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\title[Quasistatic crack growth in elasto-plastic materials]
{Quasistatic crack growth 
in elasto-plastic materials:\\
the two-dimensional case
 }
\author[Gianni Dal Maso]{Gianni Dal Maso}
\address[Gianni Dal Maso]{SISSA, Via Beirut 2-4, 34014 Trieste,
Italy}
\email[Gianni Dal Maso]{dalmaso@sissa.it}

\author[Rodica Toader]{Rodica Toader}
\address[Rodica Toader]{Universit\`a di Udine, D.I.C.A., Via delle Scienze 208, 33100
 Udine, Italy}
\email[Rodica Toader]{toader@uniud.it}


\begin{document}

\begin{abstract}
We study a variational model for the quasistatic evolution of elasto-plastic 
materials with cracks in the  case of planar small strain associative elasto-plasticity.
\end{abstract}
\maketitle
{\small

\bigskip
\keywords{\noindent {\bf Keywords:} fracture mechanics, small strain associative elasto-plasticity,  
free-discontinuity problems, quasistatic evolution problems,
rate independent processes, variational models.
}


\bigskip
\subjclass{\noindent {\bf 2000 Mathematics Subject Classification:}
35R35,  %Free boundary problems for PDE
74R10,  %Brittle fracture
74C05, %Small-strain, rate-independent theories 
%(including rigid-plastic and elasto-plastic materials
49Q10.  %Optimization of shapes other than minimal surfaces
}

\bigskip
\bigskip

\begin{section}{Introduction}
%:Introduction

\blfootnote{Preprint SISSA 62/2008/M (September 2008)}
In this paper we present a variational model for the quasistatic 
evolution of elasto-plastic materials with cracks. 
The variational approach to fracture mechanics \cite{B-F-M} 
is based on Griffith's idea \cite{Grif} that the crack growth is determined 
by the competition between the elastic energy released when the crack 
increases and the energy dissipated to produce new crack. 
Many mathematical models  have been developed starting from this idea, 
but so far, all the models where the crack path is not prescribed deal with 
the case of brittle fracture (see \cite{F-M}, \cite{DM-Toa}, \cite{Ch}, 
\cite{DM-Toa-2}, \cite{Fra-Lar}, \cite{DM-F-T}). 

Our model (Subsection \ref{quasist})  is formulated in the general 
framework of the energy method for rate independent processes 
developed by Mielke \cite{Mielke}. The dissipation (Subsection \ref{dissip}) 
is assumed to depend both on the increase in the crack length and on the 
variation of the plastic strain. Out of the cracks the evolution is governed 
by the model of  small strain linearized elasto-plasticity developed by 
Suquet \cite{Suq} and revisited in \cite{DM-DS-M}. To be precise 
(Subsection \ref{model}), we consider the case of pressure sensitive 
elasto-plastic materials  \cite{DM-D-DS}. 

A mathematical difficulty concerning the lower semicontinuity of the 
dissipation (Lemma \ref{semicont}) imposes a severe limitation to 
our model: the problem is formulated in the plane and the cracks 
must satisfy an a priori bound on the number of connected components. 
In the paper we present  the case of plane elasticity. The antiplane case  
can be treated in a similar way.

Our main result is the existence of a quasistatic evolution with prescribed 
time dependent boundary displacements (Theorem \ref{main}). 
To obtain this result we prove a new version (Theorem \ref{jumptransfer}) 
of  the jump transfer theorem of \cite{Fra-Lar}, which takes into account 
the additive decomposition of the strain into an elastic and a plastic part. 

We also prove that, under suitable regularity conditions, the variational 
solution considered in our model satisfies the usual flow rule of linearized 
elasto-plasticity (Theorem \ref{flowrule}). 
Moreover we show that the crack tips cannot be isolated from the plastic 
region (Theorem \ref{thtip}). Finally, we prove that when the plastic yield 
surface tends to infinity the solutions obtained in our model converge, 
in a very strong way, to a solution according to the model of linearly elastic 
materials with brittle cracks introduced in \cite{DM-Toa} and \cite{Ch} (Theorem \ref{convbrittle}).

\end{section}




\begin{section}{Formulation of the problem}\label{formulation}
%:Formulation of the problem

\subsection{Notation and preliminaries.}



Throughout the paper $\Ltwo$  and $\hone$ denote the Lebesgue measure in $\Rtwo$ and the 
{$1$-dimensional} Hausdorff measure, respectively. Unless otherwise specified, the 
expression {\em almost everywhere\/} (abbreviated as {\em a.e.\/}) always refers 
to~$\Ltwo$.
Given a finite-dimensional Hilbert space $\Xi$ and a locally compact subset $E$ of 
$\Rtwo$, the space of $\Xi$-valued bounded Radon measures on $E$ is denoted by
$M_b(E;\Xi)$. For every measure $\mu\in M_b(E;\Xi) $, the symbol $|\mu|$ denotes
 its  total variation,  while $\mu^a$ and $\mu^s$ denote the absolutely continuous and 
 singular parts with respect to $\Ltwo$. The former will always be identified with its density. 
 For every $|\mu|$-measurable set $A$, the measure $\mu\LLL A$ 
 is defined by $(\mu\LLL A)(B):=\mu(A\cap B)$ for every  $|\mu|$-measurable set $B$.

By the Riesz Representation Theorem (see, e.g., \cite[Theorem 6.19]{Rud}) 
$M_b(E;\Xi)$ can be identified with the dual of $C_0(E;\Xi)$, the space of continuous 
functions $\varphi\colon E\to\Xi$ such that the set $\{|\varphi|\ge\e\}$ is compact for 
every $\e>0$. The weak$^*$ topology on $M_b(E;\Xi)$ is defined using this duality. 
The dual norm, which induces the strong topology on $M_ b(E;\Xi)$, is given by 
$\|\mu\|_1=|\mu|(E)$. It coincides with the $L^1$-norm if $\mu=\mu^a$. 
If $1< r\le\infty$, we use the notation 
$\|\cdot\|_r$ or $\|\cdot\|_{r,E}$ for the $L^r$ norm on $E$ with respect to $\Ltwo$ 
or $\hone$ (or to some other measure which will be clear from the context).

It is well-known that a sequence $\mu_n$ converges to $\mu$ weakly$^*$ in $M_ b(E;\Xi)$
 if and only if $\|\mu_n\|_1$ is bounded and 
 $\langle \varphi,\mu_n\rangle\to\langle \varphi,\mu\rangle$ for every test function 
 $\varphi\in C_c(E;\Xi)$, where $\langle \cdot,\cdot\rangle$ denotes the duality product 
 between $C_0(E;\Xi)$ and $M_ b(E;\Xi)$ (as well as other pairs of spaces, according 
 to the context) and  $C_c(E;\Xi)$ denotes the dense subspace of $C_0(E;\Xi)$ 
 composed of all continuous functions $\varphi\colon E\to\Xi$ with compact support in $E$.

Let $U$ be a bounded open set in $\Rtwo$. 
The Lipschitz part $\partial_LU$  of the boundary of $U$ is the set of points $x\in\partial U$ 
such that $U$ is the sub-graph of a Lipschitz function near $x$, in the sense that there 
exist an orthogonal coordinate system $(y_1,y_2)$, a rectangle $R={]a,b[}{\times}{]c,d[}$ 
containing $x$, and a Lipschitz function $\Phi\colon{]a,b[}\to{]c,d[}$, such that 
$U\cap R=\{y\in U:y_2<\Phi(y_1)\}$.

We recall that the space $BD(U)$ of functions of bounded deformation is defined 
as the space of functions $u\in L^1(U;\R^2)$ such that the symmetric part of the gradient  
$Eu:=\frac 1 2 (Du+(Du)^T)$ is a matrix-valued bounded Radon measure  (see 
\cite{Tem}).   We refer to \cite{ACDM} for the decomposition $Eu=E^au+E^cu+E^ju$, 
as well as for other fine properties of the 
functions in $BD(U)$.  The following proposition is proved in  \cite[Theorem II.2.3]{Tem}.

\begin{proposition}\label{bdlbd1}
Assume that $U$ has a Lipschitz boundary and let $u\in L^1_{loc}(U;\R^2)$ with 
$Eu\in M_ b(U;\Msym)$. Then $u\in BD(U)$.
\end{proposition}

 By applying Proposition \ref{bdlbd1} to $U\cap R$ we obtain the following corollary.
\begin{corollary}\label{bvlbv2}
Let $u\in L^1_{loc}(U;\R^2)$ with $Eu\in M_ b(U;\Msym)$. For every $x\in\partial_LU$ 
there exists an open neighbourhood $V$ of $x$ such that $u\in BD(U\cap V)$.
\end{corollary}

Note that under the assumptions of the previous result the trace of $u$ on $\partial_LU$, 
still denoted by $u$, is well-defined and belongs to $L^1_{loc}(\partial_LU;\R^2)$.  

  
Let $\Ku$ be the set of all compact subsets
of $\overline U$.  
We recall that the {\it Hausdorff distance\/} between
$K_{1},\, K_{2}\in\Ku$ is defined by
$$
d_{H}(K_{1},K_{2}):=
\max\big\{ \sup_{x\in K_1}{\rm dist}(x,K_2),
\sup_{y\in K_2}{\rm dist}(y,K_1)\big\}\,,
$$
with the conventions  ${\rm dist}(x,\emptyset)={\rm diam}(U)$ and
$\sup\emptyset=0$, so that $d_{H}(\emptyset, K)=0$ if $K=\emptyset$
and $d_{H}(\emptyset, K)={\rm diam}(U)$ if $K\neq\emptyset$.
We say that $K_n\to K$ in the Hausdorff metric if
${d_H(K_n,K)\to0}$. It is well-known that the space $\Ku$ endowed with the 
Hausdorff metric is compact (see, e.g., 
\cite[Blaschke's Selection Theorem]{Rog}).
\subsection{Cracks, displacements, and plastic strains} \label{model}
 Let $\Om$ be a bounded connected open set in $\Rtwo$ with Lipschitz boundary
 $\partial\Om$.  The set $\overline\Om$  represents the 
 {\em reference configuration\/} of an elasto-plastic body with cracks. 
  Let us fix a compact set $\Gamma_0$ contained in $\overline\Om$, with 
  $\hone(\Gamma_0)<+\infty$ and with a finite number of connected components. 
The set $\Gamma_0\cap\Om$ plays the role of the {\em initial crack}, while 
$\Gamma_0\cap\partial\Om$ represents  the {\em traction free part of the boundary}.   
 
Throughout the paper we fix an integer $m$, 
larger than or equal to the number of connected components of $\Gamma_0$. 
The set $\Kmf$ of  {\em admissible cracks} is defined as the collection of all compact 
subsets $\Gamma$ of $\overline\Om$ containing $\Gamma_0$, having at most $m$ 
connected components, and with 
 $\hone(\Gamma)<+\infty$. The bound on the number of connected components 
 is due only to mathematical reasons and will be crucial in some compactness and 
 semicontinuity arguments. 
 
The set $\Om_0:=\Om\setminus\Gamma_0$ represents the {\em initial elasto-plastic region}. 
We will prescribe a time dependent boundary displacement on the 
{\em Dirichlet part of the boundary} $\partial_D\Om_0:=\partial\Om\setmeno\Gamma_0$. 
In this paper we do not
 study the case of cracks originated by discontinuities imposed
 on $\partial_D\Om_0$. Therefore, for every time $t\in{[0,T]}$ we consider only 
 {\em prescribed boundary displacements}
 that are traces on $\partial_D\Om_0$ of functions 
$w(t)\in H^1(\Om_0;\R^2)$, so that there is always a
 configuration with finite energy without additional cracks which satisfies the
 boundary conditions. If the displacement of the solution will develop sharp 
 discontinuities out of $\Gamma_0$, this will be the result of the minimization process.
 
 We assume
 \be\label{w}
 w\in AC({[0,T]};H^1(\Om_0;\R^2))\,,
 \ee
 so that the time derivative
 $t\mapsto\dot w(t)$ belongs to the space  
 $L^1({[0,T]}; H^{1}(\Om_0;\R^2))$ and its spatial gradient
 $t\mapsto\nabla \dot w(t)$ belongs to the space
$L^1({[0,T]};L^2(\Om_0;\mathbb M^{2{\times}2}))$.


For every compact set $\Gamma\subset\overline\Om$ we 
define
\be\label{omgamma}
\Om_\Gamma:=\Om\setmeno\Gamma\,,\quad\hat\Om_\Gamma:=
\overline\Om\setmeno\Gamma\,,\quad\hbox{and}
\quad\partial_D\Om_\Gamma:=\partial\Om\setmeno\Gamma\,,
\ee
and we
consider the space 
$$
BD_\ell(\Om_\Gamma,\partial_D\Om_\Gamma):=\{v\in BD_{loc}(\Om_\Gamma): 
Ev\in M_b(\Om_\Gamma;\Msym)\,,\ v\in L^1(\partial_D\Om_\Gamma;\R^2)\}\,. 
$$


If $\Gamma\supset\Gamma_0$ and 
$w\in H^1(\Om_0;\R^2)$, 
 we say that $u\in BD_\ell(\Om_\Gamma,\partial_D\Om_\Gamma)$,
 $e\in L^2(\Om;\Msym)$, and $p\in M_ b(\overline\Om;\Msym)$ satisfy the 
 {\em weak kinematic admissibility conditions} on $\overline\Om$ if
\begin{eqnarray}
&& Eu=e+p\qquad\qquad\quad\hbox{in }\Om_\Gamma\,,\label{adm1k}\\
&& p=(w-u)\,{\odot}\,\nu_\Om\hone\quad\hbox{ on }\partial_D\Om_\Gamma\,,
\label{adm2k}\\
&& e=0\quad\hbox{and}\quad p=0\quad\quad\hbox{on } \Gamma\,,
\label{adm3k}
\end{eqnarray}
where $\nu_\Om$ is the outer unit normal to $\partial\Om$, and 
$a\,{\odot}\,b$ is the symmetrized tensor product between two vectors $a,b\in \R^2$, 
i.e., the symmetric matrix with entries $(a_ib_j+a_jb_i)/2$. 
The function $u$ plays the role of the {\em displacement}, $w$ is the 
prescribed boundary condition, while $e$ and $p$ represent the {\em elastic} and the 
{\em plastic strain}, respectively. 
The condition on $\partial\Om$ shows that the prescribed displacement is not attained 
on $\Gamma\cap\partial_D\Om_0=\Gamma\cap\partial\Om$, nor on the part of 
$\partial_D\Om_\Gamma$ where a  plastic slip occurs. Equality \eqref{adm2k} gives 
the quantitative relation between the plastic slip at the boundary and the mismatch in 
the boundary condition.

