$\mathcal{H}^{1}(S^{i}_{\varepsilon})\geq(1-2\alpha_{\varepsilon})\mathcal{H}^{1}(\gamma^{i}_{\varepsilon})$,
we conclude that for small ${\varepsilon}$

$\frac{1}{{\varepsilon}}|\mu_{\varepsilon}|(\Omega^{\prime}){=\sum_{i\in
I_{\varepsilon}}|b_{\varepsilon}^{i}|\mathcal{H}^{1}(\gamma_{\varepsilon}^{i}\cap\Omega^{\prime})}\leq
cE.$ (5.12)

Therefore there is a subsequence such that
${\varepsilon}^{-1}\mu_{\varepsilon}$ converges to a limiting measure
$\mu\in\mathcal{M}(\Omega^{\prime};\mathbb{R}^{3\times 3})$, with
$|\mu|(\Omega^{\prime})\leq cE$. Since $\Omega^{\prime}$ was arbitrary, we
have $\mu\in\mathcal{M}(\Omega;\mathbb{R}^{3\times 3})$, and there is a
diagonal subsequence such that
${\varepsilon}_{k}^{-1}\mu_{{\varepsilon}_{k}}{\displaystyle\mathop{\rightharpoonup}^{*}}\mu$
in all $\Omega^{\prime}\subset\subset\Omega$. By [CGM15] the limit also
belongs to $\mathcal{M}_{\mathcal{B}}^{1}(\Omega)$. This concludes the proof
of (5.5) and (5.6). ∎

######  Proposition 5.4.

Let $\Omega\subset\mathbb{R}^{3}$ be a bounded, connected Lipschitz set.
Assume $W$ obeys $H^{\mathrm{W}}_{\mathrm{Finite}}$ or
$H^{\mathrm{W}}_{\mathrm{Lin}}$ for some $p\in(1,2)$. Let
$(\mu_{\varepsilon},\beta_{\varepsilon})\in\mathcal{A}^{*}_{\varepsilon}$ be
such that $\mu_{\varepsilon}$ is
$(h_{\varepsilon},\alpha_{\varepsilon})$-dilute in the sense of Definition
3.3, where $(h_{\varepsilon},\alpha_{\varepsilon})$ obeys (3.38), and

$E^{\mathrm{elast}}[\beta_{\varepsilon},\Omega]\leq
E{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}$

for some $E>0$,

Then there are a subsequence ${\varepsilon}_{k}\to 0$, ${{\eta}}\in
L^{1}_{\mathrm{loc}}(\Omega;\mathbb{R}^{3\times 3})$ with
$\mathop{\mathrm{curl}}{{\eta}}=0$,
$\mu\in\mathcal{M}_{\mathcal{B}}^{1}(\Omega)$ and $Q_{*}\in{\mathrm{SO}}(3)$
such that $(\mu_{{\varepsilon}_{k}},\beta_{{\varepsilon}_{k}})$ converges to
$(\mu,\eta,Q_{*})$ in finite kinematics with $p$ growth if
$H^{\mathrm{W}}_{\mathrm{Finite}}$ holds, and to $(\mu,\eta)$ in infinitesimal
kinematics with $p$ growth if $H^{\mathrm{W}}_{\mathrm{Lin}}$ holds, in the
sense of Definition 3.5.

###### Proof.

We first remark that Proposition 5.2 holds for this sequence (with
$\rho_{\varepsilon}={\varepsilon}$). Fix a connected Lipschitz set
$\Omega^{\prime}\subset\subset\Omega$. Then by (5.5) for ${\varepsilon}$ small
enough we have

$|\mu_{\varepsilon}|(\Omega^{\prime})\leq{cE}{\varepsilon}.$ (5.13)

Let

${d_{\varepsilon}}:={\mathrm{dist}}(\beta_{\varepsilon},{\mathrm{SO}}(3))\in
L^{1}(\Omega).$ (5.14)

From $H^{\mathrm{W}}_{\mathrm{Finite}}$ we obtain the pointwise bound
$d_{\varepsilon}^{2}\wedge d_{\varepsilon}^{p}=\Phi_{p}(d_{\varepsilon})\leq
cW(\beta_{\varepsilon})$. Since $q:=\frac{3}{2}\wedge p<2$,

$\left(\int_{\Omega\cap\\{{d_{\varepsilon}}\leq
1\\}}{d_{\varepsilon}^{q}}dx\right)^{1/q}\leq{(\mathcal{L}^{3}(\Omega))^{\frac{1}{q}-\frac{1}{2}}}\left(\int_{\Omega\cap\\{{d_{\varepsilon}}\leq
1\\}}{d_{\varepsilon}^{2}}dx\right)^{1/2}\leq
c\left({\varepsilon}^{2}\log\frac{1}{{\varepsilon}}E\right)^{1/2}.$ (5.15)

Using first $q\leq p$ and then $q<2$, for ${\varepsilon}$ small

$\left(\int_{\Omega\cap\\{{d_{\varepsilon}}>1\\}}{d_{\varepsilon}^{q}}dx\right)^{1/q}\leq\left(\int_{\Omega\cap\\{{d_{\varepsilon}}>1\\}}{d_{\varepsilon}^{p}}dx\right)^{1/q}{\leq
c\left({\varepsilon}^{2}\log\frac{1}{{\varepsilon}}E\right)^{1/2}.}$ (5.16)

Combining these two estimates,

$\|{\mathrm{dist}}(\beta_{\varepsilon},{\mathrm{SO}}(3))\|_{L^{q}(\Omega)}\leq
c{\varepsilon}\log^{1/2}\frac{1}{{\varepsilon}}E^{1/2}.$ (5.17)

By Proposition 5.1 and (5.13), there is $Q_{\varepsilon}\in{\mathrm{SO}}(3)$
such that for small ${\varepsilon}$

${\|\beta_{\varepsilon}-Q_{\varepsilon}\|_{L^{q}(\Omega^{\prime})}}\leq
c|\mu_{\varepsilon}|(\Omega^{\prime})+c\|{\mathrm{dist}}(\beta_{\varepsilon},{\mathrm{SO}}(3))\|_{L^{q}(\Omega)}\leq
c{\varepsilon}\log^{1/2}\frac{1}{{\varepsilon}}E^{1/2}$ (5.18)

where $c$ may depend on $\Omega^{\prime}$. The rotation $Q_{\varepsilon}$,
however, may be chosen not to depend on $\Omega^{\prime}$. Passing to a
further subsequence we can assume $Q_{{\varepsilon}_{k}}\to Q_{*}$ for some
$Q_{*}\in{\mathrm{SO}}(3)$. Taking a diagonal subsequence, this holds for all
subsets $\Omega^{\prime}$.

From (5.18) we see that, after extracting a further subsequence, there is
$\eta\in L^{q}(\Omega^{\prime};\mathbb{R}^{3\times 3})$ such that

$\eta_{{\varepsilon}_{k}}:=\frac{Q_{{\varepsilon}_{k}}^{T}\beta_{{\varepsilon}_{k}}-{\mathrm{Id}}}{{{\varepsilon}_{k}}\log^{1/2}\frac{1}{{\varepsilon}_{k}}}\mathop{\rightharpoonup}\eta\text{
weakly in $L^{q}(\Omega^{\prime};{\mathbb{R}^{3\times 3}})$}.$ (5.19)

Again, after taking a diagonal subsequence this holds for all
$\Omega^{\prime}$. In order to prove $\mathop{\mathrm{curl}}\eta=0$, we write

$\mathop{\mathrm{curl}}\eta_{\varepsilon}=\frac{Q_{\varepsilon}^{T}\mathop{\mathrm{curl}}\beta_{\varepsilon}}{{\varepsilon}\log^{1/2}\frac{1}{{\varepsilon}}}=\frac{1}{{\varepsilon}\log^{1/2}\frac{1}{{\varepsilon}}}Q_{\varepsilon}^{T}\mu_{\varepsilon}.$
(5.20)

As $|\mu_{\varepsilon}|(\Omega^{\prime})\leq c{\varepsilon}$,
$\mathop{\mathrm{curl}}\eta_{\varepsilon}\to 0$ distributionally in
$\Omega^{\prime}$.

The proof in the geometrically linear case is analogous. ∎

###  5.3 Lower bound

We define, given a pair
$(\mu,\beta)\in\mathcal{M}_{{\varepsilon}\mathcal{B}}^{1}(\Omega)\times
L^{1}(\Omega;\mathbb{R}^{3\times 3})$ and $\rho,{\varepsilon}>0$,

$E^{\mathrm{lb}}_{\rho,{\varepsilon}}[\mu,\beta]:=\begin{cases}\displaystyle
E^{\mathrm{elast}}[\beta,\Omega\setminus(\mathop{\mathrm{supp}}\mu)_{\rho}],&\text{
if $(\mu,\beta)\in\mathcal{A}^{\mathrm{core}}_{\rho,{\varepsilon}}$},\\\
\infty,&\text{ otherwise}.\end{cases}$ (5.21)

We recall that the class $\mathcal{A}^{\mathrm{core}}_{\rho,{\varepsilon}}$ of
admissible pairs was defined in (3.50). As
$\mathcal{A}_{\varepsilon}^{*}\subseteq\mathcal{A}_{\rho,{\varepsilon}}^{\mathrm{core}}$
for every $\rho>0$ by Lemma 3.8(iii), for any
$(\mu,\beta)\in\mathcal{M}^{1}_{{\varepsilon}\mathcal{B}}(\Omega)\times
L^{1}(\Omega;\mathbb{R}^{3\times 3})$ and all $\rho>0$ we have

$\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}E^{\mathrm{lb}}_{\rho,{\varepsilon}}[\mu,\beta]\leq
F^{\mathrm{subcr}}_{\varepsilon}[\mu,\beta].$ (5.22)

This will permit (in Section 7) to prove the lower bound in Theorem 3.6 by
means of a lower bound on $E^{\mathrm{lb}}_{\rho_{\varepsilon},{\varepsilon}}$
for a good choice of $\rho_{\varepsilon}$. Precisely, we assume that
$\rho_{\varepsilon}\to 0$ is such that

$\lim_{{\varepsilon}\to 0}\frac{\log\rho_{\varepsilon}}{\log{\varepsilon}}=1.$
(5.23)

######  Theorem 5.5.

Let $\Omega\subset\mathbb{R}^{3}$ be open, bounded, Lipschitz. Let
$\rho_{\varepsilon}\to 0$ be such that (5.23) holds true. Let
$(\mu_{\varepsilon},\beta_{\varepsilon})\in\mathcal{A}^{\mathrm{core}}_{\rho_{\varepsilon},{\varepsilon}}$
be such that $\mu_{\varepsilon}$ is
$(h_{\varepsilon},\alpha_{\varepsilon})$-dilute, where
$(h_{\varepsilon},\alpha_{\varepsilon})$ obeys (3.38).

Assume that $W$ obeys $H^{\mathrm{W}}_{\mathrm{Finite}}$ for some $p\in(1,2]$.
Assume that there are a sequence ${\varepsilon}_{k}\to 0$ and
$(\mu,\eta,Q)\in\mathcal{M}^{1}_{\mathcal{B}}(\Omega)\times L^{1}_{\rm
loc}(\Omega;\mathbb{R}^{3\times 3})\times{\mathrm{SO}}(3)$, with $\mu=b\otimes
t\mathcal{H}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to5.0pt{\vfill\hrule
width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to3.35pt{\vfill\hrule width=3.35pt,height=0.2pt}}\,}}\gamma$ and
$\mathop{\mathrm{curl}}{{\eta}}=0$, such that
$(\mu_{{\varepsilon}_{k}},\beta_{{\varepsilon}_{k}})$ converges to
$(\mu,\eta,Q)$ in finite kinematics with $p$ growth and radius
$\rho_{{\varepsilon}_{k}}$ in the sense of Definition 3.9.

Then

$\int_{\gamma}\psi_{\mathbb{C}}^{\mathrm{rel}}(b,Qt)d\mathcal{H}^{1}+\int_{\Omega}\frac{1}{2}\mathbb{C}_{Q}{{\eta}}\cdot{{\eta}}dx\leq\liminf_{k\to\infty}\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}E^{\mathrm{lb}}_{\rho_{{\varepsilon}_{k}},{\varepsilon}_{k}}[\mu_{{\varepsilon}_{k}},{{\beta}}_{{\varepsilon}_{k}}],$
(5.24)

with $\mathbb{C}:=D^{2}W({\mathrm{Id}})$ and
$\psi_{\mathbb{C}}^{\mathrm{rel}}$ defined as in (3.42). Further, if the
right-hand side of (5.24) is finite then $\eta\in
L^{2}(\Omega;\mathbb{R}^{3\times 3})$.

