2.5 for the exact statement).

Our result extends the result in [8], where the same problem is treated in
dimension $2$, i.e., $\Omega\subset\mathbb{R}^{2}$, and from which we borrow
several ideas for the proof. The main difference is that in dimension $2$ the
support of the measures in
$\mathcal{M}_{df}^{1}(\Omega;\mathcal{B}\times\mathbb{S}^{1})$ has codimension
$1$, so that the energy in (1.1) reduces to a functional defined in the space
of $SBV$ functions with values in $\mathcal{B}$ (and the measure $\mu$ is
nothing but the rotated gradient of the phase field $u$). This allow the
authors to use tools from the Calculus of Variations for functionals defined
on partitions and therefore in $SBV$ (see [1]).

In dimension larger than $2$ there is no phase field describing the admissible
configuration, so we use techniques of geometric measure theory and the
analysis of functionals defined on rectifiable currents (rephrased in terms of
the measures in
$\mathcal{M}_{df}^{1}(\Omega;\mathcal{B}\times\mathbb{S}^{2})$). Energies of
the form (1.1) have been studied in [6] where the authors give necessary and
sufficient conditions for the lower semicontinuity of such functionals.

The major difficulty in the proof of the $\Gamma$-convergence is the
construction for the upper bound. Implementing a standard density argument we
first need to reduce to the case of measures absolutely continuous with
respect to Lebesgue and having piecewise constant density. This step requires
to prove that such measures are dense in energy for the limiting functional
$E_{0}$. The second key ingredient of the proof is then an ad hoc construction
which allows to approximate divergence free piecewise constant fields with
measures concentrated on polyhedral closed curves and optimal energy. As a
byproduct of the construction for the recovery sequence we thus obtain a
different proof of the approximation of divergence free vector fields by means
of measures defined on closed curves (see Theorem 3.5). Stated by J. Bourgain
and H. Brezis in [3] in the context of solenoidal charges in the sense of
Smirnov (see [21]), this density result is proved in [15] with respect to the
strict topology of measures.

The main motivation for our analysis is the study of line defects in a
$3$-dimensional crystal, the so called dislocations. At a mesoscopic scale
(larger than the microscopic lattice spacing) they are indeed identified with
line objects carrying a vector multiplicity belonging to the lattice, so that
they can be represented as measures belonging to
$\mathcal{M}_{df}^{1}(\Omega;\mathcal{B}\times\mathbb{S}^{2})$. The divergence
free constraint is reminiscent of the topological nature of these defects. At
this level one can associate to each dislocation a line tension energy, i.e.,
an energy with the same form of (1.1). Such energies can in turn be derived
from more fundamental models. For example, in [10], [13], and [5] the authors
deduce the line tension model accounting for the elastic distortion in the
material induced by the presence of dislocations. See also [19], [20], [11],
and [18] for a similar derivation in dimension 2 for cylindrical geometry
where dislocations are viewed as point defects. Similarly, this type of line
tension model can also be derived under the assumption that the line defects
are contained in a given (slip) plane as limit of nonlocal phase transition
energies (in the spirit of the Cahn Hilliard energies for liquid-liquid phase
transtions), known in the literature of dislocations as (generalised) Nabarro-
Peirls models (see [7] and the references therein). The natural representation
of the line energy here is given by functionals defined on the space of $BV$
functions with values in a discrete group, so that dislocations are identified
by the jump set of such functions, see [9].

In this paper we are interested in the case of a large number of dislocations
on a macroscopic scale. The interest of this case lying in the fact that a
large quantity of dislocations is responsible for plastic deformation in the
material (see [8] for a more complete analysis). In particular, starting from
the rescaled version of the line tension energy defined in (1.2), we are
interested in recovering an effective energy for a large system of
dislocations on a scale at which they can be seen as diffused. In this respect
our limiting energy $E_{0}$ can be understood as the macroscopic (self) energy
associated to a continuous distribution of dislocations.

The result presented in here is also a crucial step for a derivation of a
macroscopic model for plasticity accounting for the presence of defects in the
same spirit of [12]. The derivations of such macroscopic models as limit of
elastic energies for incompatible fields, under proper energy scalings, will
appear in a forthcoming paper.

The structure of the paper is as follows: in Section 2 we give preliminary
definitions and recall some known results. In Section 2.3 we state the main
result and present the proof of the lower bound. In Section 3 we present the
approximation results needed for the upper bound, the latter being proved in
Section 4.

##  2 Preliminaries and statement of the main result

We first set all the notation needed for the statement and proof of our main
result. Unless further specified, in what follows $\Omega$ is a bounded open
set of $\mathbb{R}^{n}$ with Lipschitz boundary. In what follows, without loss
of generality we will assume $\mathcal{B}=\mathbb{Z}^{N}$, for $N\geq 1$.

###  2.1 Configurations

We start with the set of admissible configurations. In what follows we will
denote with $[\mathcal{M}(\Omega)]^{N\times n}$ the space of bounded Radon
measures with values in $\mathbb{R}^{N\times n}$. The space
$\mathcal{M}^{1}(\Omega;\mathcal{S}\times\mathbb{S}^{n-1})$ will denote the
set of measures in $[\mathcal{M}(\Omega)]^{N\times n}$ of the form

$\mu=\theta\otimes\tau\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\gamma,$ (2.1)

where $\gamma\subset\mathbb{R}^{n}$ is a $1$-rectifiable set with tangent
vector $\tau\in\mathbb{S}^{n-1}$ defined $\mathcal{H}^{1}$ a.e. on $\gamma$,
$\mathcal{S}\subset\mathbb{R}^{N}$ is a generic subset, and
$\theta\in[L^{1}(\gamma,\mathcal{H}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\gamma)]^{N}$ is such that
$\theta(x)\in\mathcal{S}$ for $\mathcal{H}^{1}$-a.e. $x\in\gamma$. We remark
that in most cases we will consider the set $\mathcal{S}$ to be a discrete
lattice that spans $\mathbb{R}^{N}$ but for notational purposes it is
convenient to give the above definition for general sets. We also observe that
although the main result, Theorem 2.5, is stated for the case $n=3$, in
Section $3$ we prove an approximation result which holds in any dimension,
hence it is convenient to give the relevant definitions for an arbitrary
dimension $n\geq 2$.

We say that a measure in $[\mathcal{M}(\Omega)]^{N\times n}$ is divergence
free if it is row-wise divergence free, i.e., if the following holds true

$\int_{\Omega}\sum_{j=1}^{n}\partial_{j}\varphi
d\mu_{ij}=0\qquad\forall\varphi\in C_{0}^{\infty}(\Omega),\qquad\forall
i=1,\cdots,N.$

The subset of $\mathcal{M}^{1}(\Omega;\mathcal{S}\times\mathbb{S}^{n-1})$ (and
$[\mathcal{M}(\Omega)]^{N\times n}$) of divergence free measures will be
denoted by $\mathcal{M}_{df}^{1}(\Omega;\mathcal{S}\times\mathbb{S}^{n-1})$
(respectively $[\mathcal{M}(\Omega)]_{df}^{N\times n}$). Note that, if
$G\in[L^{1}(\Omega)]^{N\times n}$ and
$\mu=G\mathcal{L}^{n}\in[\mathcal{M}(\Omega)]^{N\times n}$ with
$\mathcal{L}^{n}$ the $n$-dimensional Lebesgue measure, then
$\mathop{\mathrm{div}}G=0$ in the sense of distributions if and only if
$\mu\in\mathcal{[}\mathcal{M}(\Omega)]_{df}^{N\times n}$.

We say that a measure
$\mu\in\mathcal{M}^{1}(\Omega;\mathcal{S}\times\mathbb{S}^{n-1})$ is
polyhedral if its support if formed by a finite number of straight closed
segments.

######  Remark 2.1.

Given a measure $\mu=b\otimes t\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\gamma\in\mathcal{M}^{1}(\Omega;\mathbb{Z}^{N}\times\mathbb{S}^{n-1})$,
where $b\in\mathbb{Z}^{N}$ is constant and
$\gamma\in[\operatorname{Lip}([0,1])]^{n}$ is a curve such that
$\gamma^{\prime}=t$, it holds that

$\langle\mu,\nabla\varphi\rangle=\langle
b,\varphi(\gamma(1))-\varphi(\gamma(0))\rangle,\ \
\forall\varphi\in[C_{c}^{1}(\mathbb{R}^{n})]^{N},$ (2.2)

hence it must be

$\mathop{\mathrm{div}}\mu=b\delta_{\gamma(0)}-b\delta_{\gamma(1)},$ (2.3)

where $b\delta_{a}\in[\mathcal{M}(\Omega)]^{N}$ is a Dirac delta centered at
$a\in\Omega$ with multiplicity $b$. Accordingly we say that $\mu$ carries a
mass of $b$ at $\gamma(0)$ and a mass of $-b$ at $\gamma(1)$, where the sign
depends on the orientation $t$. In particular for such a measure to be
divergence free in $\Omega$, the curve $\gamma$ must not have endpoints
contained in $\Omega$.

######  Remark 2.2.

If $\Omega$ is a simply connected domain, measures in
$\mathcal{M}_{df}^{1}(\Omega;\mathbb{Z}^{N}\times\mathbb{S}^{n-1})$ can be
extended to measures on the whole of $\mathbb{R}^{n}$ that can be
characterized as measures concentrated on unions of countably many closed
Lipschitz loops with constant multiplicity in $\mathbb{Z}^{N}$ (see [6],
Theorem 2.5, for the precise statement given in terms of $1$-rectifiable
currents).

The set $\mathcal{M}_{df}^{1}(\Omega;\mathbb{Z}^{N}\times\mathbb{S}^{2})$
represents the set of admissible configurations for the class of energies
under consideration.

###  2.2 Energy densities and their main properties

Here we recall the main properties of the class of energy densities

$\psi:\mathbb{Z}^{N}\times\mathbb{S}^{2}\longrightarrow[0,+\infty).$

The $\mathcal{H}^{1}$-elliptic envelope of $\psi$ is the function $\psi^{rel}$
obtained by solving, for any $b\in\mathbb{Z}^{N}$ and $t\in\mathbb{S}^{2}$,
the cell problem

$\displaystyle\psi^{\mathrm{rel}}(b,t):=\inf\Big{\\{}$
$\displaystyle\int_{\gamma}\psi(\theta,\tau)d\mathcal{H}^{1}:\mu=\theta\otimes\tau\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\gamma\in\mathcal{M}_{df}^{1}(B_{1};\mathbb{Z}^{N}\times\mathbb{S}^{n-1}),$
(2.4) $\displaystyle\operatorname{supp}\left(\mu-b\otimes
t\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\left(\mathbb{R}t\cap
B_{1/2}\right)\right)\subset\subset B_{1/2}\Big{\\}},$

where $B_{r}=B_{r}(0)$ denotes a ball of radius $r$ and center 0. We say that
$\psi$ is $\mathcal{H}^{1}$-elliptic if $\psi^{rel}=\psi$ (see [6]).