The {\em stored elastic energy} depends only on the elastic strain and is defined by 
$$
\Q(e):=\int_\Om Q(e(x))\,dx\,,
$$ 
 where 
 $$
 Q(\xi):=\tfrac 1 2\C\xi\,{:}\,\xi\,.
 $$
 Here and henceforth $\C$ is the {\em elasticity tensor} and the colon denotes the 
 scalar product between matrices:
 $\xi\,{:}\,\zeta:={\rm tr}(\xi\zeta^T)=\sum_{ij}\xi_{ij}\zeta_{ij}$.

We assume that 
\be\label{lowerboundC}
\alpha_\C|\xi|^2\le Q(\xi)\le\beta_\C|\xi|^2\quad\hbox{for every }\xi\in\Msym\,.
\ee
for some constants $0<\alpha_\C\le\beta_\C$.
These inequalities imply
\be\label{upperboundC}
|\C\xi|\le 2\beta_\C|\xi|\quad\hbox{for every }\xi\in\Msym\,.
\ee

Let $\K$ be a closed convex set in $\Msym$, whose boundary plays the role of the 
{\em yield surface}. 
We assume that there exist two constants $r_\K$ and $R_\K$ with $0<r_\K\le R_\K<+\infty$ such that 
\be\nonumber
\{\xi\in\Msym:|\xi|\le r_\K\}\subset \K\subset \{\xi\in\Msym:|\xi|\le R_\K\}\,.
\ee
The {\em support function} $H\colon\Msym\to[0,+\infty[$ of $\K$ is defined by
\be\label{support}
H(\xi):=\sup_{\sigma\in\K}\sigma\,{:}\,\xi\,.
\ee
Then $H$ turns out to be convex and positively homogeneous of degree one. 
The previous bounds 
on $\K$ imply that
\be\label{Hrk}
r_\K|\xi|\le H(\xi)\le R_\K|\xi|\,.
\ee

For every measure $\mu\in M_ b(\overline\Om;\Msym)$ let $\tfrac{\mu}{|\mu|}$ be the 
Radon-Nikodym derivative of $\mu$ with respect to its variation $|\mu|$. 
According to the theory of convex functions of measures (see \cite{G-S}) we define the 
measure $H(\mu)\in M_ b(\overline\Om)$ by setting
\be\label{defH}
H(\mu)(B):=\int_BH\Big(\frac{\mu}{|\mu|}(x)\Big)d|\mu|(x)
\ee
for every Borel set $B\subset\overline\Om$.

From (\ref{adm1k}) and (\ref{adm2k}) it follows that the measure $p$ vanishes on every 
$\hone$-negligible set and can be decomposed in a unique way as
$$
p=p^a+p^c+p^j\,,
$$
where $p^a$ is absolutely continuous with respect to $\Ltwo$, $p^c$ is singular 
with respect to 
$\Ltwo$ and vanishes on all Borel sets $B$ with $\hone(B)<+\infty$, 
while $p^j$ is concentrated 
on a $\sigma$-finite set for the measure $\hone$. 
More precisely, from (\ref{adm1k}) and  (\ref{adm2k}) we deduce that on the set 
$\hat\Om_\Gamma$ defined in \eqref{omgamma} we have
\begin{equation}\label{p}
p^a=E^a u-e \,,\quad p^c=E^cu\,,\quad 
p^j=[u]\,{\odot}\,\nu_u\hone\LLL J_u+
(w-u)\,{\odot}\,\nu_\Om\hone\LLL\partial_D\Om_\Gamma\,,
\end{equation}
where $J_u$ is the jump set of $u$, $\nu_u$ is the approximate unit normal to $J_u$, and 
$[u]:=u^+-u^-$ is the amplitude of the jump of $u$ (see \cite{ACDM}). 
The measures $p^c$ and $p^j$ will be called the Cantor part and the jump part of $p$, 
respectively.



Given $w\in H^1(\Om_0;\R^2)$ and a compact set $\Gamma$ with 
$\Gamma_0\subset\Gamma\subset\overline\Om$, we define the set 
$A(w,\Gamma)$ of {\em admissible strains} as the set  of all  $(e,p)$, with 
$$
e\in L^2(\Om;\Msym) \quad \hbox{and}\quad
p\in  M_ b(\overline\Om;\Msym)\,, 
$$
 such that there exists $u\in BD_\ell(\Om_\Gamma,\partial_D\Om_\Gamma)$ for 
 which $u$, $e$, $p$ satisfy the weak kinematic 
admissibility conditions \eqref{adm1k}--\eqref{adm3k} on~${\overline\Om}$.
Finally, we consider the set $A(w)$ of {\em admissible triples}, defined as the set of all 
$(e,p,\Gamma)$ with $\Gamma\in\Kmf$ and $(e,p)\in A(w,\Gamma)$.

\subsection{The dissipation of an elasto-plastic process with cracks}\label{dissip}

We define the {\em dissipation  distance} on $  M_ b(\overline\Om;\Msym){\times}\Kmf$ by setting
\begin{equation}\label{D}
{\mathcal{D}} ((p_2,\Gamma_2),(p_1,\Gamma_1)):=
\begin{cases}
H(p_2-p_1)(\hat\Om_2)+
\hone(\Gamma_2\setmeno \Gamma_1) &
\hbox{ if }\Gamma_1\subset \Gamma_2\\
+\infty & \hbox{ otherwise,}
\end{cases}
\end{equation}
where $\hat\Om_2:=\overline\Om\setmeno\Gamma_2$.
Then ${\mathcal{D}} $ is a nonsymmetric semidistance, i.e., it is nonnegative and  satisfies the triangle inequality. 
The next lemma states the  lower semicontinuity properties of ${\mathcal{D}}$.
\begin{lemma}\label{semicont}
Let  $(p_n^1,\Gamma_n^1)$ and $(p_n^2,\Gamma_n^2)$ be two sequences in 
$  M_ b(\overline\Om;\Msym){\times}\Kmf$. For $i=1,2$ assume 
that 
\begin{eqnarray*}
\Gamma_n^i\to \Gamma^i\quad\hbox{in the Hausdorff metric}\,,\\ 
p_n^i\wtos p^i \quad\hbox{weakly}{}^* 
\hbox { in }  M_ b(\hat\Om_i;\Msym)\,,
\end{eqnarray*}
where $\hat\Om_i:=\overline\Om\setmeno\Gamma^i$.
Then
$$
{\mathcal{D}} ((p^2,\Gamma^2),(p^1,\Gamma^1))\le
\liminf_{n\to\infty}{\mathcal{D}} ((p_n^2,\Gamma_n^2),(p_n^1,\Gamma_n^1))\,.
$$
\end{lemma}
\begin{proof} For $i=1,2$ and  for every $n$ let us define 
$\hat\Om_n^i:=\overline\Om\setmeno \Gamma_n^i$. 
We may assume that the liminf is finite. Then $\Gamma_n^1\subset \Gamma_n^2$ 
for infinitely many indices~$n$, which gives $\Gamma^1\subset \Gamma^2$. 
Therefore the conclusion follows from the inequalities
\begin{eqnarray}
&\displaystyle H(p^2-p^1)(\hat\Om_2)\le \liminf_{n\to\infty}H(p_n^2-p_n^1)
(\hat\Om_n^2)\,,\label{semicont1}\\
&\displaystyle \hone(\Gamma^2\setmeno \Gamma^1)\le \liminf_{n\to\infty}
\hone(\Gamma_n^2\setmeno \Gamma_n^1)\,.
\label{semicont2}
\end{eqnarray}
To prove \eqref{semicont1} we consider an open set $U$ with 
$\overline U\cap \Gamma^2=\emptyset$. 
Since $\overline U\cap \Gamma_n^2=\emptyset$ for $n$ large enough, by 
\cite[Theorem 2]{Re} we obtain
$$
H(p^2-p^1)(\overline\Om\cap U)\le \liminf_{n\to\infty}H(p_n^2-p_n^1)(\overline\Om\cap U)\le
\liminf_{n\to\infty}H(p_n^2-p_n^1)(\hat\Om_n^2)\,.
$$
Taking the supremum over all open sets $U$ with 
$\overline U\cap \Gamma^2=\emptyset$ we obtain \eqref{semicont1}.

Inequality \eqref{semicont2} is a consequence of the Gol\c ab semicontinuity 
theorem and is proved in \cite[Corollary 3.4]{DM-Toa}. 
\end{proof}

Let $t\mapsto (p(t),\Gamma(t))$ be a function from $[a,b]$ into 
$  M_ b(\overline\Om;\Msym){\times}\Kmf$,
and let 
\be\label{omt}
\Om(t):=\Om\setmeno\Gamma(t)\,,\quad\hat\Om(t):=
\overline\Om\setmeno\Gamma(t)\,,\quad \hbox{and}
\quad \partial_D\Om(t):=\partial\Om\setmeno\Gamma(t)\,.
\ee
The {\em dissipation} of $(p(\cdot),\Gamma(\cdot))$ on the 
interval $[a,b]$ is defined as 
\be
 {\rm Diss}(p(\cdot),\Gamma(\cdot);a,b):=
 \sup \sum_{i=1}^k{\mathcal{D}} ((p(t_i), \Gamma(t_i)),(p(t_{i-1}),\Gamma(t_{i-1})))\,,\label{dissd}
\ee
where the supremum is taken over all finite partitions $a=t_0\le t_1\le \dots\le t_k=b$.

We say that a function $t\mapsto \Gamma(t)$ from $[a,b]$ into $\Kmf$ 
is increasing if $\Gamma(s)\subset \Gamma(t)$ for every $s<t$. 
{}From the definition of ${\mathcal{D}} $ and  ${\rm Diss}$ it follows that 
\begin{equation}\label{irrev}
 {\rm Diss}(p(\cdot),\Gamma(\cdot);a,b)<+\infty\qquad\Longrightarrow\qquad
t\mapsto \Gamma(t) \hbox{ is increasing}\,. 
\end{equation}
Moreover, for every $t\in[a,b]$  we have 
\begin{eqnarray}
& r_\K|p(t)|(\hat\Om(t))\le r_\K|p(a)|(\hat\Om(a))+ {\rm Diss}(p(\cdot),\Gamma(\cdot);a,b)\,,\label{varp}\\
& \hone(\Gamma(t))\le \hone(\Gamma(a))+ {\rm Diss}(p(\cdot),\Gamma(\cdot);a,b)\,.\label{misK}
\end{eqnarray}

\begin{theorem}\label{dotp}
Let $t\mapsto (p(t),\Gamma(t))$ be a function from $[a,b]$ into $  M_ b(\overline\Om;\Msym){\times}\Kmf$ such that 
$ {\rm Diss}(p(\cdot),\Gamma(\cdot);a,b)<+\infty$. Then for a.e.\ $t\in[a,b]$ there exists a measure 
$\dot p(t)\in   M_ b(\overline\Om;\Msym)$ with the following properties:
\begin{eqnarray}
&\displaystyle \tfrac 1 h (p(t)-p(t-h))\wtos \dot p(t)\hbox{ weakly}{}^* 
\hbox { in }  M_ b(\hat\Om(t);\Msym)\hbox{ as }h\to0+\,,\label{increm}\\
& \dot p(t)=0\hbox{ on }\Gamma(t)\,.\label{zero}
\end{eqnarray}
\end{theorem}
\begin{proof} We recall that \eqref{increm} means, by definition, that 
$$
\langle\varphi,\tfrac 1 h (p(t)-p(t-h))\rangle\to\langle\varphi, \dot p(t)\rangle
\quad\hbox{for every }\varphi\in C^0(\overline\Om;\Msym)\hbox{  with }
\varphi=0\hbox{ on }\Gamma(t)\,.
$$
For every $t_1,t_2\in[a,b]$ with $t_1\le t_2$ we have 
\be\label{rk}
|p(t_2)-p(t_1)|(\hat\Om(t_2))\le \tfrac{1}{r_\K} {\rm Diss}(p(\cdot),\Gamma(\cdot);t_1,t_2)\,.
\ee
We shall use the first part of \cite[Theorem 7.1]{DM-DS-M}, which holds also for 
functions of bounded variation.
Let $D$ be a countable dense set of $[a,b]$.  For every $\tau\in D$ we can apply 
the above mentioned theorem to the interval $[a,\tau]$ and to the set 
$\hat\Om(\tau)$, and we deduce that there exists a set $N_\tau\subset [a,\tau]$, 
with 
${\mathcal{L}}^1(N_\tau)=0$, such that for every $t\in[a,\tau]\setmeno N_\tau$ the sequence 
$ \tfrac 1 h (p(t+h)-p(t))$
converges  weakly$^*$ in $  M_ b(\hat\Om(\tau);\Msym)$ as 
$h\to0$. Let $V(t):= {\rm Diss}(p(\cdot),\Gamma(\cdot);a,t)$. 
Since $t\mapsto V(t)$ and $t\mapsto\Gamma(t)$ are increasing, by 
\cite[Proposition 6.1]{DM-Toa} there exists a set $N\subset [a,b]$, with 
${\mathcal{L}}^1(N)=0$ and containing all sets $N_\tau$ for $\tau\in D$, such that 
for every $t\in [a,b]\setmeno N$
\be\label{conv}
\tfrac 1 h (V(t+h)-V(t))\to \dot V(t)<+\infty\ 
\hbox{and}\ \Gamma(t+h)\to\Gamma(t)\ \hbox{in the Hausdorff metric}
 \ee
as $h\to0$. 