Assume $W$ obeys $H^{\mathrm{W}}_{\mathrm{Lin}}$ for some $p\in(1,2]$. Then
the same holds with respect to the convergence in infinitesimal kinematics
with $p$ growth and radius $\rho_{{\varepsilon}_{k}}$ in the sense of
Definition 3.9 with $\psi_{\mathbb{C}}^{\mathrm{rel}}(b,Qt)$ replaced by
$\psi_{\mathbb{C}}^{\mathrm{rel}}(b,t)$ and $\mathbb{C}_{Q}$ by
$\mathbb{C}=D^{2}W(0)$.

###### Proof.

We discuss the proof in the finite case, the linear case is similar. For
notational simplicity we write ${\varepsilon}$ for ${\varepsilon}_{k}$.
Without loss of generality we can assume that $\Omega$ is connected and that
the $\liminf$ is finite and, possibly passing to a further subsequence, that
it is a limit; in particular

$E^{\mathrm{lb}}_{\rho_{\varepsilon},{\varepsilon}}[\mu_{\varepsilon},\beta_{\varepsilon}]\leq
E{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}$ (5.25)

for some $E>0$. During the proof we shall extract further subsequences, which
are not made explicit in the notation.
By hypothesis on $(\mu_{\varepsilon},\beta_{\varepsilon})$ we have that

$\frac{1}{{\varepsilon}}\mu_{\varepsilon}{\displaystyle\mathop{\rightharpoonup}^{*}}\mu\hskip
14.22636pt{\text{in $\Omega^{\prime}$ for all
$\Omega^{\prime}\subset\subset\Omega$,}}$ (5.26)

and there is $Q_{\varepsilon}\in{\mathrm{SO}}(3)$ such that
$Q_{\varepsilon}\to Q$ and

$\eta_{\varepsilon}:=\frac{Q_{\varepsilon}^{T}\beta_{\varepsilon}-{\mathrm{Id}}}{{\varepsilon}(\log\frac{1}{{\varepsilon}})^{1/2}}\chi_{\Omega\setminus(\mathop{\mathrm{supp}}\mu_{{\varepsilon}})_{\rho_{{\varepsilon}}}}\mathop{\rightharpoonup}{{\eta}}\text{
weakly in }L^{q}_{\mathrm{loc}}(\Omega;\mathbb{R}^{3\times 3})$ (5.27)

for $q:=\frac{3}{2}\wedge p$.

Fix a connected Lipschitz set $\Omega^{\prime}\subset\subset\Omega$. The key
idea in the proof is to use Proposition 4.3 and the double differentiability
of $W$ near the identity to show that, writing
$\mu_{\varepsilon}={\varepsilon}b_{\varepsilon}\otimes
t_{\varepsilon}\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma_{\varepsilon}$, there are
$\sigma_{\varepsilon}\to 0$ and two disjoint subsets
$\Omega^{\prime}_{\varepsilon},\omega^{\prime}_{\varepsilon}\subset\Omega\setminus(\gamma_{\varepsilon})_{\rho_{\varepsilon}}$
such that
$\mathcal{L}^{3}(\Omega^{\prime}\setminus\Omega^{\prime}_{\varepsilon})\to 0$,

$(1-\sigma_{\varepsilon})\int_{\Omega^{\prime}\cap\gamma_{\varepsilon}}\psi_{\mathbb{C}}({b_{\varepsilon}},Q_{\varepsilon}t_{\varepsilon})d\mathcal{H}^{1}\leq\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}\int_{\omega_{\varepsilon}^{\prime}}W(\beta_{\varepsilon})dx,$
(5.28)

and

$\int_{\Omega^{\prime}_{\varepsilon}}\frac{1}{2}\mathbb{C}_{Q_{\varepsilon}}\eta_{\varepsilon}\cdot\eta_{\varepsilon}dx-\sigma_{\varepsilon}\leq\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}\int_{\Omega^{\prime}_{\varepsilon}}W(\beta_{\varepsilon})dx.$
(5.29)

We start by showing (5.28), following the setting in the proof of Proposition
5.2. Since $\mu_{\varepsilon}$ is dilute we can write
$\mu_{\varepsilon}=\sum_{i}{\varepsilon}b^{i}_{\varepsilon}\otimes
t_{\varepsilon}^{i}{\cal H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma^{i}_{\varepsilon}$, with
$\gamma_{\varepsilon}=\cup_{i}\gamma_{\varepsilon}^{i}$
$(h_{\varepsilon},\alpha_{\varepsilon})$-dilute, and we can assume
$\mathcal{H}^{1}(\gamma_{\varepsilon}^{i})\in[h_{\varepsilon},2h_{\varepsilon}]$.
We define $R_{\varepsilon}$, $S_{\varepsilon}^{i}$, $T_{\varepsilon}^{i}$,
$\hat{T}^{i}_{\varepsilon}$, $I_{\varepsilon}$ as in the proof of Proposition
5.2 where now
$r_{\varepsilon}:=\max\\{\rho_{\varepsilon},{\varepsilon}h_{\varepsilon}^{-2}\\}$.
We set

$\gamma^{\prime}_{\varepsilon}:=\bigcup_{i\in
I_{\varepsilon}}\gamma^{i}_{\varepsilon}\hskip 14.22636pt\text{ and }\hskip
14.22636pt\omega^{\prime}_{\varepsilon}:=\bigcup_{i\in
I_{\varepsilon}}T^{i}_{\varepsilon}\setminus\hat{T}^{i}_{\varepsilon}\subseteq(\gamma^{\prime}_{\varepsilon})_{R_{\varepsilon}}\setminus(\gamma_{\varepsilon})_{\rho_{\varepsilon}}.$
(5.30)

Clearly
$\Omega^{\prime}\cap\gamma_{\varepsilon}\subseteq\gamma_{\varepsilon}^{\prime}$.
As in Proposition 5.2, for ${\varepsilon}$ sufficiently small the
$T^{i}_{\varepsilon}$ are disjoint and $T^{i}_{\varepsilon}\subseteq\Omega$
for all $i\in I_{\varepsilon}$. By Corollary 4.4 with
$h=\mathcal{H}^{1}(S_{\varepsilon}^{i})$, $R=R_{\varepsilon}$,
$r=r_{\varepsilon}$, for every index $i\in I_{\varepsilon}$ we get

$E^{\mathrm{elast}}[\beta_{\varepsilon},T_{{\varepsilon}}^{i}\setminus\hat{T}_{\varepsilon}^{i}]\geq{(1-\tilde{\sigma}^{i}_{\varepsilon})}{\varepsilon}^{2}\mathcal{H}^{1}(S_{\varepsilon}^{i})\log\frac{R_{\varepsilon}}{r_{\varepsilon}}\psi_{\mathbb{C}}(b_{\varepsilon}^{i},Q_{\varepsilon}t_{\varepsilon}^{i}),$
(5.31)

where

$\begin{split}{\tilde{\sigma}_{\varepsilon}^{i}:=\tilde{\sigma}}\left({\frac{r_{\varepsilon}}{R_{\varepsilon}}}+\frac{\|\beta_{\varepsilon}-Q_{\varepsilon}\|_{L^{1}(\Omega^{\prime}\setminus(\gamma_{\varepsilon})_{r_{\varepsilon}})}}{R_{\varepsilon}^{3}}+{\frac{{\varepsilon}|b_{\varepsilon}^{i}|}{r_{\varepsilon}}}\right),\end{split}$
(5.32)

and $\tilde{\sigma}\colon(0,\infty)\to(0,\infty)$ nondecreasing and such that
$\lim_{s\to 0}\tilde{\sigma}(s)=0$. We show that the argument of
$\tilde{\sigma}$ converges uniformly in $i$ to zero. From the definitions of
$R_{\varepsilon}$ and $r_{\varepsilon}$, the diluteness condition (3.38) and
(5.23) we obtain $r_{\varepsilon}/R_{\varepsilon}\to 0$. By (5.27) we have
that
$\sup_{\varepsilon}\|\eta_{\varepsilon}\|_{L^{1}(\Omega^{\prime})}<\infty$,
and recalling the definition of $R_{\varepsilon}$ and the diluteness
assumption,

$\begin{split}\lim_{{\varepsilon}\to
0}\frac{\|\beta_{\varepsilon}-Q_{\varepsilon}\|_{L^{1}(\Omega^{\prime}\setminus(\gamma_{\varepsilon})_{r_{\varepsilon}})}}{R_{\varepsilon}^{3}}\leq\lim_{{\varepsilon}\to
0}\frac{\|\eta_{\varepsilon}\|_{L^{1}(\Omega^{\prime})}{\varepsilon}\log^{1/2}\frac{1}{{\varepsilon}}}{\alpha_{\varepsilon}^{6}h_{\varepsilon}^{6}}=0.\end{split}$
(5.33)

We remark that (5.5) implies that for sufficiently small ${\varepsilon}$ one
has

${\sum_{i\in
I_{\varepsilon}}|b_{\varepsilon}^{i}|\mathcal{H}^{1}(\gamma_{i}^{\varepsilon})}\leq{cE},$
(5.34)

and using $\mathcal{H}^{1}(\gamma_{i}^{\varepsilon})\geq h_{\varepsilon}$ and
$r_{\varepsilon}\geq{\varepsilon}h_{\varepsilon}^{-2}$, we obtain

${\lim_{{\varepsilon}\to 0}\sup_{i\in
I_{\varepsilon}}\frac{{\varepsilon}|b_{\varepsilon}^{i}|}{r_{\varepsilon}}\leq\lim_{{\varepsilon}\to
0}h_{\varepsilon}\sum_{i\in
I_{\varepsilon}}|b_{\varepsilon}^{i}|\mathcal{H}^{1}(\gamma_{i}^{\varepsilon})\leq\lim_{{\varepsilon}\to
0}ch_{\varepsilon}=0.}$ (5.35)

Therefore for all ${i\in I_{\varepsilon}}$

$\begin{split}\tilde{\sigma}_{\varepsilon}^{i}\leq{\hat{\sigma}_{\varepsilon}:=\tilde{\sigma}}\left({\frac{r_{\varepsilon}}{R_{\varepsilon}}}+\frac{c{\varepsilon}\log^{1/2}\frac{1}{{\varepsilon}}}{\alpha_{\varepsilon}^{6}h_{\varepsilon}^{6}}+ch_{\varepsilon}\right)\end{split}$
(5.36)

with $\hat{\sigma}_{\varepsilon}\to 0$. In turn, summing (5.31) over $i\in
I_{\varepsilon}$ and using (5.7) we obtain

$\begin{split}\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}E^{\mathrm{elast}}[\beta_{\varepsilon},\bigcup_{i\in
I_{\varepsilon}}T_{{\varepsilon}}^{i}\setminus\hat{T}_{\varepsilon}^{i}]\geq&(1-\hat{\sigma}_{\varepsilon})\sum_{i\in
I_{\varepsilon}}\mathcal{H}^{1}(S_{\varepsilon}^{i})\frac{\log\frac{R_{\varepsilon}}{r_{\varepsilon}}}{\log\frac{1}{{\varepsilon}}}\psi_{\mathbb{C}}(b_{\varepsilon}^{i},Q_{\varepsilon}t_{\varepsilon}^{i})\\\
\geq&(1-\hat{\sigma}_{\varepsilon})\frac{\log\frac{R_{\varepsilon}}{r_{\varepsilon}}}{\log\frac{1}{{\varepsilon}}}(1-2\alpha_{\varepsilon})\int_{\gamma_{\varepsilon}^{\prime}}\psi_{\mathbb{C}}(b_{\varepsilon},Q_{\varepsilon}t_{\varepsilon})d\mathcal{H}^{1}.\end{split}$
(5.37)

By

$\lim_{{\varepsilon}\to
0}\frac{\log\frac{R_{\varepsilon}}{r_{\varepsilon}}}{\log\frac{1}{{\varepsilon}}}=\lim_{{\varepsilon}\to
0}\frac{\log\frac{\alpha_{\varepsilon}^{2}h_{\varepsilon}^{4}}{{\max\\{\rho_{\varepsilon},{\varepsilon}h_{\varepsilon}^{-2}\\}}}}{\log\frac{1}{{\varepsilon}}}=1,$
(5.38)

the factor in front of the integral converges to 1. Therefore (5.28) follows.
In turn, from (5.28) we obtain