We will assume that $\psi$ is $\mathcal{H}^{1}$-elliptic and we extend it to
the whole of $\mathbb{R}^{N}\times\mathbb{S}^{2}$ by setting
$\psi(b,t)=+\infty$ for all $b\in\mathbb{R}^{N}\smallsetminus\mathbb{Z}^{N}$.
Further, we assume that

$\psi(b,t)\geq c\,|b|\qquad\forall\
b\,\in\mathbb{Z}^{N}\smallsetminus\\{0\\}\hbox{ and }t\in\mathbb{S}^{2}.$
(2.5)

Moreover we recall that $\mathcal{H}^{1}$-ellipticity implies that $\psi$ is
subattidive and has linear growth at infinity in the first entry, i.e., for
all $b,b^{\prime}\in\mathbb{R}^{N}$ and $t\in\mathbb{S}^{2}$ it satisfies

$\psi(b+b^{\prime},t)\leq\psi(b,t)+\psi(b^{\prime},t)\qquad\hbox{and
}\qquad\psi(b,t)\leq\overline{c}\,|b|,$ (2.6)

for some positive constant $\overline{c}$, see [6, Lemma 3.2 (iii) and (iv)].

The recession function
$\psi_{\infty}:\mathbb{R}^{N}\times\mathbb{S}^{2}\rightarrow$ $[0,\infty]$ of
$\psi$ is given by

$\psi_{\infty}(b,t):=\liminf_{s\rightarrow+\infty}\frac{1}{s}\psi(sb,t).$
(2.7)

This is a crucial ingredient in order to determine the effective energy
density of the limiting energy $E_{0}$. For the readers’ convenience we now
recall some of the main properties of $\psi_{\infty}$ as they are proved in
[8].

######  Proposition 2.3.

Let $\psi:\mathbb{R}^{N}\times\mathbb{S}^{2}\to\mathbb{R}\cup\\{+\infty\\}$ be
$\mathcal{H}^{1}$-elliptic, satifying (2.5) and such that $\psi(b,t)=+\infty$
if $b\in\mathbb{R}^{N}\setminus\mathbb{Z}^{N}$. Let $\psi_{\infty}$ be its
recession function. Then the following hold:

  1. (i)

$\psi_{\infty}(\cdot,t)$ is positively $1$-homogeneous for all
$t\in\mathbb{S}^{2}$;

  2. (ii)

Let $\mathcal{Q}=\left\\{\lambda z:\lambda>0,z\in\mathbb{Z}^{N}\right\\}$,
then $\psi_{\infty}(b,t)=+\infty$ if
$b\in\mathbb{R}^{N}\smallsetminus\mathcal{Q}$, and $\psi_{\infty}(b,t)\leq
c\,|b|$ for $b\in\mathcal{Q}$;

  3. (iii)

Let $b\in\mathbb{R}^{N},t\in\mathbb{S}^{2}$, for any sequence
$z_{j}\in\mathbb{Z}^{N}$ such that $\left|z_{j}\right|\rightarrow+\infty$ and
$z_{j}/\left|z_{j}\right|\rightarrow b/|b|$ one has

$\lim_{j\rightarrow+\infty}\frac{1}{\left|z_{j}\right|}\psi\left(z_{j},t\right)=\frac{1}{|b|}\psi_{\infty}(b,t).$

The proof of property (i) is immediate, while properties (ii) and (iii) can be
found in [8], Lemma 3.3 and 3.4 respectively.

Finally we define the function $g:\mathbb{R}^{N\times
3}\longrightarrow[0,+\infty)$, which provides the effective energy density, as
the convex envelope of

$g_{\infty}(A)=\begin{cases}\psi_{\infty}(b,t)\quad&\hbox{if }A=b\otimes t,\ \
b\in\mathbb{R}^{N},\ \ t\in\mathbb{S}^{2},\\\
{+\infty}&\hbox{otherwise},\end{cases}$ (2.8)

i.e., $g(A)=g^{**}_{\infty}(A)$. Some important properties of $g$ are
described in the following

######  Lemma 2.4.

The function $g$ is continuous, $1$-homogeneous, and there are $c_{1},c_{2}>0$
such that

$c_{1}|A|\leq g(A)\leq c_{2}|A|,$

for all matrices $A\in\mathbb{R}^{N\times 3}$.

###### Proof.

Every matrix $A\in\mathbb{R}^{N\times 3}$ can be decomposed as a convex
combination of rank 1 matrices on which $g_{\infty}$ is finite, namely

$A=\sum_{i=1}^{N}\sum_{j=1}^{3}\dfrac{|A_{ij}|}{3\|Ae_{j}\|_{1}}\dfrac{3A_{ij}\|Ae_{j}\|_{1}}{|A_{ij}|}e_{j}\otimes
e_{i}=\sum_{i=1}^{N}\sum_{j=1}^{3}\lambda_{j}^{i}q_{j}^{i}\otimes e_{i},$

where $\lambda_{j}^{i}:=3^{-1}|A_{ij}|\|Ae_{j}\|_{1}^{-1}$,
$\sum_{j}\sum_{i}\lambda_{j}^{i}=1$, and
$q^{i}_{j}:=3A_{ij}\|Ae_{j}\|_{1}|A_{ij}|^{-1}e_{j}\in\mathcal{Q}$. Then by
convexity we have

$g(A)\leq\sum_{i,j}\dfrac{1}{3}\dfrac{|A_{ij}|}{\|Ae_{j}\|_{1}}\psi_{\infty}\left(q^{i}_{j}\otimes
e_{i}\right)\leq\sum_{i,j}\dfrac{1}{3}\dfrac{|A_{ij}|}{\|Ae_{j}\|_{1}}c\,|q^{i}_{j}|\leq
c|A|.$

Being convex and finite, $g$ is continuous. Now from the $1$-homogeneity of
$\psi_{\infty}$ we infer that also $g_{\infty}$ is positively $1$-homogeneous.
By Caratheodory’s Theorem for every $\xi\in\mathbb{R}^{N\times 3}$ and
$\eta>0$, there exist $3N+1$ vectors $\xi_{i}\in\mathbb{R}^{N\times 3}$, and
numbers $t_{i}\geq 0$, such that $\sum_{i=1}^{3N+1}t_{i}=1$ and
$\xi=\sum_{i=1}^{3N+1}t_{i}\xi_{i}$ and
$g(\xi)+\eta\geq\sum_{i=1}^{3N+1}t_{i}g_{\infty}(\xi_{i})$; hence we have that

$\lambda
g(\xi)+\lambda\eta\geq\lambda\sum_{i=1}^{3N+1}t_{i}g_{\infty}(\xi_{i})=\sum_{i=1}^{3N+1}t_{i}g_{\infty}(\lambda\xi_{i})\geq
g(\lambda\xi),$

and then $\lambda g(\xi)\geq g(\lambda\xi)$. The opposite inequality is
finally obtained similarly replacing $\lambda$ and $\xi$ by $\lambda^{-1}$ and
$\lambda\xi$. The bound from below is a consequence of continuity and
$1$-homogeneity. ∎

###  2.3 The $\Gamma$-convergence result

We now have all the ingredients in order to state the main result of the
paper.

######  Theorem 2.5.

Let $\psi:\mathbb{Z}^{N}\times\mathbb{S}^{2}\longrightarrow[0,+\infty)$ be
$\mathcal{H}^{1}$-elliptic and obey $\frac{1}{c}|b|\leq\psi(b,t)$ for all
$b\in\mathbb{Z}^{N}$ and $t\in\mathbb{S}^{2}$. Let
$\Omega\subset\mathbb{R}^{3}$ be an open bounded set, uniformly Lipschitz and
simply connected. Then the functionals

$E_{\sigma}(\mu):=\begin{cases}\displaystyle{\int_{\gamma}{\sigma\psi\left(\frac{\theta}{\sigma},\tau\right)d\mathcal{H}^{1}}}&\textit{if
}\mu=\theta\otimes\tau\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\gamma\in\mathcal{M}_{df}^{1}(\Omega;\sigma\mathbb{Z}^{N}\times\mathbb{S}^{2}),\\\
+\infty&otherwise,\end{cases}$ (2.9)

$\Gamma$-converge, as $\sigma\to 0$, with respect to the weak$*$ topology of
$[\mathcal{M}(\Omega)]^{N\times 3}$, to

$E_{0}(\mu)=\begin{cases}\displaystyle{\int_{\Omega}{g\left(\frac{d\mu}{d\|\mu\|}\right)d\|\mu\|}}&\textit{if
}\mu\in[\mathcal{M}(\Omega)]_{df}^{N\times 3},\\\
+\infty&otherwise,\end{cases}$ (2.10)

where $g:\mathbb{R}^{N\times 3}\longrightarrow[0,+\infty)$ is the convex
envelope of $g_{\infty}$ as defined in (2.8).

######  Remark 2.6.

Notice that from the lower bound on the density $\psi$ one immediately deduces
that a sequence with equi-bounded energy has also bounded total variation.
Therefore the compactness part of the $\Gamma$-convergence result is
immediate.

The proof of Theorem 2.5 will be a consequence of Proposition 2.7 and
Proposition 4.1 (respectively the lower and the upper bound).

As for the lower bound it is quite straightforward and it is a consequence of
the definition of the energy density $g$. We give its proof with the
proposition below.

######  Proposition 2.7.

Let $\psi:\mathbb{Z}^{N}\times\mathbb{S}^{2}\longrightarrow[0,+\infty)$ be
$\mathcal{H}^{1}$-elliptic and obey $\frac{1}{c}|b|\leq\psi(b,t)$ for all
$b\in\mathbb{Z}^{N}$ and $t\in\mathbb{S}^{2}$, let
$\Omega\subset\mathbb{R}^{3}$ be open and bounded. Then for every sequence
$\sigma_{j}\to 0$ and
$\mu_{j}\in\mathcal{M}_{df}^{1}(\Omega;\sigma_{j}\mathbb{Z}^{N}\times\mathbb{S}^{2})$
converging weakly$*$ to some divergence free measure
$\mu\in[\mathcal{M}(\Omega)]^{N\times 3}$ we have

$\liminf_{j\to+\infty}E_{\sigma_{j}}(\mu_{j})\geq E_{0}(\mu).$

###### Proof.

By the subadditivity of $\psi$ we have that for all
$t\in\mathbb{S}^{2},b\in\mathbb{R}^{N},s>0,k\geq 1$ it holds

$\dfrac{\psi(sb,t)}{s}=\dfrac{k\psi(sb,t)}{sk}\geq\dfrac{\psi(ksb,t)}{sk},$

hence, by the definitions of $\psi_{\infty}$ and $g$, we have

$\dfrac{\psi(sb,t)}{s}\geq\liminf_{k\rightarrow+\infty}\dfrac{\psi(ksb,t)}{sk}\geq\psi_{\infty}(b,t)\geq
g(b\otimes t).$ (2.11)

Let
$\mu_{j}\in\mathcal{M}_{df}^{1}(\Omega;\sigma_{j}\mathbb{Z}^{N}\times\mathbb{S}^{2})$
be converging to $\mu\in[\mathcal{M}(\Omega)]_{df}^{N\times 3}$ and be such
that $\liminf_{j}E_{\sigma_{j}}(\mu_{j})<+\infty$. Then from (2.11) we obtain

$E_{\sigma_{j}}(\mu_{j})=\int_{\gamma_{j}}\sigma_{j}\psi(b_{j}\sigma_{j}^{-1},t_{j})d\mathcal{H}^{1}\geq\int_{\gamma_{j}}g(b_{j}\otimes
t_{j})d\mathcal{H}^{1}=E_{0}(\mu_{j}).$

To conclude we use the fact that $E_{0}$ is weakly lower semicontinuous by
Reshetnyak’s Theorem and Lemma 2.4, so that

$\liminf_{j\rightarrow+\infty}E_{\sigma_{j}}(\mu_{j})\geq E_{0}(\mu).$ (2.12)

∎

The upper bound instead represents the core of the paper. It requires a
technical construction and will be presented in the next section, where we
will first show an approximation result for divergence free measures and then
make an explicit construction with optimal energy.

