# A general compactness theorem in $G(S)BD$

Antonin Chambolle  CEREMADE, CNRS, Université Paris-Dauphine PSL, France and
Mokaplan (INRIA/CNRS/PSL<EMAIL_ADDRESS>and  Vito
Crismale  Dipartimento di Matematica Guido Castelnuovo, Piazzale Aldo Moro 5,
00185 Roma, Italy<EMAIL_ADDRESS>
###### Abstract.

We give a new, simpler proof of a compactness result in $GSBD^{p}$, $p>1$, by
the same authors, which is also valid in $GBD$ (the case $p=1$), and shows
that bounded sequences converge a.e., after removal of a suitable sequence of
piecewise infinitesimal rigid motions, subject to a fixed partition.

##  1\. Introduction

Generalized (special) functions with bounded deformation ($G(S)BD$) have been
introduced by G. Dal Maso [11] in order to properly tackle free discontinuity
problems [12, 2] in linearized elasticity, and in particular the minmization
of the Griffith functional

$\int_{\Omega\setminus K}\mathbb{C}e(u):e(u)dx+\gamma\mathcal{H}^{d-1}(K),$
(1.1)

introduced in [14] to model and approximate brittle fracture growth in linear
elastic materials. In this functional, $\Omega\subset\mathbb{R}^{d}$ is a
bounded $d$-dimensional domain (in practice $d\in\\{2,3\\}$), $u$ a vectorial
displacement, expected to be smooth, with symmetrized gradient
$e(u)=(Du+Du^{T})/2$, except across a $(d-1)$-dimensional fracture set $K$.
The tensor $\mathbb{C}$ contains the physical constants of the problem, and
defines a positive definite quadratic form on symmetric tensors, while
$\gamma>0$ is the toughness of the material. Showing existence to minimizers
to this functional has been a difficult task, developed over many years. The
situation mostly evolved after [11] introduced for the first time a reasonable
energy space for a weak form of (1.1), where $K$ is replaced with $J_{u}$, the
intrinsic jump set of $u$ [13]. Existence results could then be proved [3, 15,
8, 10, 6, 7] for weak, then “strong” minimizers (that is, for the original
problem in $(u,K)$). Most of these works rely upon a rigidity result for
displacements with small jumps, established in [5].

In particular, the main result in [8] is a compactness result in $GSBD^{p}$,
the subspace of $GSBD$ (which is defined precisely in Section 2.1) of
displacements with $p$-integrable symmetrized gradient and jump set of finite
surface. In this result, a sequence which is bounded in energy (roughly,
(1.1), with the Lagrangian replaced with $|e(u)|^{p}$) will converge up to
subsequences either to a $GSBD^{p}$ limit $u(x)$ or to $+\infty$ (with some
appropriate semicontinuity properties). This is not really an issue for the
study of (1.1), since replacing $u$ with $0$ where it is infinite, we recover
that the limit of a minimizing sequence is a minimizer.

However, it was observed in [16, 9] that this compactness result is not
sufficient for studying more general, non-homogeneous variational problems,
where the Lagrangian is not minimal at $0$. In that case, one has to study
more finely what happens in the “infinity” set. Following similar (yet far
more precise) results in the scalar case [16], the authors could show in [9] a
more complete compactness result, and in particular the existence of a
Caccioppoli partition where, on each set of the partition, the sequence
converges to a finite limit after substraction of a suitable sequence of
infinitesimal rigid motions (affine functions with skew-symmetric gradients).

In addition, S. Almi and E. Tasso [1] recently extended [8], with a different
proof, to sequences merely bounded in $GBD$ (roughly, the case $p=1$ in [8]),
while the proof in [9], relying on a fine result of [4] valid only for $p>1$,
would not work in $GBD$.

The purpose of this note is to give an alternative proof of the main
compactness result of [9], which does not rely on [4] and is also valid in
$GBD$, thus permitting to deal with non-homogeneous problems also in this
framework. Precisely, we show the compactness Theorem 1.1 below (the notation
is made precise in Section 2.1). The proof of this result is quite simpler, in
a sense, than in [9], yet also more interesting. It only relies on a suitable
version of the approximate Poincaré-Korn inequality of [5] proven in Theorem
2.3, which asserts that the energy controls how far a function is to rigid
motions (hence to finite-dimensional), combined with a multiscale
construction. We hope that this scheme can be useful for other purposes. We
observe for instance that, combined with the celebrated extension method of
Nitsche [18], a simplified version or this proof allows to easily deduce
Rellich-type theorems in $BD$ [21, 19, 20, 17].

######  Theorem 1.1.

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain and let $u_{k}\in
GBD(\Omega)$ be such that

$\sup_{k\in\mathbb{N}}\widehat{\mu}_{u_{k}}(\Omega)<+\infty.$ (1.2)

Then there exist a subsequence, not relabelled, a Caccioppoli partition
$\mathcal{P}=(P_{n})_{n}$ of $\Omega$, a sequence of piecewise rigid motions
$(a_{k})_{k}$ with

$a_{k}=\sum_{n\in\mathbb{N}}a_{k}^{n}\chi_{P_{n}},$ (1.3a)
$|a_{k}^{n}(x)-a_{k}^{n^{\prime}}(x)|\to+\infty\quad\text{for
}{\mathcal{L}}^{d}\text{-a.e.\ }x\in\Omega,\text{ for all }n\neq{n^{\prime}},$
(1.3b)

and $u\in GBD(\Omega)$ such that

$\displaystyle u_{k}-a_{k}$ $\displaystyle\to
u\quad{\mathcal{L}}^{d}\text{-a.e.\ in }\Omega,$ (1.4a)
$\displaystyle\mathcal{H}^{d-1}(\partial^{*}\mathcal{P}\cap\Omega)$
$\displaystyle\leq\lim_{\sigma\to+\infty}\liminf_{k\to\infty}\,\mathcal{H}^{d-1}(J^{\sigma}_{u_{k}}).$
(1.4b)

If in addition $(u_{k})_{k}$ is bounded in $GSBD^{p}(\Omega)$, $p>1$ (that is,
(3.18) below holds), following [9] one obtains in addition to the last
estimate:

$\displaystyle\mathcal{H}^{d-1}((\partial^{*}\mathcal{P}\cup
J_{u})\cap\Omega)\leq\liminf_{k\to\infty}\,\mathcal{H}^{d-1}(J_{u_{k}}),$
(1.4c)

see Remark 3.1 in Section 3.3.

The plan of the note is as follows: we first define properly the notions which
are useful for this work (Sec. 2.1). Then, in Section 2.2 we show that a
partial rigidity result of [5] is also valid in $GBD$, without further
integrability assumption. The following section is devoted to the proof of
Theorem 1.1. Thanks to the rigidity result, we build an appropriate
Caccioppoli partition which will satisfy the thesis of the Theorem in Section
3.1. We end up proving the compactness (1.4a) (Sec. 3.2) and the lower-
semicontinuity (1.4b) (Sec. 3.3).

##  2\. Preliminaries

###  2.1. Notation

Given $\Omega\subset{\mathbb{R}}^{d}$ open, we use the notation
$L^{0}(\Omega;\mathbb{R}^{m})$ for the space of ${\mathcal{L}}^{d}$-measurable
functions $v\colon\Omega\to\mathbb{R}^{m}$, endowed with the topology of
convergence in measure. For any locally compact subset
$B\subset{\mathbb{R}}^{d}$, (i.e. any point in $B$ has a neighborhood
contained in a compact subset of $B$), the space of bounded
$\mathbb{R}^{m}$-valued Radon measures on $B$ [respectively, the space of
$\mathbb{R}^{m}$-valued Radon measures on $B$] is denoted by
$\mathcal{M}_{b}(B;\mathbb{R}^{m})$ [resp., by
$\mathcal{M}(B;\mathbb{R}^{m})$]. If $m=1$, we write $\mathcal{M}_{b}(B)$ for
$\mathcal{M}_{b}(B;\mathbb{R})$, $\mathcal{M}(B)$ for
$\mathcal{M}(B;\mathbb{R})$, and $\mathcal{M}^{+}_{b}(B)$ for the subspace of
positive measures of $\mathcal{M}_{b}(B)$. For every
$\mu\in\mathcal{M}_{b}(B;\mathbb{R}^{m})$, its total variation is denoted by
$|\mu|(B)$.