$\liminf_{{\varepsilon}\to
0}\int_{\gamma_{\varepsilon}^{\prime}}\psi_{\mathbb{C}}(b_{\varepsilon},Q_{\varepsilon}t_{\varepsilon})d\mathcal{H}^{1}\leq\liminf_{{\varepsilon}\to
0}\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}\int_{\omega_{\varepsilon}^{\prime}}W(\beta_{\varepsilon})dx.$
(5.39)

By $Q_{\varepsilon}\to Q$, (2.36) and the relaxation result of [CGM15, Theorem
3.1], this implies

$\int_{\Omega^{\prime}\cap\gamma}\psi_{\mathbb{C}}^{\mathrm{rel}}(b,Qt)d\mathcal{H}^{1}\leq\liminf_{{\varepsilon}\to
0}\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}\int_{\omega_{\varepsilon}^{\prime}}W(\beta_{\varepsilon})dx.$
(5.40)

We next show (5.29). Since $W$ is twice differentiable at the identity, there
is $\omega:[0,\infty)\to[0,\infty)$, monotone and continuous in $0$, with
$\omega(0)=0$, such that for any $\delta>0$ sufficiently small we have

$W({{\mathrm{Id}}+F})\geq\frac{1}{2}\mathbb{C}F\cdot
F-\omega({\delta})|F|^{2}\text{ for all }F\in\mathbb{R}^{3\times 3}\text{ with
}|F|\leq\delta.$ (5.41)

We shall apply this with
$F=(Q_{\varepsilon}^{T}\beta_{\varepsilon}(x)-{\mathrm{Id}})\chi_{\Omega\setminus(\gamma_{\varepsilon})_{\rho_{\varepsilon}}}{(x)}={\varepsilon}\log^{1/2}\frac{1}{{\varepsilon}}\eta_{\varepsilon}(x)$,
away from the boundary, the singularity, and the parts where
$\eta_{\varepsilon}$ is exceptionally large. To make this precise, we first
define

$\hat{\Omega}^{\prime}_{\varepsilon}:=\Omega^{\prime}\setminus(\gamma^{\prime}_{\varepsilon})_{R_{\varepsilon}}\setminus(\partial\Omega^{\prime})_{R_{\varepsilon}}\subseteq\Omega^{\prime}\setminus(\gamma_{\varepsilon})_{R_{\varepsilon}}.$
(5.42)

As $\Omega^{\prime}$ is Lipschitz, $\gamma_{\varepsilon}$ is dilute and
$\mathcal{H}^{1}(\gamma^{i}_{\varepsilon})\in[h_{\varepsilon},2h_{\varepsilon}]$,
using (5.34) and $|b_{i}|\geq c>0$ we obtain

$\mathcal{L}^{3}(\Omega^{\prime}\setminus\hat{\Omega}^{\prime}_{\varepsilon})\leq
c(\Omega^{\prime})R_{\varepsilon}+c\sum_{i\in
I_{\varepsilon}}(R_{\varepsilon}^{3}+h_{\varepsilon}R_{\varepsilon}^{2})\leq
cR_{\varepsilon}+cR_{\varepsilon}^{2}\sum_{i\in
I_{\varepsilon}}\mathcal{H}^{1}(\gamma_{\varepsilon}^{i})\to 0.$ (5.43)

We then choose $L_{\varepsilon}\to\infty$ such that
$L_{\varepsilon}{\varepsilon}\log^{1/2}\frac{1}{{\varepsilon}}+L_{\varepsilon}^{2}\omega(L_{\varepsilon}{\varepsilon}\log^{1/2}{\frac{1}{{\varepsilon}}})\to
0$ and define

$\begin{split}\Omega^{\prime}_{{\varepsilon}}:=&\\{x\in\hat{\Omega}^{\prime}_{\varepsilon}:|\beta_{\varepsilon}(x)-Q_{\varepsilon}|\leq
L_{\varepsilon}{\varepsilon}\log^{1/2}\frac{1}{{\varepsilon}}\\}=\\{x\in\hat{\Omega}^{\prime}_{\varepsilon}:|\eta_{\varepsilon}(x)|\leq
L_{\varepsilon}\\}.\end{split}$ (5.44)

Since $\eta_{\varepsilon}$ is bounded in
$L^{q}(\Omega^{\prime};{\mathbb{R}^{3\times 3}})$,
$\mathcal{L}^{3}(\hat{\Omega}^{\prime}_{\varepsilon}\setminus\Omega^{\prime}_{\varepsilon})\to
0$, and recalling (5.43) we obtain
$\mathcal{L}^{3}(\Omega^{\prime}\setminus\Omega^{\prime}_{\varepsilon})\to 0$.
For any $x\in\Omega^{\prime}_{{\varepsilon}}$ we have

$\begin{split}W(\beta_{\varepsilon}(x))=&W(\beta_{\varepsilon}(x)Q_{\varepsilon}^{T})\\\
\geq&\frac{1}{2}\mathbb{C}(\beta_{\varepsilon}(x)Q_{\varepsilon}^{T}-{\mathrm{Id}})\cdot(\beta_{\varepsilon}(x)Q_{\varepsilon}^{T}-{\mathrm{Id}})\\\
&\hskip
28.45274pt-|\beta_{\varepsilon}(x)Q_{\varepsilon}^{T}-{\mathrm{Id}}|^{2}\omega({L_{\varepsilon}{\varepsilon}\log^{1/2}(1/{\varepsilon})})\\\
\geq&{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}\left(\frac{1}{2}\mathbb{C}_{Q_{\varepsilon}}\eta_{\varepsilon}(x)\cdot\eta_{\varepsilon}(x)-L_{\varepsilon}^{2}\omega({L_{\varepsilon}{\varepsilon}\log^{1/2}(1/{\varepsilon})})\right).\end{split}$
(5.45)

Therefore

$\begin{split}\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}\int_{\Omega^{\prime}_{{\varepsilon}}}W(\beta_{\varepsilon})dx\geq&\int_{\Omega^{\prime}_{{\varepsilon}}}\frac{1}{2}\mathbb{C}_{Q_{\varepsilon}}\eta_{\varepsilon}\cdot\eta_{\varepsilon}dx-
L_{\varepsilon}^{2}\omega({L_{\varepsilon}{\varepsilon}\log^{1/2}(1/{\varepsilon})})\mathcal{L}^{3}(\Omega)\end{split}$
(5.46)

which is (5.29). Moreover by (5.27) and
$\mathcal{L}^{3}(\Omega^{\prime}\setminus\Omega^{\prime}_{\varepsilon})\to 0$
we obtain that

$\eta_{\varepsilon}\chi_{\Omega^{\prime}_{\varepsilon}}\mathop{\rightharpoonup}\eta\text{
in }L^{1}(\Omega^{\prime};{\mathbb{R}^{3\times 3}})$ (5.47)

and, by lower semicontinuity and
$\mathbb{C}_{Q_{\varepsilon}}\to\mathbb{C}_{Q}$,

$\int_{\Omega^{\prime}}\frac{1}{2}\mathbb{C}_{Q}\eta\cdot\eta
dx\leq\liminf_{{\varepsilon}\to
0}\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}\int_{\Omega^{\prime}_{{\varepsilon}}}W(\beta_{\varepsilon})dx.$
(5.48)

Combining (5.40) and (5.48), recalling that
$\omega^{\prime}_{\varepsilon}\cap\Omega^{\prime}_{\varepsilon}=\emptyset$, we
obtain

$\
\int_{\Omega^{\prime}\cap\gamma}\psi_{\mathbb{C}}^{\mathrm{rel}}(b,Qt)d\mathcal{H}^{1}+\int_{\Omega^{\prime}}\frac{1}{2}\mathbb{C}_{Q}{{\eta}}\cdot{{\eta}}dx\leq\liminf_{k\to\infty}\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}E^{\mathrm{lb}}_{\rho_{{\varepsilon}_{k}},{\varepsilon}_{k}}[\mu_{{\varepsilon}_{k}},{{\beta}}_{{\varepsilon}_{k}}]$
(5.49)

for all Lipschitz sets $\Omega^{\prime}\subset\subset\Omega$; as the right-
hand side does not depend on $\Omega^{\prime}$ the same holds on $\Omega$.
This proves (5.24).

If the right-hand side is finite, then $\eta+\eta^{T}\in
L^{2}(\Omega;\mathbb{R}^{3\times 3})$. By Korn’s inequality, recalling that
$\mathop{\mathrm{curl}}\eta=0$, this implies $\eta\in
L^{2}(\Omega;\mathbb{R}^{3\times 3})$. ∎

##  6 Upper bound

A key technical result of this section, which permits a refinement of the
argument used for the upper bound in [CGO15, GMS21] is the following
statement. It shows that the upper bound for the unrelaxed problem can be
taken at constant $\beta$.

######  Proposition 6.1.

Let $\mathbb{C}$ obey (2.1). Let $\mu=b\otimes
t\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma\in\mathcal{M}_{\mathcal{B}}^{1}(\mathbb{R}^{3})$
be polygonal, and let ${{\beta}}\in L^{3/2}(\mathbb{R}^{3};\mathbb{R}^{3\times
3})$ be the solution to

$\begin{cases}\mathop{\mathrm{curl}}{{\beta}}=\mu,\\\
\mathop{\mathrm{div}}\mathbb{C}{{\beta}}=0.\end{cases}$ (6.1)

Let $\nu_{\varepsilon}$ be the positive measure defined by

$\nu_{\varepsilon}:=\frac{1}{\log\frac{1}{{\varepsilon}}}\frac{1}{2}\mathbb{C}{{\beta}}\cdot{{\beta}}\mathcal{L}^{3}{\mathchoice{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to5.0pt{\vfill\hrule
width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}(\mathbb{R}^{3}\setminus(\gamma)_{\varepsilon}).$

Then $\nu_{\varepsilon}$ converges weakly-$*$ to

$\nu_{0}:=\psi_{\mathbb{C}}(b,t)\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to5.0pt{\vfill\hrule
width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to3.35pt{\vfill\hrule width=3.35pt,height=0.2pt}}\,}}\gamma.$ (6.2)

In particular, if $\Omega\subset\mathbb{R}^{3}$ is an open set with
$|\mu|(\partial\Omega)=0$, then

$\lim_{{\varepsilon}\to
0}\frac{1}{\log\frac{1}{{\varepsilon}}}\int_{\Omega\setminus(\gamma)_{\varepsilon}}\frac{1}{2}\mathbb{C}{{\beta}}\cdot{{\beta}}dx=\int_{\Omega\cap\gamma}\psi_{\mathbb{C}}(b,t)d\mathcal{H}^{1}.$
(6.3)

Further, the positive measures

$\hat{\nu}_{\varepsilon}:=\frac{1}{\log\frac{1}{{\varepsilon}}}|{{\beta}}|^{2}\mathcal{L}^{3}{\mathchoice{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to5.0pt{\vfill\hrule
width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}(\mathbb{R}^{3}\setminus(\gamma)_{\varepsilon})$
(6.4)

are uniformly locally bounded, in the sense that for all $R>0$ one has

$\limsup_{{\varepsilon}\to 0}\hat{\nu}_{\varepsilon}(B_{R})<\infty,$ (6.5)

and for any $s\in(0,1)$ obey

$\limsup_{{\varepsilon}\to
0}\hat{\nu}_{\varepsilon}((\gamma)_{{\varepsilon}^{s}})\leq c(1-s).$ (6.6)

###### Proof.

Since $\mu$ is polygonal, we have that $\gamma=\cup_{i}\gamma_{i}$, with the
$\gamma_{i}$ being finitely many pairwise disjoint segments in
$\mathbb{R}^{3}$. For the same reason, $b\in
L^{\infty}(\gamma,\mathcal{H}^{1};\mathcal{B})$. By Theorem 2.2(iii), for any
$x\not\in\overline{\gamma}$ we have

$|{{\beta}}|(x)\leq\frac{c}{{\mathrm{dist}}(x,\gamma)}=c\max_{i}f_{i}(x)$

where

$f_{i}(x):=\frac{1}{{\mathrm{dist}}(x,\gamma_{i})}\,.$ (6.7)