##  3 Approximation of divergence free measures

In this section we show two approximation results which are crucial for the
$\Gamma$-limsup inequality. We will show that any divergence free measure
$\mu\in[\mathcal{M}(\Omega)]^{N\times n}$ can be approximated, strictly and
therefore in energy, with measures that are absolutely continuous with respect
to the Lebesgue measure, piecewise constant and divergence free. Further we
will show that, for $n=3$ the latters can be approximated with measures in
$\mathcal{M}_{df}^{1}(\Omega;\mathcal{Q}\times\mathbb{S}^{2})$. The case of
dimension $n>3$ presents some difficulties due to specific construction
contained in Lemma 3.10, we nevertheless expect that the approach followed in
this paper can be adapted to the general case.

###  3.1 Piecewise constant approximation

Here we consider the general case of functions and measures in
$\Omega\subseteq\mathbb{R}^{n}$, for $n\geq 2$. First we introduce a class of
admissible sequences of triangulations
$\mathcal{T}:=\\{T^{i}\\}_{i=1,\ldots,M}$ of $\Omega\subseteq\mathbb{R}^{n}$.

######  Definition 3.1.

We say that a family $\mathcal{T}:=\\{T^{i}\\}_{i=1,\ldots,M}$ of simplexes is
an _admissible sequence of triangulations (of aspect ratio $C_{0}$)_ if the
closed tetrahedra $T^{i}$, with $1\leq i\leq M$, satisfy

  1. (i)

$\Omega\subset\subset\bigcup_{i=1}^{M}T^{i}$;

  2. (ii)

$\operatorname{int}(T^{i})\cap\operatorname{int}(T^{j})=\emptyset$ for
$i\not=j$ (here $\operatorname{int}(T)$ is the topological interior part of
the set $T$);

  3. (iii)

there exists a positive constant $C_{0}>0$ such that for every $j$ there
exists a point $x^{j}$ so that

$B_{C_{0}r}(x^{j})\subseteq T^{j}\subseteq B_{r}(x^{j}),$

where $r=\max_{i}{\operatorname{diam}}(T_{i})$ is the size of the
triangulation.

We say that a function $f$ is _piecewise constant relatively to $\mathcal{T}$_
if $f$ is constant in $\operatorname{int}(T^{j})$ for every
$j\in\\{1,\ldots,M\\}$.

######  Theorem 3.2 (Piecewise constant approximation).

Let $\Omega$ be a simply connected open set with Lipschitz boundary, and let
$\mu\in[\mathcal{M}(\Omega)]^{N\times n}$ with $\mathop{\mathrm{div}}\mu=0$ be
given. Then there exists a sequence of measures
$\mu_{k}\in[\mathcal{M}(\Omega)]^{N\times n}$ such that
$\mu_{k}\overset{\ast}{\rightharpoonup}\mu$, with
$\mu_{k}=A_{k}\mathcal{L}^{n}$ and $A_{k}$ is piecewise constant relatively to
a sequence of admissible triangulation $\mathcal{T}_{k}$, and
$\mathop{\mathrm{div}}\mu_{k}=0$ in $\Omega$. Furthermore
$\lim_{k}\|\mu_{k}\|(\Omega)=\|\mu\|(\Omega)$.

The proof of Theorem 3.2 will be given essentially in two steps. At first, in
Lemma 3.3, we approximate $\mu$ via measures having smooth densities up to the
boundary. In the second step we reduce to measures which are piecewise
constant with respect to a triangulation of $\Omega$. In both cases the main
difficulty is given by the free divergence constraint, and in order to modify
the measures while preserving this constraint it will be convenient to
interpret $\mu$ as a current, since the push forward of a current preserves
solenoidality.

We start by regularizing the measures. The following lemma is an adaptation to
the present context of Proposition 6, Chapter 5 in [14].

######  Lemma 3.3 (Smoothing).

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open Lipschitz set, let
$\mu\in[\mathcal{M}(\Omega)]^{N\times n}$ be divergence free, then it exists a
family of functions $F_{\varepsilon}\in[L^{1}(\Omega)]^{N\times
n}\cap[C^{\infty}(\Omega)]^{N\times n}$ such that

  * i)

$F_{\varepsilon}\mathcal{L}^{n}\overset{\ast}{\rightharpoonup}\mu$ in the
sense of measures;

  * ii)

$\|F_{\varepsilon}\mathcal{L}^{n}\|(\Omega)\rightarrow\|\mu\|(\Omega)$;

  * iii)

$\mathop{\mathrm{div}}F_{\varepsilon}=0$ in $\Omega$.

###### Proof.

Let $d\in C^{\infty}(\Omega)$ be a smooth version of $d(\cdot,\Omega^{c})$,
namely a function satisfying $0<d(x)<d(x,\Omega^{c})$ and $|\nabla d(x)|\leq
1$ for all $x\in\Omega$. We define, for every $z\in B_{1}(0)$,
$H_{z}(x):=x+d(x)z$ and observe that $H_{z}(\Omega)=\Omega$,
$\nabla_{x}H_{z}(x)=Id+z\otimes\nabla d(x)$. Let $\rho_{\varepsilon}$ be a
convolution kernel and define the regularization of
$\varphi\in[C_{0}(\Omega)]^{N\times n}$ for all $x\in\Omega$ as follows

$\displaystyle\varphi_{\varepsilon}(x)$
$\displaystyle:=\int_{B_{1}(0)}\rho_{\varepsilon}(-z)\varphi(x+zd(x))\nabla_{x}H_{z}(x)dz$
$\displaystyle=\int_{\Omega}{d(x)^{-n}\rho_{\varepsilon}\left(\frac{x-y}{d(x)}\right)\varphi(y)\left(Id+\frac{y-x}{d(x)}\otimes\nabla
d(x)\right)dy},$

where we performed the change of variable $y=x+d(x)z$. We then define by
duality the mollification of $\mu$ to be
$\langle\mu_{\varepsilon},\varphi\rangle:=\langle\mu,\varphi_{\varepsilon}\rangle$,
hence

$\displaystyle\langle\mu_{\varepsilon},\varphi\rangle$
$\displaystyle=\int_{\Omega}\left(\int_{B_{1}(0)}\rho_{\varepsilon}(-z)\varphi(x+zd(x))\nabla_{x}H_{z}(x)dz\right)d\mu(x)$
$\displaystyle=\int_{\Omega}\langle\int_{\Omega}d(x)^{-n}\rho_{\varepsilon}\left(\frac{x-y}{d(x)}\right)\varphi(y)\left(Id+\frac{y-x}{d(x)}\otimes\nabla
d(x)\right)dy,G(x)\rangle d\|\mu\|(x),$

where $G\in[L^{1}(\Omega,\|\mu\|)]^{N\times n}$ is such that $|G|=1$
$\|\mu\|$-a.e. and $G\|\mu\|=\mu$.

Rearranging the integrals in the definition of $\mu_{\varepsilon}$ and using
Fubini’s Theorem it is easy to see that
$\mu_{\varepsilon}<\\!\\!<\mathcal{L}^{n}$ with density function defined for
$y\in\Omega$ by

$F_{\varepsilon}(y):=\int_{\Omega}d(x)^{-n}\rho_{\varepsilon}\left(\frac{x-y}{d(x)}\right)G(x)(Id+\nabla
d(x)\otimes\frac{y-x}{d(x)})d\|\mu\|(x).$ (3.1)

Clearly $F_{\varepsilon}\in[C^{\infty}(\Omega)]^{N\times n}$.

By uniform continuity, $\varphi_{\varepsilon}$ converges uniformly in $\Omega$
to $\varphi$, and thus $\mu_{\varepsilon}\overset{\ast}{\rightharpoonup}\mu$
in $[\mathcal{M}(\Omega)]^{N\times n}$, which proves i). Furthermore it holds

$\displaystyle|\langle\mu_{\varepsilon},\varphi\rangle|$
$\displaystyle=\left|\int_{\Omega}\langle\int_{B_{\varepsilon}(0)}\rho_{\varepsilon}(-z)(Id+\nabla
d(x)\otimes z)\varphi(x+zd(x))dz,G(x)\rangle d\|\mu\|(x)\right|$
$\displaystyle\leq\int_{\Omega}(1+2\varepsilon)\|\varphi\|_{\infty}d\|\mu\|(x),$

hence

$\|\mu_{\varepsilon}\|(\Omega)\leq(1+2\varepsilon)\|\mu\|(\Omega),$ (3.2)

thus $\lim_{\varepsilon\rightarrow
0}\|\mu_{\varepsilon}\|(\Omega)=\|\mu\|(\Omega)$, and therefore ii) holds.

Finally we now prove that $\mu_{\varepsilon}$ is row-wise divergence free,
i.e.,

$\sum_{j=1}^{n}\int_{\Omega}F_{\varepsilon}^{ij}(y)\partial_{j}\psi(y)dy=0,\ \
\ \ \forall\psi\in C_{c}^{1}(\Omega),\ \ i=1,\cdots,N.$ (3.3)

To see this, we denote with $G^{i}$ the $i$-th row of $G$, then we compute

$\displaystyle\sum_{j=1}^{n}\int_{\Omega}F_{\varepsilon}^{ij}(y)\partial_{j}\psi(y)dy$
$\displaystyle=\sum_{j,l=1}^{n}\int_{B(0,1)}\int_{\Omega}\rho_{\varepsilon}(z)G_{il}(x)\partial_{l}(H_{z})^{j}(x)\partial_{j}\psi(x+d(x)z)d\|\mu\|(x)dz$
$\displaystyle=\int_{B(0,1)}\rho_{\varepsilon}(z)\int_{\Omega}\langle
G^{i},\nabla(\psi\circ H_{z})(x)\rangle d\|\mu\|(x)dz,$

where we used the change of variables $y=x+d(x)z$ and, in the last equality,
we used the fact that

$\sum_{j=1}^{n}\partial_{l}(H_{z})^{j}(x)\partial_{j}\psi(x+d(x)z)=\partial_{l}(\psi\circ
H_{z})(x).$

The claim now follows since $\psi\circ H_{z}\in C_{c}^{1}(\Omega)$ and $\mu$
is divergence free.