Let us fix $t\in [a,b]\setmeno N$. Then there exists a Radon measure  on 
$\hat\Om(t)$, that we denote by $\dot p(t)$, such that for every $\tau\in D$ with 
$\tau>t$ we have
\begin{equation}\label{weaklim1}
\tfrac 1 h (p(t+h)-p(t))\wtos \dot p(t)\quad\hbox{ weakly}{}^* 
\hbox { in }  M_ b(\hat\Om(\tau);\Msym)\quad\hbox{ as }h\to0\,.
\end{equation}
By \eqref{rk} for every $h>0$ and every $\tau\in D$ with $\tau>t$ we have 
\be\label{diseq1}
\tfrac {1} {h} |p(t)-p(t-h)|(\hat\Om(\tau))\le\tfrac {1} {h} |p(t)-p(t-h)|(\hat\Om(t))\le  
\tfrac {1} {r_\K h} (V(t)-V(t-h))\,,
\ee
so that \eqref{conv} and \eqref{weaklim1} imply $r_\K|\dot p|(\hat\Om(\tau))\le \dot V(t)$. 
By \eqref{conv}, passing to the limit we obtain
\be\label{dotpdotv}
|\dot p(t)|(\hat\Om(t))\le \tfrac{1}{r_\K} \dot V(t)<+\infty\,.
\ee
We now extend the measure $\dot p(t)$ to $\overline\Om$ by setting 
$\dot p(t)=0$ on $\Gamma(t)$, and we obtain that 
$\dot p(t)\in   M_ b(\overline\Om;\Msym)$. It remains to prove \eqref{increm}. 
Let us fix $\varphi\in C_0(\hat\Om(t);\Msym)$. By \eqref{conv}, for every $\e>0$ 
there exist $\tau_\e\in D$, with $\tau_\e>t$, and 
$\varphi_\e\in C_c(\hat\Om(\tau_\e);\Msym)$ such that
$\|\varphi_\e-\varphi\|_\infty<\e$. By \eqref{diseq1} and \eqref{dotpdotv} for 
every $h>0$ we have
\begin{eqnarray*}
&\displaystyle  |\langle \varphi,\tfrac 1 h (p(t)-p(t-h))\rangle - \langle \varphi,\dot p(t)\rangle |\le
 |\langle \varphi_\e,\tfrac 1 h (p(t)-p(t-h))\rangle - \langle \varphi_\e,\dot p(t)\rangle |+\\
&\displaystyle+\tfrac {\e }{r_\K h} (V(t)-V(t-h))+\tfrac {\e }{r_\K} \dot V(t)\,.
\end{eqnarray*}
From \eqref{weaklim1} we obtain
$$
\limsup_{h\to0+}|\langle \varphi,\tfrac 1 h (p(t)-p(t-h))\rangle - 
\langle \varphi,\dot p(t)\rangle |\le\tfrac {2\e }{r_\K} \dot V(t)\,.
$$
Since $\e>0$ is arbitrary, this proves \eqref{increm}.
\end{proof}


The following result is an immediate consequence of Lemma~\ref{semicont}.
\begin{lemma}\label{lsemicont2}
Let $t\mapsto (p_n(t),\Gamma_n(t))$ be a sequence of functions from $[a,b]$ into 
$  M_ b(\overline\Om;\Msym){\times}\Kmf$. Assume that for every $t\in[a,b]$
\begin{eqnarray*}
& \Gamma_n(t)\to \Gamma(t)\quad\hbox{in the Hausdorff metric,}\\ 
& p_n(t)\wtos p(t)\quad\hbox{weakly}{}^* 
\hbox { in }  M_ b(\hat\Om(t);\Msym)\,.
\end{eqnarray*}
Then
$$
{\rm Diss}(p(\cdot),\Gamma(\cdot);a,b)\le
\liminf_{n\to\infty}{\rm Diss}(p_n(\cdot),\Gamma_n(\cdot);a,b).
$$
\end{lemma}

\subsection{The notion of quasistatic evolution.}\label{quasist}
We now introduce  a notion of quasistatic evolution 
in the framework of the energetic formulation of rate-independent 
processes developed by Mielke (see~\cite{Mielke}).
\begin{definition}\label{defquasist} Assume that $w\in AC([0,T];H^1(\Om_0;\R^2))$.
A {\em quasistatic evolution} with boundary condition $w$ is a function 
$t\mapsto(e(t),p(t),\Gamma(t))$ from $[0,T]$ into 
$L^2(\Om;\Msym){\times}  M_ b(\overline\Om;\Msym){\times}\Kmf$ 
which satisfies the following conditions:
\begin{itemize}
\item[(a)] {\em global stability}: for every $t\in[0,T]$ we have $(e(t),p(t),\Gamma(t))\in 
A(w(t))$ and
\be\label{glstab}
\Q(e(t))\le \Q(e^*)+{\mathcal{D}}((p^*\!,\Gamma^*),(p(t),\Gamma(t)))
\quad\hbox{for every }\,(e^*\!,p^*\!,\Gamma^*)\in A(w(t))\,;
\ee
\item[(b)] {\em energy balance}: the function $t\mapsto e(t)$ belongs to 
$L^\infty([0,T];L^2(\Om;\Msym))$ and for every $t\in{[0,T]}$
\be\label{eneq}
\Q(e(t))+{\rm Diss}(p(\cdot),\Gamma(\cdot);0,t)=
\Q(e(0))+\int_0^t\langle \sigma(s),E\dot w(s)\rangle\,ds\,,
\ee
where $\langle\cdot,\cdot\rangle$ is the scalar product in 
$L^2(\Om_0;\Msym)$, and 
$\sigma(t):=\C e(t)$.
\end{itemize}
\end{definition}

\begin{remark} It is enough to verify the inequality in \eqref{glstab} when 
$\Gamma(t)\subset\Gamma^*$. Indeed, if this inclusion is not satisfied, then 
${\mathcal{D}}((p^*\!,\Gamma^*),(p(t),\Gamma(t)))=+\infty $ by \eqref{D}. 
Moreover, the energy balance and \eqref{irrev} imply  the irreversibility of 
the crack growth, i.e.,  the fact that 
$t\mapsto \Gamma(t)$ is increasing.
\end{remark}

The integral term in \eqref{eneq} represents the energy put into the system 
by the prescribed boundary displacement in the interval $[0,T]$.

We shall study the existence of a quasistatic evolution with prescribed 
{\em initial data}\break 
$(e_0,p_0,\Gamma_0)$. By condition (a) (global stability) we must assume that  
\be\label{min000}
\begin{cases}
(e_0,p_0,\Gamma_0)\in A(w(0))\,,\vspace{2pt}\\
\Q(e_0)\le Q(e^*)+{\mathcal{D}}((p^*\!,\Gamma^*),(p_0,\Gamma_0))
\quad\hbox{for every }(e^*\!,p^*\!,\Gamma^*)\in A(w(0))\,.
\end{cases}
\ee

We conclude this section by stating the main result of the paper.
\begin{theorem}\label{main}
Let $w\in AC([0,T];H^1(\Om_0;\R^2))$ and $(e_0,p_0,\Gamma_0)\in A(w(0))$. 
Assume that \eqref{min000} holds. Then there exists a quasistatic evolution with 
boundary condition $w$ and with $(e(0),p(0),\Gamma(0))\allowbreak=(e_0,p_0,\Gamma_0)$.
\end{theorem}

\end{section}

\begin{section}{Preliminary results}\label{prelim}
%:Preliminary results

Throughout the paper we fix a  bounded open set $\tilde\Om$ containing 
$\overline\Om$ and with smooth boundary. 
As usual in the study of functionals with linear growth, the (relaxed) boundary condition 
is obtained by imposing a constraint on 
$\tilde\Om\setmeno\overline\Om$.
 
 Due to the presence of the cracks that might disconnect part of the domain 
 from the Dirichlet boundary, it is not easy to control the displacements through the strains. 
 We overcome these difficulties using the following two lemmas.
\begin{lemma}\label{4.1}
Let $\Gamma_n$ be a sequence of compact subsets of $\overline\Om$ and let 
$v_n\in BD_{loc}(\tilde\Om\setmeno \Gamma_n)$ with $v_n=0$ a.e.\ on 
$\tilde\Om\setmeno\overline\Om$. Let $\Gamma$ be a compact subset of 
$\overline\Om$ and let $\mu\in   M_ b(\tilde\Om\setmeno \Gamma;\Msym)$. 
Assume that $\Gamma_n \to \Gamma$ in the Hausdorff metric and 
$Ev_n\wtos\mu$ weakly$^*$ in  $  M_ b(U;\Msym)$ 
for every open set $U\subset\subset\tilde\Om\setmeno \Gamma$.
Then there exists a function $v\in BD_{loc}(\tilde\Om\setmeno \Gamma)$ such that 
$v=0$ a.e.\ on $\tilde\Om\setmeno\overline\Om$ and 
$Ev=\mu$ on $\tilde\Om\setmeno \Gamma$.
\end{lemma}
\begin{proof} It is enough to prove the result on each connected component $V$ of 
$\tilde\Om\setmeno \Gamma$. It is easy to see that $V$ can be expressed as the 
union of a sequence of connected open sets $V_k$ with Lipschitz boundary such 
that $V_k\subset\subset V_{k+1}$ for every $k$. As $\Gamma_n\to \Gamma$ 
in the Hausdorff metric, for every $k$ the inclusion 
$V_k \subset\subset\tilde\Om\setmeno \Gamma_n$ holds for $n$ large enough. 
Under our assumptions we have $v_n\in BD(V_k)$, $|Ev_n|(V_k)$ is bounded 
uniformly with respect to~$n$, and $v_n=0$ a.e.\ on $V_k\setmeno \overline\Om$. 

If $V\setmeno\overline\Om\neq\emptyset$,  for $k$ large enough we have 
$V_k\setmeno\overline\Om\neq\emptyset$, and therefore, there exist a subsequence, 
still denoted $v_n$, and a function 
 $v\in BD(V_k)$ such that $v_n\to v$ in $L^1(V_k;\R^2)$ and $Ev_n\wtos Ev$ 
 weakly$^*$ in  $  M_ b(V_k;\Msym)$. This implies $v=0$ a.e.\ on 
 $V_k\setmeno\overline\Om$ and $Ev=\mu$ on $V_k$. 
 Repeating this argument  for every $k$ large enough, we construct
  $v\in BD_{loc}(V)$ such that $v=0$ a.e.\ on $V\setmeno\overline\Om$ and $Ev=\mu$ on $V$.

If, instead, $V\setmeno\overline\Om=\emptyset$, 
by the Poincar\'e-Wirtinger inequality in $BD$ 
(see \cite[Chapter II, Proposition 2.2]{Tem})  for every $k$ there is a constant $M_k$ such that 
$$
\int_{V_k}|v_n-R_n|\,dx\le M_k\,|Ev_n|(V_k)\qquad\hbox{for every }n\,,
$$
where $R_n$ is a rigid motion depending on $v_n$. It is easy to see that one 
can choose the same $R_n$ for every $k$.
It follows that $v_n-R_n$ is bounded in $BD(V_k)$ uniformly with respect to $n$,
therefore, there exist a subsequence, still denoted $v_n$, and a function 
 $v\in BD(V_k)$ such that $v_n-R_n\to v$ in $L^1(V_k;\R^2)$ and $Ev_n\wtos Ev$
 weakly$^*$ in  $  M_ b(V_k;\Msym)$. This implies  $Ev=\mu$ on $V_k$.  
 Repeating this argument  for every $k$, we construct $v\in BD_{loc}(V)$ 
 such that $Ev=\mu$ on $V$.
\end{proof}


\begin{lemma}\label{closure} For every $n$ let 
$w_n\in H^1(\Om_0;\R^2)$, let $\Gamma_n$ be a compact set with 
$\Gamma_0\subset \Gamma_n\subset\overline\Om$, and let 
$(e_n,p_n)\in A(w_n,\Gamma_n)$. Let   
$w\in H^1(\Om_0;\R^2)$,  $e\in L^2(\Om;\Msym)$, $p\in  M_ b(\overline\Om;\Msym)$, 
and let $\Gamma$ be a compact set contained in $\overline\Om$. Assume that
\begin{eqnarray}
&& w_n\wto w\qquad\,\hbox{weakly in }H^1(\Om_0;\R^2)\,,\nonumber\\
&& e_n\wto e\qquad\,\,\hbox{ weakly in }L^2(\Om;\Msym)\,,\nonumber\\
&& p_n\wtos p\qquad\,\hbox{ weakly$^*$ in }   M_ b(\hat\Om_\Gamma;\Msym)
\hbox{ and }p=0\hbox{ on }\Gamma\,,\nonumber\\
&& \vphantom{L^2(\Om;\Msym)}\Gamma_n\to \Gamma 
\qquad\hbox{ in the Hausdorff metric}\,,\nonumber\\
&& \vphantom{L^2(\Om;\Msym)}\hbox{either } \Ltwo(\Gamma)=0
\hbox{ or }\Gamma\subset\Gamma_n \hbox{ for every }n\,.\label{3.1}
\end{eqnarray}
Then $(e,p)\in A(w,\Gamma)$.
\end{lemma}
\begin{proof}
As $(e,p)\in A(w,\Gamma)$ if and only if $(e-Ew,p)\in A(0,\Gamma)$, 
it is not restrictive to assume $w_n=w=0$. We proceed as 
in the proof of \cite[Lemma 2.1]{DM-DS-M}.
Let $\Om_n:=\Om_{\Gamma_n}$, $\partial_D\Om_n:=\partial\Om\setmeno\Gamma_n$, 
and $u_n\in BD_\ell(\Om_n,\partial_D\Om_n)$ be such that 
\eqref{adm1k}--\eqref{adm3k} are satisfied by $u_n$, $e_n$, $p_n$, and $\Gamma_n$. 
The functions $u_n$, $e_n$, and $e$, as well as the measures $p_n$ and $p$, 
are extended to $0$ on  $\tilde\Om\setmeno\overline\Om$, keeping the same notation.
Our assumptions imply that 
\begin{eqnarray}
&& e_n\wto e\qquad\,\,\hbox{ weakly in }L^2(\tilde\Om;\Msym)\,,\nonumber\\
&& p_n\wtos p\qquad\,\hbox{ weakly$^*$ in }   M_ b(\tilde\Om\cap U;\Msym)\,,\nonumber
\end{eqnarray}
for every open set $U\subset\subset\Rtwo\setmeno \Gamma$.
By \eqref{adm1k} and \eqref{adm2k} we have  $Eu_n=e_n+p_n$ on 
$\tilde\Om\setmeno \Gamma_n$.
Therefore 
$$
Eu_n\wtos e+p\qquad\,\hbox{ weakly$^*$ in }   M_ b(\tilde\Om\cap U;\Msym)
$$
for every open set $U\subset\subset\Rtwo\setmeno \Gamma$.
By Lemma \ref{4.1}, there exists $u\in BD_{loc}(\tilde\Om\setmeno \Gamma)$ 
such that $u=0$ a.e.\ on $\tilde\Om\setmeno\overline\Om$ and 
$Eu=e+p$ on $\tilde\Om\setmeno \Gamma$.
This implies $u\in BD_\ell(\Om_\Gamma,\partial_D\Om_\Gamma)$ and 
\begin{eqnarray*}
&& Eu=e+p\qquad\,\hbox{ in }\Om_\Gamma\,,\label{adm1U}\\
&& p=-u\,{\odot}\,\nu_\Om\hone\qquad\hbox{on }\partial_D\Om_\Gamma \,,\label{adm2U}
\end{eqnarray*}
where, in the last line $u$ denotes the trace on $\partial_D\Om_\Gamma$ 
from $\Om_\Gamma$. As $p=0$ on $\Gamma$ by assumption, and $e=0$ a.e.\ 
on $\Gamma$ by \eqref{3.1}, these equalities imply that  $(e,p)\in A(0,\Gamma)$. 
\end{proof}