We say that $v\in L^{1}(\Omega)$ is a _function of bounded variation_ on
$\Omega$, and we write $v\in BV(\Omega)$, if
$\mathrm{D}_{i}v\in\mathcal{M}_{b}(\Omega)$ for $i=1,\dots,n$, where
$\mathrm{D}v=(\mathrm{D}_{1}v,\dots,\mathrm{D}_{n}v)$ is its distributional
derivative. A vector-valued function $v\colon\Omega\to\mathbb{R}^{m}$ is in
$BV(\Omega;\mathbb{R}^{m})$ if $v_{j}\in BV(\Omega)$ for every $j=1,\dots,m$.
The space $BV_{\mathrm{loc}}(\Omega)$ is the space of $v\in
L^{1}_{\mathrm{loc}}(\Omega)$ such that
$\mathrm{D}_{i}v\in\mathcal{M}(\Omega)$ for $i=1,\dots,d$.

We call _infinitesimal rigid motion_ any affine function with skew-symmetric
gradient and _piecewise rigid motion_ any function of the form
$\sum_{j\in\mathbb{N}}a_{j}\chi_{P_{j}}$, where $(P_{j})_{j}$ is a Caccioppoli
partition of $\Omega$ (that is, a partition into sets of finite perimeters,
with finite total perimeter) and any $a_{j}$ is an infinitesimal rigid motion.

Fixed $\xi\in{\mathbb{S}^{d-1}}$, we let

$\Pi^{\xi}:=\\{y\in{\mathbb{R}}^{d}\colon y\cdot\xi=0\\},\qquad
B^{\xi}_{y}:=\\{t\in\mathbb{R}\colon y+t\xi\in B\\}\ \ \ \text{ for any
$y\in{\mathbb{R}}^{d}$ and $B\subset{\mathbb{R}}^{d}$},$ (2.1)

and for every function $v\colon B\to{\mathbb{R}}^{d}$ and $t\in B^{\xi}_{y}$,
let

$v^{\xi}_{y}(t):=v(y+t\xi),\qquad\widehat{v}^{\xi}_{y}(t):=v^{\xi}_{y}(t)\cdot\xi.$
(2.2)

Moreover, let
$\Pi^{\xi}(x):=x-(x\cdot\xi)\xi\in\xi^{\perp}=\\{y\in{\mathbb{R}}^{d}\colon
y\cdot\xi=0\\}$ for every $x\in{\mathbb{R}}^{d}$.

######  Definition 2.1 (“$GBD$” [11]).

Let $\Omega\subset{\mathbb{R}}^{d}$ be a bounded open set, and let $v\in
L^{0}(\Omega;{\mathbb{R}}^{d})$. Then $v\in GBD(\Omega)$ if there exists
$\lambda_{v}\in\mathcal{M}^{+}_{b}(\Omega)$ such that one of the following
equivalent conditions holds true for every $\xi\in{\mathbb{S}^{d-1}}$:

  * (a)

for every $\tau\in C^{1}(\mathbb{R})$ with
$-\tfrac{1}{2}\leq\tau\leq\tfrac{1}{2}$ and $0\leq\tau^{\prime}\leq 1$, the
partial derivative
$\mathrm{D}_{\xi}\big{(}\tau(v\cdot\xi)\big{)}=\mathrm{D}\big{(}\tau(v\cdot\xi)\big{)}\cdot\xi$
belongs to $\mathcal{M}_{b}(\Omega)$, and for every Borel set $B\subset\Omega$

$\big{|}\mathrm{D}_{\xi}\big{(}\tau(v\cdot\xi)\big{)}\big{|}(B)\leq\lambda_{v}(B);$

  * (b)

$\widehat{v}^{\xi}_{y}\in BV_{\mathrm{loc}}(\Omega^{\xi}_{y})$ for
$\mathcal{H}^{d-1}$-a.e. $y\in\Pi^{\xi}$, and for every Borel set
$B\subset\Omega$

$\int_{\Pi^{\xi}}\Big{(}\big{|}\mathrm{D}{\widehat{v}}_{y}^{\xi}\big{|}\big{(}B^{\xi}_{y}\setminus
J^{1}_{{\widehat{v}}^{\xi}_{y}}\big{)}+\mathcal{H}^{0}\big{(}B^{\xi}_{y}\cap
J^{1}_{{\widehat{v}}^{\xi}_{y}}\big{)}\Big{)}\,\mathrm{d}\mathcal{H}^{d-1}(y)\leq\lambda_{v}(B),$

where $J^{1}_{{\widehat{u}}^{\xi}_{y}}:=\left\\{t\in
J_{{\widehat{u}}^{\xi}_{y}}:|[{\widehat{u}}_{y}^{\xi}]|(t)\geq 1\right\\}$.

The function $v$ belongs to $GSBD(\Omega)$ if $v\in GBD(\Omega)$ and
$\widehat{v}^{\xi}_{y}\in SBV_{\mathrm{loc}}(\Omega^{\xi}_{y})$ for every
$\xi\in{\mathbb{S}^{d-1}}$ and for $\mathcal{H}^{d-1}$-a.e. $y\in\Pi^{\xi}$.

For $v\in GBD(\Omega)$, denoting by

$(\widehat{\mu}_{v})^{\xi}_{y}(B):=|\mathrm{D}{\widehat{v}}_{y}^{\xi}\big{|}\big{(}B\setminus
J^{1}_{{\widehat{v}}^{\xi}_{y}}\big{)}+\mathcal{H}^{0}\big{(}B\cap
J^{1}_{{\widehat{v}}^{\xi}_{y}}\big{)}\quad\text{for every
$B\subset\Omega^{\xi}_{y}$ Borel}$ (2.3)

($(\widehat{\mu}_{v})^{\xi}_{y}\in\mathcal{M}_{b}^{+}(\Omega^{\xi}_{y})$ for
every $\xi\in{\mathbb{S}^{d-1}}$ and $\mathcal{H}^{d-1}$-a.e. $y\in\Pi_{\xi}$)
and by

$(\widehat{\mu}_{v})^{\xi}(B):=\int_{\Pi^{\xi}}(\widehat{\mu}_{v})^{\xi}_{y}(B^{\xi}_{y})\,\mathrm{d}\mathcal{H}^{d-1}(y)\quad\text{for
every $B\subset\Omega$ Borel},$ (2.4)

it holds that $(\widehat{\mu}_{v})^{\xi}\in\mathcal{M}_{b}^{+}(\Omega)$,
$(\widehat{\mu}_{v})^{\xi}\leq\lambda_{v}$ for any $\lambda_{v}$ satisfying
condition (b) of Definition 2.1 and that

$\widehat{\mu}_{v}(B):=\sup_{k}\sup\Big{\\{}\sum_{i=1}^{k}(\widehat{\mu}_{v})^{\xi_{i}}(B_{i})\colon(\xi_{i})_{i}\subset{\mathbb{S}^{d-1}},\,B_{1},\dots,B_{k}\subset
B,B_{i}\cap B_{j}=\emptyset\,\forall i\neq j\Big{\\}}$ (2.5)

is the smallest measure $\lambda_{v}$ that satisfies condition (b) of
Definition 2.1.