In particular, this shows that

$\nu_{\varepsilon}\leq
c\hat{\nu}_{\varepsilon}\leq\frac{c}{\log\frac{1}{{\varepsilon}}}\sum_{i}f_{i}^{2}\mathcal{L}^{3}{\mathchoice{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to5.0pt{\vfill\hrule
width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}(\mathbb{R}^{3}\setminus(\gamma)_{\varepsilon})\leq
c\sum_{i}\nu_{\varepsilon}^{i},$ (6.8)

where

$\nu_{\varepsilon}^{i}:=\frac{1}{\log\frac{1}{{\varepsilon}}}f_{i}^{2}\mathcal{L}^{3}{\mathchoice{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to5.0pt{\vfill\hrule
width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}(\mathbb{R}^{3}\setminus(\gamma_{i})_{\varepsilon}).$

We fix an index $i$ and estimate $\nu_{\varepsilon}^{i}$ separately near
$\gamma_{i}$ and far away. Let $\delta,R>0$. Since $f_{i}\leq 1/\delta$
outside $(\gamma_{i})_{\delta}$,

$\nu_{\varepsilon}^{i}(B_{R}\setminus(\gamma_{i})_{\delta})\leq\frac{\delta^{-2}}{\log\frac{1}{{\varepsilon}}}\mathcal{L}^{3}(B_{R}).$
(6.9)

We recall that we denote by $B_{r}(x)$ and $B^{\prime}_{r}(x)$ the balls of
radius $r>0$ centered at $x$ in $\mathbb{R}^{3}$ and $\mathbb{R}^{2}$
respectively; for $x=0$ we simply write $B_{r}$ and $B^{\prime}_{r}$. Let now
$x\in\overline{\gamma_{i}}$, $r>0$. Then

$\begin{split}\nu_{\varepsilon}^{i}(B_{r}(x))&\leq\frac{1}{\log\frac{1}{{\varepsilon}}}\left(\int_{B_{r}\setminus
B_{\varepsilon}}\frac{1}{|y|^{2}}dy+2r\int_{B_{r}^{\prime}\setminus
B_{\varepsilon}^{\prime}}\frac{1}{|y^{\prime}|^{2}}dy^{\prime}\right)\\\
&=\frac{1}{\log\frac{1}{{\varepsilon}}}(4\pi(r-{\varepsilon})+2r\cdot
2\pi\cdot\log\frac{r}{{\varepsilon}})\\\ &\leq\frac{4\pi r+4\pi
r\log{(r/{\varepsilon})}}{\log(1/{\varepsilon})}.\end{split}$ (6.10)

Note that the first integral in the first inequality of (6.10) is needed to
estimate $\nu_{\varepsilon}^{i}(B_{r}(x))$ in the case that $x$ is near the
endpoints of $\gamma_{i}$. For any $\delta\in(0,\mathcal{H}^{1}(\gamma_{i}))$
we can cover $(\gamma_{i})_{\delta}$ with at most
$3\mathcal{H}^{1}(\gamma_{i})/\delta$ balls $B_{2\delta}(x^{k})$, with
$x^{k}\in\overline{\gamma}_{i}$. Therefore, using (6.10) with $r=2\delta$ and
(6.9),

$\nu_{\varepsilon}^{i}(B_{R})\leq\frac{3\mathcal{H}^{1}(\gamma_{i})}{\delta}\frac{8\pi\delta+{8\pi\delta\log(2\delta/{\varepsilon})}}{\log(1/{\varepsilon})}+\frac{\delta^{-2}}{\log(1/{\varepsilon})}\mathcal{L}^{3}(B_{R}),$

which implies

$\limsup_{{\varepsilon}\to 0}\nu_{\varepsilon}^{i}(B_{R})\leq
24\pi\mathcal{H}^{1}(\gamma_{i})<\infty$ (6.11)

for all $R>0$. Recalling (6.8), this proves (6.5).

For any $s\in(0,1)$ the same computation, using $\delta={\varepsilon}^{s}$ in
(6.10), shows that

$\limsup_{{\varepsilon}\to
0}\nu_{\varepsilon}^{i}((\gamma_{i})_{{\varepsilon}^{s}})\leq
3\mathcal{H}^{1}(\gamma_{i})\limsup_{{\varepsilon}\to
0}\frac{8\pi+8\pi\log(2{\varepsilon}^{s-1})}{\log\frac{1}{{\varepsilon}}}=24\pi\mathcal{H}^{1}(\gamma_{i})(1-s),$
(6.12)

and similarly $\limsup_{{\varepsilon}\to
0}\nu_{\varepsilon}^{i}((\gamma_{j})_{{\varepsilon}^{s}})=0$ for $j\neq i$.
This proves (6.6).

By (6.11), after taking a subsequence we can assume that for each $i$
$\nu_{\varepsilon}^{i}{\displaystyle\mathop{\rightharpoonup}^{*}}\nu_{0}^{i}$
for some Radon measure $\nu_{0}^{i}$. By (6.9) we obtain
$\nu_{0}^{i}(B_{R}\setminus(\gamma_{i})_{\delta})=0$ for any $\delta$ and $R$;
taking $\delta\to 0$ and $R\to\infty$ this gives
$\nu^{i}_{0}(\mathbb{R}^{3}\setminus\overline{\gamma_{i}})=0$. By (6.10) we
obtain

$\nu_{0}^{i}(B_{r}(x))\leq 4\pi r\leq
4\pi\mathcal{H}^{1}(B_{r}(x)\cap\gamma_{i})\text{ for any
}x\in\overline{\gamma_{i}},r\in(0,\mathcal{H}^{1}(\gamma_{i})).$

Therefore $\nu_{0}^{i}\ll\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to5.0pt{\vfill\hrule
width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to3.35pt{\vfill\hrule width=3.35pt,height=0.2pt}}\,}}\gamma_{i}$. Summing over
$i$ (and possibly taking a further subsequence),
$\nu_{\varepsilon}\mathop{\rightharpoonup}\nu_{0}$ with
$\nu_{0}\ll\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule width=3.35pt,height=0.2pt}}\,}}\gamma$.

In order to conclude the proof it suffices to show that for any $i$ we have

$\frac{d\nu_{0}}{d\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma}(x)=\psi_{\mathbb{C}}(b_{i},t_{i})\hskip
14.22636pt\text{ for $\mathcal{H}^{1}$-almost every }x\in\gamma_{i},$ (6.13)

where we recall that $b_{i}\in\mathcal{B}$ and $t_{i}\in S^{2}$ are defined by
$\mu=\sum_{i}b_{i}\otimes t_{i}\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to5.0pt{\vfill\hrule
width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to3.35pt{\vfill\hrule width=3.35pt,height=0.2pt}}\,}}\gamma_{i}$.

For $x\in\gamma_{i}$ we denote by $T_{r}^{x}$ a cylinder with axis parallel to
$t_{i}$, radius $r>0$ and height $2r$ centered at $x$,

$T_{r}^{x}:=x+Q_{t_{i}}(B_{r}^{\prime}\times(-r,r)),$

where as usual $Q_{t_{i}}$ is an element of ${\mathrm{SO}}(3)$ with
$Q_{t_{i}}e_{3}=t_{i}$. By the Radon-Nikodým theorem, for
$\mathcal{H}^{1}$-almost every $x\in\gamma_{i}$ there is a sequence $r_{k}\to
0$ such that $\nu_{0}(\partial T^{x}_{r_{k}})=0$ and

$\frac{d\nu_{0}}{d\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma_{i}}(x)=\lim_{k\to\infty}\frac{\nu_{0}(T_{r_{k}}^{x})}{\mathcal{H}^{1}(\gamma_{i}\cap
T_{r_{k}}^{x})}.$

We can assume that $x$ is not an endpoint of $\gamma_{i}$. For sufficiently
large $k$, $\mathcal{H}^{1}(\gamma_{i}\cap T_{r_{k}}^{x})=2r_{k}$. Recalling
that $\nu_{\varepsilon}(T_{r_{k}}^{x})\to\nu_{0}(T_{r_{k}}^{x})$, the
definition of $\nu_{\varepsilon}$, and ${\mathrm{dist}}(x,\gamma_{j})>0$ for
$j\neq i$,

$\frac{d\nu_{0}}{d\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma_{i}}(x)=\lim_{k\to\infty}\frac{1}{2r_{k}}\lim_{{\varepsilon}\to
0}\frac{1}{\log\frac{1}{{\varepsilon}}}\int_{T_{r_{k}}^{x}\setminus(\gamma_{i})_{\varepsilon}}\frac{1}{2}\mathbb{C}{{\beta}}\cdot{{\beta}}dy.$
(6.14)

By Theorem 2.2(ii)

${{\beta}}(y)=\int_{\mathbb{R}^{3}}N(y-z)d\mu(z)=\sum_{j}\int_{\gamma_{j}}N(y-z)b_{j}\otimes
t_{j}d\mathcal{H}^{1}(z)=:\sum_{j}{{\beta}}_{j}(y).$

Assume first $j\neq i$, so that ${\mathrm{dist}}(x,\gamma_{j})>0$. Since
$|N(z)|\leq c/|z|^{2}$, we have $|{{\beta}}_{j}|\leq c$ in $T_{r_{k}}^{x}$ for
sufficiently large $k$. The term $\beta_{i}$ instead is compared with
${{\beta}}_{b_{i},t_{i}}$. By Lemma 2.4 we have that

${{\beta}}_{b_{i},t_{i}}(y-x)-{{\beta}}_{i}(y)=\int_{(x+\mathbb{R}t_{i})\setminus\gamma_{i}}N(y-z)(b_{i}\otimes
t_{i})d\mathcal{H}^{1}(z)$

which implies, using again $|N(z)|\leq c/|z|^{2}$,

$|{{\beta}}_{b_{i},t_{i}}(y-x)-{{\beta}}_{i}(y)|\leq c\hskip 14.22636pt\text{
for }y\in T_{r_{k}}^{x}$

for sufficiently large $k$ (with $c$ depending on $x$ and $i$).

As for any $\delta{\in(0,1]}$

$\mathbb{C}{{\beta}}(y)\cdot{{\beta}}(y)\leq(1+\delta)\mathbb{C}{{\beta}}_{b_{i},t_{i}}(y-x)\cdot{{\beta}}_{b_{i},t_{i}}(y-x)+\frac{c}{\delta}|{{\beta}}(y)-{{\beta}}_{b_{i},t_{i}}(y-x)|^{2}$

we obtain, recalling (2.30) in order to evaluate the integral,

$\begin{split}\int_{T_{r_{k}}^{x}\setminus(\gamma_{i})_{\varepsilon}}\frac{1}{2}\mathbb{C}{{\beta}}\cdot{{\beta}}dy&\leq(1+\delta)\int_{T_{r_{k}}^{0}\setminus(\mathbb{R}t_{i})_{\varepsilon}}\frac{1}{2}\mathbb{C}{{\beta}}_{b_{i},t_{i}}\cdot{{\beta}}_{b_{i},t_{i}}dy+\frac{c}{\delta}r_{k}^{3}\\\
&=(1+\delta)2r_{k}\log\frac{r_{k}}{{\varepsilon}}\psi_{\mathbb{C}}(b_{i},t_{i})+\frac{c}{\delta}r_{k}^{3}.\end{split}$

Therefore (6.14) gives

$\frac{d\nu_{0}}{d\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma_{i}}(x)\leq(1+\delta)\lim_{k\to\infty}\lim_{{\varepsilon}\to
0}\frac{\log\frac{r_{k}}{{\varepsilon}}}{\log\frac{1}{{\varepsilon}}}\psi_{\mathbb{C}}(b_{i},t_{i}){=(1+\delta)\psi_{\mathbb{C}}(b_{i},t_{i})}$

for any $\delta>0$, so that

$\frac{d\nu_{0}}{d\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma_{i}}(x)\leq\psi_{\mathbb{C}}(b_{i},t_{i}).$

In order to prove the converse bound we start from (6.14) and notice that
$\beta$ is an admissible test function in (2.37) for
$\Theta(\mathbb{C},b_{i},t_{i},{2r_{k}},r_{k},{\varepsilon})$. Therefore

$\frac{d\nu_{0}}{d\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma_{i}}(x)\geq\liminf_{k\to\infty}\liminf_{{\varepsilon}\to
0}\frac{\log\frac{r_{k}}{{\varepsilon}}}{\log\frac{1}{{\varepsilon}}}\Theta(\mathbb{C},b_{i},t_{i},{2r_{k}},r_{k},{\varepsilon})=\psi_{\mathbb{C}}(b_{i},t_{i}),$
(6.15)

where in the last step we used Lemma 2.5(i). This concludes the proof. ∎

######  Remark 6.2.

The same argument also yields

$\limsup_{{\varepsilon}\to
0}\int_{(\gamma)_{2{\varepsilon}}}|\beta\ast\varphi_{\varepsilon}|^{2}dx<\infty.$
(6.16)

To see this, we observe that $|\beta|\leq c\sum_{i}f_{i}$ implies
$|\beta\ast\varphi_{\varepsilon}|\leq
c\sum_{i}(f_{i}\ast\varphi_{\varepsilon})$. With (6.7) we obtain
$|f_{i}\ast\varphi_{\varepsilon}|\leq c/{\varepsilon}$ everywhere, and
therefore $\|\beta\ast\varphi_{\varepsilon}\|_{L^{\infty}}\leq
c/{\varepsilon}$. For each segment $\gamma_{i}$ one computes
$\mathcal{L}^{3}((\gamma_{i})_{\varepsilon})=\pi{\varepsilon}^{2}\mathcal{H}^{1}(\gamma_{i})+\frac{4\pi}{3}{\varepsilon}^{3}$,
and since there are finitely many of them we have
$\mathcal{L}^{3}((\gamma)_{2{\varepsilon}})\leq
c{\varepsilon}^{2}+c{\varepsilon}^{3}$. Condition (6.16) follows.