The next theorem is the building block in the construction of the discrete time 
approximations obtained by solving incremental minimum problems.

\begin{theorem}\label{esistmin}
Let $w_1\in H^1(\Om_0;\R^2)$, $p_1\in M_b(\overline\Om;\Msym)$, and 
 $\Gamma_1\in\Kmf$. Then  the minimum problem 
\begin{equation}\label{min}
\min\, \{\Q(e)+{\mathcal{D}}((p,\Gamma),(p_1,\Gamma_1)):  (e,p,\Gamma)\in A(w_1)\}
\end{equation}
has a solution.
\end{theorem}
\begin{proof}
Since $(Ew_1,0,\Gamma_1)\in A(w_1)$ and 
${\mathcal{D}}((0,\Gamma_1),(p_1,\Gamma_1))<+\infty$, the infimum is finite.
Let $(e^n,p^n,\Gamma^n)$ be a minimizing sequence.
By the definition of ${\mathcal{D}}$, there exists  a constant $C>0$ such that
$$
\Q(e^n)+H(p^n-p_1)(\hat\Om^n)+\hone(\Gamma^n\setmeno \Gamma_1)\le C
$$
for every $n$, where $\hat\Om^n:=\overline\Om\setmeno \Gamma^n$. 
Since $|p^n|(\Gamma^n)=0$ we obtain 
$$
\Q(e^n)+r_\K|p^n|(\overline\Om)+\hone(\Gamma^n)\le C+
r_\K|p_1|(\overline\Om)+\hone(\Gamma_1)\,.
$$
Then there exists a subsequence, still denoted by $(e^n,p^n,\Gamma^n)$,
such that $e^n$ converges to a function $e$ weakly in $L^2(\Om;\Msym)$,  
$p^n$ converges to a measure $\tilde p$ weakly$^*$ in $  M_ b(\overline\Om;\Msym)$, and 
$\Gamma^n$ converges  to a set $\Gamma\in \Kmf$ in the Hausdorff metric 
(see, e.g. \cite[Corollary 3.3]{DM-Toa}).  
We define $p:=\tilde p\LLL\hat\Om_\Gamma$. Then $p=0$ on $\Gamma$ and 
$p^n\wtos p$ weakly$^*$ in $  M_ b(\hat\Om_\Gamma;\Msym)$.

By Lemma~\ref{semicont} 
it follows that 
\begin{equation}\label{semi3}
\Q(e)\le\liminf_{n\to\infty}\Q(e^n)\quad\hbox{and}\quad
{\mathcal{D}}((p,\Gamma),(p_1,\Gamma_1))\le
\liminf_{n\to\infty}{\mathcal{D}}((p^n,\Gamma^n),(p_1,\Gamma_1))\,.
\end{equation}
By Lemma~\ref{closure} we have  $(e,p,\Gamma)\in A(w_1)$,
and from \eqref{semi3} we conclude that $(e,p,\Gamma)$ solves the minimum 
problem~\eqref{min}.
\end{proof}

\begin{remark}\label{unique}
If $(e,p,\Gamma)$ and $(\tilde e,\tilde p,\Gamma)$ are solutions of the minimum 
problem~\eqref{min}, then $e=\tilde e$ a.e.\ in $\Om$. Indeed, from the minimality  we have 
$\Q(e)+H(p-p_1)(\hat\Om_\Gamma)=\Q(\tilde e)+H(\tilde p-p_1)(\hat\Om_\Gamma)$. 
If the equality $e=\tilde e$ a.e.\ in $\Om$ is not satisfied, by the strict convexity of $\Q$ we 
obtain 
$\Q(\frac 12 e+\frac 12 \tilde e)+H(\frac 1 2p+\frac 1 2\tilde p-p_1)(\hat\Om_\Gamma)<
\Q(e)+H(p-p_1)(\hat\Om_\Gamma)$. 
Since $(\frac 12 e+\frac 12 \tilde e,\frac 1 2p+\frac 1 2\tilde p,\Gamma)\in A(w_1)$, 
this contradicts the minimality of $(e,p,\Gamma)$.

In the same way it can be shown that if $\Gamma_1$ is a compact subset of 
$\overline\Om$ containing $\Gamma_0$, and  $(e,p)$ and $(\tilde e,\tilde p)$ solve the minimum problem
\begin{equation}\label{minred}
\min\, \{\Q(e)+H(p-p_1)(\hat\Om_{\Gamma_1}):  (e,p)\in A(w_1,\Gamma_1)\}\,,
\end{equation}
 then  $e=\tilde e$ a.e.\ in $\Om$.

\end{remark}

Let us recall that given any bounded open subset $U$ in $\R^2$, 
the symbol $LD(U)$ denotes 
the space of functions $u\in L^1(U;\R^2)$ whose symmetrized gradient 
$Eu$ belongs to $L^1(U;\Msym)$. 
For every compact set $\Gamma\subset\overline\Om$ we consider  the space 
$$
LD_\ell(\Om_\Gamma,\partial_D\Om_\Gamma):=\{v\in LD_{loc}(\Om_\Gamma) : 
Ev\in L^1(\Om_\Gamma;\Msym),\ v\in L^1(\partial_D\Om_\Gamma;\R^2)\}\,.
$$

In the following lemma we introduce the normal component of $\sigma$ on 
$\partial_D\Om_\Gamma$ under very weak regularity assumptions, 
and we prove some of its properties. 
\begin{lemma}\label{l666}
Let $\Gamma$ be a compact subset of $\overline\Om$ and let 
$\sigma\in L^\infty(\Om_\Gamma;\Msym)$ with 
${\rm div}\sigma\in L^2(\Om_\Gamma;\R^2)$. Then there exists a unique function 
$[\sigma\nu_\Om]\in L^\infty(\partial_D\Om_\Gamma;\R^2)$ such that 
\be\label{5550}
\langle [\sigma\nu_\Om],v\rangle_{\partial_D\Om_\Gamma}=
\langle {\rm div}\sigma,v\rangle_{\Om_\Gamma}+
\langle \sigma, Ev\rangle_{\Om_\Gamma}
\ee
for every $v\in LD(\Om)$ with $v=0$ a.e.\ in a neighbourhood of $\Gamma$, and 
there exists a constant $c>0$, depending on $\Om$, but independent of $\Gamma$ 
and $\sigma$, such that 
\be\label{5551}
\| [\sigma\nu_\Om]\|_\infty\le c(\|\sigma\|_\infty+\|{\rm div}\sigma\|_2)\,.
\ee
The definition of $ [\sigma\nu_\Om]$ is local, i.e., 
if $\hat\sigma$ satisfies the same hypotheses as $\sigma$ and coincides with 
$\sigma$ a.e.\ on $U\cap\Om_\Gamma$, with $U$ an open set,  then 
\be\label{sigma12}
 [\sigma\nu_\Om]= [\hat\sigma\nu_\Om]\quad\hone\hbox{-a.e.\ on }
 U\cap\partial_D\Om_\Gamma\,.
\ee
Moreover, 
\be\label{5552}
[\sigma_k\nu_\Om]\wtos[\sigma\nu_\Om]\quad
\hbox{ weakly$^*$ in }L^\infty(\partial_D\Om_\Gamma;\R^2)\,,
\ee
whenever $\sigma_k\wtos\sigma$ weakly$^*$ in 
$L^\infty(\Om_\Gamma;\Msym)$ and ${\rm div}\sigma_k\wto{\rm div}\sigma$ 
weakly in $L^2(\Om_\Gamma;\R^2)$.  Finally, if $\sigma\in C^0(\hat\Om_\Gamma;\Msym)$ we have also
\be\label{5553}
[\sigma\nu_\Om](x)=\sigma(x)\,\nu_\Om(x)\quad\hbox{for $\hone$-a.e.\ }x\in\partial_D\Om_\Gamma\,.
\ee
\end{lemma}
\begin{proof}Since $\Om\subset\R^2$ is bounded and has a Lipschitz boundary, 
we have the embedding $LD(\Om)\subset L^2(\Om;\R^2)$ 
(see \cite[Chapter II, Theorem 1.2]{Tem}) so that the right-hand side of \eqref{5550} 
is well-defined. Moreover, it vanishes for every 
$v\in C^1_c(\Om_\Gamma;\R^2)$. By approximation we can show that  it vanishes 
for every $v\in LD(\Om)$, with $v=0$ a.e.\ in a neighbourhood of $\Gamma$, 
whose trace on  $\partial_D\Om_\Gamma$ is identically zero. It follows that the right-hand 
side of \eqref{5550}  depends only on the trace of $v$ on $\partial_D\Om_\Gamma$ whenever
 $v\in LD(\Om)$ and $v=0$ a.e.\ in a neighbourhood of $\Gamma$. 
 Therefore, it defines  a distribution  on $\partial_D\Om_\Gamma$, denoted by 
 $[\sigma\nu_\Om]$. 

In order to prove that $[\sigma\nu_\Om]\in L^\infty(\partial_D\Om_\Gamma;\R^2)$ 
and satisfies \eqref{5551} it is enough to show that  every point of 
$\partial_D\Om_\Gamma$ is contained in an open rectangle $R$  such that 
$[\sigma\nu_\Om]\in L^\infty(R\cap \partial_D\Om_\Gamma;\R^2)$ and 
\eqref{5551} holds for the restriction of $[\sigma\nu_\Om]$ to 
$R\cap \partial_D\Om_\Gamma$. We may assume that $R\cap\Om$ is the subgraph 
of a Lipschitz function, and that $R\cap\Gamma=\emptyset$ so that 
$R\cap\partial_D\Om_\Gamma=R\cap\partial\Om$. 
Given $\varphi\in L^1(R\cap\partial\Om;\R^2)$ with compact support in 
$R\cap\partial\Om$,  by the Gagliardo Extension Theorem \cite{Ga} we can find 
$v\in W^{1,1}(R\cap\Om;\R^2)$ such that $v=\varphi$ $\hone$-a.e.\ on 
$R\cap\partial\Om$, $v=0$ $\hone$-a.e.\ on $\Om\cap\partial R$, and 
$\|v\|_1+\|Ev\|_1\le c_1\|\varphi\|_1$, where the constant $c_1$ depends only on $\Om$, 
but not on $R$. 
We can extend $v$ to $\Om$ by setting $v=0$ on $\Om\setmeno R$ and the 
previous inequality is still satisfied. By using the continuity of the embedding 
$LD(\Om)\subset L^2(\Om;\R^2)$ we obtain the estimate $\|v\|_2+\|Ev\|_1\le c_2\|\varphi\|_1$ 
for a suitable constant $c_2$ depending only on $\Om$. 
By \eqref{5550} this implies that 
$|\langle [\sigma\nu_\Om],\varphi\rangle_{\partial_D\Om_\Gamma}|\le 
c_2(\|\sigma\|_\infty+\|{\rm div}\sigma\|_2)\|\varphi\|_1$, and hence
$[\sigma\nu_\Om]\in L^\infty(R\cap \partial\Om;\R^2)$ and \eqref{5551} holds for 
the restriction of $[\sigma\nu_\Om]$ to $R\cap \partial\Om$. 
We conclude that $[\sigma\nu_\Om]\in L^\infty(\partial_D\Om_\Gamma;\R^2)$ 
and satisfies \eqref{5551}.