Every $v\in GBD(\Omega)$ has an _approximate symmetric gradient_ $e(v)\in
L^{1}(\Omega;{\mathbb{M}^{d\times d}_{sym}})$ such that for every
$\xi\in{\mathbb{S}^{d-1}}$ and $\mathcal{H}^{d-1}$-a.e. $y\in\Pi^{\xi}$ there
holds

$e(v)(y+t\xi)\xi\cdot\xi=(\widehat{v}^{\xi}_{y})^{\prime}(t)\quad\text{for
}\mathcal{L}^{1}\text{-a.e.\ }t\in\Omega^{\xi}_{y};$ (2.6)

the _approximate jump set_ $J_{v}$ is still countably
$(\mathcal{H}^{d-1},d{-}1)$-rectifiable (cf. [11, Theorem 6.2] and [13]) and
may be reconstructed from its slices through the identity

$(J^{\xi}_{v})^{\xi}_{y}=J_{\widehat{v}^{\xi}_{y}}\quad\text{and}\quad
v^{\pm}(y+t\xi)\cdot\xi=(\widehat{v}^{\xi}_{y})^{\pm}(t)\ \text{ for
}t\in(J_{v})^{\xi}_{y},$ (2.7)

where $J^{\xi}_{v}:=\\{x\in J_{v}\colon[v]\cdot\xi\neq 0\\}$ (it holds that
$\mathcal{H}^{d-1}(J_{v}\setminus J_{v}^{\xi})=0$ for $\mathcal{H}^{d-1}$-a.e.
$\xi\in{\mathbb{S}^{d-1}}$). For every $\sigma>0$ we also denote

$J_{v}^{\sigma}:=\\{x\in J_{v}\colon|[v]|<\sigma\\}.$ (2.8)

By (2.7), for every $\sigma>0$, every $\xi\in{\mathbb{S}^{d-1}}$, and
$\mathcal{H}^{d-1}$-a.e. $y\in\Pi_{\xi}$

$J_{\widehat{v}^{\xi}_{y}}^{\sigma}\subset(J_{v}^{\sigma})^{\xi}_{y},$ (2.9)

where $J_{\widehat{v}^{\xi}_{y}}^{\sigma}=\\{t\in
J_{\widehat{v}^{\xi}_{y}}\colon|[\widehat{v}^{\xi}_{y}]|<\sigma\\}$.

We recall from [9] the following lemma on piecewise rigid motions.

######  Lemma 2.2.

Let $(\mathcal{P}_{j})_{j}$ be a Caccioppoli partition and let $(a_{h})_{h}$
be a sequence of piecewise rigid motions such that (1.3a) and (1.3b) hold.
Then for $\mathcal{H}^{d-1}$-a.e. $\xi\in{\mathbb{S}^{d-1}}$

$|(a_{h}^{j}-a_{h}^{i})(x)\cdot\xi|\to+\infty\quad\text{as
}h\to+\infty\quad\text{for }{\mathcal{L}}^{d}\text{-a.e.\ }x\in\Omega,\text{
for all }i\neq j.$ (2.10)

###  2.2. Rigidity in $GBD$

The following result is obtained in the footsteps of Proposition 2.1 in [5].
Let $Q_{\delta}=(-\delta/2,\delta/2)^{d}$.

######  Theorem 2.3.

There exist $c>0$ such that for any $\delta>0$, $u\in GSBD(Q_{\delta})$, there
exists $\omega\subset Q_{\delta}$ with $|\omega|\leq
c\delta\mathcal{H}^{d-1}(J_{u}^{1})$ and an infinitesimal rigid motion $a$
such that

$\int_{Q_{\delta}\setminus\omega}|u-a|\,\mathrm{d}x\leq
c\delta\,\widehat{\mu}_{u}(Q_{\delta}\setminus J_{u}^{1}).$

###### Proof.

We sketch the proof, highlighting the modifications with respect to [5,
Proposition 2.1].

As in [5], we may assume by a rescaling argument $\delta=1$ (and write $Q$ for
$Q_{1}$), and that $\mathcal{H}^{d-1}(J_{u}^{1})\leq\frac{1}{32d^{3}}$,
otherwise it is enough to take $\omega=Q$, $a=0$, $c=32d^{3}$. We define the
function
$T\colon{\mathbb{R}}^{d}\times{\mathbb{S}^{d-1}}\times\mathbb{R}\to\mathbb{R}$
by

$T(x,\xi,t):=\begin{dcases}1\quad&\text{if }x\in Q,\,x+t\xi\in Q\text{ and
}x+[0,t]\xi\cap J^{1}_{u}\neq\emptyset,\\\
0\quad&\text{otherwise}.\end{dcases}$ (2.11)

By definition of distributional derivative it holds that

$\xi\cdot\big{(}u(x+t\xi)-u(x)\big{)}=\int_{x\cdot\xi}^{x\cdot\xi+t}\mathrm{D}\widehat{u}^{\xi}_{y}(s)\,\mathrm{d}s,\quad
y:=\Pi^{\xi}(x)$ (2.12)

and $\mathrm{D}\widehat{u}^{\xi}_{y}\leq(\widehat{\mu}_{u})^{\xi}_{y}$ on
$[x\cdot\xi,x\cdot\xi+t]$ if $T(x,\xi,t)=0$ and $x,\,x+t\xi\in Q$ (recall
(2.9)) at least for a.e. $x\in Q$ and $t\in\mathbb{R}$. We remark that (2.11)
is the analogue of [5, definition (2.6)] when replacing $J_{u}$ with
$J_{u}^{1}$, and that (2.12) is the analogue of [5, equation (2.5)].
Therefore, exactly as in [5], one obtains that

$\int_{{\mathbb{R}}^{d}}\int_{\mathbb{R}}T(x,\xi,t)\,\mathrm{d}t\,\mathrm{d}x\leq
4d\mathcal{H}^{d-1}(J_{u}^{1})\quad\text{for any }\xi\in{\mathbb{S}^{d-1}}$

and that there exists $t_{*}\in(1/2,1)$ (fixed for the rest of the proof) and
$q_{1}\subset q$ with ${\mathcal{L}}^{d}(q_{1})/{\mathcal{L}}^{d}(q)>3/4$ such
that, defining

$z_{i}(z_{0}):=z_{0}+t_{*}e_{i}\quad\text{for all }i=1,\dots,d,\qquad
E_{z_{0}}:=\Big{(}\bigcup_{i=1}^{d}[z_{0},z_{i}(z_{0})]\cup\bigcup_{1\leq
i<j\leq d}[z_{i}(z_{0}),z_{j}(z_{0})]\Big{)}$ (2.13a) it holds $\text{for
$z_{0}\in q_{1}$: }E_{z_{0}}\cap J_{u}^{1}=\emptyset,\,E_{z_{0}}\subset Q.$
(2.13b)

For $t_{*}$ fixed above and any $z_{0}\in q$ let us consider

$\begin{split}F(z_{0})&:=\sum_{0\leq i<j\leq
n}|\mathrm{D}\widehat{u}^{\xi_{i,j}}_{y_{i,j}(z_{0})}|([z_{i}(z_{0})\cdot\xi_{i,j},z_{j}(z_{0})\cdot\xi_{i,j}]),\\\
&\xi_{i,j}:=\frac{z_{i}(z_{0})-z_{j}(z_{0})}{|z_{i}(z_{0})-z_{j}(z_{0})|},\,y_{i,j}(z_{0}):=\Pi^{\xi_{i,j}}(z_{i}(z_{0}))=z_{i}(z_{0})-(z_{i}(z_{0})\cdot\xi_{i,j})\xi_{i,j}.\end{split}$

We notice that $\xi_{i,j}=\frac{e_{i}-e_{j}}{\sqrt{2}}$ if $i\neq 0$, while
$\xi_{i,j}=e_{j}$ if $i=0$. Fixed $i\neq j$, we integrate for $z_{0}\in q$ as
$\widetilde{z}_{0}=\Pi^{\xi_{i,j}}(z_{0})$ ranges in $\xi_{i,j}^{\perp}$ and
$z^{\prime}_{0}=z_{0}\cdot\xi_{i,j}$ ranges in
$q^{\xi_{i,j}}_{\Pi^{\xi_{i,j}}(z_{0})}$, using Fubini’s Theorem. Moreover, if
$\Pi^{\xi_{i,j}}(z_{0})$ is fixed to a value
$\widetilde{z}_{0}\in\Pi^{\xi_{i},j}$, also $y_{i,j}(z_{0})$ is fixed and
equal to

$\widehat{z}_{0}:=\Pi^{\xi_{i,j}}(z_{0})+t_{*}\Pi^{\xi_{i,j}}(e_{i})=\widetilde{z}_{0}+t_{*}\Pi^{\xi_{i,j}}(e_{i}),$

so that in such a case
$[z_{i}(z_{0})\cdot\xi_{i,j},z_{j}(z_{0})\cdot\xi_{i,j}]\subset\widehat{z}_{0}+\mathbb{R}\xi_{i,j}\cap
Q$ and