We next deal with a technical issue related to the intersection of $\gamma$
and the boundary of $\Omega$. Indeed, in the above proposition one has
integrated over the complement of an ${\varepsilon}$-neighbourhood of the
entire curve $\gamma\subset\mathbb{R}^{3}$, whereas in
$E^{\mathrm{ub}}_{\varepsilon}$ one integrates over the complement of an
${\varepsilon}$-neighbourhood of the part of $\gamma$ which is contained
within $\Omega$. It is apparent that this may generate difficulties if
$\gamma$ is tangential to $\partial\Omega$, including the extreme case in
which one of the segments composing $\gamma$ is contained in $\partial\Omega$.
This situation is obviously not generic, and indeed we show in the next result
that in the generic situation this boundary effect vanishes in the limit.

Let $\Omega\subseteq\mathbb{R}^{3}$ be open and Lipschitz,
$\gamma\subset\mathbb{R}^{3}$ a finite union of closed segments. We say that
$\gamma$ is transversal to $\partial\Omega$ if $\gamma\cap\partial\Omega$
consists of finitely many points, each of them belongs to only one of the
segments, is a point where $\partial\Omega$ has a normal, and the normal is
not orthogonal to $\gamma$.

######  Lemma 6.3.

Let $\Omega\subseteq\mathbb{R}^{3}$ be an open Lipschitz set,
$\gamma\subset\mathbb{R}^{3}$ a finite union of closed segments.

  1. (i)

For ($\mathcal{L}^{3}$-)almost all $a\in\mathbb{R}^{3}$ the curve $a+\gamma$
is transversal to $\partial\Omega$.

  2. (ii)

If $\gamma$ is transversal to $\partial\Omega$, then in the setting of
Proposition 6.1

$\limsup_{{\varepsilon}\to
0}\frac{1}{\log\frac{1}{{\varepsilon}}}\int_{\Omega\cap(\gamma)_{{\varepsilon}}\setminus(\Omega\cap\gamma)_{\varepsilon}}|\beta|^{2}dx=0.$
(6.17)

The proof of the second part uses the following geometric fact.

######  Lemma 6.4.

Let $\Omega\subseteq\mathbb{R}^{n}$ be an open Lipschitz set,
$x\in\partial\Omega$ with normal $\nu\in S^{n-1}$, $v\in S^{n-1}$ with
$v\cdot\nu>0$. Then there are $c,c^{\prime},\rho>0$ such that

$t\leq c\,{\mathrm{dist}}(x+tv,\Omega)\hskip 14.22636pt\text{for all
$t\in[0,\rho]$}$ (6.18)

and

$|x-y|\leq c^{\prime}|x+tv-y|\hskip 14.22636pt\text{for all $t\in[0,\rho]$,
$y\in\Omega$}.$ (6.19)

###### Proof.

After a change of variables we can assume that $x=0$, $\nu=e_{n}$, $v_{n}>0$,
and there is a Lipschitz function $f:\mathbb{R}^{n-1}\to\mathbb{R}$ with
$f(0)=0$, $Df^{\prime}(0)=0$, and

$B_{r}\cap\Omega=B_{r}\cap\\{y_{n}<f(y^{\prime})\\}.$ (6.20)

for some $r>0$. We write $y=(y^{\prime},y_{n})$, with
$y^{\prime}\in\mathbb{R}^{n-1}$ and $y_{n}\in\mathbb{R}$, and the same for the
other vectors. As $Df^{\prime}(0)=0$ and $v_{n}>0$, we can choose
$\rho\in(0,\frac{1}{2}r]$ such that
$|f(y^{\prime})|\leq\frac{1}{2}v_{n}|y^{\prime}|$ for all
$|y^{\prime}|\leq\rho$. For $t\in[0,\rho]$ we have $tv\in
B_{r}\setminus\Omega$ and

${\mathrm{dist}}(tv,\Omega)={\mathrm{dist}}(tv,\Omega\cap
B_{r})={\mathrm{dist}}(tv,\partial\Omega\cap B_{r}),$ (6.21)

which implies

${\mathrm{dist}}(tv,\Omega)\geq\min_{y^{\prime}\in\mathbb{R}^{n-1}}|tv-(y^{\prime},f(y^{\prime}))|.$
(6.22)

We then estimate $|y^{\prime}|\leq t+|y^{\prime}-tv^{\prime}|$ and, with
$|v^{\prime}|\leq 1$,

$|tv_{n}-f(y^{\prime})|\geq
tv_{n}-\frac{1}{2}v_{n}\,|y^{\prime}|\geq\frac{1}{2}v_{n}t-\frac{1}{2}v_{n}\,|tv^{\prime}-y^{\prime}|$
(6.23)

for all $t\in[0,\rho]$, which together with (6.22) implies

${\mathrm{dist}}(tv,\Omega)\geq\min_{y^{\prime}\in\mathbb{R}^{n-1}}\frac{1}{2}|tv_{n}-f(y^{\prime})|+\frac{1}{2}|tv^{\prime}-y^{\prime}|\geq\frac{1}{4}v_{n}t,$
(6.24)

which is (6.18) with $c=4/v_{n}$.

To prove (6.19), let $y\in\Omega$, $t\in[0,\rho]$. Then $|v|=1$, (6.18) and
${\mathrm{dist}}(x+tv,\Omega)\leq|x+tv-y|$ give

$|x-y|\leq t+|x+tv-y|\leq(c+1)|x+tv-y|$ (6.25)

which proves (6.19). ∎

###### Proof of Lemma 6.3.

(i): There are finitely many points in $\gamma$ which belong to more than one
segment, and for almost all choices of $a$ none of them belongs to the
2-dimensional set $\partial\Omega$. For the rest of the argument we can
consider each of the segments composing $\gamma$ individually. Analogously,
since $\partial\Omega$ is covered by finitely many Lipschitz graphs, it
suffices to prove the assertion for one of them. Therefore it suffices to show
the following: if $v\in S^{2}$, $F:\mathbb{R}^{2}\to\mathbb{R}^{3}$ is
Lipschitz, and $r>0$, then for $\mathcal{L}^{3}$-almost every
$a\in\mathbb{R}^{3}$ the following holds: the equation

$F(y^{\prime})\in a+v\mathbb{R}:y^{\prime}\in B_{r}^{\prime}$ (6.26)

has finitely many solutions $y^{\prime}_{1},\dots,y^{\prime}_{K}$, at each
$y_{k}^{\prime}$ the function $F$ is differentiable, with $v$ not contained in
the space spanned by $\partial_{1}F(y^{\prime}_{k})$ and
$\partial_{2}F(y^{\prime}_{k})$. We remark that one can replace
$\mathcal{L}^{3}$-almost every $a\in\mathbb{R}^{3}$ with
$\mathcal{H}^{2}$-almost every $a\in v^{\perp}$.

To see this, one considers the Lipschitz function $G:B_{r}^{\prime}\to
v^{\perp}\subseteq\mathbb{R}^{3}$ defined by
$G(y^{\prime}):=P_{v}F(y^{\prime})$, where $P_{v}:={\mathrm{Id}}-v\otimes v$
is the projection onto the space $v^{\perp}$. For $a\in v^{\perp}$, equation
(6.26) can be rewritten as $G(y^{\prime})=a$. As $F$ is Lipschitz, it is
almost everywhere differentiable, and $G(\\{y^{\prime}\in
B_{r}^{\prime}:DF(y^{\prime})\text{ does not exist}\\})$ is an
$\mathcal{H}^{2}$-null set. By Sard’s Lemma, $G(\\{y^{\prime}\in
B_{r}^{\prime}:{\rm rank\,}DG(y^{\prime})\leq 1\\})$ is also an
$\mathcal{H}^{2}$-null set. We observe that, since $DG=({\mathrm{Id}}-v\otimes
v)DF$, $DG$ has full rank exactly when $v$ is not contained in the space
spanned by $\partial_{1}F$ and $\partial_{2}F$. By the area formula [AFP00,
Th. 2.71] applied to the function $G$ we obtain

$\int_{B_{r}^{\prime}}\sqrt{\det
DG^{T}DG}d\mathcal{L}^{2}=\int_{v^{\perp}}\mathcal{H}^{0}(B_{r}^{\prime}\cap
G^{-1}(a))d\mathcal{H}^{2}(a),$ (6.27)

so that $B_{r}^{\prime}\cap G^{-1}(a)$ is a finite set for
$\mathcal{H}^{2}$-almost all $a$. This proves the claim.

(ii): Let $\\{x_{1},\dots,x_{M}\\}=\gamma\cap\partial\Omega$. By Lemma 6.4
there are $c^{\prime}$, $\rho>0$ such that (6.19) holds for all of them.
Possibly reducing $\rho$, $\gamma\cap B_{\rho}(x_{i})$ is a diameter of
$B_{\rho}(x_{i})$ for all $i$. Pick $c_{*}\in(0,1)$ with $c_{*}c^{\prime}<1$.
We claim that for ${\varepsilon}$ sufficiently small one has

$\Omega\cap(\gamma)_{c_{*}{\varepsilon}}\setminus(\Omega\cap\gamma)_{\varepsilon}=\emptyset.$
(6.28)

We prove (6.28) by contradiction. If not, there are ${\varepsilon}_{k}\to 0$,
$y_{k}\in\Omega$, $z_{k}\in\gamma$ with $|y_{k}-z_{k}|<c_{*}{\varepsilon}_{k}$
and ${\mathrm{dist}}(y_{k},\Omega\cap\gamma)\geq{\varepsilon}_{k}$. As
$c_{*}<1$ the last two conditions imply $z_{k}\in\gamma\setminus\Omega$. By
compactness of $\bar{\Omega}$, passing to a subsequence we can assume that
$y_{k}$ and $z_{k}$ converge to the same limit, which belongs to
$\overline{\Omega}\cap(\gamma\setminus\Omega)=\partial\Omega\cap\gamma$.
Therefore there is $i\in\\{1,\dots,M\\}$ with $z_{k}\to x_{i}$, which means
that for sufficiently large $k$ we have $z_{k}\in\gamma\cap B_{\rho}(x_{i})$
and therefore $z_{k}=x_{i}+t_{k}v_{i}$ for some $t_{k}\to 0$. By (6.19) we
have

$|y_{k}-x_{i}|\leq c^{\prime}|y_{k}-z_{k}|\leq
c^{\prime}c_{*}{\varepsilon}_{k}<{\varepsilon}_{k}.$ (6.29)

This contradicts
${\varepsilon}_{k}\leq{\mathrm{dist}}(y_{k},\Omega\cap\gamma)\leq|y_{k}-x_{i}|$
and concludes the proof of (6.28).