The locality property \eqref{sigma12} can be obtained easily from \eqref{5550} 
using  test functions $v$ with compact support in $U$.
The convergence property \eqref{5552} follows directly from \eqref{5550}.
To prove \eqref{5553}  it is enough to show that each point of 
$\partial_D\Om_\Gamma$ is contained in an open rectangle $R$  such that 
\be\label{5554}
[\sigma\nu_\Om](x)=\sigma(x)\,\nu_\Om(x) \quad\hone\hbox{-a.e.\ }x\in R\cap \partial\Om\,. 
\ee
 As above we assume that $R\cap\Om$ is the subgraph of a Lipschitz function, and that $R\cap\Gamma=\emptyset$.
 If 
$\sigma\in C^1(\overline{R\cap\Om};\Msym)$, and $v\in C^1(\overline\Om;\R^2)$ 
with ${\rm supp} \,v\subset R$, then 
\eqref{5550} is satisfied with $[\sigma\nu_\Om](x)=\sigma(x)\,\nu_\Om(x)$. 
This proves \eqref{5554} in this case. If $\sigma\in C^0(\overline{R\cap\Om};\Msym)$, 
by  mollifying suitable translations of $\sigma$  (see, e.g.,  \cite[Lemma 2.3]{DM-DS-M}),  
we can construct a sequence $\sigma_k\in C^1(\overline{R\cap\Om};\Msym)$ 
converging to $\sigma$ uniformly and  such that ${\rm div}\sigma_k\to{\rm div }\sigma$ 
strongly in $L^2(R\cap \Om;\R^2)$.  Since $[\sigma_k\nu_\Om](x)=\sigma_k(x)\,\nu_\Om(x)$ 
$\hone$-a.e.\ $x\in R\cap \partial\Om$, \eqref{5554} follows from \eqref{5552}. 
This concludes the proof of \eqref{5553}.
\end{proof}

The following approximation lemma will be used to derive a suitable form of the 
Euler equation for the minimum problem \eqref{minred}.

\begin{lemma}\label{l667}
Let $\Gamma$ be a compact subset of $\overline\Om$, let $\e>0$ and let 
$\varphi\in LD_\ell(\Om_\Gamma,\partial_D\Om_\Gamma)$. Then there exists a function 
$\varphi_\e\in LD_\ell(\Om_\Gamma,\partial_D\Om_\Gamma)$ whose trace on 
$\partial_D\Om_\Gamma$ has compact support in $\partial_D\Om_\Gamma$, such that
\begin{equation}\label{999}
 \|E\varphi_\e-E\varphi\|_{1,\Om_\Gamma}<\e\qquad\hbox{and}\qquad
 \|\varphi_\e-\varphi\|_{1,\partial_D\Om_\Gamma}<\e\,.
 \end{equation}
\end{lemma}
\begin{proof} There exists a sequence $\psi_k\in L^1(\partial_D\Om_\Gamma;\R^2)$ 
with compact support in 
$\partial_D\Om_\Gamma$ such that $\psi_k\to\varphi$ strongly in $L^1(\partial_D\Om_\Gamma;\R^2)$. 
By the Gagliardo Extension Theorem there exists a sequence 
$v_k\in W^{1,1}(\Om;\R^2)$ such that the trace of $v_k$ agrees with $\psi_k-\varphi$ 
$\hone$-a.e.\ on $\partial_D\Om_\Gamma$ and $v_k\to 0$ strongly in $W^{1,1}(\Om;\R^2)$.
Then $\varphi_k:=\varphi+v_k$ belongs to  $LD_\ell(\Om_\Gamma,\partial_D\Om_\Gamma)$, 
the trace of $\varphi_k$ agrees with $\psi_k$ on  $\partial_D\Om_\Gamma$, and 
$E\varphi_k\to E\varphi$ strongly in $L^1(\Om_\Gamma;\Msym)$. The conclusion follows.
\end{proof}

We are now in a position to prove the Euler conditions for the solutions $(e,p)$ 
of the minimum problem \eqref{minred} which satisfy the additional condition $p=p_1$.

\begin{theorem}\label{2222}
Let $w_1\in H^1(\Om_0;\R^2)$, let $(e_1,p_1,\Gamma_1)\in A(w_1)$, and let 
$\sigma_1:=\C e_1$, $\Om_1:=\Om\setmeno\Gamma_1$, $\hat\Om_1:=\overline\Om\setmeno\Gamma_1$, 
and $\partial_D\Om_1:=\partial\Om\setmeno\Gamma_1$. 
Assume that $(e_1,p_1)$ is a solution of the minimum problem
\be\label{min99}
\min\{\Q(e)+H(p-p_1)(\hat\Om_1) : (e,p)\in A(w_1,\Gamma_1)\}\,.
\ee
Then 
\begin{eqnarray}
& \sigma_1(x)\in \K\qquad\hbox{for a.e.\ }x\in\Om_1\,,\label{sK}\\
&\langle\sigma_1,E\varphi\rangle_{\Om_1}=\langle[\sigma_1\nu_\Om],\varphi\rangle_{\partial_D\Om_1}
\quad\hbox{for every }\varphi\in LD_\ell(\Om_1,\partial_D\Om_1)\,.\label{div0}
\end{eqnarray}
In particular we have 
\be\label{divs}
{\rm div}\sigma_1=0\quad\hbox{in }\Om_1\,.
\ee
If $\sigma_1\in C(\hat\Om_1;\Msym)$ we have also
\begin{eqnarray}
& \sigma_1(x)\in \K\qquad\hbox{for every }x\in\hat\Om_1\,,\label{sK1}\\
& \langle\sigma_1,E\varphi\rangle_{\Om_1}=
\langle\sigma_1,\varphi\odot\nu_\Om\rangle_{\partial_D\Om_1} \quad
\hbox{for every }\varphi\in BD_\ell(\Om_1,\partial_D\Om_1)\,.\label{divs3}
\end{eqnarray}
\end{theorem}

\begin{proof}
Let us fix $\xi\in \Msym$ and a Borel set $B\subset\Om_1$, whose characteristic 
function is denoted by $1_B$. Then 
$(e_1-\e1_B\xi,p_1+\e 1_B\xi)\in A(w_1,\Gamma_1)$ for every $\e>0$, so that
$\Q(e_1)\le \Q(e_1-\e1_B\xi)+\e H(1_B\xi)(\hat\Om_1)$. Therefore
$$
0\le -\int_B\sigma_1(x)\,{:}\,\xi\,dx +\e\int_BQ(\xi)\,dx+\int_BH(\xi)\,dx\quad\hbox{for every }\e>0\,.
$$
As $\e$ and $B$ are arbitrary we obtain $\sigma_1(x)\,{:}\,\xi\le H(\xi)$ for a.e.\ $x\in\Om_1$ and 
every $\xi\in \Msym$, hence $\sigma_1(x)\in \partial H(0)$ for a.e.\ $x\in\Om_1$. 
Inclusion \eqref{sK} follows from the equality $\partial H(0)=\K$ (see \cite[Corollary 23.5.3]{Ro}).

Let us fix  $\varphi\in H^1(\Om_1;\R^2)$ with $\varphi=0$ $\hone$-a.e.\  on $\partial_D\Om_1$. 
It is easy to see that $(e_1+\e E\varphi,p_1\in A(w_1, \Gamma_1)$ for every $\e\in\R$, hence 
$\Q(e_1)\le \Q(e_1+\e E\varphi)$. This implies $0\le\langle \sigma_1,E\varphi\rangle_{\Om_1}+\e\Q(E\varphi)$ for 
$\e>0$ and $0\ge\langle \sigma_1,E\varphi\rangle_{\Om_1}+\e\Q(E\varphi)$ for $\e<0$. As $\e\to0$ we obtain 
\be\label{sigma1}
\langle\sigma_1,E\varphi\rangle_{\Om_1}=0\quad\hbox{ 
for every }\varphi\in H^1(\Om_1;\R^2)\quad\hbox{with }\varphi=0\quad\hone\hbox{-a.e.\  on }\partial_D\Om_1\,,
\ee which implies \eqref{divs}.

Let us prove now  that 
\be\label{997}
\langle\sigma_1,E\varphi\rangle_{\Om_1}=0\quad 
\hbox{for every }\varphi\in LD_\ell(\Om_1,\partial_D\Om_1)\quad\hbox{with }\varphi=0\ 
\hone\hbox{-a.e.\  on }\partial_D\Om_1\,. 
\ee
This will be done by approximating the set $\Om_1$ by more regular sets.
Let $\Gamma^k$ be a decreasing sequence of compact sets with intersection 
$\Gamma_1$, and let $\Om^k:=\Om\setmeno\Gamma^k$, 
$\hat\Om^k:=\overline\Om\setmeno\Gamma^k$, and 
$\partial_D\Om^k:=\partial\Om\setmeno\Gamma^k=\partial\Om^k\setmeno\Gamma^k$. 
We may choose $\Gamma^k$ so that $\Om^k$ has Lipschitz boundary and $\partial_D\Om^k$ 
is composed of a finite number of disjoint arcs. 
For every $e\in L^2(\Om^k;\Msym)$ and  $p\in M_b(\hat\Om^k;\Msym)$ we define 
$$
\Q^k(e):=\int_{ \Om^k} Q(e(x))\,dx\quad\hbox{and}\quad 
{\mathcal{H}}^k(p):=\int_{\hat\Om^k}H(\frac{p}{|p|}(x))\,d|p|(x)
\,. 
$$

Since $A(w_1,\Gamma^k)$ is weakly$\times$weakly$^*$ closed in 
$L^2(\Om;\Msym){\times}M_b(\overline\Om;\Msym)$ by Lemma \ref{closure}, 
a straightforward application of the direct method  of the calculus of variations 
implies that for every $k$  the minimum problem 
\be\label{varphik}
\min\{\Q^k(e)+{\mathcal{H}}^k(p-p_1) : (e,p)\in A(w_1,\Gamma^k) \}
\ee 
has a solution $(e^k,p^k)$. Let us define $\sigma^k:=\C e^k$. 
Arguing as in the first part of the proof we show that $\sigma^k(x)\in\K$ for 
a.e.\ $x\in\Om^k$ and that 
\be\label{sigmak}
\langle\sigma^k,E\varphi\rangle_{\Om^k}=0\quad\hbox{ 
for every }\varphi\in H^1(\Om^k;\R^2)\quad\hbox{with }\varphi=0\quad\hone
\hbox{-a.e.\  on }\partial_D\Om^k\,,
\ee
 which implies 
${\rm div}\sigma^k=0$ in $\Om^k$.
 By Lemma \ref{l666} with $\Om$,  $\Gamma$, and $\sigma$ replaced by $\Om^k$,  
 $\emptyset$, and $\sigma^k$, respectively, we obtain
$$
\langle[\sigma^k\nu_{\Om^k}],\varphi\rangle_{\partial\Om^k}=
\langle\sigma^k,E\varphi\rangle_{\Om^k}\quad\hbox{for every }\varphi\in LD(\Om^k)\,,
$$
and therefore \eqref{sigmak} and the regularity of $\partial_D\Om^k$ give  
 $[\sigma^k\nu_{\Om^k}]=0$ $\hone$-a.e.\  on $\partial\Om^k\setmeno\partial_D\Om^k$. 
 We conclude that
\be\label{96}
 \langle\sigma^k,E\varphi\rangle_{\Om^k}=0\quad\hbox{for every }\varphi\in LD(\Om^k)\quad
 \hbox{with }\varphi=0\quad\hone\hbox{-a.e.\  on }\partial_D\Om^k\,.
\ee
From \eqref{varphik} it follows that 
$$
\Q^k(e^k)+{\mathcal{H}}^k(p^k-p_1)\le\Q(e_1)\,.
$$
From the lower bounds on $Q$ and $H$ we deduce that there exist 
$e^*\in L^2_{loc}(\Om_1;\Msym)$ and $p^*\in M(\hat\Om_1;\Msym)$ such that 
$e^k\wto e^*$ weakly in  $L^2(U\cap\Om_1;\Msym)$ and $p^k\wtos p^*$ weakly$^*$ in 
$M_b(U\cap\hat\Om_1;\Msym)$ 
for every bounded open set $U$ with $\overline U\cap\Gamma_1=\emptyset$, and by 
lower semicontinuity we obtain that $e^*\in L^2(\Om_1;\Msym)$, 
$p^*\in M_b(\hat\Om_1;\Msym)$, and 
$$
\Q(e^*)+H(p^*-p_1)(\hat\Om_1)\le\Q(e_1)\,.
$$
By Lemma \ref{closure} we have $(e^*,p^*)\in A(w_1,\Gamma_1)$, so that 
$(e^*,p^*)$ is a solution of the minimum problem \eqref{min99}. By Remark \ref{unique} 
we conclude that $e^*=e_1$ a.e.\ on $\Om_1$.

It follows that 
\be\label{conv98}
\sigma^k\wto \sigma_1\quad\hbox{ weakly in }L^2(U\cap\Om_1;\Msym)
\ee 
for every bounded open set $U$ with $\overline U\cap\Gamma_1=\emptyset$.
Since $\K$ is bounded and $\sigma^k(x)\in \K$ for a.e.\ $x\in\Om^k$, we deduce that 
\be\label{97}
 \langle\sigma^k,E\varphi\rangle_{\Om^k}\to \langle\sigma_1,E\varphi\rangle_{\Om_1}\quad
 \hbox{for every }\varphi\in LD_\ell(\Om_1,\partial_D\Om_1)\,.
\ee

Let us fix now $\varphi\in LD_\ell(\Om_1,\partial_D\Om_1)$ with $\varphi=0$ $\hone$-a.e.\  on 
$\partial_D\Om_1$. Since $\Om^k$ has Lipschitz boundary and $\partial_D\Om^k\subset\partial_D\Om_1$,  
we have $\varphi\in LD(\Om^k)$ and $\varphi=0$ $\hone$-a.e.\  on 
$\partial_D\Om^k$. By \eqref{96} we have $\langle\sigma^k,E\varphi\rangle_{\Om^k}=0$, 
so that \eqref{97} gives $\langle\sigma_1,E\varphi\rangle_{\Om_1}=0$ which concludes the proof of \eqref{997}.

Let us fix 
$\varphi\in LD_\ell(\Om_1,\partial_D\Om_1)$ and $\e>0$. By Lemma \ref{l667} there exists 
$\varphi_\e\in LD_\ell(\Om_1,\partial_D\Om_1)$ satisfying \eqref{999}, whose trace on 
$\partial_D\Om_1$ has compact support in $\partial_D\Om_1$. Using a cut-off function we can find 
$\psi_\e\in LD(\Om_1)$ whose trace on $\partial_D\Om_1$ 
coincides with $\varphi_\e$ and whose support does not intersect $\Gamma_1$.
By \eqref{divs} and  Lemma \ref{l666}  we have 
$\langle\sigma_1,E\psi_\e\rangle_{\Om_1}=
\langle[\sigma_1\nu_\Om],\psi_\e\rangle_{\partial_D\Om_1}$. 
By construction  
$\langle[\sigma_1\nu_\Om],\psi_\e\rangle_{\partial_D\Om_1}=
\langle[\sigma_1\nu_\Om],\varphi_\e\rangle_{\partial_D\Om_1}$. 
By \eqref{997} we deduce that also 
$\langle\sigma_1,E\varphi_\e\rangle_{\Om_1}=
\langle\sigma_1,E\psi_\e\rangle_{\Om_1}$, 
hence 
$\langle\sigma_1,E\varphi_\e\rangle_{\Om_1}=
\langle[\sigma_1\nu_\Om],\varphi_\e\rangle_{\partial_D\Om_1}$.
By \eqref{999}
it follows that 
$| \langle\sigma_1,E\varphi\rangle_{\Om_1}-
\langle[\sigma_1\nu_\Om],\varphi\rangle_{\partial_D\Om_1}|<
(\|\sigma_1\|_\infty+\|[\sigma_1\nu_\Om]\|_\infty)\e$.
Since $\e>0$ is arbitrary we obtain \eqref{div0}.