∎

We say that a mapping $\phi$ is a potential for a divergence free matrix field
$F$ if $\mathcal{R}\phi=F$ for some first order linear differential operator
$\mathcal{R}$ satisfying $\operatorname{div}\mathcal{R}\psi=0$ for every
$\psi\in[C^{\infty}(\Omega)]^{N\times m}$ where $m=n(n-1)/2$. For example if
$n=3$ then $\mathcal{R}=\operatorname{curl}$. We observe that if $\Omega$ is
simply connected then such an operator $\mathcal{R}$ always exists as a direct
consequence of Poincaré’s Lemma.

Thanks to Lemma 3.3, we simply need to prove Theorem 3.2 for divergence free
measures of the form $\mu=F\mathcal{L}^{n}$. To do so we would like to pass to
a potential $\phi$ of $F$, and then approximate $\phi$ with piecewise affine
functions by linear interpolating over a sequence of triangulations of
$\Omega$. In order to show the convergence of the interpolating sequence to
$\phi$ we need boundedness of its derivatives (see for instance [17]), while
from Lemma 3.3 we can only infer $\phi\in[C^{\infty}(\Omega)]^{N\times m}$. To
obtain such bound we will modify $F$, since clearly a uniform bound for the
derivatives of $F$ implies bounds for the derivatives of $\phi$. With this
goal in mind we state below an extension lemma for $F$ whose proof follows
closely the one of a similar extension lemma proved in [6, Lemma 2.3] in the
context of $1$-rectifiable currents.

######  Lemma 3.4 (Extension).

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded Lipschitz set. There exist an
open bounded set $\hat{\Omega}$ compactly containing $\Omega$ such that for
every $F\in[L^{1}(\Omega)]^{N\times n}$ with $\mathop{\mathrm{div}}F=0$ in
$\Omega$, there is a function $\hat{F}\in[L^{1}(\hat{\Omega})]^{N\times n}$,
with $\mathop{\mathrm{div}}\hat{F}=0$ in $\hat{\Omega}$, such that $\hat{F}=F$
in $\Omega$. In particular the measure
$\hat{\mu}=\hat{F}\mathcal{L}^{n}\in[\mathcal{M}(\hat{\Omega})]_{df}^{N\times
n}$ extends the measure $\mu=F\mathcal{L}^{n}$.

###### Proof.

We will prove the lemma, first in the case of $F\in[L^{1}(\Omega)]^{n}$ with
$\mathop{\mathrm{div}}F=0$ in $\Omega$, then the matrix valued case will
follow simply by extending row-wise.
Choose a function $N\in C^{1}(\partial\Omega;\mathbb{S}^{n-1})$, such that
$N(x)\cdot\nu(x)\geq\alpha>0$ for almost all $x\in\partial\Omega$, where $\nu$
is the outer normal to $\partial\Omega$ (see [16] for details). Consider the
mapping $g:\partial\Omega\times(-\rho_{0},\rho_{0})\rightarrow\mathbb{R}^{n}$
defined by $g(x,t)=x+tN(x)$, then there exists $\rho_{0}$ sufficiently small
such that $g$ is bijective and bi-Lipschitz: indeed, by Local Invertibility
Theorem for Lipschitz mappings (see [4]), there exists $\rho_{0}$ small enough
such that $g$ is locally invertible with Lispchitz inverse function,
furthermore, since $g(x,0)=x$, we also get global invertibility (see [2] Lemma
33). Let $D_{0}=g(\partial\Omega\times(-\rho_{0},\rho_{0}))$ and
$h:D_{0}\rightarrow D_{0}$ be defined by $h(g(x,t))=g(x,-t)$. Then $h$ is bi-
Lipschitz and coincides with its inverse. We then set
$\hat{\Omega}:=\Omega\cup D_{0}$ and we define the extension
$\hat{F}(y)=\chi_{\Omega}F(y)+\chi_{D_{0}\smallsetminus\Omega}\nabla
h(h^{-1}(y))F(h^{-1}(y))|\nabla h^{-1}|$, for all $y\in\hat{\Omega}$.

We now show that $\hat{F}$ is divergence free. We compute, for every $\psi\in
C_{c}^{1}(\hat{\Omega})$,

$\begin{split}\int_{\hat{\Omega}}\langle\nabla\psi,\hat{F}\rangle
dy&=\int_{\Omega}\langle\nabla\psi,F\rangle
dy+\int_{D_{0}\setminus\Omega}\langle\nabla\psi,\nabla
h(h^{-1}(y))F(h^{-1}(y))\rangle|\nabla h^{-1}|dy\\\
&=\int_{\Omega}\langle\nabla\psi,F\rangle dy+\int_{\Omega\cap
D_{0}}\langle\nabla[\psi\circ h](x),F(x)\rangle dx.\end{split}$

We then observe that, by direct computation, it holds

$D[(\chi_{D_{0}\smallsetminus\Omega}\psi)\circ h]=\chi_{\Omega\cap
D_{0}}(x)\nabla[\psi\circ h],$

in the sense of distributions in $\Omega$, thus in particular
$(\chi_{D_{0}\smallsetminus\Omega}\psi)\circ h\in W^{1,\infty}(\Omega)$.
Therefore we obtain

$\int_{\hat{\Omega}}\langle\nabla\psi,\hat{F}\rangle
dy=\int_{\Omega}\langle\nabla[\psi-(\chi_{D_{0}\setminus\Omega}\psi)\circ
h],F\rangle dy.$

The right hand side is zero since
$\varphi=\psi-(\chi_{D_{0}\smallsetminus\Omega}\psi)\circ h\in
W^{1,\infty}_{0}(\Omega)$: indeed it is zero for all $\varphi\in
C^{\infty}_{0}(\Omega)$ and then also in $W^{1,\infty}_{0}(\Omega)$ by density
of smooth and compactly supported functions with respect to the weak$*$
topology in $W^{1,\infty}(\Omega)$.

The general case is obtained by using the above construction row-wise. ∎

We are ready to prove Theorem 3.2.

###### Proof of Theorem 3.2.

Thanks to Lemma 3.3, we first find a sequence of fields
$F_{h}\in[L^{1}(\Omega)]^{N\times n}\cap[C^{\infty}(\Omega)]^{N\times n}$ such
that $\mu_{h}=F_{h}\mathcal{L}^{n}\overset{\ast}{\rightharpoonup}\mu$,
$\mathop{\mathrm{div}}F_{h}=0$ in $\Omega$ and
$\lim_{h\to+\infty}\|\mu_{h}\|(\Omega)=\|\mu\|(\Omega)$.

Using Lemma 3.4 we can find a set $\hat{\Omega}$ compactly containing $\Omega$
and a sequence of functions $\hat{F}_{h}$ extending $F_{h}$ such that
$\mathop{\mathrm{div}}\hat{F}_{h}=0$ in $\hat{\Omega}$. We then choose
$\Omega^{\prime}$ such that
$\Omega\subset\subset\Omega^{\prime}\subset\subset\hat{\Omega}$ and by
(standard) convolution we can now find a sequence
$G_{h}\in[L^{1}(\Omega^{\prime})]^{N\times
n}\cap[C^{\infty}(\Omega^{\prime})]^{N\times n}$, with
$\mathop{\mathrm{div}}G_{h}=0$ in $\Omega^{\prime}$, such that
$G_{h}\mathcal{L}^{n}$ converges strictly to $\mu$. We then apply Poincaré
Lemma to $G_{h}$ and find a potential
$\phi^{h}\in[C^{\infty}(\Omega^{\prime})]^{N\times n(n-1)/2}$, such that
$\mathcal{R}\phi^{h}=G_{h}$.

Now we fix a sequence of admissible triangulations

$\mathcal{T}_{k}:=\\{T_{k}^{i}\\}_{i=1,\ldots,M(k)},$

of aspect ratio $C_{0}$ and size $1/k$, such that
$\Omega\subset\subset\bigcup_{i=1}^{M(k)}T_{k}^{i}\subset\subset\Omega^{\prime}$,
and for every $h$ we construct a sequence of piecewise affine functions
$\phi^{h}_{k}$ obtained interpolating linearly the values of $\phi^{h}$ on the
vertices of the tetrahedra $T_{k}^{i}$. By classical discretization arguments
(see for instance [17], Theorem 11.40) we have that

$\|\nabla\phi^{h}_{k}-\nabla\phi^{h}\|_{L^{1}(\Omega)}\leq\frac{C}{k^{2}}\sup_{\Omega}|\nabla^{2}\phi^{h}|,$

where the constant $C$ depends on $C_{0}$. Hence
$G^{h}_{k}:=\mathcal{R}\phi^{h}_{k}$ converges to $G_{h}$ in $L^{1}$ for every
$h$. Furthermore $\mathop{\mathrm{div}}G^{h}_{k}=0$ in $\Omega$ and
$G^{h}_{k}$ is piecewise constant.

In conclusion by a diagonal procedure we obtain a sequence of piecewise
constant divergence free fields that converge strictly to $\mu$ in $\Omega$. ∎

###  3.2 Optimal construction via polygonal supported measures

From now on we focus on the special case $n=3$. Indeed the constructions we
perform in Lemma 3.6 and Lemma 3.10 depend on the dimension: while in the case
$n=3$ the shared face of two simplex is $2$ dimensional, in the generic case
two neighbouring simplex share a $n-1$ dimensional face. Nevertheless we
expect that our construction can be adapted to any dimension.

We now show a second approximation result that is closely related with our
energies. We will need to show that any piecewise constant divergence free
measure can be obtained as a limit of a sequence of measures (concentrated on
lines) with equi-bounded energy. This density result requires a rather
technical construction, a by-product of which is the theorem stated below and
proved at the end of this section.

######  Theorem 3.5.

Given a divergence free measure $\mu\in[\mathcal{M}(\Omega)]^{N\times 3}$,
there exists a sequence of polyhedral measures
$\mu_{k}\in\mathcal{M}^{1}_{df}(\Omega;\mathcal{Q}\times\mathbb{S}^{2})$ such
that $\mu_{k}\overset{\ast}{\rightharpoonup}\mu$.

A result of this type can actually be obtained as a consequence of the
celebrated result of Smirnov [21] which shows that every normal current
without boundary in $\mathbb{R}^{n}$ can be decomposed in elementary
solenoids. As Bourgain and Brezis pointed out in [3], this decomposition
implies an approximation for divergence free vector fields in terms of
measures supported on curves. The proof of such approximation was given in
[15], where the authors show the existence of the approximating sequence by
means of an argument that doesn’t allow to choose the curves in the
approximation. This feature clashes with our need to control the energy of the
approximating sequence. Our result is instead constructive (see Lemma 3.6) and
will imply the $\Gamma$-limsup inequality in our main result.