$|\mathrm{D}\widehat{u}^{\xi_{i,j}}_{y_{i,j}(z_{0})}|([z_{i}(z_{0})\cdot\xi_{i,j},z_{j}(z_{0})\cdot\xi_{i,j}])\leq(\widehat{\mu}_{u})^{\xi_{i,j}}_{\widehat{z}_{0}}((Q\setminus
J_{u}^{1})^{\xi_{i,j}}_{\widehat{z}_{0}})$

regardless of the value of $z^{\prime}_{0}=z_{0}\cdot\xi_{i,j}$ (satisfying
$\widetilde{z}_{0}+z^{\prime}_{0}\xi_{i,j}\in q$ since we integrate over
$z_{0}\in q$). It follows that (notice that
$\mathcal{L}^{1}(\\{s\in\mathbb{R}\colon\widetilde{z}_{0}+z^{\prime}_{0}\xi_{i,j}\in
Q\\})\leq\sqrt{2}$)

$\begin{split}\int_{q}|\mathrm{D}\widehat{u}^{\xi_{i,j}}_{y_{i,j}(z_{0})}|([z_{i}(z_{0})\cdot\xi_{i,j},&z_{j}(z_{0})\cdot\xi_{i,j}])\,\mathrm{d}z_{0}\\\
&=\int\limits_{\widetilde{z}_{0}=\Pi^{\xi_{i,j}}_{z_{0}}\in\xi_{i,j}^{\perp}}\,\mathrm{d}\mathcal{H}^{d-1}(\widetilde{z}_{0})\int\limits_{(\widetilde{z}_{0}+\mathbb{R}\xi_{i,j})\cap
q}|\mathrm{D}\widehat{u}^{\xi_{i,j}}_{y_{i,j}(z_{0})}|([z_{i}(z_{0})\cdot\xi_{i,j},z_{j}(z_{0})\cdot\xi_{i,j}])\,\mathrm{d}z^{\prime}_{0}\\\
&\leq\sqrt{2}\int\limits_{\widetilde{z}_{0}\in\xi_{i,j}^{\perp}}(\widehat{\mu}_{u})^{\xi_{i,j}}_{\widehat{z}_{0}}((Q\setminus
J_{u}^{1})^{\xi_{i,j}}_{\widehat{z}_{0}})\,\mathrm{d}\mathcal{H}^{d-1}(\widetilde{z}_{0})\\\
&\leq\sqrt{2}(\widehat{\mu}_{u})^{\xi_{i,j}}(Q\setminus
J_{u}^{1}).\end{split}$ (2.14)

Summing (2.14) over $0\leq i<j\leq d$, we get

$\int_{q}F(z_{0})\,\mathrm{d}z_{0}\leq\sqrt{2}(d+1)^{2}\widehat{\mu}_{u}(Q\setminus
J_{u}^{1}).$

and there exists $q_{2}\subset q$ with
${\mathcal{L}}^{d}(q_{2})/{\mathcal{L}}^{d}(q)>3/4$ such that for every
$z_{0}\in q_{2}$

$F(z_{0})\leq 4\sqrt{2}(d+1)^{2}\widehat{\mu}_{u}(Q\setminus J_{u}^{1}).$
(2.15)

This is the analogue of [5, condition (2.8)]. At this stage, following [5], it
holds that for any $z_{0}$ satisfying (2.13b) and (2.15) the affine map
$a\colon{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ such that
$a(z_{i}(z_{0}))=u(z_{i}(z_{0}))$ for all $i=0,\dots,d$ satisfies

$|e(a)|\leq c\widehat{\mu}_{u}(Q\setminus J_{u}^{1}).$ (2.16)

Arguing exactly as in Step 3 of the proof of Proposition 2.1 in [5] we find a
set $q_{3}\subset q$ with ${\mathcal{L}}^{d}(q_{3})/{\mathcal{L}}^{d}(q)>3/4$
such that if $z_{0}\in q_{3}$ then

$\begin{split}&\text{ \eqref{analoga25} holds for any $[z_{i}(z_{0}),y]$ for
$y\in Q\setminus\omega$, where }\\\
&\omega=\bigcup_{i=0}^{n}\omega_{(i)},\qquad\omega_{(i)}:=\\{y\in Q\colon
y=z_{i}+t\xi\text{ with }T(z_{i},\xi,t)=1\\}.\end{split}$ (2.17)

With (2.12) and the fact that for any $y$ there are $d$ points in
$\\{z_{0},\dots,z_{d}\\}$ such that the simplex generated by those and $y$ has
volume at least $t_{*}/(d+1)!$, (2.17) implies that

$\begin{split}|w(y)|\leq&c\sum_{i=0}^{n}|\mathrm{D}\widehat{w}^{\xi_{i,y}}_{x_{i,y}(z_{0})}|([z_{i}(z_{0})\cdot\xi_{i,y},y\cdot\xi_{i,y}])\quad\text{for
all }y\in Q\setminus\omega,\text{ where }\\\
&w:=u-a,\quad\xi_{i,y}:=\frac{y-z_{i}(z_{0})}{|y-z_{i}(z_{0})|},\quad
x_{i,y}(z_{0}):=\Pi^{\xi_{i,y}}(z_{i}(z_{0}))\end{split}$ (2.18)

Let us consider the quantity

$H(z_{0}):=\sum_{i=0}^{n}H_{i}(z_{0}),\quad
H_{i}(z_{0}):=\int_{Q\setminus\omega}|\mathrm{D}\widehat{w}^{\xi_{i,y}}_{x_{i,y}(z_{0})}|([z_{i}(z_{0})\cdot\xi_{i,y},y\cdot\xi_{i,y}])\,\mathrm{d}y.$
(2.19)

In the following we prove that there exists $q_{4}\subset q_{3}$ with
${\mathcal{L}}^{d}(q_{4})/{\mathcal{L}}^{d}(q_{3})>3/4$ such that $H(z_{0})$
is controlled by $\widehat{\mu}_{u}(Q\setminus J_{u}^{1})$ times a constant
$c$ depending only on $d$, for every $z_{0}\in q_{4}$. Together with (2.16)
and (2.18) this will conclude the proof.

By Fubini’s Theorem, we have that for every $i=0,\dots,d$

$\int_{q_{3}}H_{i}(z_{0})\,\mathrm{d}z_{0}=\int_{Q\setminus\omega}\int_{q_{3}}|\mathrm{D}\widehat{w}^{\xi_{i,y}}_{x_{i,y}(z_{0})}|([z_{i}(z_{0})\cdot\xi_{i,y},y\cdot\xi_{i,y}])\,\mathrm{d}z_{0}\,\mathrm{d}y.$

For fixed $y\in Q\setminus\omega$ we argue similarly to what done to prove
(2.14), namely we integrate for $z_{0}\in q_{3}$ as
$\widetilde{z}_{0}=\Pi^{\xi_{i,y}}(z_{0})$ ranges in $\xi_{i,y}^{\perp}$ and
$z^{\prime}_{0}=z_{0}\cdot\xi_{i,y}$ ranges in
$(q_{3})_{\widetilde{z}_{0}}^{\xi_{i,y}}$, using Fubini’s Theorem. Then, given
$\widetilde{z}_{0}$, we have that
$x_{i,y}(z_{0})=\widetilde{z}_{0}+t_{*}\Pi^{\xi_{i,y}}(e_{i})=:\widehat{z}_{0}$
and that