Assume now $\sigma_1\in C(\hat\Om_1;\Msym)$. Then \eqref{sK1} follows from 
\eqref{sK} and \eqref{divs3} holds for every $\varphi\in LD_\ell(\Om_1,\partial_D\Om_1)$ 
by \eqref{5552} and \eqref{div0}. The case 
$\varphi\in BD_\ell(\Om_1,\partial_D\Om_1)$ follows by approximation arguing as in 
\cite[Theorem II.3.3]{Tem}.
\end{proof} 


\end{section}

\begin{section}{Crack transfer}
%:Crack transfer

In this section we adapt to $BD$ the jump transfer theorem proved in \cite{Fra-Lar} for $BV$. 
To this end
we need an extension result for $BD$ functions with a precise estimate of the constant 
in the energy bound.
\begin{lemma}\label{lemmaext}
Let $U_1$ and $U_2$ be two bounded open subsets of $\R^2$ with $U_1\subset U_2$.
Assume that $U_1$ has Lipschitz boundary. Then there exists a constant $c=c(U_1,U_2)>0$ 
such that for every $u\in BD(U_1)$ there exists $v\in BD(U_2)$ with $v=u$ on $U_1$ and
\begin{equation}\label{902}
|Ev|(U_2)\le c|Eu|(U_1)\,.
\end{equation}
The constant $c$ is invariant under homotheties, i.e., 
$c(\lambda U_1,\lambda U_2)=c(U_1,U_2)$ for every $\lambda>0$.
\end{lemma}
\begin{proof}
By the trace theorem in $BD(U_1)$ (see \cite[Chapter II, Theorem 2.1]{Tem}) 
there exists a constant $a=a(U_1)>0$ such that
$$
\int_{\partial U_1}|u|\,d\hone\le a\int_{U_1}|u|\,dx+a|Eu|(U_1)\,.
$$
Since $E$ vanishes on the class $\mathcal R$ of infinitesimal rigid motions in $\R^2$, the previous 
inequality gives
$$
\int_{\partial U_1}|u-R|\,d\hone\le a\int_{U_1}|u-R|\,dx+a|Eu|(U_1)
$$
for every $R\in {\mathcal{R}}$. By the Poincar\'e-Wirtinger inequality in $BD$ 
(see \cite[Chapter II, Proposition 2.2]{Tem}) there exists a constant $b=b(U_1)$ 
such that for every $u\in BD(U_1)$ there exists $R\in {\mathcal{R}}$ satisfying 
$$
\int_{ U_1}|u-R|\,dx\le b|Eu|(U_1)\,,
$$
which, together with the previous inequality gives
\be\label{900}
\int_{\partial U_1}|u-R|\,d\hone\le (ab+a)|Eu|(U_1)\,.
\ee
By the Gagliardo Extension Theorem (see \cite{Ga}) there exists a constant $G=G(U_1,U_2)>0$  
such that for every $\zeta\in L^1(\partial U_1;\R^2)$ there exists a function 
$z\in W^{1,1}(U_2\setmeno U_1;\R^2)$ such that $z=\zeta$ $\hone$-a.e.\ on $\partial U_1$ and 
\be\label{901}
\int_{U_2\setmeno U_1}|Dz|\,dx\le G\int_{\partial U_1}|\zeta|\,d\hone\,.
\ee
If we apply this result to $\zeta=u-R$ and define $v:=z+R$, we obtain the result with 
$c:=(ab+a)G+1$.
The invariance under homotheties in \eqref{902} can be easily obtained by a change of variables.
\end{proof}


\begin{theorem}[Crack transfer]\label{jumptransfer}
Let $\Gamma_n,\Gamma^*\in\Kmf$ with $\Gamma_n\to \Gamma^*$ in the Hausdorff metric, 
and let $v\in BD_\ell(\Om^*,\partial_D\Om^*)$, with $\Om^*:=\Om\setmeno\Gamma^*$ and
$\partial_D\Om^*:=\partial\Om\setmeno\Gamma^*$.
Then there exist a sequence of closed sets $C_n\subset\Rtwo$,
a sequence  $\Gamma^*_n\in\Kmf$, and a sequence 
$v_n\in BD_\ell(\Om^*_n,\partial_D\Om^*_n)$, with 
$\Om^*_n:=\Om\setmeno \Gamma^*_n$ and $\partial_D\Om^*_n:=\partial\Om\setmeno\Gamma^*_n$,
such that
\begin{eqnarray}
&& \Gamma^*_n\to \Gamma^*\quad\hbox{ in the Hausdorff metric}\,,\label{[(a)]}\\
&& \Gamma^*_n\supset \Gamma_n\cup(\Gamma^*\setmeno C_n)\,,\label{[(b)]}\\
&& \hone(\Gamma^*_n\setmeno \Gamma_n)\to0\,,\label{[(c)]}\\
&& \hone(\Gamma^*\setmeno C_n)\to 0\,,\label{[(d)]}\\
&& C_n\cup \Gamma^*\to \Gamma^*\quad\hbox{ in the Hausdorff metric}\,,\label{[(e)]}\\
&& \Ltwo(C_n)\to0\,,\label{[(e')]}\\
&& v_n=v\quad\hbox{ a.e.\ on }\Om^*_n\setmeno C_n\,,\label{[(f)]}\\
&& v_n=v\quad\hone\hbox{-a.e.\ on }\partial_D\Om^*_n\setmeno C_n\,,\label{[(f')]}\\
&& |E v_n|(\Om^*_n\cap C_n)\to0\,,\label{[(g)]}\\
&& \displaystyle\int_{\partial_D\Omega^*_n\cap C_n}|v_n|\,d\hone
\to0\,.\label{[(h)]}
\end{eqnarray}
\end{theorem}

\begin{proof} 
We extend $v$ by setting $v=0$ on $\tilde\Om\setmeno\overline\Om$. Since 
$v\in L^1(\partial_D\Om^*;\R^2)$, it turns out that  $v\in BD_{loc}(\tilde\Om\setmeno \Gamma^*)$ and 
$|Ev|(\tilde\Om\setmeno \Gamma^*)<+\infty$.

Since every connected set $C$ of $\Rtwo$ with $\hone(C)<+\infty$ is $(\hone,1)$-countably rectifiable 
(see \cite[Theorem~2]{EH} or \cite[Theorem 4.4.5]{Amb-Til}), the set $\Gamma^*$ is $(\hone,1)$-countably rectifiable 
and, therefore, for $\hone$-a.e.\ $x\in \Gamma^*$ we can define an approximate unit normal $\nu^*(x)$,  
an approximate unit tangent $\tau^*(x)$, and we can consider the squares
$$
Q_\rho(x):=\{y\in\Rtwo:|(y-x)\nu^*(x)|<\rho\,,\  |(y-x)\tau^*(x)|<\rho\}\,,
$$
defined for every $\rho>0$. 
By the rectifiability of $\Gamma^*$, for $\hone$-a.e.\ $x\in \Gamma^*$ we have
\begin{equation}\label{lo*}
\hone\LLL(x+\tfrac 1 \rho(\Gamma^*-x))\wtos
\hone\LLL T(x)\quad\hbox{weakly$^*$ in } Q_1(x)
\end{equation}
as $\rho\to0$, where 
$T(x):=\{y\in\Rtwo:(y-x)\nu^*(x)=0\}$ is the approximate tangent line.
We can also prove (see \cite{Gia}) that
\begin{equation}\label{dilat}
(x+\tfrac 1 \rho(\Gamma^*-x))\cap\overline Q_1(x)\to T(x)\cap \overline Q_1(x)
\end{equation}
in the Hausdorff metric as $\rho\to0$.

Let $\nu_\Om(x)$ be the outer unit normal to $\partial\Om$ at $x$, which is defined for 
$\hone$-a.e.\ $x\in \partial\Om$. For $\hone$-a.e.\ $x\in \Gamma^*\cap \partial\Om$ 
we may assume $\nu^*(x)=\nu_\Om(x)$, and that
\begin{equation}\label{dom}
(x+\tfrac 1 \rho(\overline\Om-x))\cap\overline Q_1(x)\to T^-(x)\cap \overline Q_1(x)
\end{equation}
in the Hausdorff metric, where $T^-(x):=\{y\in\Rtwo:(y-x)\nu_\Om(x)\le0\}$.

Let us fix $0<\delta<\frac 1 4$. For $\hone$-a.e.\ $x\in \Gamma^*$ and for every 
$\rho>0$ we define the rectangles
\begin{eqnarray*}
& Q_{\rho,\delta}^+(x):=\{y\in\Rtwo:\delta\rho<
(y-x)\nu^*(x)<\rho\,,\ |(y-x)\tau^*(x)|<\rho\}\,,\\
& M_{\rho,\delta}(x):=\{y\in\Rtwo:|(y-x)\nu^*(x)|<\delta\rho\,,
\ |(y-x)\tau^*(x)|<\rho\}\,,\\
& Q_{\rho,\delta}^-(x):=\{y\in\Rtwo:-\rho<(y-x)\nu^*(x)<
-\delta\rho\,,\ |(y-x)\tau^*(x)|<\rho\}\,.
\end{eqnarray*}
The lateral boundary of $M_{\rho,\delta}(x)$ is given by 
$$
L_{\rho,\delta}(x):=\{y\in\Rtwo: |(y-x)\nu^*(x)|\le\delta\rho\,,
\ |(y-x)\tau^*(x)|=\rho\}\,,
$$

By~\eqref{lo*} and~\eqref{dilat}, arguing as in \cite[Theorem~5.1]{AP}, 
we can prove that for $\hone$-a.e.\ $x\in \Gamma^*$ there exists $0<r_\delta(x)<\delta$  
with the following properties:
for every 
$0<\rho< r_\delta(x)$ there exists a connected compact set
$A_{\rho,\delta}(x)\subset M_{\rho,\delta}(x)\cap\overline\Om$ 
(a segment if $x\in\Om$) such that
\begin{eqnarray*}
&& \hone(\Gamma^*\cap Q_\rho(x))>\rho\,,\\
&& \Gamma^*\cap (\overline Q{}^-_{\rho,\delta}(x)\cup \overline Q{}^+_{\rho,\delta}(x))=
\emptyset\,,\\
&& (\Gamma^*\cap Q_\rho(x))\cup L_{\rho,\delta}(x)\cup A_{\rho,\delta}(x)\quad
\hbox{is connected}\,,\\
&& \hone(A_{\rho,\delta}(x))<c_1\delta\rho\,,
\end{eqnarray*}
where $c_1>0$ is a constant depending only on the geometry of 
$\partial\Om$.
For $\hone$-a.e.\ $x\in \Gamma^*\cap\Omega$ we may assume that $Q_{r_\delta(x)}(x)\subset\Om$. 
By \eqref{dom} we may also assume that for 
$\hone$-a.e.\ $x\in \Gamma^*\cap\partial\Omega$ 
and  for every 
$0<\rho< r_\delta(x)$
we have $Q_\rho(x)\subset\tilde\Om$,  
$Q_{\rho,\delta}^-(x)\subset\Om$, $Q_{\rho,\delta}^+(x)\cap\overline\Om=\emptyset$,
and $Q_\rho(x)\setmeno\overline\Om$ is connected. 

By a variant of the Besicovitch covering theorem proved in \cite[Theorem 5.13]{Mor} 
(see also the Errata in \cite{Ble-Mor} and a new proof in \cite[Section~1.2]{Fon-Leo}), 
there exists a finite family of pairwise disjoint squares 
$$
\overline Q_{\rho^\delta_1}(x^\delta_1),\dots, 
\overline Q_{\rho^\delta_{N_\delta}}(x^\delta_{N_\delta})\,,\hbox{ with }
x^\delta_i\in \Gamma^*\hbox{ and }
0<\rho^\delta_i<r_\delta(x_i^\delta)<\delta\,,
$$
satisfying the properties listed above and whose union  covers $\Gamma^*$ up 
to a set of $\hone$ measure $\delta$. Before stating these properties in a 
more precise manner, let us simplify the notation and set 
$$
Q^\delta_i:=Q_{\rho^\delta_i}(x^\delta_i)\,, \
Q_{i,\delta}^\pm:=Q_{\rho^\delta_i,\delta}^\pm(x^\delta_i)\,,\  
M_{i,\delta}:=M_{\rho^\delta_i,\delta}(x^\delta_i)\,,\ 
L_{i,\delta}:=L_{\rho^\delta_i,\delta}(x^\delta_i)\,,\  
A_{i,\delta}:=A_{\rho^\delta_i,\delta}(x^\delta_i)\,.
$$
Finally, we define
\be\label{Cdelta}
C^\delta:=\cup_{i} \overline Q{}^\delta_i\,.
\ee
Then for $i=1,\dots,N_\delta$ we have
\begin{eqnarray}
&& \hone(\Gamma^*\cap Q^\delta_i)>\rho^\delta_i\,,\label{2.6}\\
&& \Gamma^*\cap (\overline Q{}^-_{i,\delta}\cup \overline Q{}^+_{i,\delta})=\emptyset
\,,\label{2.10}\\
&&\Gamma^*_{i,\delta}:= (\Gamma^*\cap Q^\delta_i)\cup L_{i,\delta}
\cup A_{i,\delta}\quad
\hbox{is connected}\,,\label{2.9}\\
&& \hone(A_{i,\delta})<c_1\delta\rho^\delta_i\,,\label{new*}\\
&& \hone(\Gamma^*\setmeno C^\delta)<\delta\,,\label{2.13}\\
&&x^\delta_i\in \Gamma^*\cap\Omega\  \Longrightarrow\   
Q^\delta_i\subset\Om\,,\label{2.11}\\ 
&&x^\delta_i\in \Gamma^*\cap\partial\Omega\  \Longrightarrow\  
 Q^\delta_i\subset\tilde\Om\,,\  Q_{i,\delta}^-\subset\Om\,,\  Q_{i,\delta}^+\cap\overline\Om
 =\emptyset\,,\ Q^\delta_i\setmeno\overline\Om
\quad\hbox{is connected}\,.\label{2.12} 
\end{eqnarray}
For every $0<\delta<\tfrac14$ it is not restrictive to assume that  each connected component of $\Gamma^*$ with
positive Hausdorff measure contains at least one of the points $x^\delta_i$.
Note that 
\be\label{Gammadelta}
C^\delta\subset \Gamma^*(\delta):=\{y\in\Rtwo:{\rm dist}(y,\Gamma^*)\le\sqrt2\delta\}\,,
\ee
and  hence $\Ltwo(C^\delta)\to0$ 
as $\delta\to0$.