More precisely we approximate $\mu$ by piecewise constant fields using Theorem
3.2, then on each tetrahedron of the triangulation we construct measures in
$\mathcal{M}_{df}^{1}(\Omega,\mathcal{Q}\times\mathbb{S}^{2})$ and then we
glue these local approximations obtaining the result. This is the most
delicate passage of the construction: indeed gluing while preserving the
divergence free constraint presents some difficulities, to overcome which it
is important to choose the right boundary condition on each tetrahedron, see
(v) of Lemma 3.6 and Remark 3.9.

We first start with a single tetrahedron. To this aim we need to introduce
some notation.

Given a tetrahedron $T\subset\mathbb{R}^{3}$, we perform the following
subdivision of its boundary $\partial T$ in closed triangles: consider a face
of $T$ with edges of length $l_{1}\leq l_{2}\leq l_{3}$, we divide each of
these edges in $k$ segments of length ${l_{i}}/{k}$, $i\in\\{1,2,3\\}$, and
consequently we obtain a division of that face in $k^{2}$ (closed) triangles,
denoted by $\Delta(h,k)$, with $h=1,\ldots,4k^{2}$, see Figure 1. Hence

$\partial T=\bigcup_{h=1}^{4k^{2}}\Delta(k,h).$ (3.4)

Note that there exists a universal constant $C>0$ such that for all $(h,k)$ we
have

$\mathcal{H}^{2}(\Delta(h,k))\leq C\frac{({\rm diam}(T))^{2}}{k^{2}}.$ (3.5)

We denote with $d(T,k,h)$ the baricenter of each triangle $\Delta(k,h)$, and
we call $n_{h}$ the outer unit normal, with respect to $T$, in $\Delta(k,h)$.

Figure 1: An example of the subdivision of the tetrahedron $T$, with two faces
explicitly subdivided for $k=2$, where the blue dots are the baricenters of
the triangles obtained with the subdivision.

######  Lemma 3.6.

Let $T\subset\mathbb{R}^{3}$ be a $3$-simplex and $\mathcal{F}$ a finite union
of planes in $\mathbb{R}^{3}$. Let $A\in\mathbb{R}^{N\times 3}$ and assume
that $A=\sum_{j=1}^{M}b^{j}\otimes t^{j}$, with $b^{j}\in\mathcal{Q}$,
$t^{j}\in\mathbb{S}^{2}$. Then there exist sequences of polyhedral measures
$\mu_{k},\nu_{k},\omega_{k},\rho_{k}\in\mathcal{M}^{1}_{{\rm
loc}}(\mathbb{R}^{3};\mathcal{Q}\times\mathbb{S}^{2})$ such that
$\mu_{k}=\nu_{k}+\omega_{k}+\rho_{k}$ and satisfy the following:

  1. (i)

$\nu_{k}=\frac{1}{k^{4}}\sum_{j=1}^{M}b^{j}\otimes
t^{j}\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(\Gamma^{j}_{k}\cap T^{k}),$

where $\Gamma^{j}_{k}$ are straight lines parallel to $t^{j}$, satisfying

$\frac{1}{k^{4}}\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Gamma^{j}_{k}\overset{\ast}{\rightharpoonup}\mathcal{L}^{3},$
(3.6)

and $T_{k}\subset\subset T$ is a tetrahedron satisfying ${\rm
dist}(T_{k},\partial T)\leq{\rm diam}(T)/k^{2}$;

  2. (ii)

$\mu_{k}\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}T=\nu_{k}+\omega_{k}$,
$\mu_{k}\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}T^{c}=\rho_{k}$,
$\|\mu_{k}\|(\partial T)=0$,
$\mathcal{H}^{1}(\operatorname{supp}\rho_{k}\cap\mathcal{F})=0$;

  3. (iii)

for every compact set $K\subset\mathbb{R}^{3}$ it holds
$\lim_{k\to+\infty}\|\omega_{k}\|(K)+\|\rho_{k}\|(K)=0$;

  4. (iv)

for every $\varphi\in[C_{c}(\mathbb{R}^{3})]^{N\times 3}$ it holds

$\lim_{k\rightarrow\infty}\int_{T}\varphi
d\mu_{k}=\int_{T}\langle\varphi,A\rangle dx;$

  5. (v)

for every $\varphi\in[C_{c}^{\infty}(\mathbb{R}^{3})]^{N}$ it holds

$\langle\mu_{k},\nabla\varphi\rangle=\sum_{h=1}^{4k^{2}}\langle
An_{h},\varphi(d(T,k,h))\rangle\mathcal{H}^{2}(\Delta(k,h)).$

###### Proof.

Without loss of generality we may assume that the baricenter of $T$ coincides
with the origin and we define $T_{k}:=(1-\frac{1}{k^{2}})T$. Therefore $T_{k}$
is a tetrahedron similar to $T$ and ${\rm dist}(T_{k},\partial T)\leq{\rm
diam}(T)/k^{2}$.

The construction is quite natural but somewhat involved, therefore we shall
present it in several steps.

_Step 1. The segments inside $T_{k}$._

We define the approximating measures inside $T_{k}$. For every
$j\in\\{1,\ldots,M\\}$ we consider the $2$-dimensional vector space $\Pi^{j}$
whose normal is $t^{j}$. Let $v_{1}^{j},v_{2}^{j}$ be an orthonormal base of
this plane, and consider the square lattice on $\Pi^{j}$ defined by
$\mathcal{G}^{j}_{k}=\operatorname{span}_{\mathbb{Z}}(\frac{1}{k^{2}}v_{1}^{j},\frac{1}{k^{2}}v_{2}^{j})$.
We then set $\Gamma_{k}^{j}:=\mathcal{G}^{j}_{k}+\mathbb{R}t^{j}$ and define

$\nu_{k}^{j}:=\frac{b^{j}}{k^{4}}\otimes t^{j}\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Gamma_{k}^{j}.$ (3.7)

It is easy to check that $\nu_{k}^{j}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}T_{k}\in[\mathcal{M}(\mathbb{R}^{3})]^{N\times
3}$ and that for all $\varphi\in[C_{c}(\mathbb{R}^{3})]^{N\times 3}$ it holds

$\displaystyle\lim_{k\rightarrow+\infty}\langle\nu_{k}^{j}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}T_{k},\varphi\rangle=\lim_{k\rightarrow+\infty}\langle\nu_{k}^{j}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}T,\varphi\rangle$
$\displaystyle=\langle b^{j}\otimes t^{j}\mathcal{L}^{3}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}T,\varphi\rangle.$ (3.8)

The measure inside $T_{k}$ is then

$\nu_{k}:=\sum_{j=1}^{M}\nu_{k}^{j}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}T_{k},$ (3.9)

which then converges weakly$*$ to $A\,\mathcal{L}^{3}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}T$.

We now subdivide $\partial T$ in triangles that we call ${\Delta(h,k)}$, for
$h=1,\cdots,4k^{2}$ as specified in (3.4), i.e., $\partial
T=\bigcup_{h=1}^{4k^{2}}\Delta(k,h)$ and $d(T,k,h)$ is the baricenter of
$\Delta(k,h)$. This subdivision induces in turns a subdivision of $\partial
T_{k}$ in triangles $\delta(h,k)=(1-\frac{1}{k^{2}})\Delta(h,k)$ by projecting
from the origin the $\Delta(k,h)$ onto $\partial T_{k}$. Hence we can write
$\partial T_{k}=\bigcup_{h=1}^{4k^{2}}\delta(k,h)$ with $|x-y|\less
C\operatorname{diam}(T)k^{-1}$ for every $x\in\Delta(h,k),y\in\delta(k,h)$.

Up to removing a negligible number of lines (so that (3.8) still holds), we
can assume that $\Gamma^{j}_{k}$ intersects $\partial T_{k}$ only on isolated
points and that none of these points belong to more than one triangle
$\delta(k,h)$. Indeed for each one of the $4k^{2}$ triangles $\delta(k,h)$
there are at most $\mathcal{O}(k)$ lines that intersect its contour, hence
there are at most $\mathcal{O}(k^{3})$ lines that ought to be discarded but
each line has a mass of $\mathcal{O}(k^{-4})$, hence the total error is
negligible in the limit. We can also assume that
$\mathcal{H}^{1}(\Gamma_{k}^{j}\cap\Gamma_{k}^{i})=0$ for every $j\not=i$.

_Step 2. Definition in $T\smallsetminus T_{k}$_

For each $1\leq h\leq 4k^{2}$ we now want to connect the lines of $\nu_{k}$,
which end on $\delta(k,h)$, with the baricenter of $\Delta(k,h)$. We define

$R(k,j,h):=\Gamma_{k}^{j}\cap\delta(k,h),\qquad N(k,j,h):=\\#R(k,j,h).$ (3.10)

We observe that
$\mathcal{H}^{2}(\delta(k,h))=(1-\frac{1}{k^{2}})^{2}\mathcal{H}^{2}(\Delta(k,h))$
and hence

$\begin{split}&|\mathcal{H}^{2}(\Delta(k,h))-\mathcal{H}^{2}(\delta(k,h))|\leq
C\frac{(\operatorname{diam}(T))^{2}}{k^{4}},\\\ &N(k,j,h)\leq
Ck^{4}\mathcal{H}^{2}(\delta(k,h))\leq
C(\operatorname{diam}(T))^{2}k^{2}.\end{split}$ (3.11)

In order to concentrate the mass in the baricenter $d(T,k,h)$ of each
$\Delta(k,h)$ we connect each point $p\in R(k,j,h)$ to $d(T,k,h)$ using a
small straight segment $[p,d(T,k,h)]$. On each of these segments we then
define the measure

$\omega(k,j,h,p):=\frac{1}{k^{4}}b^{j}\otimes
t_{p}\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}[p,d(T,k,h)],$ (3.12)

where $t_{p}$ is the unit tangent vector in the direction
$(d(T,k,h)-p)\operatorname{sign}(\langle t_{j},n_{h}\rangle)$ and $n_{h}$ is
the outward normal vector of $\delta(k,h)$.

We then define

$\omega_{k}:=\sum_{j=1}^{M}\sum_{h=1}^{4k^{2}}\sum_{p\in
R(k,j,h)}\omega(k,j,h,p).$ (3.13)

Since $\|\omega(k,j,h,p)\|(T)\leq C\operatorname{diam}(T)|b^{j}|k^{-5}$ from
(3.11) we infer

$\|\omega_{k}\|(\mathbb{R}^{n})\leq
C\frac{1}{k}(\operatorname{diam}(T))^{3}\sum_{j=1}^{M}|b^{j}|,$ (3.14)

and then

$\lim_{k\rightarrow+\infty}\|\omega_{k}\|(\mathbb{R}^{n})=0.$ (3.15)

$\nu_{k}$
---
$\omega_{k}$
---
$\rho_{k}$
---
$T_{k}$
---
$T$
---
Figure 2: A portion of $\mu_{k}$, for a single $j$ and in a two dimensional
schematization. The blue dots are the baricenter of the
$\Delta(k,h)^{\prime}s$.