$|\mathrm{D}\widehat{w}^{\xi_{i,y}}_{x_{i,y}(z_{0})}|([z_{i}(z_{0})\cdot\xi_{i,y},y\cdot\xi_{i,y}])\leq(\widehat{\mu}_{w})^{\xi_{i,y}}_{\widehat{z}_{0}}((Q\setminus
J_{u}^{1})^{\xi_{i,y}}_{\widehat{z}_{0}}),$

regardless of the value of $z^{\prime}_{0}$. Therefore

$\begin{split}\int_{q_{3}}|\mathrm{D}\widehat{w}^{\xi_{i,y}}_{x_{i,y}(z_{0})}|([z_{i}(z_{0})\cdot\xi_{i,y},y\cdot\xi_{i,y}])\,\mathrm{d}z_{0}&\leq\sqrt{2}\int_{\xi_{i,y}^{\perp}}(\widehat{\mu}_{w})^{\xi_{i,y}}_{\widehat{z}_{0}}((Q\setminus
J_{u}^{1})^{\xi_{i,y}}_{\widehat{z}_{0}})\,\mathrm{d}\mathcal{H}^{d-1}(\widetilde{z}_{0})\\\
&\leq\sqrt{2}(\widehat{\mu}_{w})^{\xi_{i,y}}(Q\setminus
J_{u}^{1})=\sqrt{2}(\widehat{\mu}_{u})^{\xi_{i,y}}(Q\setminus
J_{u}^{1})\end{split}$ (2.20)

where the equality above follows from the fact that $a$ is an infinitesimal
rigid motion. Summing (2.20) over $i$ (and arguing as done for (2.15)) we get
that

$\int_{q_{3}}H(z_{0})\,\mathrm{d}z_{0}\leq c\widehat{\mu}_{u}(Q\setminus
J_{u}^{1}),$

thus we find $q_{4}\subset q_{3}$ with
${\mathcal{L}}^{d}(q_{4})/{\mathcal{L}}^{d}(q_{3})>3/4$ and $H(z_{0})\leq
c\widehat{\mu}_{u}(Q\setminus J_{u}^{1})$ for every $z_{0}\in q_{4}$; we
conclude the proof by picking $z_{0}$ in $q_{1}\cap q_{2}\cap q_{4}$ (which
has positive measure) and integrating (2.18) over $y\in Q\setminus\omega$. ∎

##  3\. Proof of the compactness theorem

In this section we prove Theorem 1.1. In Subsection 3.1 we construct a
suitable partition $\mathcal{P}=(P_{n})_{n}$ of $\Omega$ and a sequence
piecewise rigid functions $(a_{k})_{k}$ satisfying (1.3); in the next two
subsections we prove the existence of $u\in GBD(\Omega)$ satisfying (1.4a) and
the lower semicontinuity condition (1.4b) on the surface measure of
$\partial^{*}\mathcal{P}$, respectively.

###  3.1. Construction of a partition

For $\delta>0$, let
$\mathcal{Q}^{\delta}=\\{z+(-\delta/2,\delta/2]^{d}:z\in\delta\mathbb{Z}^{d},\
z+(-\delta/2,\delta/2)^{d}\subset\Omega\\}$. Let $\eta>0$, small, and define

$\mathcal{B}^{\delta}(u_{k})=\\{Q\in\mathcal{Q}^{\delta}:\mathcal{H}^{d-1}(J^{1}_{u_{k}}\cap
Q)>\eta\delta^{d-1}\\},\quad\mathcal{G}^{\delta}(u_{k})=\mathcal{Q}^{\delta}\setminus\mathcal{B}^{\delta}(u_{k}).$

By construction, one has

$\left|{\textstyle\bigcup_{Q\in\mathcal{B}^{\delta}(u_{k})}Q}\right|\leq\frac{\delta}{\eta}\mathcal{H}^{d-1}(J_{u_{k}}^{1}).$
(3.1)

We fix a first value $\delta_{0}>0$, small enough so that
$\mathcal{G}^{\delta_{0}}(u_{k})\neq\emptyset$ for all $k\geq 1$, and for
$j\geq 0$, denote $\delta_{j}=\delta_{0}2^{-j}$. Upon extracting a
subsequence, we may assume that $\mathcal{B}^{\delta_{0}}(u_{k})$ is not
depending on $k$. By a diagonal argument, we may (and will) assume even that
for any $j\geq 0$, $\mathcal{B}^{\delta_{j}}(u_{k})$ does not depend on $k$ if
$k$ is large enough (depending on $j$). We denote then
$\mathcal{B}^{\delta_{j}}$ (and $\mathcal{G}^{\delta_{j}}$) the corresponding
limiting sets, dropping the dependence in $u_{k}$.

Thanks to Theorem 2.3, for any $j\geq 0$ and each
$Q\in\mathcal{Q}^{\delta_{j}}$, there is $\omega_{k}^{Q}\subset Q$ and
$a_{k}^{Q}$, an infinitesimal rigid motion, such that

$\int_{Q\setminus\omega_{k}^{Q}}|u_{k}-a_{k}^{Q}|dx\leq
c\delta_{j}\widehat{\mu}_{u_{k}}(Q\setminus J_{u_{k}}^{1})$ (3.2)

and $|\omega_{k}^{Q}|\leq c\delta_{j}\mathcal{H}^{d-1}(J_{u_{k}}^{1}\cap Q)$.
In particular:

$Q\in\mathcal{G}^{\delta_{j}}(u_{k})\quad\Rightarrow\quad|\omega_{k}^{Q}|\leq
c\eta|Q|.$ (3.3)

Considering the finite family of sequences of infinitesimal rigid motions
$(a_{k}^{Q}-a_{k}^{Q^{\prime}})_{k}$,
$\\{Q,Q^{\prime}\\}\subset\mathcal{G}^{\delta_{0}}$ (which does not depend on
$k$), we may assume, upon extracting a subsequence, that either
$|a_{k}^{Q}(x)-a_{k}^{Q^{\prime}}(x)|\to\infty$ a.e., or
$\sup_{k}\sup_{|x|\leq 1}|a_{k}^{Q}(x)-a_{k}^{Q^{\prime}}(x)|<+\infty$. By a
diagonal argument, similarly, for $j\geq 1$, considering
$(a_{k}^{Q}-a_{k}^{Q^{\prime}})_{k}$,
$\\{Q,Q^{\prime}\\}\in\mathcal{G}^{\delta_{j}}(u_{k})$, which for $k$ large
enough is $\mathcal{G}^{\delta_{j}}$, not depending on $k$, we may assume the
same.

We let for $j\geq 0$

$B_{j}=\Big{(}\Omega\setminus\bigcup_{Q\in\mathcal{Q}^{\delta_{j}}}Q\Big{)}\cup\bigcup_{l\geq
j}\bigcup_{Q\in\mathcal{B}^{\delta_{l}}}Q.$

Thanks to (3.1), it holds that

$|B_{j}|\leq\left|\Omega\setminus{\textstyle\bigcup_{Q\in\mathcal{Q}^{\delta_{j}}}Q}\right|+\frac{2\delta_{j}}{\eta}\sup_{k}\widehat{\mu}_{u_{k}}(\Omega),$
(3.4)

so that $\lim_{j}|B_{j}|=0$; moreover $(B_{j})_{j}$ is decreasing, that is
$B_{j+1}\subseteq B_{j}$ for all $j\geq 0$.

We define a partition $(P^{j}_{n})_{n=1}^{N_{j}}$ of $\Omega\setminus B_{j}$
as follows: the sequences $(a_{k}^{Q})$, $Q\in\mathcal{G}^{\delta_{j}}$, for
$k$ large enough so that
$\mathcal{G}^{\delta_{j}}(u_{k})=\mathcal{G}^{\delta_{j}}$, can be grouped in
equivalent classes for the relationship $a_{k}^{Q}\sim a_{k}^{Q^{\prime}}$
when $\sup_{k}\sup_{|x|\leq 1}|a_{k}^{Q}(x)-a_{k}^{Q^{\prime}}(x)|<+\infty$.
Then, we say that $Q\sim Q^{\prime}$ whenever $a_{k}^{Q}\sim
a_{k}^{Q^{\prime}}$. We define, for each equivalence class $\mathcal{C}_{n}$
in $\mathcal{G}^{\delta_{j}}$, $n=1,\dots,N_{j}$, the set
$P^{j}_{n}=\bigcup_{Q\in\mathcal{C}_{n}}Q\setminus B_{j}$.