We now observe that (6.28) implies
$\Omega\setminus(\Omega\cap\gamma)_{\varepsilon}\subseteq\Omega\setminus(\gamma)_{c_{*}{\varepsilon}}$
and therefore

$\Omega\cap(\gamma)_{{\varepsilon}}\setminus(\Omega\cap\gamma)_{\varepsilon}\subseteq\Omega\cap(\gamma)_{{\varepsilon}}\setminus(\gamma)_{c_{*}{\varepsilon}}\subseteq(\gamma)_{{\varepsilon}}\setminus(\gamma)_{c_{*}{\varepsilon}}.$
(6.30)

For any $s\in(0,1)$, if ${\varepsilon}$ is sufficiently small then
$(c_{*}{\varepsilon})^{s}\geq{\varepsilon}$. Using first (6.4) and then (6.6),

$\begin{split}\limsup_{{\varepsilon}\to
0}\frac{1}{\log\frac{1}{{\varepsilon}}}\int_{(\gamma)_{{\varepsilon}}\setminus(\gamma)_{c_{*}{\varepsilon}}}|\beta|^{2}dx\leq&\limsup_{{\varepsilon}\to
0}\frac{\log\frac{1}{c_{*}{\varepsilon}}}{\log\frac{1}{{\varepsilon}}}\hat{\nu}_{c_{*}{\varepsilon}}((\gamma)_{(c_{*}{\varepsilon})^{s}})\\\
=&\limsup_{{\varepsilon}\to
0}\hat{\nu}_{{\varepsilon}}((\gamma)_{{\varepsilon}^{s}})\leq
c(1-s).\end{split}$ (6.31)

As $s\in(0,1)$ was arbitrary, this concludes the proof of (6.17). ∎

We next come to the key construction for the proof of the upper bound. We
define, given a pair
$(\mu,\beta)\in\mathcal{M}^{1}_{{\varepsilon}\mathcal{B}}(\Omega)\times
L^{1}(\Omega;\mathbb{R}^{3\times 3})$,

$E^{\mathrm{ub}}_{\varepsilon}[\mu,\beta]:=\begin{cases}\displaystyle
E^{\mathrm{elast}}[\beta,\Omega\setminus(\mathop{\mathrm{supp}}\mu)_{{\varepsilon}}],&\text{
if $(\mu,\beta)\in\mathcal{A}^{*}_{\varepsilon}$, }\\\ \infty,&\text{
otherwise,}\end{cases}$ (6.32)

where $E^{\mathrm{elast}}$ and $\mathcal{A}^{*}_{\varepsilon}$ are defined in
(3.37) and (3.36) respectively.

######  Theorem 6.5.

Let $\Omega\subseteq\mathbb{R}^{3}$ be a bounded Lipschitz set. Assume that
$W$ obeys $H^{\mathrm{W}}_{\mathrm{Finite}}$ for some $p\in(1,2]$. Let
$(\mu,\eta,Q)\in\mathcal{M}_{\mathcal{B}}^{1}(\Omega)\times
L^{2}(\Omega;\mathbb{R}^{3\times 3})\times{\mathrm{SO}}(3)$ with $\mu=b\otimes
t\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule width=3.35pt,height=0.2pt}}\,}}\gamma$
and $\mathop{\mathrm{curl}}\eta=0$. Then for every sequence
${\varepsilon}_{k}\to 0$ and any
$(h_{{\varepsilon}_{k}},\alpha_{{\varepsilon}_{k}})$ satisfying (3.38) there
is $(\mu_{k},\beta_{k})\in\mathcal{A}_{{\varepsilon}_{k}}^{*}$, with $\mu_{k}$
$(h_{{\varepsilon}_{k}},\alpha_{{\varepsilon}_{k}})$-dilute, such that

$\frac{1}{{\varepsilon}_{k}}\mu_{k}{\displaystyle\mathop{\rightharpoonup}^{*}}\mu\hskip
14.22636pt{\text{ in $\Omega^{\prime}$, for all
$\Omega^{\prime}\subset\subset\Omega$}}$ (6.33)

and

$\frac{Q^{T}\beta_{k}-{\mathrm{Id}}}{{\varepsilon}_{k}(\log\frac{1}{{\varepsilon}_{k}})^{1/2}}\mathop{\rightharpoonup}{{\eta}}\text{
weakly in }L^{q}_{\mathrm{loc}}(\Omega;\mathbb{R}^{3\times 3})$ (6.34)

for all $q\in[1,2)$. Moreover

$\limsup_{k\to\infty}\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}E^{\mathrm{ub}}_{{\varepsilon}_{k}}[\mu_{k},\beta_{k}]\leq\int_{\gamma}{\psi_{\mathbb{C}}^{\mathrm{rel}}(b,Qt)}d\mathcal{H}^{1}+\int_{\Omega}\frac{1}{2}{\mathbb{C}_{Q}}{{\eta}}\cdot{{\eta}}dx$
(6.35)

where $\mathbb{C}:=D^{2}W({\mathrm{Id}})$, $\psi_{\mathbb{C}}^{\mathrm{rel}}$
is defined as in (3.42), $\mathbb{C}_{Q}$ as in (3.28) and

$\limsup_{k\to\infty}\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}\int_{\Omega\cap(\mathop{\mathrm{supp}}\mu_{k})_{{\varepsilon}_{k}}}{\mathrm{dist}}^{q}(\beta_{k},{\mathrm{SO}}(3))dx=0$
(6.36)

for all $q\in[1,2)$.

Assume $W$ obeys $H^{\mathrm{W}}_{\mathrm{Lin}}$ for some $p\in(1,2]$. Let
$(\mu,\eta)\in\mathcal{M}_{\mathcal{B}}^{1}(\Omega)\times
L^{2}(\Omega;\mathbb{R}^{3\times 3})$ with $\mu=b\otimes
t\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule width=3.35pt,height=0.2pt}}\,}}\gamma$
and $\mathop{\mathrm{curl}}\eta=0$. Then the same holds with

$\frac{\beta_{k}}{{\varepsilon}_{k}(\log\frac{1}{{\varepsilon}_{k}})^{1/2}}\mathop{\rightharpoonup}{{\eta}}\text{
weakly in }L^{q}_{\mathrm{loc}}(\Omega;\mathbb{R}^{3\times 3})$ (6.37)

in place of (6.34), $\psi_{\mathbb{C}}^{\mathrm{rel}}(b,t)$ in place of
$\psi_{\mathbb{C}}^{\mathrm{rel}}{(b,Qt)}$ and $\mathbb{C}$ in place of
$\mathbb{C}_{Q}$ in (6.35) and $|\beta_{k}+\beta_{k}^{T}|^{q}$ in place of
${\mathrm{dist}}^{q}(\beta_{k},{\mathrm{SO}}(3))$ in (6.36).

###### Proof.

We prove the thesis in the finite case as the linear case can be treated
similarly. As in [CGO15, Proposition 6.8], the proof is divided in two steps.
In the first one we provide an explicit construction attaining the unrelaxed
limiting energy, and under the additional assumption that $\eta$ is
essentially bounded. In the second step we conclude by using a diagonal
argument.

Step 1: We assume additionally that $\mu=b\otimes
t\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma\in\mathcal{M}^{1}_{\mathcal{B}}(\mathbb{R}^{3})$
is polyhedral, transversal to $\partial\Omega$ in the sense of Lemma 6.3,
which implies $|\mu|(\partial\Omega)=0$, and that $\eta\in
L^{\infty}(\mathbb{R}^{3};\mathbb{R}^{3\times 3})$ with
$\mathop{\mathrm{curl}}\eta=0$. We first sketch the treatment of the rotation
$Q$, ignoring $\eta$ at first. Our aim is to define $\beta_{k}$ as
$Q({\mathrm{Id}}+{\varepsilon}_{k}\xi)$ for some matrix-valued field $\xi$,
and then to expand the energy, using the invariance (3.23) and then a Taylor
series, as

$\begin{split}W(Q+{\varepsilon}_{k}Q\xi)&=W((Q+{\varepsilon}_{k}Q\xi)Q^{T})=W({\mathrm{Id}}+{\varepsilon}_{k}Q\xi
Q^{T})\\\ &\simeq\frac{1}{2}{\varepsilon}_{k}^{2}\mathbb{C}(Q\xi
Q^{T})\cdot(Q\xi
Q^{T})=\frac{1}{2}{\varepsilon}_{k}^{2}\mathbb{C}_{Q}\xi\cdot\xi,\end{split}$
(6.38)

with $\mathbb{C}_{Q}$ defined as in (3.28). In particular, $\xi$ should be
chosen to solve the linear problem with coefficients $\mathbb{C}_{Q}$, and to
obey $\mathop{\mathrm{curl}}(Q\xi)=\mu$.

We now start the construction. Let $\xi\in
L^{3/2}(\mathbb{R}^{3};\mathbb{R}^{3\times 3})$ be the solution to

$\begin{cases}\mathop{\mathrm{curl}}{\xi}={Q^{T}\mu},\\\
{\mathop{\mathrm{div}}\mathbb{C}_{Q}\xi}=0,\end{cases}$ (6.39)

as in Theorem 2.2. Clearly $Q^{T}\mu=Q^{T}b\otimes
t\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma\in\mathcal{M}^{1}_{\mathcal{B}}(\mathbb{R}^{3})$
is also transversal to $\partial\Omega$. We define

$\mu_{k}:={\varepsilon}_{k}\mu{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\Omega\quad\text{ and
}\quad{\beta_{k}:=Q+{\varepsilon}_{k}\log^{1/2}\frac{1}{{\varepsilon}_{k}}Q\eta+{\varepsilon}_{k}{Q\xi.}}$
(6.40)

Then clearly (6.33) holds and
$(\mu_{k},\beta_{k})\in\mathcal{A}^{*}_{{\varepsilon}_{k}}$. Since $\mu$ is
polyhedral, by Theorem 2.2(iii) for all $q\in[1,2)$ we have $\xi\in
L^{q}(\Omega;\mathbb{R}^{3\times 3})$ which implies

$\frac{Q^{T}\beta_{k}-{\mathrm{Id}}}{{\varepsilon}_{k}\log^{1/2}\frac{1}{{\varepsilon}_{k}}}={{\eta}}+\frac{{\xi}}{\log^{1/2}\frac{1}{{\varepsilon}_{k}}}\to{{\eta}}\
\text{ strongly in }L^{q}(\Omega;\mathbb{R}^{3\times 3}),$ (6.41)

and in particular (6.34) holds. Similarly,

$\begin{split}\limsup_{k\to\infty}\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}\int_{\Omega\cap(\gamma)_{{\varepsilon}_{k}}}&{\mathrm{dist}}^{q}(\beta_{k},{\mathrm{SO}}(3)){\,dx}\\\
&\leq\limsup_{k\to\infty}\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}\int_{\Omega\cap(\gamma)_{{\varepsilon}_{k}}}|\beta_{k}-Q|^{q}{\,dx}\\\
&=\limsup_{k\to\infty}\frac{1}{\log\frac{1}{{\varepsilon}_{k}}}\int_{\Omega\cap(\gamma)_{{\varepsilon}_{k}}}|\beta|^{q}{\,dx}=0.\end{split}$

Let $s\in(0,1)$ be fixed. By the growth condition on $W$ it holds

$\begin{split}\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}E^{\mathrm{ub}}_{{\varepsilon}_{k}}[\mu_{k},\beta_{k}]\leq\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}&E^{\mathrm{elast}}[\beta_{k},\Omega\setminus(\gamma)_{{\varepsilon}_{k}^{s}}]\\\
&+\frac{c}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}\int_{\Omega\cap(\gamma)_{{\varepsilon}_{k}^{s}}\setminus{(\gamma\cap\Omega)_{{\varepsilon}_{k}}}}|\beta_{k}-Q|^{2}{\,dx}.\end{split}$
(6.42)

By Theorem 2.2(iii) and $\eta\in L^{\infty}(\Omega;\mathbb{R}^{3\times 3})$,
for any $x\in\Omega\setminus(\gamma)_{{\varepsilon}_{k}^{s}}$ we have

${|Q^{T}\beta_{k}(x)-{\mathrm{Id}}|=|{\varepsilon}_{k}\log^{1/2}\frac{1}{{\varepsilon}_{k}}\eta(x)+{\varepsilon}_{k}\xi(x)|\leq
c({\varepsilon}_{k}\log^{1/2}\frac{1}{{\varepsilon}_{k}}+{\varepsilon}_{k}^{1-s})=:\delta_{k}.}$