See Figure 2 for a representation of a portion of $\nu^{j}_{k}$ and
$\omega(k,j,h,p)$.

_Step 3. Definition outside $T$._
We define the mass at the baricenter $d(T,k,h)$ to be the following vector
valued quantity:

$B(T,k,h):=\sum_{j=1}^{M}N(k,j,h)\operatorname{sign}(\langle
t^{j},n_{h}\rangle)\frac{b^{j}}{k^{4}}.$ (3.16)

This definition makes sense: indeed we observe that
$\operatorname{supp}(\nu_{k}+\omega_{k})$ is given by a finite family of
piecewise straight lines connecting different baricenters, hence using Remark
2.1 one can show that for every $\varphi\in[C_{c}^{1}(\mathbb{R}^{3})]^{N}$ it
holds

$\langle\nu_{k}+\omega_{k},\nabla\varphi\rangle=\sum_{h=1}^{4k^{2}}\langle
B(T,k,h),\varphi(d(T,k,h))\rangle,$ (3.17)

which means precisely that the vector valued mass carried by
$\nu_{k}+\omega_{k}$ at each baricenter on $\partial T$ is given by
$B(T,k,h)$.

For every $\Delta(k,h)$ with $1\leq h\leq 4k^{2}$ we define the measures

$\rho_{k}(h):=(\mathcal{H}^{2}(\Delta(k,h))An_{h}-B(T,k,h))\otimes\tau(k,h)\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\gamma(k,h),$ (3.18)

where $\gamma(k,h)$ is an arbitrary half line with direction $\tau(k,h)$, not
intersecting $\mathcal{F}$ and having endpoint in $d(T,k,h)$ (see Remark 3.9
for the heuristics).
Again recalling Remark 2.1, for every
$\varphi\in[C_{c}^{\infty}(\mathbb{R}^{3})]^{N}$ it holds

$\langle\rho_{k}(h),\nabla\varphi\rangle=\langle\mathcal{H}^{2}(\Delta(k,h))An_{h}-B(T,k,h),\varphi(d(T,k,h))\rangle.$
(3.19)

We then define the total average error as follows

$\rho_{k}:=\sum_{h=1}^{4k^{2}}\rho_{k}(h).$ (3.20)

We now want to show that, for every compact set $K$, $\|\rho_{k}\|(K)$ tends
to zero. With that aim in mind we first prove the following claim.

_Claim: there exists a universal constant $C$ such that_

$|B(T,k,h)-\mathcal{H}^{2}(\delta(k,h))An_{h}|\leq
C\frac{1}{k^{3}}\operatorname{diam}(T)\sum_{j=1}^{M}|b_{j}|.$ (3.21)

From the definition of $B(T,k,h)$ and from the fact that
$An_{h}=\sum_{j=1}^{M}b_{j}\langle t_{j},n_{h}\rangle$ it is clear that we
only need to consider $1\leq h\leq 4k^{2}$ and $1\leq j\leq M$ such that
$\langle t^{j},n_{h}\rangle\not=0$. Let then $j,h$ be as above and consider
the elementary cell of the lattice $\mathcal{G}_{k}^{j}$, i.e.,
$\mathcal{C}_{k}^{j}:=\Pi_{j}\cap\\{\frac{s_{1}}{k^{2}}v_{1}^{j}+\frac{s_{2}}{k^{2}}v_{2}^{j}\,:\,0\leq
s_{1},s_{2}\leq 1\\}$. Let also $\Sigma_{h}$ be the plane that contains
$\delta(k,h)$, then if we denote by $P^{h,j}_{k}$ the elementary cell of the
(planar) lattice $\Sigma_{h}\cap\Gamma^{j}_{k}$ we have that

$\mathcal{H}^{2}(P^{h,j}_{k})=\mathcal{H}^{2}(\mathcal{C}_{k}^{j})\dfrac{1}{|\langle
t^{j},n_{h}\rangle|}=\frac{1}{k^{4}|\langle t^{j},n_{h}\rangle|},$ (3.22)

since $\mathcal{C}_{k}^{j}$ is obtained by orthogonal projection of a
translation of $P^{h,j}_{k}$ on $\Pi_{j}$.
From the previous equality and the fact that $A=\sum_{j=1}^{M}b_{j}\otimes
t_{j}$ we obtain that

$\mathcal{H}^{2}(\delta(k,h))An_{h}=\sum_{j=1}^{M}\dfrac{\mathcal{H}^{2}(\delta(k,h))}{\mathcal{H}^{2}(P^{h,j}_{k})}\operatorname{sign}(\langle
t^{j},n_{h}\rangle)\frac{b^{j}}{k^{4}}.$ (3.23)

Therefore in order to show (3.21), in view of the definition of $B(T,k,h)$, it
is enough to show that

$\left|\dfrac{\mathcal{H}^{2}(\delta(k,h))}{\mathcal{H}^{2}(P^{h,j}_{k})}-N(k,h,j)\right|\leq
C\operatorname{diam}(T)k.$ (3.24)

This inequality clearly holds true, since
$\mathcal{H}^{2}(\delta(k,h))/\mathcal{H}^{2}(P^{h,j}_{k})$ counts the number
of points in $R(k,h,j)$ up to an error due to those points of the lattice
$\Sigma_{h}\cap\Gamma^{j}_{k}$ that are close to the contour of $\delta(k,h)$.
The number of such points can in turn be estimated by
$C\operatorname{diam}(T)k$, which proves (3.24).

Finally from (3.21), (3.11) and (3.20) we obtain that for every compact set
$K\subset\mathbb{R}^{n}$ we have

$\|\rho_{k}\|(K)\leq\sum_{h=1}^{4k^{2}}\|\rho_{k}(h)\|(K)\leq\frac{C}{k}\operatorname{diam}(T){\rm
diam}(K)\sum_{j=1}^{M}|b_{j}|,$ (3.25)

which then tends to zero as $k\to+\infty$.

_Step 4. Conclusions._

We now combine all these constructions and define $\mu_{k}$ in the whole of
$\mathbb{R}^{n}$, namely

$\mu_{k}:=\nu_{k}+\omega_{k}+\rho_{k}.$ (3.26)

We claim that $\mu_{k}$ satisfies the thesis. Indeed (i) follows directly from
the definition of $\nu_{k}$ in Step 1, while (ii) from the definition of
$\omega_{k}$ and $\rho_{k}$ in Step 2 and Step 3 respectively. Property (iii)
follows from (3.14) and (3.25). As for (iv) it is a consequence of (3.8), (ii)
and (iii).

To see (v) we simply write
$\langle\mu_{k},\nabla\varphi\rangle=\langle\nu_{k}+\omega_{k},\nabla\varphi\rangle+\langle\rho_{k},\nabla\varphi\rangle$
and recall (3.17) and (3.19).

∎

######  Remark 3.7.

We observe that each of the approximating measures $\mu_{k}$ constructed in
the previous lemma are such that

$\|\mu_{k}\|(K)\leq
C\left[\mathcal{L}^{3}(T)+\frac{1}{k}\operatorname{diam}(T){\rm
diam}(K)+\frac{1}{k}(\operatorname{diam}(T))^{3}\right]\sum_{j=1}^{M}|b_{j}|,$
(3.27)

for every compact set $K\subset\mathbb{R}^{3}$.

######  Remark 3.8.

Let $A(x)=\sum_{i=1}^{M}A_{i}\chi_{T^{i}}(x)$ be a piecewise constant function
with null divergence, where $A_{i}\in\mathbb{R}^{N\times 3}$ and the $T^{i}$’s
are tetrahedra. If $T^{i}$ and $T^{j}$ share a face, then, by integrating over
a small cube across the common face, it is easy to see that it must hold

$A_{i}\nu=A_{j}\nu,$ (3.28)

where $\nu$ is the unit normal vector of the common face.

######  Remark 3.9.

A few comments on the measures constructed in Lemma 3.6 are in order. Recall
the definition of $\Gamma^{j}_{k}$ and $\delta(k,h)$ given in Lemma 3.6. With
a direct computation one can see that, on average, the number of lines in
$\Gamma^{j}_{k}$ intersecting $\delta(k,h)$ is
$\hat{N}(k,j,h):=\mathcal{H}^{2}(\delta(k,h))k^{4}|\langle
t^{j},n_{h}\rangle|$. Consequently, given that $A=\sum_{j}b^{j}\otimes t^{j}$,
the _averaged mass_ on each baricenter is

$\hat{B}(T,k,h)=\sum_{j=1}^{M}\hat{N}(k,j,h)\operatorname{sign}(\langle
t^{j},n_{h}\rangle)\frac{b^{j}}{k^{4}}=\mathcal{H}^{2}(\delta(k,h))An_{h}.$

Furthermore, since (3.28) holds, it is clear how the averaged mass is a more
convenient boundary datum than the exact mass $B(T,k,h)$, as defined in
(3.16). Indeed, in Lemma 3.10, the averaged mass will allow us to glue
together the local construction of Lemma 3.6 performed in different tetrahedra
preserving the divergence free constraint.

In this sense, the $\rho_{k}$’s in (3.26) are to be considered just a small
correction necessary to pass from $B$ to $\hat{B}$.

We now glue together the local construction of Lemma 3.6 to obtain the global
approximating sequence.

######  Lemma 3.10.

Let $\Omega\subset\mathbb{R}^{3}$ be an open set,
$\mathcal{T}=\\{T^{1},\cdots,T^{M}\\}$ be an admissible triangulaiton of
$\Omega$ and $A=\sum_{i=1}^{M}A_{i}\chi_{T^{i}}$ be a divergence free
piecewise constant function, with respect to $\mathcal{T}$, where
$A_{i}\in\mathbb{R}^{N\times 3}$. Assume that
$A_{i}=\sum_{j=1}^{M^{i}}b_{j}^{i}\otimes t_{j}^{i}$, for some
$b_{j}^{i}\in\mathcal{Q}$, $t_{j}^{i}\in\mathbb{S}^{2}$, then there exists a
sequence of polyhedral measures
$\mu_{k}\in\mathcal{M}_{df}^{1}(\Omega;\mathcal{Q}\times\mathbb{S}^{2})$ such
that

  1. (i)

$\mu_{k}\overset{\ast}{\rightharpoonup}\operatorname{A}\mathcal{L}^{3}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Omega$,

  2. (ii)

there exists a sequence of measures
$\eta_{k}\in\mathcal{M}^{1}(\Omega;\mathcal{Q}\times\mathbb{S}^{2})$ such that
$\|\eta_{k}\|(K)\to 0$, for every compact set $K$, and

$\mu_{k}=\frac{1}{k^{4}}\sum_{i=1}^{M}\sum_{j=1}^{M^{i}}b_{j}^{i}\otimes
t_{j}^{i}\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(\Gamma^{j,i}_{k}\cap
T^{i}_{k})+\eta_{k},$ (3.29)

where $T^{i}_{k}\subset\subset T^{i}$ with ${\rm dist}(T^{i}_{k},\partial
T^{i})\leq{\rm diam}(T^{i})/k^{2}$, and $\Gamma_{k}^{j,i}$ is a union of
straight lines parallel to $t^{i}_{j}$. Furthermore
$\mathcal{H}^{1}(\operatorname{supp}(\mu_{k})\cap\partial T^{i})=0$ for every
$1\leq i\leq M$.