Observe that for any $j\geq 1$, if $Q\in\mathcal{G}^{\delta_{j}}$,
$Q^{\prime}\in\mathcal{G}^{\delta_{j+1}}$ with $Q^{\prime}\subset Q$ (and $k$
is large enough), then one has:

$\int_{Q^{\prime}\setminus(\omega_{k}^{Q}\cup\omega_{k}^{Q^{\prime}})}|a_{k}^{Q}-a_{k}^{Q^{\prime}}|dx\leq
c\delta\widehat{\mu}_{u_{k}}(Q\setminus J_{u_{k}}^{1}),$

so that, provided $\eta>0$ was chosen small enough (to ensure for instance
that $|\omega_{k}^{Q}\cup\omega_{k}^{Q^{\prime}}|\leq|Q^{\prime}|/2$, which is
guaranteed if $\eta<2^{-d}/(4c)$, cf. (3.3)), $\sup_{k}\sup_{|x|\leq
1}|a_{k}^{Q}(x)-a_{k}^{Q^{\prime}}(x)|<+\infty$.

Hence given $Q\in Q^{\delta_{l}}$, $Q^{\prime}\in Q^{\delta_{l^{\prime}}}$,
for $l^{\prime}\geq l\geq j$, and such that $Q^{\prime}\subset Q$ and
$|Q^{\prime}\setminus B_{j}|>0$, one obtains by induction,
$\sup_{k}\sup_{|x|\leq 1}|a_{k}^{Q}(x)-a_{k}^{Q^{\prime}}(x)|<+\infty$ (as all
the intermediate cubes are all “good” at their respective scale).

It follows that: for any smaller scales $l,l^{\prime}\geq j$, and any
$Q\in\mathcal{G}^{\delta_{l}}$,
$Q^{\prime}\in\mathcal{G}^{\delta_{l^{\prime}}}$ with both $Q\setminus B_{j}$
and $Q^{\prime}\setminus B_{j}$ of positive measure and contained in the same
component $P_{n}^{j}$, one finds that $\sup_{k}\sup_{|x|\leq
1}|a_{k}^{Q}(x)-a_{k}^{Q^{\prime}}(x)|<+\infty$. Indeed, there are
$\tilde{Q},\tilde{Q}^{\prime}\in\mathcal{G}^{\delta_{j}}$ with
$Q\subset\tilde{Q}$, $Q^{\prime}\subset\tilde{Q}^{\prime}$ and
$\tilde{Q}\sim\tilde{Q}^{\prime}$.

In particular this shows that for $j^{\prime}\geq j$ and
$n\in\\{1,\dots,N_{j}\\}$, there is
$n^{\prime}\in\\{1,\dots,N_{j^{\prime}}\\}$ such that $P^{j}_{n}\subset
P^{j^{\prime}}_{n^{\prime}}$. We may always number the sets $(P^{j}_{n})_{n}$,
$j\geq 1$, according to the numbering of $(P^{j-1}_{n})_{n}$, so that in fact
$P^{j}_{n}\subset P^{j^{\prime}}_{n}$ for any $j^{\prime}\geq j$ and any
$n\in\\{1,\dots,N_{j}\\}$. As a consequence, we may define, for $1\leq
n<1+\sup_{j}N_{j}\in\mathbb{N}\cup\\{+\infty\\}$, the set
$P_{n}=\bigcup_{j}P^{j}_{n}$ (where the union starts at the first $j$ such
that $n\leq N_{j}$). These sets partition $\Omega\setminus\bigcap_{j}B_{j}$,
hence, up to a Lebesgue-negligible set, $\Omega$.

For each $n$, we choose an arbitrary $Q\in\mathcal{G}^{\delta_{j}}$, at some
arbitrary scale $j\geq 0$, with $|Q\setminus B_{j}|>0$ and $Q\setminus
B_{j}\subset P_{n}$, and we associate to $P_{n}$ the subsequence
$(a_{k}^{Q})_{k}$, hence denoted $(a_{k}^{n})_{k}$. It follows that for any
other such cube $Q$ at any other scale $j$, one has $\sup_{k}\sup_{|x|\leq
1}|a_{k}^{n}(x)-a_{k}^{Q}(x)|<+\infty$, while
$\lim_{k}|a_{k}^{n^{\prime}}(x)-a_{k}^{Q}(x)|=+\infty$ a.e. if $n^{\prime}\neq
n$.

###  3.2. Compactness

We introduce the smooth, one-Lipschitz truncations
$t_{\sigma}(x):=\sigma\tanh(x/\sigma)$, for $\sigma>0$. We also let
$v_{k}=\sum_{n}a_{k}^{n}\chi_{P_{n}}$. Note that at the scale $j\geq 0$, one
has that

$v_{k}|_{\Omega\setminus B_{j}}=\sum_{n}a_{k}^{n}\chi_{P_{n}\setminus
B_{j}}=\sum_{n=1}^{N_{j}}a_{k}^{n}\chi_{P_{n}^{j}},$

showing that $v_{k}|_{\Omega\setminus B_{j}}$ is built up of $N_{j}$
infinitesimal rigid motions.

For each scale $j\geq 0$ let

$w_{k}^{j}=\Bigg{(}\sum_{Q\in\mathcal{G}^{\delta_{j}}(u_{k})}a_{k}^{Q}\chi_{Q}-v_{k}\Bigg{)}(1-\chi_{B_{j}}).$

By construction, $\sup_{k}\sup_{x\in\Omega\setminus
B_{j}}|w_{k}^{j}(x)|<+\infty$ and since $v_{k}|_{\Omega\setminus B_{j}}$ is
built up of a bounded number of affine maps, the sequence of functions
$(w_{k}^{j})_{k}$ is finite-dimensional, and we may extract a subsequence such
that it converges to some limit $w^{j}$. By a diagonal argument, we may assume
this is true for all $j\geq 0$.

For $e\in\mathbb{R}^{d}$, $|e|=1$, $\sigma>0$, we consider the sequences of
functions $u^{e,\sigma}_{k}:=t_{\sigma}(e\cdot(u_{k}-v_{k}))$. We let
$\omega_{k}^{j}:=\bigcup_{Q\in\mathcal{G}^{\delta_{j}}(u_{k})}\omega_{k}^{Q}$,
then $|\omega_{k}^{j}|\leq c\delta_{j}\mathcal{H}^{d-1}(J_{u_{k}}^{1})$. Thus:

$\displaystyle\int_{\Omega}|u^{e,\sigma}_{k}-t_{\sigma}(e\cdot
w_{k}^{j})|dx\leq$
$\displaystyle\sigma\left|B_{j}\cup\omega_{k}^{j}\right|+\int_{\Omega\setminus(B_{j}\cup\omega_{k}^{j})}|e\cdot(u_{k}-w_{k}^{j})|dx$
(3.5) $\displaystyle\leq$
$\displaystyle\sigma\eta_{j}+\sum_{Q\in\mathcal{G}^{\delta_{j}}(u_{k})}\int_{Q\setminus\omega_{k}^{Q}}|u_{k}-a_{k}^{Q}|dx$
$\displaystyle\leq$ $\displaystyle\sigma\eta_{j}+C\delta_{j}$

where we have let
$\eta_{j}=|B_{j}|+c\delta_{j}\sup_{k}\mathcal{H}^{d-1}(J_{u_{k}}^{1})$, and
$C=c\sup_{k}\widehat{\mu}_{u_{k}}(\Omega\setminus J_{u_{k}}^{1})<+\infty$, and
used (3.2). Using that $w_{k}^{j}-w_{l}^{j}\to 0$ as $k,l\to\infty$ in
$L^{1}(\Omega)$, we find that:

$\limsup_{k,l\to\infty}\int_{\Omega}|u^{e,\sigma}_{k}-u^{e,\sigma}_{l}|dx\leq
2(\sigma\eta_{j}+C\delta_{j}).$

Sending $j\to+\infty$ we find that
$\limsup_{k,l\to\infty}\int_{\Omega}|u^{e,\sigma}_{k}-u^{e,\sigma}_{l}|dx=0$,
that is, $(u^{e,\sigma}_{k})_{k}$ are Cauchy sequences and converge to some
limit $u^{e,\sigma}$ in $L^{1}(\Omega)$.