By frame indifference and Taylor expansion near the identity there is $\omega$
with $\omega(\delta)\to 0$ as $\delta\to 0$ such that

$\begin{split}W(\beta_{k})&=W(\beta_{k}Q^{T})=W({\mathrm{Id}}+(\beta_{k}Q^{T}-{\mathrm{Id}}))\\\
\leq&\frac{1}{2}\mathbb{C}(\beta_{k}Q^{T}-{\mathrm{Id}})\cdot(\beta_{k}Q^{T}-{\mathrm{Id}})+\omega(\delta_{k})|\beta_{k}Q^{T}-{\mathrm{Id}}|^{2}\\\
=&{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}\frac{1}{2}\mathbb{C}_{Q}\eta\cdot\eta+{\varepsilon}_{k}^{2}\frac{1}{2}\mathbb{C}_{Q}\xi\cdot\xi\\\
&+{\varepsilon}_{k}\log^{1/2}\frac{1}{{\varepsilon}_{k}}\mathbb{C}_{Q}\eta\cdot\xi+\omega(\delta_{k})|Q^{T}\beta_{k}-{\mathrm{Id}}|^{2}\end{split}$

pointwise, and

$\begin{split}&\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}E^{\mathrm{elast}}[\beta_{k},\Omega\setminus(\gamma)_{{\varepsilon}_{k}^{s}}]\leq\int_{\Omega\setminus(\gamma)_{{\varepsilon}_{k}^{s}}}\frac{1}{2}{\mathbb{C}_{Q}}\eta\cdot\eta{\,dx}+\frac{1}{\log\frac{1}{{\varepsilon}_{k}}}\int_{\Omega\setminus(\gamma)_{{\varepsilon}_{k}^{s}}}\frac{1}{2}\mathbb{C}_{Q}\xi\cdot\xi{\,dx}\\\
&\quad+\frac{c}{\log^{1/2}\frac{1}{{\varepsilon}_{k}}}\int_{\Omega\setminus(\gamma)_{{\varepsilon}_{k}^{s}}}|{{\eta}}|\,|\xi|{\,dx}+\int_{\Omega\setminus(\gamma)_{{\varepsilon}_{k}^{s}}}\omega(\delta_{k})\frac{|\log^{1/2}\frac{1}{{\varepsilon}_{k}}{{\eta}}+{\xi}|^{2}}{\log\frac{1}{{\varepsilon}_{k}}}{\,dx}.\end{split}$
(6.43)

Now (6.3) in Proposition 6.1 yields

$\begin{split}\limsup_{k\to\infty}\frac{1}{\log\frac{1}{{\varepsilon}_{k}^{s}}}\int_{\Omega\setminus(\gamma)_{{\varepsilon}_{k}^{s}}}\frac{1}{2}\mathbb{C}_{Q}\xi\cdot\xi{\,dx}\leq&\int_{\Omega\cap\gamma}{\psi_{\mathbb{C}_{Q}}(Q^{T}b,t)}d\mathcal{H}^{1}\\\
=&\int_{\Omega\cap\gamma}{\psi_{\mathbb{C}}(b,Qt)}d\mathcal{H}^{1}\end{split}$
(6.44)

where in the last step we used Lemma 3.1(iii). Since $\eta\in
L^{\infty}(\Omega;\mathbb{R}^{3\times 3})$ and $\xi\in
L^{1}(\Omega;\mathbb{R}^{3\times 3})$ we derive that

$\lim_{k\to\infty}\frac{1}{\log^{1/2}\frac{1}{{\varepsilon}_{k}}}\int_{\Omega\setminus(\gamma)_{{\varepsilon}_{k}^{s}}}|{{\eta}}|\,|\xi|{\,dx}=0.$
(6.45)

Additionally, recalling (6.5)

$\begin{split}&\limsup_{k\to\infty}\int_{\Omega\setminus(\gamma)_{{\varepsilon}_{k}^{s}}}\frac{|\log^{1/2}\frac{1}{{\varepsilon}_{k}}Q{{\eta}}|^{2}+|\xi|^{2}}{\log\frac{1}{{\varepsilon}_{k}}}{\,dx}\\\
&=\lim_{k\to\infty}\int_{\Omega\setminus(\gamma)_{{\varepsilon}_{k}^{s}}}{|{{\eta}}|^{2}}+\frac{|{\xi}|^{2}}{\log\frac{1}{{\varepsilon}_{k}}}{\,dx}<\infty.\end{split}$
(6.46)

As $\omega(\delta_{k})\to 0$, also the last error term vanishes in the limit.

Finally, again from $\eta\in L^{\infty}(\Omega;\mathbb{R}^{3\times 3})$ and
(6.6) in Proposition 6.1 we obtain

$\begin{split}&\lim_{k\to\infty}\frac{1}{{{\varepsilon}_{k}^{2}}\log\frac{1}{{\varepsilon}_{k}}}\int_{\Omega\cap(\gamma)_{{\varepsilon}_{k}^{s}}\setminus(\gamma)_{{\varepsilon}_{k}}}|\beta_{k}-Q|^{2}{\,dx}\\\
&\leq
2\lim_{k\to\infty}\int_{\Omega\cap(\gamma)_{{\varepsilon}_{k}^{s}}\setminus(\gamma)_{{\varepsilon}_{k}}}|\eta|^{2}{\,dx}+2\lim_{k\to\infty}\frac{1}{\log\frac{1}{{\varepsilon}_{k}}}\int_{\Omega\cap(\gamma)_{{\varepsilon}_{k}^{s}}\setminus(\gamma)_{{\varepsilon}_{k}}}|{\xi}|^{2}{\,dx}\\\
&\leq c(1-s).\end{split}$ (6.47)

By Lemma 6.3(ii) the same holds with the integration domain
${\Omega\cap(\gamma)_{{\varepsilon}_{k}^{s}}\setminus(\gamma\cap\Omega)_{{\varepsilon}_{k}}}$.
Gathering together (6.42)–(6.47), we infer

$\limsup_{k\to\infty}\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}E^{\mathrm{ub}}_{{\varepsilon}_{k}}[\mu_{k},\beta_{k}]\leq\int_{\Omega}\frac{1}{2}\mathbb{C}_{Q}{{\eta}}\cdot{{\eta}}{\,dx}+s\int_{\Omega\cap\gamma}\psi_{\mathbb{C}}(b,Qt)d\mathcal{H}^{1}+c(1-s)$

for all $s\in(0,1)$, and hence for $s=1$.

Step 2: Let $\mu=b\otimes t\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to5.0pt{\vfill\hrule
width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma\in{\mathcal{M}}^{1}_{\mathcal{B}}(\Omega)$,
$\eta\in L^{2}(\Omega;\mathbb{R}^{3\times 3})$ with
$\mathop{\mathrm{curl}}\eta=0$. We first approximate $\eta$, and then $\mu$.

We first show that we can construct a smooth sequence $\eta_{j}$ with
$\mathop{\mathrm{curl}}\eta_{j}=0$ such that $\eta_{j}\to\eta$ strongly in
$L^{2}(\Omega;\mathbb{R}^{3\times 3})$. To see this, observe that since
$\Omega$ is Lipschitz there are an open set $\omega$ with
$\partial\Omega\subset\omega$ and a bilipschitz map $\Phi:\omega\to\omega$
such that $\Phi(x)=x$ for $x\in\partial\Omega$ and
$\Phi(\Omega\cap\omega)=\omega\setminus\overline{\Omega}$. We extend $\eta$ to
$\Omega\cup\omega$ by

$\eta(x):=\eta(\Phi(x))D\Phi(x)\text{ for }x\in\omega\setminus\Omega.$ (6.48)

Obviously $\eta\in L^{2}(\Omega\cup\omega;\mathbb{R}^{3\times 3})$. For every
ball $B\subset\omega$ such that $B\cap\Omega$ is simply connected there is
$u\in W^{1,2}(B\cap\Omega;\mathbb{R}^{3})$ such that $\eta=Du$ in
$B\cap\Omega$, and $u\circ\Phi$ gives an extension of $u$ to
$W^{1,2}(B\cup\Phi(B);\mathbb{R}^{3})$. As $\eta=D(u\circ\Phi)$ on
$\Phi(B)\setminus\Omega$ we obtain $\mathop{\mathrm{curl}}\eta=0$ on
$\Omega\cup\omega$. We then define $\eta_{j}$ as the mollification on scale
$1/j$ of the extension. For every $j$ we have $\eta_{j}\in
C^{\infty}(\overline{\Omega};\mathbb{R}^{3\times 3})$,
$\mathop{\mathrm{curl}}\eta_{j}=0$, and

$\lim_{j\to\infty}\|\eta_{j}-\eta\|_{L^{2}(\Omega)}=0$ (6.49)

which implies

$\lim_{j\to\infty}\int_{\Omega}\frac{1}{2}\mathbb{C}_{Q}\eta_{j}\cdot\eta_{j}dx=\int_{\Omega}\frac{1}{2}\mathbb{C}_{Q}\eta\cdot\eta
dx.$ (6.50)

We now turn to the measure $\mu$. By [CGM15, Theorem 3.1 and Lemma 2.3] (see
also [CGO15, Theorem 6.5 and Lemma 6.2]) we can find a sequence
$\nu_{j}=\tilde{b}_{j}\otimes\tilde{t}_{j}\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to5.0pt{\vfill\hrule
width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\tilde{\gamma}_{j}\in{\mathcal{M}}^{1}_{\mathcal{B}}(\mathbb{R}^{3})$
such that $\nu_{j}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\Omega{\displaystyle\mathop{\rightharpoonup}^{*}}\mu$,

$\lim_{\delta\to 0}|\nu_{j}|((\partial\Omega)_{\delta})=0\text{ for all $j$,
}$ (6.51)

and

$\limsup_{j\to\infty}\int_{\tilde{\gamma}_{j}\cap\Omega}\psi_{\mathbb{C}}(\tilde{b}_{j},Q\tilde{t}_{j})d\mathcal{H}^{1}\leq\int_{\gamma}\psi_{\mathbb{C}}^{\mathrm{rel}}(b,Qt)d\mathcal{H}^{1}.$
(6.52)

By [CGO15, Lemma 6.4], which we can apply by Lemma 2.5(iii), for every $j$
there are a polyhedral measure $\mu_{j}=b_{j}\otimes
t_{j}\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma_{j}\in{\mathcal{M}}^{1}_{\mathcal{B}}(\mathbb{R}^{3})$
and a bijective map $f^{j}\in C^{1}(\mathbb{R}^{3};\mathbb{R}^{3})$ such that

$|f^{j}_{\sharp}\mu_{j}-\nu_{j}|(\mathbb{R}^{3})\leq\frac{1}{j},$
$|Df^{j}(x)-{\mathrm{Id}}|+|f^{j}(x)-x|\leq\frac{1}{j}\quad\text{for all
}x\in\mathbb{R}^{3},$

and, for some $r_{j}>0$,

$\int_{\gamma_{j}\cap(\Omega)_{r_{j}}}\psi_{\mathbb{C}}(b_{j},Qt_{j})d\mathcal{H}^{1}\leq(1+c\frac{1}{j})\int_{\tilde{\gamma}_{j}\cap\Omega}\psi_{\mathbb{C}}(\tilde{b}_{j},Q\tilde{t}_{j})d\mathcal{H}^{1}+c\frac{1}{j},$
(6.53)

where $(\Omega)_{r_{j}}=\\{x:{\mathrm{dist}}(x,\Omega)<r_{j}\\}$. Here
$f^{j}_{\sharp}\mu_{j}$ denotes the push-forward of the current corresponding
to the measure $\mu_{j}$, in the sense that

$f^{j}_{\sharp}(b_{j}\otimes t_{j}\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to5.0pt{\vfill\hrule
width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}\gamma_{j})=\left(b_{j}\otimes\frac{D_{t_{j}}f_{j}}{|D_{t_{j}}f_{j}|}\right)\circ
f_{j}^{-1}\mathcal{H}^{1}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to5.0pt{\vfill\hrule width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to3.35pt{\vfill\hrule
width=3.35pt,height=0.2pt}}\,}}f_{j}(\gamma_{j}).$ (6.54)