###### Proof.

For every tetrahedron $T^{i}\in\mathcal{T}$ we apply Lemma 3.6 on $A_{i}$ to
find four sequences of measures
$\mu_{k}^{i},\nu_{k}^{i},\omega^{i}_{k},\rho^{i}_{k}\in\mathcal{M}^{1}_{{\rm
loc}}(\mathbb{R}^{3};\mathcal{Q}\times\mathbb{S}^{2})$. Define
$\mu_{k}:=\sum_{i=1}^{M}\mu_{k}^{i}$ and
$\eta_{k}:=\sum_{i=1}^{M}\omega_{k}^{i}+\rho^{i}_{k}$. Set
$\Omega^{\prime}:={\rm int}(\cup_{T^{i}\in\mathcal{T}}T^{i})$.

We observe that from (ii) of Lemma 3.6 , by taking an appropriate family of
plane $\mathcal{F}$ (containing all the boundaries $\partial T^{i}$), without
loss of generality we can assume that
$\mathcal{H}^{1}(\operatorname{supp}\mu_{k}^{i}\cap\operatorname{supp}\mu_{k}^{l})=0$
for $i\not=l$, and $\mathcal{H}^{1}(\operatorname{supp}(\mu_{k})\cap\partial
T^{i})=0$ for every $1\leq i\leq M$. In particular
$\mu_{k}\in\mathcal{M}^{1}(\Omega;\mathcal{Q}\times\mathbb{S}^{2})$ since each
$\mu_{k}^{i}$ does and they have disjoint support.

We first show that, with this definition, $\mu_{k}$ is divergence free in
$\Omega^{\prime}$. Indeed from (v) of Lemma 3.6 we get that for every
$\varphi\in[C^{\infty}_{c}(\Omega^{\prime})]^{N}$

$\langle\mu_{k},\nabla\varphi\rangle=\sum_{i=1}^{M}\langle\mu_{k}^{i},\nabla\varphi\rangle=\sum_{i=1}^{M}\sum_{h=1}^{4k^{2}}\langle\mathcal{H}^{2}(\Delta_{i}(k,h))A_{i}n^{i}_{h},\varphi(d(T_{i},k,h))\rangle,$

where $n_{h}^{i}$ is the outer unit normal with respect to $T_{i}$, and
$\Delta_{i}(k,h)$ is one of the $4k^{2}$ triangles that tile $\partial T_{i}$
(as defined in (3.4)). Then using that for any pair $i,j\in\\{1,\ldots,M\\}$
and $h,h^{\prime}\in\\{1,\ldots,4k^{2}\\}$ we have that either
$\mathcal{H}^{2}(\Delta_{i}(k,h)\cap\Delta_{j}(k,h^{\prime}))=0$ or
$\Delta_{i}(k,h)=\Delta_{j}(k,h^{\prime})$ we denote the set

$\mathcal{A}_{k}=\\{(i,j,h,h^{\prime}):\ i<j\,,\
\mathcal{H}^{2}(\Delta_{i}(k,h)\cap\Delta_{j}(k,h^{\prime}))>0\\},$

and we can rewrite

$\langle\mu_{k},\nabla\varphi\rangle=\sum_{\mathcal{A}_{k}}\langle\mathcal{H}^{2}(\Delta_{i}(k,h))(A_{i}n^{i}_{h}+A_{j}n_{h^{\prime}}^{j}),\varphi(d(T_{i},k,h))\rangle=0,$
(3.30)

since $n_{h}^{i}=-n_{h^{\prime}}^{j}$ and (3.28) holds.

Furthermore, from (iv) of Lemma 3.6 and the fact that
$\lim_{k}\|\mu_{k}^{i}\|(\operatorname{int}T^{i})=\lim_{k}\|\mu_{k}^{i}\|(\Omega)$,
we have that $\mu_{k}\overset{\ast}{\rightharpoonup}A\mathcal{L}^{n}$ in
$\Omega$.

Finally property (ii) is a direct consequence of choice of $\mathcal{F}$, the
definition of $\mu_{k}^{i}$ and $\rho_{k}^{i}$, and (i) of Lemma 3.6. ∎

###### Proof of Theorem 3.5.

Let $\mu\in[\mathcal{M}(\Omega)]^{N\times 3}$ be divergence free. From Theorem
3.2 we obtain a sequence $\mathcal{T}_{k}=\\{T_{k}^{i}\\}_{i=1,\cdots,M(k)}$
of admissible triangulations of $\Omega$ and a sequence
$A_{k}:\Omega\to\mathbb{R}^{N\times 3}$ of piecewise constant functions
relatively to $\mathcal{T}_{k}$, such that
$A_{k}\mathcal{L}^{3}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Omega$ approximates strictly
$\mu$. For each $k$, let $A_{k}(\Omega)=\\{A_{k}^{i}\\}_{i=1,\cdots,M(k)},$
and write $A_{k}^{i}=\sum(A_{k}^{i})_{lm}e_{l}\otimes e_{m}$. From Lemma 3.10
applied to each $A_{k}$ we then get a sequence of measures
$\mu_{k}^{h}\in\mathcal{M}_{df}^{1}(\Omega;\mathcal{Q}\times\mathbb{S}^{2})$
approximating $A_{k}\mathcal{L}^{3}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Omega$ and such that, recalling
(3.27), $\|\mu_{k}^{h}\|(\Omega)\leq C\|A_{k}\|_{L^{1}}\leq C\|\mu\|(\Omega)$,
hence we can find a diagonal sequence
$\mu_{k}^{h(k)}\overset{\ast}{\rightharpoonup}\mu$. ∎

##  4 The upper bound

The approximation results proved above provide a local construction which is
the crucial ingredient for the proof of the upper bound, which is stated and
proved below.

######  Proposition 4.1.

Let $\psi:\mathbb{Z}\times\mathbb{S}^{2}\longrightarrow[0,+\infty)$ be
$\mathcal{H}^{1}$-elliptic and obey $\frac{1}{c}|b|\leq\psi(b,t)$ for all
$b\in\mathbb{Z}$ and $t\in\mathbb{S}^{2}$. Let $\Omega\subset\mathbb{R}^{3}$
be a bounded open set, simply connected with Lipschitz boundary. Then for
every $\mu\in[\mathcal{M}(\Omega)]^{N\times 3}$ with null divergence there
exists a sequence
$\mu_{\sigma}\in\mathcal{M}_{df}^{1}(\Omega;\sigma\mathbb{Z}^{N}\times\mathbb{S}^{2})$
converging weakly$*$ to $\mu$ such that

$\liminf_{\sigma\to 0}E_{\sigma}(\mu_{\sigma})\leq E_{0}(\mu).$

###### Proof.

The strategy of the proof follows closely the one in [8]. It consists of a
first step in which, using the approximation results, one reduces to
divergence free measures concentrated on polyhedral curves whose limiting
energy resolves the convexification procedure in the definition of the
$\Gamma$-limit. With this we reduce the analysis to the construction of a
recovery sequence for the auxiliary functional $F_{\infty}$ defined as
follows:

$F_{\infty}(\mu,\omega):=\begin{cases}\displaystyle{\int_{\Gamma\cap\omega}\psi_{\infty}(\theta,t)d\mathcal{H}^{1}}\
\ &\hbox{if }\mu=\theta\otimes t\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Gamma\in\mathcal{M}_{df}^{1}(\omega;\mathcal{Q}\times\mathbb{S}^{2}),\\\
&\mu\hbox{ polyhedral};\\\ +\infty&\hbox{otherwise},\end{cases}$ (4.1)

where $\omega$ is an open set, and $F_{\infty}(\mu):=F_{\infty}(\mu,\Omega)$.

Step 1: Reduction from $E_{0}$ to $F_{\infty}$.

We now prove that $E_{0}$ is the relaxation of $F_{\infty}$ with respect to
the weak$*$ topology, i.e., $E_{0}=\bar{F}_{\infty}$. Note that from the
definition of $E_{0}$ we have that $E_{0}\leq F_{\infty}$, and hence
$E_{0}\leq\bar{F}_{\infty}$, therefore we just have to prove the upper bound.
From Theorem 3.2 and Reshetnyak Continuity Theorem we obtain that divergence
free piecewise constant measures are dense in energy for $E_{0}$, i.e., for
every divergence free measure $\mu\in[\mathcal{M}(\Omega)]^{N\times 3}$ there
exists a sequence of divergence free piecewise constant measures $\nu_{k}$
such that $\nu_{k}\overset{\ast}{\rightharpoonup}\mu$ and

$\lim_{k\rightarrow+\infty}E_{0}(\nu_{k})=E_{0}(\mu).$ (4.2)

Thus, since the weak$*$ convergence is metrizable on bounded set of
$\mathcal{M}(\Omega)$, without loss of generality we can now assume $\mu$ to
be a divergence free piecewise constant measure of the form

$\mu=\sum_{i=1}^{M}\chi_{T^{i}}A_{i}\mathcal{L}^{3},$ (4.3)

where $\mathcal{T}=\\{T^{1},\cdots,T^{M}\\}$ is an admissible triangulation of
$\Omega$. We thus construct the recovery sequence in the relaxation of
$F_{\infty}$ for a measure $\mu$ as in (4.3).