We show now that the limit $u^{e,\sigma}$ is $t_{\sigma}(e\cdot u)$ for some
well-defined measurable function $u$. Let us consider a subsequence such that
$u^{e,1}_{k}\to u^{e,1}$ a.e. In that case:

  * •

either $|u^{e,1}(x)|=1$, which happens if and only if
$\lim_{k\to\infty}|e\cdot(u_{k}(x)-v_{k}(x))|=+\infty$, and in particular for
any $\sigma>0$, $|u^{e,\sigma}(x)|=\sigma$;

  * •

or, by continuity, $e\cdot(u_{k}(x)-v_{k}(x))\to\tanh^{-1}(u^{e,1}(x))$, and
we also have that $u^{e,\sigma}(x)=t_{\sigma}(\tanh^{-1}(u^{e,1}(x)))$ for any
$\sigma>0$.

Let $A=\\{x:|u^{e,1}(x)|=1\\}=\\{x:|u^{e,\sigma}(x)|=\sigma\\}$: then, for
$j\geq 0$,

$\int_{A\setminus B_{j}}|u^{e,\sigma}_{k}-t_{\sigma}(e\cdot
w_{k}^{j})|dx\geq\int_{A\setminus B_{j}}|u^{e,\sigma}_{k}|dx-\int_{A\setminus
B_{j}}|w_{k}^{j}|dx\stackrel{{\scriptstyle
k\to\infty}}{{\longrightarrow}}\sigma|A\setminus B_{j}|-\int_{A\setminus
B_{j}}|w^{j}|dx.$

On the other hand thanks to (3.5):

$\int_{A\setminus B_{j}}|u^{e,\sigma}_{k}-t_{\sigma}(e\cdot
w_{k}^{j})|dx\leq\sigma\eta_{j}+C\delta_{j},$

hence:

$\sigma|A\setminus B_{j}|\leq\int_{A\setminus
B_{j}}|w^{j}|dx+\sigma\eta_{j}+C\delta_{j}.$

Dividing by $\sigma$ and letting $\sigma\to\infty$, we deduce:

$|A\setminus B_{j}|\leq\eta_{j},$

so that $|A|=0$. It follows that $\tanh^{-1}u^{e,1}$ is finite a.e.

To sum up, we have shown that for any $e$ with $|e|=1$, there is a measurable
function $u^{e}$ such that $t_{\sigma}(e\cdot(u_{k}-v_{k}))\to
t_{\sigma}(u^{e})$ in $L^{1}(\Omega)$ for any $\sigma>0$. It is then obvious
to check that $u^{e}=e\cdot u$ for some measurable vector-valued function $u$,
and, up to a subsequence, to deduce that $u_{k}-v_{k}\to u$ a.e. in $\Omega$.

###  3.3. Lower semicontinuity

We argue similarly to what done in [9, Step 2 in Section 3] to prove the
$GSBD^{p}$ analogue of (1.4b), that is [9, equation (1.5d)].

Let us fix $\sigma>1$, $\xi\in{\mathbb{S}^{d-1}}$ in a set of full
$\mathcal{H}^{d-1}$-measure of ${\mathbb{S}^{d-1}}$ for which (2.10) holds
(cf. Lemma 2.2), and define

$\mathrm{I}^{\sigma,\xi}_{y}(u_{k}):=|\mathrm{D}(\widehat{u}_{k})^{\xi}_{y}|\big{(}\Omega^{\xi}_{y}\setminus
J^{\sigma}_{(\widehat{u}_{k})^{\xi}_{y}}\big{)}\,.$ (3.6)

Since

$\mathrm{I}^{\sigma,\xi}_{y}(u_{k})\leq(\widehat{\mu}_{u_{k}})^{\xi}_{y}(\Omega^{\xi}_{y})+(\sigma-1)\mathcal{H}^{0}\big{(}\Omega^{\xi}_{y}\cap(J^{1}_{(\widehat{u}_{k})^{\xi}_{y}}\setminus
J^{\sigma}_{(\widehat{u}_{k})^{\xi}_{y}})\big{)}\leq\sigma(\widehat{\mu}_{u_{k}})^{\xi}_{y}(\Omega^{\xi}_{y}),$

it holds that

$\int_{\Pi^{\xi}}\mathrm{I}^{\sigma,\xi}_{y}(u_{k})\,\mathrm{d}\mathcal{H}^{d-1}(y)\leq\sigma\widehat{\mu}^{\xi}_{u_{k}}(\Omega)\leq\sigma\sup_{k\in\mathbb{N}}\widehat{\mu}_{u_{k}}(\Omega).$
(3.7)

Following exactly [9, Step 2 in Section 3] for $\sigma>1$ fixed (using (3.7)
in place of [9, estimate (3.11)]), we get that all the [9, (3.12)-(3.18)] hold
for $J_{v}$ replaced by $J^{\sigma}_{v}$ and $\widehat{I}^{\xi}_{y}$ replaced
by $\mathrm{I}^{\sigma,\xi}_{y}$. In particular, for $\mathcal{H}^{d-1}$-a.e.
$\xi\in{\mathbb{S}^{d-1}}$ and $\mathcal{H}^{d-1}$-a.e. $y\in\Pi_{\xi}$, along
a suitable subsequence $(\cdot)_{j}$ depending on $\sigma$, $\xi$,
$\varepsilon\in(0,\sigma^{-2})$ fixed as in [9, (3.13)], and $y$, it holds
that

$(\widehat{u}_{j}-\widehat{a}_{j})^{\xi}_{y}\to\widehat{u}^{\xi}_{y}\quad\mathcal{L}^{1}\text{-a.e.\
in }\Omega^{\xi}_{y},$ (3.8)
$|(\widehat{a}^{i_{1}}_{j}-\widehat{a}^{i_{2}}_{j})^{\xi}_{y}(t)|=|(\widehat{a}^{i_{1}}_{j}-\widehat{a}^{i_{2}}_{j})^{\xi}_{y}(0)|\to+\infty\quad\text{for
}t\in\Omega^{\xi}_{y}\text{ and }i_{1}\neq i_{2},$ (3.9)
$\lim_{j\to\infty}\Big{(}\mathcal{H}^{0}\big{(}J^{\sigma}_{(\widehat{u}_{j})^{\xi}_{y}}\big{)}+\varepsilon\mathrm{I}^{\sigma,\xi}_{y}(u_{j})\Big{)}=\liminf_{m\to\infty}\Big{(}\mathcal{H}^{0}\big{(}J^{\sigma}_{(\widehat{u}_{m})^{\xi}_{y}}\big{)}+\varepsilon\mathrm{I}^{\sigma,\xi}_{y}(u_{m})\Big{)}=M(y)\in\mathbb{R}$
(3.10)

for $(\cdot)_{m}$ a subsequence of $(\cdot)_{k}$ independent of $y$ such that

$H^{\sigma}_{\varepsilon,\xi}:=\lim_{m\to\infty}\int_{\Pi_{\xi}}\Big{(}\mathcal{H}^{0}\big{(}J^{\sigma}_{(\widehat{u}_{m})^{\xi}_{y}}\big{)}+\varepsilon\mathrm{I}^{\sigma,\xi}_{y}(u_{m})\Big{)}\,\mathrm{d}\mathcal{H}^{d-1}(y)\in\mathbb{R}$
(3.11)

and

$\limsup_{\varepsilon\to
0}\int_{{\mathbb{S}^{d-1}}}H^{\sigma}_{\varepsilon,\xi}\,\mathrm{d}\mathcal{H}^{d-1}(\xi)\leq\liminf_{k\to\infty}\mathcal{H}^{d-1}(J^{\sigma}_{u_{k}}).$
(3.12)

Denoting

$\partial\mathcal{P}^{\xi}_{y}:=\bigcup_{n\in\mathbb{N}}\partial\left((P_{n})^{\xi}_{y}\right)\cap\Omega^{\xi}_{y}\subset\Omega^{\xi}_{y},$

by (3.10) we may assume, up to a further subsequence, that
$\mathcal{H}^{0}\big{(}J^{\sigma}_{(\widehat{u}_{j})^{\xi}_{y}}\big{)}=N_{y}\in\mathbb{N}$
for every $j$ and so that there are $\widehat{N}_{y}\leq N_{y}$ cluster points
in the limit, denoted by