By \eqref{2.10} the function $v$ belongs to 
$BD(Q_{i,\delta}^+)$ and to $BD(Q_{i,\delta}^-)$. 
By Lemma~\ref{lemmaext} for every $\delta>0$ and every $i=1,\dots,N_\delta$
there exist two functions 
$z_{i,\delta}^+\,,z_{i,\delta}^-\in BD(Q^\delta_i)$ such that
\begin{eqnarray}
z_{i,\delta}^+=v\hbox{ a.e.\ on }Q_{i,\delta}^+,\qquad 
|Ez_{i,\delta}^+|(Q^\delta_i)\le c_2 |Ev|(Q_{i,\delta}^+)\,,\label{ext+}
\\
z_{i,\delta}^-=v\hbox{ a.e.\ on }Q_{i,\delta}^-,\qquad
|Ez_{i,\delta}^-|(Q^\delta_i)\le 
c_2 |Ev|(Q_{i,\delta}^-)\,,\label{ext-}
\end{eqnarray}
with a constant $c_2>1$ independent of $i$ and $\delta$.
This follows easily from the translation invariance of the constants and from the inclusions
\begin{eqnarray*}
&&R^\delta_i:=\{y\in\Rtwo:\delta\rho^\delta_i<
(y-x^\delta_i)\nu^\delta_i<(\delta+\tfrac34) \rho^\delta_i\,,\ |(y-x^\delta_i)\tau^\delta_i|<\rho^\delta_i\}
\subset Q_{i,\delta}^+\,,
\\
&&S^\delta_i:=\{y\in\Rtwo:(\delta-\tfrac 5 4)\rho^\delta_i<
(y-x^\delta_i)\nu^\delta_i<(\delta +\tfrac 3 4)\rho^\delta_i\,,\ |(y-x^\delta_i)\tau^\delta_i|<\rho^\delta_i\}\supset Q^\delta_i\setmeno Q_{i,\delta}^+\,,
\end{eqnarray*}
where $\nu^\delta_i=\nu^*(x^\delta_i)$ and $\tau^\delta_i=\tau^*(x^\delta_i)$. 
Indeed, one can first consider the extension from
$R^\delta_i$ to 
$S^\delta_i$, and then restrict it to
$Q^\delta_i\setmeno Q_{i,\delta}^+$.
Note that \eqref{2.12} implies that 
\be
z_{i,\delta}^+=0\hbox{ a.e.\ on }Q^\delta_i\setmeno\Om\,. \label{ext0+}
\ee

As $\Gamma_n\to \Gamma^*$ in the Hausdorff metric, there exists an integer
$m_\delta>0$ such that 
$\Gamma_n\cap \overline Q{}^\delta_i\subset  M_{i,\delta}\cup L_{i,\delta}$ for 
$n\ge m_\delta$. Note that the closed set 
$(\Gamma_n\cap \overline Q{}^\delta_i)\cup L_{i,\delta} \cup A_{i,\delta}$
has at most $m+3$ connected components for every
$n\ge m_\delta$ and converges to the connected set $\Gamma^*_{i,\delta}$
in the Hausdorff metric.
By \cite[Lemma 3.6]{DM-Toa} there exist closed connected sets $\Gamma^\delta_{i,n}$,
with
\be\label{star}
(\Gamma_n\cap \overline Q{}^\delta_i)\cup L_{i,\delta}\cup A_{i,\delta}\subset 
\Gamma^\delta_{i,n}
\subset \overline M_{i,\delta}\cap\overline\Om\,,
\ee
such that $\Gamma^\delta_{i,n}\to \Gamma^*_{i,\delta}$ in the Hausdorff metric  and  
\begin{equation}\label{jin}
\hone(\Gamma^\delta_{i,n}\setmeno 
(\Gamma_n\cup L_{i,\delta} \cup A_{i,\delta}) )\to0\qquad\hbox{as $n\to\infty$.}
\end{equation}


For  $n\ge m_\delta$
let $U_{i,n}^\delta$ be the connected component of $Q^\delta_i\setmeno 
\Gamma^\delta_{i,n}$ 
which contains $Q_{i,\delta}^+$, and let 
$V_{i,n}^\delta=Q^\delta_i\setmeno(\Gamma^\delta_{i,n}\cup U_{i,n}^\delta)$.
Since $\Gamma^\delta_{i,n}$ is connected and contains $L_{i,\delta}$, the set
$V_{i,n}^\delta$ contains $Q_{i,\delta}^-$.

We now define a function $v^\delta_n$ in $Q^\delta_i\setmeno \Gamma^\delta_{i,n}$ by setting
$$
v^\delta_n:=\begin{cases}
z_{i,\delta}^+\quad\hbox{ on }U_{i,n}^\delta\,, \\
z_{i,\delta}^-\quad\hbox{ on }V_{i,n}^\delta\,,
\end{cases}
$$
where the functions $z_{i,\delta}^\pm$ satisfy \eqref{ext+} and \eqref{ext-}. 
Then $v^\delta_n\in BD(Q^\delta_i\setmeno \Gamma^\delta_{i,n})$, 
$v^\delta_n=v$ a.e.\ on $Q^\delta_i\setmeno M_{i,\delta}$, and
$|E v^\delta_n|(Q^\delta_i\setmeno\Gamma^\delta_{i,n})\le c_2|Ev|(Q^\delta_i\setmeno\Gamma^*)|$. Moreover, 
\begin{equation}\label{vdn0}
v^\delta_n=0\quad\hbox{a.e.\ on }Q^\delta_i\setmeno\Om
\end{equation}
by \eqref{2.12} and \eqref{ext0+}.

Let $(y_j)$ be the finite family of isolated points of $\Gamma^*$. 
As $\Gamma_n\to \Gamma^*$ in the Hausdorff metric, for every $j$ and $n$ 
there exists a connected compact set $B_{j,n}\subset\overline\Om$ joining $y_j$ with 
$\Gamma_n$ and such that
\begin{equation}\label{sjn}
\hone(B_{j,n})\to0\qquad\hbox{as }n\to\infty\,.
\end{equation}
We define 
\begin{equation}\label{kdn}
\Gamma_n^\delta=\Gamma_n\cup (\cup_i \Gamma^\delta_{i,n})
\cup (\Gamma^*\setmeno C^\delta)\cup(\cup_jB_{j,n})\,,
\end{equation}
where $C^\delta$ is defined in \eqref{Cdelta}.
By \eqref{2.10} and \eqref{star} the sets $\Gamma_n^\delta$ are compact.
Using again \eqref{2.10} and \eqref{star} 
it is easy to see that the number of connected components of 
$\Gamma_n^\delta$ does not exceed the number of connected components of 
$\Gamma_n$, hence $\Gamma_n^\delta\in {\mathcal{C}}_m$.

We complete the definition of $v^\delta_n$ by setting 
 $v^\delta_n=v$ a.e.\ on $(\tilde\Om\setmeno \Gamma_n^\delta)\setmeno C^\delta$.
It is clear that $v^\delta_n\in BD_{loc}(\tilde\Om\setmeno \Gamma_n^\delta)$ and 
$|Ev^\delta_n|(\tilde\Om\setmeno \Gamma_n^\delta)<+\infty$.
 By \eqref{vdn0} we 
have
$v^\delta_n=0$ a.e.\ on $\tilde\Om\setmeno\Om$, so that its trace from the 
interior of $\Om$ satisfies $v^\delta_n\in L^1(\partial\Om\setmeno \Gamma_n^\delta;\R^2)$.

By \eqref{ext+}, \eqref{ext-}, and \eqref{kdn} we have
\begin{eqnarray}
&\displaystyle
\vphantom{\sum_i}|E v^\delta_n|({C^\delta\setmeno \Gamma^\delta_n})\le 
c_2|E v|(C^\delta\setmeno \Gamma^*)\,,
\label{200}
\\
&\displaystyle\begin{array}{c}
\displaystyle\hone(\Gamma^\delta_n\setmeno \Gamma_n)\le 
\sum_i\hone(\Gamma^\delta_{i,n}\setmeno (\Gamma_n\cup
L_{i,\delta}\cup A_{i,\delta}))+\sum_i \hone(L_{i,\delta})+{}\\
\displaystyle{}+
\sum_i \hone(A_{i,\delta})
+\hone(\Gamma^*\setmeno C^\delta)+
\sum_j\hone(B_{j,n}).
\end{array}
\label{201}
\end{eqnarray}
By \eqref{jin} and \eqref{sjn}, the first and the last  term in the right-hand side of 
\eqref{201} tend to $0$ as $n\to\infty$. 
Therefore, there exists $n_\delta\ge m_\delta$ such that
they are less than $\delta$ for $n\ge n_\delta$.
By \eqref{2.6}, \eqref{new*}, and \eqref{2.13} we have
$$
\sum_i \hone(L_{i,\delta})+
\sum_i \hone(A_{i,\delta})+
\hone(\Gamma^*\setmeno C^\delta)
\le
4\delta\sum_i \rho^\delta_i+
c_1\delta\sum_i \rho^\delta_i+\delta
\le  \delta(4+c_1)\hone(\Gamma^*)+\delta\,.
$$
 We conclude that
\begin{equation}\label{205}
\hone(\Gamma^\delta_n\setmeno \Gamma_n)\le
\delta(4+c_1)\hone(\Gamma^*)+3\delta\,.
\end{equation}

{}From \eqref{ext+} and  \eqref{ext-}  in particular we obtain
\begin{equation}\label{203}
 \int_{(J_{v^\delta_n}\setmeno \Gamma^\delta_n)\cap \overline Q{}^\delta_i}|[v^\delta_n]|\,d\hone
 \le c_2\sqrt2 |E v|(\overline Q{}^\delta_i\setmeno \Gamma^*)\,.
\end{equation}
Using the fact that $v^\delta_n$ vanishes on $\tilde\Om\setmeno\overline\Om$,  we deduce that 
\begin{equation}\label{23}
\int_{(\partial\Om\setmeno \Gamma^\delta_n)\cap
 \overline Q{}^\delta_i}|v^\delta_n|\,d\hone\le 
c_2\sqrt2 |E v|(\overline Q{}^\delta_i\setmeno \Gamma^*)\,.
\end{equation}

We now fix a sequence $\delta_k\to0$ with $\delta_k>0$ for every $k$. It is not restrictive to assume that 
$n_{\delta_k}$ is increasing. For every  $n\ge n_{\delta_1}$ there is a unique 
$k$ such that $n_{\delta_{k}}\le n< n_{\delta_{k+1}}$ and we set
$\Gamma^*_n:=\Gamma^{\delta_k}_n$, $v_n:=v^{\delta_k}_n$, and 
$C_n:=C^{\delta_k}$. 

Using \eqref{kdn} and  the inequality
$\sum_j\hone(B_{j,n})\le\delta_k$ we obtain that
$\Gamma^*_n\subset \Gamma_n\cup \Gamma^*(\delta_k)$, where
$\Gamma^*(\delta_k)$ is defined in \eqref{Gammadelta}. This implies that 
$\Gamma^*_n\to \Gamma^*$ 
in the Hausdorff metric and proves \eqref{[(a)]}. Inclusion \eqref{[(b)]} follows 
from \eqref{kdn}, while \eqref{[(c)]} follows from \eqref{205}. Condition \eqref{[(d)]} 
follows from \eqref{2.13}, condition \eqref{[(e)]} follows form the fact that 
$C_n\subset \Gamma^*(\delta_k)$, and condition \eqref{[(f)]} is fulfilled by the definition of $v_n$. 
As $\Ltwo(\Gamma^*)=0$, \eqref{[(e)]} implies \eqref{[(e')]}. Equality \eqref{[(f')]} 
follows from \eqref{[(f)]} by the properties of the trace operator. 
Condition \eqref{[(g)]} follows from \eqref{[(e)]} and \eqref{200}. By  \eqref{[(e)]}
and \eqref{23} we obtain
\begin{eqnarray*}
&\displaystyle 
 \int_{(\partial\Om\setminus \Gamma^*_n)\cap C_n}|v_n|\,d\hone\le
c_2\sqrt2 |E v|(C_n\setmeno \Gamma^*)
 \to0\,,
\end{eqnarray*}
which proves \eqref{[(h)]}. 
\end{proof}

 We shall use the following corollary.
 