First recall that $g$ is the convex envelope of $g_{\infty}$. Moreover we know
(see Lemma 2.4) that $g$ is finite and that $\psi_{\infty}(b,t)=+\infty$ if
and only if $b\in\mathbb{R}^{N}\smallsetminus\mathcal{Q}$. Therefore for any
fixed $\varepsilon>0$ and for every matrix $A_{i}$, with
$i\in\\{1,\ldots,M\\}$, we find $M^{i}_{\varepsilon}\leq 3N+1$ rank one
matrices of the form $\tilde{b}^{j,i}_{\varepsilon}\otimes
t^{j,i}_{\varepsilon}$, with $\tilde{b}^{j,i}_{\varepsilon}\in\mathcal{Q}$,
$t^{j,i}_{\varepsilon}\in\mathbb{S}^{2}$, and coefficients
$\lambda^{j,i}_{\varepsilon}>0$ such that
$\sum_{j=1}^{M^{i}_{\varepsilon}}\lambda^{j,i}_{\varepsilon}=1$,
$\sum_{j=1}^{M^{i}_{\varepsilon}}\lambda^{j,i}_{\varepsilon}\tilde{b}^{j,i}_{\varepsilon}\otimes
t^{j}_{\varepsilon}=A_{i}$, and

$g(A_{i})+\varepsilon\geq\sum_{j=1}^{M^{i}_{\varepsilon}}\lambda^{j,i}_{\varepsilon}\psi_{\infty}(\tilde{b}^{j,i}_{\varepsilon},t^{j,i}_{\varepsilon})=\sum_{j=1}^{M^{i}_{\varepsilon}}\psi_{\infty}(\lambda^{j,i}_{\varepsilon}\tilde{b}^{j,i}_{\varepsilon},t^{j,i}_{\varepsilon}),$
(4.4)

Setting
$b^{j}_{\varepsilon}:=\lambda^{j}_{\varepsilon}\tilde{b_{\varepsilon}}^{j}\in\mathcal{Q}$
we then have

$(g(A_{i})+\varepsilon)\mathcal{L}^{3}(T^{i}\cap\Omega)\geq\sum_{j=1}^{M^{i}_{\varepsilon}}\psi_{\infty}(b^{j,i}_{\varepsilon},t^{j,i}_{\varepsilon})\mathcal{L}^{3}(T^{i}\cap\Omega).$
(4.5)

We now apply Lemma 3.10 to $\mu=\sum_{i=1}^{M}\chi_{T^{i}}A_{i}$, with
$A_{i}=\sum_{j=1}^{M_{\varepsilon}^{i}}b^{j,i}_{\varepsilon}\otimes
t^{j,i}_{\varepsilon}$ satisfying (4.4), to find a sequence of measures
$\mu_{k}^{\varepsilon}\in\mathcal{M}_{df}^{1}(\Omega;\mathcal{Q}\times\mathbb{S}^{2})$
converging to $\mu$, and $\eta_{k}^{\varepsilon}$ vanishing as $k\to+\infty$.

On the other hand, from (ii) of Lemma 3.6 and using that
$\psi_{\infty}(\theta,t)\leq C|\theta|$ for all $\theta\in\mathcal{Q}$ it is
easy to see that
$F_{\infty}(\mu_{k}^{\varepsilon},\operatorname{int}(T^{i})\cap\Omega)<+\infty$
and

$F_{\infty}(\mu_{k}^{\varepsilon},\operatorname{int}(T^{i})\cap\Omega)\leq\sum_{j=1}^{M^{i}_{\varepsilon}}\psi_{\infty}(b^{j,i}_{\varepsilon},t^{j,i}_{\varepsilon})\frac{1}{k^{4}}\mathcal{H}^{1}(T_{k}^{i}\cap\Omega\cap\Gamma^{j,i,\varepsilon}_{k})+C\|\eta_{k}^{\varepsilon}\|(T^{i}\cap\Omega).$
(4.6)

Hence from (3.6) of Lemma 3.6 we get

$\limsup_{k\rightarrow+\infty}F_{\infty}(\mu_{k}^{\varepsilon},\operatorname{int}(T^{i})\cap\Omega)\leq\sum_{j=1}^{M^{i}_{\varepsilon}}\psi_{\infty}(b^{j,i}_{\varepsilon},t^{j,i}_{\varepsilon})\mathcal{L}^{3}(T^{i}\cap\Omega).$
(4.7)

Recalling (4.5) we then get

$\displaystyle E_{0}(\mu)+\varepsilon\mathcal{L}^{3}(\Omega)$
$\displaystyle=\sum_{i=1}^{M}(g(A_{i})+\varepsilon)\mathcal{L}^{3}(T^{i}\cap\Omega)\geq\sum_{i=1}^{M}\limsup_{k\rightarrow\infty}F_{\infty}(\mu_{k}^{\varepsilon},\operatorname{int}(T^{i})\cap\Omega)$
$\displaystyle\geq\limsup_{k\rightarrow\infty}\sum_{i=1}^{M}F_{\infty}(\mu_{k}^{\varepsilon},\operatorname{int}(T^{i})\cap\Omega)=\limsup_{k\rightarrow\infty}F_{\infty}(\mu^{\varepsilon}_{k},\Omega),$

where in the last line we used the fact that
$\mathcal{H}^{1}(\operatorname{supp}(\mu_{k})\cap\partial T^{i})=0$ for all
$i$.

Now (iii) of Proposition 2.3 implies $\psi_{\infty}(b,t)\geq c|b|$ for
$b\in\mathcal{Q}$, hence, from estimate (4.4) we deduce the existence of a
universal constant $C>0$ such that

$\sum_{j=1}^{N_{\varepsilon}}|b_{\varepsilon}^{j,i}|\leq
C(|A_{i}|+\varepsilon).$ (4.8)

In particular, given (3.27), we have that the family of measures
$\mu^{\varepsilon}_{k}$ is uniformly bounded. Therefore, since the weak$*$
topology is metrizable on bounded set, via a diagonal argument we infer that
for every divergence free measure $\mu\in[\mathcal{M}(\Omega)]^{N\times 3}$
there exists a sequence $\tilde{\mu}_{h}$ weakly$*$ converging to $\mu$, such
that

$\limsup_{h\to+\infty}F_{\infty}(\tilde{\mu}_{h})\leq E_{0}(\mu).$ (4.9)

In particular this sequence satisfies $F_{\infty}(\tilde{\mu}_{h})<+\infty$.

Step 2: Recovery sequence for $F_{\infty}$.

We now prove that for a given $\tilde{\mu}=\theta\otimes
t\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Gamma\in\mathcal{M}_{df}^{1}(\Omega;\mathbb{R}^{N}\times\mathbb{S}^{2})$,
with $\Gamma$ polyhedral we can construct a sequence
$\hat{\mu}_{\sigma}\in\mathcal{M}_{df}^{1}(\Omega;\sigma\mathbb{Z}^{N}\times\mathbb{S}^{2})$
converging to $\tilde{\mu}$ and such that

$\limsup_{\sigma\rightarrow
0}E_{\sigma}(\hat{\mu}_{\sigma})=F_{\infty}(\tilde{\mu}).$ (4.10)

Without loss of generality we can assume $F_{\infty}(\tilde{\mu})<+\infty$.
First we observe that, since $\Gamma$ is composed by a finite number of
straight segments and $\tilde{\mu}$ is divergence free, $\theta$ must be
constant on each segment. In particular $\theta\in\mathcal{Q}$ attains a
finite number of values. Therefore we can apply Theorem $2.5$ of [6] to $\mu$,
deducing that there exists a finite number of polyhedral closed loops
$\Gamma_{i}$ with constant Burgers vector $\theta_{i}$ such that

$\tilde{\mu}=\sum_{i}\theta_{i}\otimes t_{i}\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Gamma_{i}.$

Here the $\theta_{i}$ do not necessary belong to $\sigma\mathbb{Z}^{N}$.
Therefore we define the following approximation of $\tilde{\mu}$ with measures

$\hat{\mu}_{\sigma}:=\sum_{i}\sigma\left\lfloor\dfrac{\theta_{i}}{\sigma}\right\rfloor\otimes
t_{i}\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Gamma_{i}\in\mathcal{M}_{df}^{1}(\Omega;\sigma\mathbb{Z}^{N}\times\mathbb{S}^{2}),$
(4.11)

where we denote
$\left\lfloor\frac{b}{\sigma}\right\rfloor=(\big{\lfloor}\frac{b_{1}}{\sigma}\big{\rfloor},\cdots,\big{\lfloor}\frac{b_{N}}{\sigma}\big{\rfloor})$.
These measures have the same support of $\mu$, and satisfy
$E_{\sigma}(\hat{\mu}_{\sigma})<+\infty$. We observe that $\hat{\mu}_{\sigma}$
is a finite sum of closed loops with constant multiplicity, therefore, again
by Theorem $2.5$ in [6], it is divergence free.

Since $\sigma\left\lfloor\frac{\theta_{i}}{\sigma}\right\rfloor$ converges to
$\theta_{i}$, we have that
$\hat{\mu}_{\sigma}\overset{\ast}{\rightharpoonup}\tilde{\mu}$. Let now
consider an arbitrary sequence $\sigma_{j}$ converging to $0$, then

$E_{\sigma_{j}}(\hat{\mu}_{\sigma_{j}})=\int_{\Gamma}\sigma_{j}\psi(z_{j},t)d\mathcal{H}^{1},$
(4.12)

where
$z_{j}(x):=\sum_{i}\left\lfloor\frac{\theta_{i}}{\sigma_{j}}\right\rfloor\chi_{\Gamma_{i}}(x)\in\mathbb{Z}^{N}$.
Since, clearly, for $\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Gamma$ a.e. $x$ it holds that
$\lim_{j\to+\infty}|z_{j}|=+\infty$ and
$\lim_{j\to+\infty}z_{j}/|z_{j}|=\theta/|\theta|$, thanks to (iii) of Lemma
2.3 we deduce

$\lim_{j\to+\infty}\frac{\psi\left(z_{j},t\right)}{|z_{j}|}=\frac{\psi_{\infty}\left(\theta,t\right)}{|\theta|}.$

On the other hand clearly it holds that
$\lim_{j\to+\infty}\sigma_{j}|z_{j}|=|\theta|$, hence by rewriting

$\sigma_{j}\psi\left(z_{j},t\right)=\sigma_{j}|z_{j}|\frac{\psi\left(z_{j},t\right)}{|z_{j}|},$
(4.13)

we deduce that

$\lim_{j\rightarrow+\infty}\sigma_{j}\psi(z_{j},t)=\psi_{\infty}\left(\theta,t\right),\
\ \ \ \mathcal{H}^{1}\textit{-a.e. }x\in\Gamma.$ (4.14)

Finally, since $|\sigma_{j}\psi(z_{j},t_{i})|\leq c|\sigma_{j}z_{j}|\leq
C(|\theta|+1)$, we conclude via dominated convergence theorem and get

$\lim_{j\to+\infty}E_{\sigma_{j}}(\hat{\mu}_{\sigma_{j}})=\int_{\Gamma}\psi_{\infty}(\theta,t)d\mathcal{H}^{1}=F_{\infty}(\tilde{\mu}).$
(4.15)

Step 3: Conclusion

We conclude the proof by combining Step 1 and Step 2. Without loss of
generality we can assume that the sequence $\tilde{\mu}_{h}$ constructed in
Step 1 satisfies

$\limsup_{h\to+\infty}F_{\infty}(\tilde{\mu}_{h})=\lim_{h\to+\infty}F_{\infty}(\tilde{\mu}_{h})\leq
E_{0}(\mu)<+\infty.$ (4.16)

Then for every $h$ we obtain via Step 2 a sequence $\hat{\mu}_{\sigma}^{h}$
weakly$*$ converging to $\tilde{\mu}_{h}$ such that

$\lim_{\sigma\to 0}E_{0}(\hat{\mu}_{\sigma}^{h})=F_{\infty}(\tilde{\mu}_{h}).$
(4.17)

A further diagonal argument provides the wanted recovery sequence and
concludes the proof. ∎

## Fundings

The present paper benefits from the support of the PRIN 2017, Variational
methods for stationary and evolution problems with singularities and
interfaces, 2017BTM7SN-004. Also MF acknowledges the hospitality given by the
HIM, Hausdorff Research Institute for Mathematics.

Declarations of interest: none.

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