$t_{1},\dots,t_{\widehat{N}_{y}}.$

Therefore we have that $K\cap
J^{\sigma}_{(\widehat{u}_{j})^{\xi}_{y}}=\emptyset$ for any $K$ compact subset
of $(t_{l},t_{l+1})$, so
$|\mathrm{D}(\widehat{u}_{k})^{\xi}_{y}|\big{(}K\setminus
J^{\sigma}_{(\widehat{u}_{j})^{\xi}_{y}}\big{)}=|\mathrm{D}(\widehat{u}_{j})^{\xi}_{y}|(K)$;
with (3.10) and the Fundamental Theorem of Calculus, this gives that, for
$\mathcal{L}^{1}$-almost any choice of $\overline{t}\in(t_{l},t_{l+1})$,

$t\mapsto(\widehat{u}_{j})^{\xi}_{y}(t)-(\widehat{u}_{j})^{\xi}_{y}(\overline{t})\text{
are equibounded w.r.t.\ $j$ in }BV_{\mathrm{loc}}(t_{l},t_{l+1}),$ (3.13)

so the bound above is also in $L^{\infty}_{\mathrm{loc}}(t_{l},t_{l+1})$. At
this stage we prove, as in [9, (3.20)], that

$\partial\mathcal{P}^{\xi}_{y}\subset\\{t_{1},\dots,t_{\widehat{N}_{y}}\\}:$
(3.14)

in fact, assuming by contradiction that there exists $l\in\\{1,\dots,M_{y}\\}$
and $i_{1}\neq i_{2}$ such that
$\partial(P_{i_{1}})^{\xi}_{y}\cap(t_{l},t_{l+1}),\partial(P_{i_{2}})^{\xi}_{y}\cap(t_{l},t_{l+1})\neq\emptyset$,
by (3.8) there are two corresponding sequences of infinitesimal rigid motions
$(a_{j}^{i_{1}})_{j}$, $(a_{j}^{i_{2}})_{j}$ for which

$\begin{split}&(\widehat{u}_{j}-\widehat{a}_{j}^{i_{1}})^{\xi}_{y}\to\widehat{u}^{\xi}_{y}\quad\mathcal{L}^{1}\text{-a.e.\
in }(P_{i_{1}})^{\xi}_{y}\cap(t_{l},t_{l+1}),\\\
&(\widehat{u}_{j}-\widehat{a}_{j}^{i_{2}})^{\xi}_{y}\to\widehat{u}^{\xi}_{y}\quad\mathcal{L}^{1}\text{-a.e.\
in }(P_{i_{2}})^{\xi}_{y}\cap(t_{l},t_{l+1}),\end{split}$ (3.15)

with
$\mathcal{L}^{1}\big{(}(P_{i_{1}})^{\xi}_{y}\cap(t_{l},t_{l+1})\big{)},\,\mathcal{L}^{1}\big{(}(P_{i_{2}})^{\xi}_{y}\cap(t_{l},t_{l+1})\big{)}>0$;
this gives (with (3.13) and since $\widehat{a}_{j}^{i}$ are infinitesimal
rigid motions and $\widehat{u}^{\xi}_{y}\colon\Omega^{\xi}_{y}\to\mathbb{R}$)
that $(\widehat{a}^{i_{1}}_{j}-\widehat{a}^{i_{2}}_{j})^{\xi}_{y}$ is constant
in $\Omega^{\xi}_{y}$ and uniformly bounded w.r.t. $j$, in contradiction with
(3.9). Therefore, (3.14) is proven.

Integrating (3.14) over $y\in\Pi^{\xi}$ and using Fatou’s lemma with (3.10),
(3.11) we deduce that

$\begin{split}\int\limits_{\Pi^{\xi}}\mathcal{H}^{0}(\partial\mathcal{P}^{\xi}_{y})\,\mathrm{d}\mathcal{H}^{d-1}(y)\leq\lim_{m\to\infty}\int\limits_{\Pi^{\xi}}\Big{(}\mathcal{H}^{0}\big{(}J^{\sigma}_{(\widehat{u}_{m})^{\xi}_{y}}\big{)}+\varepsilon\,\mathrm{I}^{\sigma,\xi}_{y}(u_{m})\Big{)}\,\mathrm{d}\mathcal{H}^{d-1}(y)\end{split}$
(3.16)

for every $\sigma>1$ and $\mathcal{H}^{d-1}$-a.e. $\xi\in{\mathbb{S}^{d-1}}$
(in view of the choice of the subsequences, see [9]). This implies that
$\mathcal{P}$ is a Caccioppoli partition.

Moreover, integrating over $\xi\in{\mathbb{S}^{d-1}}$ we get that

$\begin{split}\mathcal{H}^{d-1}(\partial^{*}\mathcal{P}\cap\Omega)&\leq
C\varepsilon\sigma\sup_{k\in\mathbb{N}}\widehat{\mu}_{u_{k}}(\Omega)+\int_{{\mathbb{S}^{d-1}}}H^{\sigma}_{\varepsilon,\xi}\,\mathrm{d}\mathcal{H}^{d-1}(\xi)\\\
&\leq
C\sqrt{\varepsilon}\sup_{k\in\mathbb{N}}\widehat{\mu}_{u_{k}}(\Omega)+\int_{{\mathbb{S}^{d-1}}}H^{\sigma}_{\varepsilon,\xi}\,\mathrm{d}\mathcal{H}^{d-1}(\xi)\end{split}$
(3.17)

for a universal constant $C>0$ and every $\sigma>1$,
$\varepsilon\in(0,\sigma^{-2})$. Letting $\varepsilon\to 0$, in view of (3.12)
and the arbitrariness of $\sigma>1$ we conclude (1.4b).

Let us now confirm that $u\in GBD(\Omega)$. For any $\xi\in{\mathbb{S}^{d-1}}$
and $\mathcal{H}^{d-1}$-a.e. $y\in\Pi^{\xi}$, setting
$\tilde{u}_{k}:=u_{k}-a_{k}$ for any $k\in\mathbb{N}$, it holds that

$(\widehat{\mu}_{\tilde{u}_{k}})^{\xi}_{y}(B)\leq(\widehat{\mu}_{u_{k}})^{\xi}_{y}(B)+\mathcal{H}^{0}(\partial^{*}\mathcal{P}^{\xi}_{y}\cap
B)\quad\text{for every $B\subset\Omega^{\xi}_{y}$ Borel},$

since $a_{k}$ is a piecewise rigid motion constant on every $P_{j}$,
$j\in\mathbb{N}$, where $\mathcal{P}=(P_{j})_{j}$. Integrating over
$\Pi_{\xi}$ and recalling (2.5) we have that

$\widehat{\mu}_{\tilde{u}_{k}}(B)\leq\widehat{\mu}_{u_{k}}(B)+\mathcal{H}^{d-1}(\partial^{*}\mathcal{P}\cap
B)\quad\text{for every $B\subset\Omega$ Borel}.$

Since $\tilde{u}_{k}$ pointwise converges ${\mathcal{L}}^{d}$-a.e. to $u$,
there is an increasing function
$\psi_{0}\colon\mathbb{R}^{+}\to\mathbb{R}^{+}$ with
$\lim_{s\to+\infty}\psi_{0}(s)=+\infty$ such that
$\|\psi_{0}(\tilde{u}_{k})\|_{L^{1}(\Omega)}$ is uniformly bounded w.r.t.
$k\in\mathbb{N}$ (see e.g. [16, Lemma 2.1]). Then we may apply [11, Corollary
11.2] to deduce that $u\in GBD(\Omega)$.

This concludes the proof of Theorem 1.1.

######  Remark 3.1.

Let us consider a sequence $(u_{k})_{k}$ such that

$\int_{\Omega}|e(u_{k})|^{p}\,\mathrm{d}x+\mathcal{H}^{d-1}(J_{u_{k}})\leq
M,\quad p>1$ (3.18)

for $M>0$ independent of $k\in\mathbb{N}$. Applying Theorem 1.1 we obtain the
compactness part of [9, Theorem 1.1] (that is [9, (1.5b)] for $u\in
GBD(\Omega)$) without using the Korn-type inequality in [4]. Combining this
with the last part of the proof ([9, Steps 2-3 in proof of Theorem 1.1]) we
obtain [9, Theorem 1.1], and in particular (1.4c). Nevertheless, the result in
[4] is crucial for [9, Theorem 1.2].

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