By Lemma 6.3(i) we can choose $f_{j}$ and $\mu_{j}$ such that $\mu_{j}$ is
transversal to $\partial\Omega$ (possibly replacing $f_{j}$ by $f_{j}+a_{j}$,
for some $a_{j}\to 0$). In particular it turns out that
$\mu_{j}{\displaystyle\mathop{\rightharpoonup}^{*}}\mu$ in $\Omega$. Moreover
since $\mu_{j}$ is polyhedral it is
$(h_{{\varepsilon}_{k}},\alpha_{{\varepsilon}_{k}})$-dilute for sufficiently
large $k$. By step 1 for every $j$ there is
$(\mu_{k}^{j},\beta_{k}^{j})\in\mathcal{A}_{{\varepsilon}_{k}}^{*}$
constructed as in (6.40) such that ${\varepsilon}_{k}^{-1}\mu_{k}^{j}=\mu_{j}$
and

$\frac{Q^{T}\beta_{k}^{j}-{\mathrm{Id}}}{{\varepsilon}_{k}\log^{1/2}\frac{1}{{\varepsilon}_{k}}}\to\eta_{j}\quad\text{
strongly in }L^{q}(\Omega;\mathbb{R}^{3\times 3}),$

for all $q\in[1,2)$. Moreover

$\limsup_{k\to\infty}\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}E^{\mathrm{ub}}_{{\varepsilon}_{k}}[\mu_{k}^{j},\beta^{j}_{k}]\leq\int_{\Omega}\frac{1}{2}\mathbb{C}_{Q}{{\eta}}_{j}\cdot{{\eta}}_{j}{\,dx}+\int_{\Omega\cap\gamma_{j}}\psi_{\mathbb{C}}(b_{j},Qt_{j})d\mathcal{H}^{1},$
(6.55)

and

$\begin{split}\limsup_{k\to\infty}\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}\int_{\Omega\cap(\mathop{\mathrm{supp}}\mu_{k}^{j})_{{\varepsilon}_{k}}}{\mathrm{dist}}^{q}(\beta^{j}_{k},{\mathrm{SO}}(3)){\,dx}=0.\end{split}$
(6.56)

Further, from (6.49), (6.53) and (6.52) it follows that

$\limsup_{j\to\infty}\int_{\Omega}\frac{1}{2}\mathbb{C}_{Q}{{\eta}}_{j}\cdot{{\eta}}_{j}{\,dx}+\int_{\Omega\cap\gamma_{j}}\psi_{\mathbb{C}}(b_{j},Qt_{j})d\mathcal{H}^{1}\leq\int_{\Omega}\frac{1}{2}\mathbb{C}_{Q}{{\eta}}\cdot{{\eta}}{\,dx}+\int_{\gamma}\psi_{\mathbb{C}}^{\mathrm{rel}}(b,Qt)d\mathcal{H}^{1}.$

To conclude it remains to construct a diagonal sequence. Notice first that for
any fixed $j$ there is $k_{j}\in\mathbb{N}$ such that for $k\geq k_{j}$ the
following properties are satisfied:

  1. (i)

$\mu_{k}^{j}{\mathchoice{\,\hbox{\vrule width=0.2pt\vbox to7.0pt{\vfill\hrule
width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to7.0pt{\vfill\hrule width=7.0pt,height=0.2pt}}\,}{\,\hbox{\vrule
width=0.2pt\vbox to5.0pt{\vfill\hrule
width=5.0pt,height=0.2pt}}\,}{\,\hbox{\vrule width=0.2pt\vbox
to3.35pt{\vfill\hrule width=3.35pt,height=0.2pt}}\,}}\Omega$ is
$(h_{{\varepsilon}_{k}},\alpha_{{\varepsilon}_{k}})$-dilute;

  2. (ii)

$\left\|\frac{Q^{T}\beta_{k}^{j}-{\mathrm{Id}}}{{\varepsilon}_{k}\log^{1/2}\frac{1}{{\varepsilon}_{k}}}-{{\eta}}_{j}\right\|_{L^{q}(\Omega;\mathbb{R}^{3\times
3})}\leq\frac{1}{j}\quad\text{ for all }q\in[1,2-\frac{1}{j}];$

  3. (iii)

$\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}\int_{\Omega\cap(\mathop{\mathrm{supp}}\mu_{k}^{j})_{{\varepsilon}_{k}}}{\mathrm{dist}}^{q}(\beta^{j}_{k},{\mathrm{SO}}(3)){\,dx}\leq\frac{1}{j}\quad\text{
for all }q\in[1,2-\frac{1}{j}];$

  4. (iv)

$\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}E^{\mathrm{ub}}_{{\varepsilon}_{k}}[\mu_{k}^{j},\beta^{j}_{k}]\leq\int_{\Omega}\frac{1}{2}\mathbb{C}_{Q}{{\eta}}_{j}\cdot{{\eta}}_{j}{\,dx}+\int_{\Omega\cap\gamma_{j}}\psi_{\mathbb{C}}(b_{j},Qt_{j})d\mathcal{H}^{1}+\frac{1}{j}.$

Without loss of generality, $k_{j}>k_{j-1}$. Finally for every $k>0$ we take
$\tilde{\mu}_{k}:=\mu_{k}^{j}$ and $\tilde{\beta}_{k}:=\beta_{k}^{j}$ if
$k\in[k_{j},k_{j+1})\cap\mathbb{N}$, and this concludes the proof. ∎

######  Remark 6.6.

The same result as in Theorem 6.5 holds if $E^{\mathrm{ub}}_{\varepsilon}$ is
replaced by

$E^{\mathrm{moll}}_{\varepsilon}[\mu,\beta]:=\begin{cases}\displaystyle
E^{\mathrm{elast}}[\beta,\Omega],&\text{ if
$(\mu,\beta)\in\mathcal{A}^{\mathrm{moll}}_{\varepsilon}$, }\\\ \infty,&\text{
otherwise,}\end{cases}$ (6.57)

where $E^{\mathrm{elast}}$ and $\mathcal{A}^{\mathrm{moll}}_{\varepsilon}$ are
defined in (3.37) and (3.51) respectively. To see this, it suffices to replace
the definition of $\beta_{k}$ in (6.40) with

$\hat{\beta}_{k}:=Q+{\varepsilon}_{k}\log^{1/2}\frac{1}{{\varepsilon}_{k}}Q\eta+{\varepsilon}_{k}Q\xi*\varphi_{\varepsilon}.$
(6.58)

Then one applies (6.3) in Proposition 6.1 to a larger set $(\Omega)_{r}$, for
some $r>0$. For almost all $r$ we have $|\mu|(\partial((\Omega)_{r}))=0$, so
that one obtains instead of (6.44)

$\limsup_{k\to\infty}\frac{1}{\log\frac{1}{{\varepsilon}_{k}^{s}}}\int_{(\Omega)_{r}\setminus(\gamma)_{{\varepsilon}_{k}^{s}}}\frac{1}{2}\mathbb{C}_{Q}\xi\cdot\xi{\,dx}\leq\int_{(\Omega)_{r}\cap\gamma}\psi_{\mathbb{C}}(b,Qt)d\mathcal{H}^{1}.$
(6.59)

By convexity this gives

$\limsup_{k\to\infty}\frac{1}{\log\frac{1}{{\varepsilon}_{k}^{s}}}\int_{\Omega\setminus(\gamma)_{2{\varepsilon}_{k}^{s}}}\frac{1}{2}\mathbb{C}_{Q}(\varphi_{{\varepsilon}_{k}}\ast\xi)\cdot(\varphi_{{\varepsilon}_{k}}\ast\xi){\,dx}\leq\int_{(\Omega)_{r}\cap\gamma}\psi_{\mathbb{C}}(b,Qt)d\mathcal{H}^{1}$
(6.60)

for almost all $r>0$, and since $|\mu|(\partial\Omega)=0$ also with the
integral over $\Omega\cap\gamma$ in the right-hand side. In the rest of the
estimates we need to use $2{\varepsilon}_{k}^{s}$ instead of
${\varepsilon}_{k}^{s}$, without any significant change. The estimate in
(6.47) carries over by convexity, if one restricts the domain by
${\varepsilon}_{k}$. As the exponent $s$ was generic, this is only relevant on
the “inner” side, and it remains to deal with $(\gamma)_{2{\varepsilon}_{k}}$.
This is finally treated by Remark 6.2, which shows that

$\limsup_{k\to\infty}\frac{1}{{\varepsilon}_{k}^{2}\log\frac{1}{{\varepsilon}_{k}}}\int_{(\gamma)_{2{\varepsilon}_{k}}}W(\hat{\beta}_{k})dx\leq\limsup_{k\to\infty}\frac{c}{\log\frac{1}{{\varepsilon}_{k}}}\int_{(\gamma)_{2{\varepsilon}_{k}}}(|\eta|^{2}+|\xi\ast\varphi_{{\varepsilon}_{k}}|^{2})dx=0.$
(6.61)

Step 2 is unchanged.

##  7 Proofs of the main results

We start with the proof of our main result, Theorem 3.6.

###### Proof of Theorem 3.6.

Compactness follows from Proposition 5.4. For the lower bound we use Theorem
5.5 with $\rho_{\varepsilon}={\varepsilon}$ and (5.22). For the upper bound we
use Theorem 6.5, observing that

$F_{\varepsilon}^{\mathrm{subcr}}[\mu,\beta]\leq\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}E^{\mathrm{ub}}_{\varepsilon}[\mu,\beta]+\frac{c}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}\
\int_{\Omega\cap(\mathop{\mathrm{supp}}\mu)_{\varepsilon}}{\mathrm{dist}}^{p}(\beta,{\mathrm{SO}}(3))dx.$

The geometrically linear case is proven analogously. ∎

We now turn to Theorem 3.10.

###### Proof of Theorem 3.10.

(i): As in the proof of Theorem 3.6, the lower bound follows from Theorem 5.5
and the fact that

$\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}E^{\mathrm{lb}}_{\rho,{\varepsilon}}[\mu,\beta]\leq
F_{\varepsilon}^{*}[\mu,\beta]$

for every $\rho\geq\rho_{\varepsilon}$ which follows as above from
$\mathcal{A}_{\varepsilon}^{*}\subseteq\mathcal{A}_{\rho,{\varepsilon}}^{\mathrm{core}}$
for every $\rho>0$.

For the upper bound we use Theorem 6.5, observing that
$\rho_{\varepsilon}\geq{\varepsilon}$ implies

$F_{\varepsilon}^{*}[\mu,\beta]\leq\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}E^{\mathrm{ub}}_{\varepsilon}[\mu,\beta].$

(ii): For the lower bound we observe that

$\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}E^{\mathrm{lb}}_{{\rho_{\varepsilon},{\varepsilon}}}[\mu,\beta]\leq
F_{\varepsilon}^{\mathrm{core}}[\mu,\beta].$

Hence the thesis follows again by Theorem 5.5.

For the upper bound we use Theorem 6.5, observing that

$F_{\varepsilon}^{\mathrm{core}}[\mu,\beta]\leq\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}E^{\mathrm{ub}}_{\varepsilon}[\mu,\beta]$

which again is a consequence of
$\mathcal{A}^{*}_{\varepsilon}\subseteq{\mathcal{A}_{\rho_{\varepsilon},{\varepsilon}}^{\mathrm{core}}}$
(see Lemma 3.8(iii)).

(iii): For lower bound we use Theorem 5.5 and the fact that

$\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}E^{\mathrm{lb}}_{\rho,{\varepsilon}}[\mu,\beta]\leq
F_{\varepsilon}^{\mathrm{moll}}[\mu,\beta]$

for every $\rho\geq{\varepsilon}$ which follows from
$\mathcal{A}_{\varepsilon}^{\mathrm{moll}}\subseteq\mathcal{A}_{\rho,{\varepsilon}}^{\mathrm{core}}$
(see Lemma 3.8(ii)).

For the upper bound we use the same argument as in Theorem 6.5, as discussed
in Remark 6.6 observing that

$F_{\varepsilon}^{\mathrm{moll}}[\mu,\beta]=\frac{1}{{\varepsilon}^{2}\log\frac{1}{{\varepsilon}}}E^{\mathrm{moll}}_{\varepsilon}[\mu,\beta].$

∎

## Acknowledgements

This work was partially funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) via project 211504053 - SFB 1060, project
390685813 - GZ 2047/1 - HCM3, and project 2044-390685587 - “Mathematics
Münster: Dynamics–Geometry–Structure” and by PRIN 2017BTM7SN$\\_$004
“Variational methods for stationary and evolution problems with singularities
and interfaces”.

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