\begin{corollary}\label{jump} Under the hypotheses of Theorem \ref{jumptransfer}, 
let $q$ and $q_n\in   M_ b(\overline\Om;\Msym)$ be defined by
\be\label{4678}
q:=\begin{cases}
Ev &  \hbox{ on }\Om^*\,,\\
-v\odot\nu_\Om\hone & \hbox{ on }\partial_D\Om^*\,,\\
0 & \hbox{ on }\Gamma^*\,,
\end{cases}
\qquad
q_n:=\begin{cases}
Ev_n &  \hbox{ on }\Om^*_n\,,\\
-v_n\odot\nu_\Om\hone & \hbox{ on }\partial_D\Om^*_n\,,\\
0 & \hbox{ on }\Gamma^*_n\,.
\end{cases}
\ee
Then
\begin{eqnarray}
&& q_n\to q\quad\hbox{  strongly}\hbox{ in }  M_ b(\overline\Om;\Msym)\,.\label{(c)}
\end{eqnarray} 
\end{corollary}

\begin{proof} 
By \eqref{[(b)]} we have $\Gamma^*\setmeno C_n\subset \Gamma^*_n\setmeno C_n$. 
By \eqref{[(f)]} and \eqref{[(f')]} it follows that $q_n=q$ on $\hat\Om^*_n\setmeno C_n$,
where $\hat\Om^*_n:=\overline\Om\setmeno\Gamma^*_n$. 
Therefore the definitions of $q$ and $q_n$, together with \eqref{[(a)]}, give
$$
|q_n-q|(\overline\Om\setmeno C_n)\le |q|(\Gamma^*_n\setmeno \Gamma^*)\to 0\,.
$$
On the other hand, by \eqref{[(e)]}, \eqref{[(g)]}, and \eqref{[(h)]} we have also
$$
|q_n|(C_n)\to0\quad\hbox{and}\quad
|q|(C_n)=|q|(C_n\cup\Gamma^*)\to|q|(\Gamma^*)=0\,,
$$
which concludes the proof of \eqref{(c)}.
\end{proof}


In order to prove the global stability we shall use the following strong approximation result.

\begin{lemma}[Approximation]\label{approx}
Let  $w,\, w_n\in H^1(\Om_0;\R^2)$,  $(e,p,\Gamma)\in A(w)$, and 
$(e_n,p_n,\Gamma_n)\allowbreak\in A(w_n)$. 
Assume that 
\begin{eqnarray}
&& \Gamma_n\to \Gamma \qquad\hbox{in the Hausdorff metric}\,,\label{3332}\\
&& e_n\wto e\qquad\,\,\hbox{ weakly in }L^2(\Om;\Msym)\,.\label{3333}
\end{eqnarray}
Let $(e^*\!,p^*\!,\Gamma^*)\in A(w)$ with $\Gamma^*\supset \Gamma$. Then there 
exists a sequence
$(e^*_n,p^*_n,\Gamma^*_n)\in A(w_n)$
 such that 
\begin{eqnarray}
&& \Gamma^*_n\to \Gamma^* \qquad\hbox{in the Hausdorff metric}\,,\label{KnK}\\
&& \Gamma^*_n\supset \Gamma_n\,,\label{new**}\\
&& \hone(\Gamma^*_n\setmeno \Gamma_n)\to 
\hone(\Gamma^*\setmeno \Gamma)\,,\label{K*K}\\
&& e^*_n\wto e^*\qquad\,\,\hbox{ weakly in }L^2(\Om;\Msym)\,,\label{e*e1}\\
&& e^*_n- e_n\to e^*-e\qquad\,\,\hbox{ strongly in }L^2(\Om;\Msym)\,,\label{e*e}\\
&& (p^*_n-p_n)\LLL\hat\Om_n^*\to (p^*-p)\LLL\hat\Om^*
\quad\hbox{ strongly in }   M_ b(\overline\Om;\Msym)\label{pp*1}\,,
\end{eqnarray}
where $\hat\Om_n^*:=\overline\Om\setmeno \Gamma^*_n$ and $\hat\Om^*:=
\overline\Om\setmeno \Gamma^*$.
\end{lemma}

\begin{proof} By \cite[Lemma 3.5]{DM-Toa} there exists a sequence 
$\tilde \Gamma_n\in\Kmf$ converging to $ \Gamma^*$ in the Hausdorff metric
and such that
\be\label{a*}
\tilde \Gamma_n\supset \Gamma_n
\quad\hbox{and}\quad
\hone(\tilde \Gamma_n\setmeno \Gamma_n)\to \hone(\Gamma^*\setmeno \Gamma)\,.
\ee

Let  $\Om^*:=\Om\setminus \Gamma^*$ and 
$\partial_D\Om^*:=\partial\Om\setminus\Gamma^*$.
Since $(e^*\!,p^*\!,\Gamma^*)\in A(w)$, there exists $u^*\in BD_\ell(\Om^*,\partial_D\Om^*)$ such 
that $u^*$, $e^*$, $p^*$, $\Gamma^*$, and $w$ satisfy \eqref{adm1k}--\eqref{adm3k}.
Since $(e,p,\Gamma)\in A(w)$ there exists $u\in BD_\ell(\Om_\Gamma,\partial_D\Om_\Gamma)$ 
such that $u$, $e$, $p$, $\Gamma$, and $w$ satisfy \eqref{adm1k}--\eqref{adm3k}. 
By Theorem~\ref{jumptransfer}, applied to $v:=u^*-u$, $\Gamma^*$, and $\tilde \Gamma_n$,
there exist a sequence $\Gamma^*_n\in \Kmf$, 
with $\Gamma^*_n\supset \tilde \Gamma_n$, and 
a sequence of functions $v_n\in BD_\ell(\Om^*_n,\partial_D\Om^*_n)$, with 
$\Om_n^*:= \Om\setmeno \Gamma^*_n$ and $\partial_D\Om_n^*:=\partial\Om\setmeno\Gamma^*_n$, 
which satisfy \eqref{[(a)]}--\eqref{[(h)]}. Condition \eqref{KnK} follows from \eqref{[(a)]}, 
condition \eqref{new**} is a consequence of \eqref{[(b)]} and \eqref{a*}, while
\eqref{K*K} can be deduced from \eqref{[(c)]} and \eqref{a*}. 
Let $q_n$ and $q$ be defined by \eqref{4678}.
We now  define $e^*_n$ and $p^*_n$ by 
\begin{eqnarray*}
&& e^*_n:=e_n+e^*-e\qquad\hbox{a.e.\ in }\Om\,,\\
&& p^*_n:=q_n+p_n\LLL\hat\Om_n^*-e^*+e\qquad\hbox{in }\overline\Om\,.
\end{eqnarray*}
In order to prove that $(e^*_n,p^*_n,\Gamma^*_n)\in A(w_n)$, we consider  functions  
$u_n\in BD_\ell(\Om_n,\partial_D\Om_n)$, where $\Om_n:=\Om\setmeno \Gamma_n$ 
and $\partial_D\Om_n:=\partial\Om\setmeno\Gamma_n$, such that $u_n$, $e_n$, $p_n$, 
$\Gamma_n$, and $w_n$ satisfy 
\eqref{adm1k}--\eqref{adm3k}. Let $u^*_n:=v_n+u_n$, defined a.e.\ in $\Om^*_n$. 
We easily check that $u^*_n\in BD_{loc}(\Om^*_n)$, $|Eu^*_n|(\Om^*_n)<+\infty$, 
$u^*_n\in L^1(\partial_D\Om^*_n;\R^2)$, $Eu^*_n=e^*_n+p^*_n$ in $\Om^*_n$, 
and $p^*_n=(w_n-u^*_n)\odot\nu_\Om\hone$ on $\partial_D\Om^*_n$, 
hence $(e^*_n,p^*_n,\Gamma^*_n)\in A(w_n)$. Moreover, 
\begin{eqnarray}
&& e^*_n-e_n=e^*-e\qquad\hbox{a.e.\ in }\Om\,,\label{7000}\\
&& (p^*_n-p_n)\LLL\hat\Om_n^* =q_n-e^*+e\qquad\hbox{in }\overline\Om\,.\label{7001}
\end{eqnarray}
Equality \eqref{7000} implies \eqref{e*e}, which, together with \eqref{3333}, 
gives \eqref{e*e1}. 
As $q_n\to q$ strongly in $M_b(\overline\Om;\Msym)$ by Corollary \ref{jump}, 
equality \eqref{7001} gives
$$
(p^*_n-p_n)\LLL\hat\Om^*_n \to q-e^*+e\qquad\hbox{strongly in }M_b(\overline\Om;\Msym)\,.
$$
This implies \eqref{pp*1} by the definition \eqref{4678} of $q$ and by the weak 
kinematic admissibility conditions for $u$, $e$, $p$, $\Gamma$, and $w$,
and for $u^*$, $e^*$, $p^*$, $\Gamma^*$, and~$w$.
\end{proof}

\begin{lemma}[Stability of global minimality]\label{approx1}
 Let  $w,\, w_n\in H^1(\Om_0;\R^2)$, $(e,p,\Gamma)\in A(w)$, and 
$(e_n,p_n,\Gamma_n)\in A(w_n)$. 
Assume that 
\begin{eqnarray}
&& \Gamma_n\to \Gamma \qquad\hbox{in the Hausdorff metric}\,,\label{3334}\\
&& e_n\wto e\qquad\,\,\hbox{ weakly in }L^2(\Om;\Msym)\,,\label{3335}
\end{eqnarray}
and that  for every $n$
\begin{equation}\label{ej}
\Q(e_n)\le \Q(e^*)+{\mathcal{D}}((p^*\!,\Gamma^*),(p_n,\Gamma_n))
\quad\hbox{for every }(e^*\!,p^*\!,\Gamma^*)\in A(w_n)\,.
\end{equation}
Then
\begin{equation}\label{111}
\Q(e)\le \Q(e^*)+{\mathcal{D}}((p^*\!,\Gamma^*),(p,\Gamma))
\quad\hbox{for every }(e^*\!,p^*\!,\Gamma^*)\in A(w)\,.
\end{equation}
\end{lemma}
\begin{proof} Let us fix $(e^*\!,p^*\!,\Gamma^*)\in A(w)$ and let 
$(e^*_n,p^*_n,\Gamma^*_n)\in A(w_n)$ be the sequence given by 
Lemma~\ref{approx}. Since $\Q(e_n)\le \Q(e^*_n)+
{\mathcal{D}}((p^*_n,\Gamma^*_n),(p_n,\Gamma_n))$ by \eqref{ej},
 the definition of ${\mathcal{D}}$ gives
$$
0\le \Q(e^*_n)-\Q(e_n)+H(p^*_n-p_n)(\hat\Om^*_n)+
\hone(\Gamma^*_n\setmeno \Gamma_n)\,,
$$
where $\hat\Om^*_n:=\overline\Om\setminus \Gamma^*_n$.
Since $e^*_n-e_n\to e^*-e$  strongly in $L^2(\Om;\Msym)$ by \eqref{e*e}, 
and $e^*_n+e_n\wto e^*+e$ weakly in $L^2(\Om;\Msym)$ by \eqref{e*e1}, we have
$$
\Q(e^*_n)-\Q(e_n)=\tfrac 1 2\langle\C (e^*_n-e_n),e^*_n+
e_n\rangle\to\tfrac 1 2\langle \C(e^*-e),e^*+e\rangle=\Q(e^*)-\Q(e)\,.
$$
Therefore \eqref{K*K} and \eqref{pp*1} give
\begin{eqnarray*}
&& \displaystyle \Q(e^*)-\Q(e)+H(p^*-p)(\hat\Om^*)+
\hone(\Gamma^*\setmeno \Gamma)=\\
&&\displaystyle =\lim_{n\to\infty}(\Q(e^*_n)-\Q(e_n)+ 
H(p^*_n-p_n)(\hat\Om^*_n)+\hone(\Gamma^*_n\setmeno \Gamma_n))\ge0\,,
\end{eqnarray*}
where $\hat\Om^*:=\overline\Om\setmeno \Gamma^*$. 
This inequality implies \eqref{111}.
\end{proof}


\end{section}

\begin{section}{The discrete time problems}\label{discrete}
%:The discrete time problems

Let us fix $T>0$ and a sequence of subdivisions $(\tni)_{0\le i\le n}$
 of the interval $[0,T]$, with 
\begin{eqnarray} 
& 0=t_n^0<t_n^1<\cdots<t_n^{n-1}<t_n^{n}=T\,, 
\label{subdiv5}\\ 
&\displaystyle 
\lim_{n\to\infty}\,
\max_{1\le i\le n} (\tni-\tnim)= 0\,. 
\label{fine5} 
\end{eqnarray} 

Let $w\in AC([0,T];H^1(\Om_0;\R^2))$ and let $(e_0,p_0,\Gamma_0)$ be an initial 
configuration that satisfies the minimality property~(\ref{min000}). 



For $i=1,\ldots,n$ we set $\wni:=w(\tni)$. 
For every $n$ we define $\eni$, $\pni$ and $\Kni$ by induction. 
We set $(e_n^0, p_n^0,\Gamma_n^0):=(e_0,p_0,\Gamma_0)$, and for $i=1,\ldots,n$ 
we define $(\eni,\pni,\Kni)$ as a solution of the problem
\begin{equation}\label{2110}
\min\, \{\Q(e)+{\mathcal{D}}((p,\Gamma),(\pnim,\Knim)):  (e,p,\Gamma)\in A(\wni)\}\,.
\end{equation}
The existence of a solution to this problem is proved in Theorem~\ref{esistmin}. We define
$\sigma^i_n:=\C\eni$.
To prove the minimality in the continuous time problem we shall use the following result.
\begin{lemma}\label{5.1} For every $(e,p,\Gamma)\in A(\wni)$ we have
\begin{equation*}
\Q(\eni)\le \Q(e) +{\mathcal{D}}((p,\Gamma),(\pni,\Kni))\,.
\end{equation*}  
\end{lemma}

\begin{proof}
By \eqref{2110} we have
\begin{eqnarray*}
\Q(\eni)+{\mathcal{D}}((\pni,\Kni),(\pnim,\Knim)) \le
 \Q(e) +{\mathcal{D}}((p,\Gamma),(\pnim,\Knim))  \,.
\end{eqnarray*}
 The conclusion follows from the triangle inequality for the dissipation distance ${\mathcal{D}}$.
\end{proof}

Let us consider the piecewise constant interpolations of $\eni$, $\pni$, $\Kni$, $\sigma^i_n$, 
and $\wni$  defined for every  $t\in[0,T]$ by
\begin{equation}\label{kt}
\en(t):=\eni\,,\quad\pn(t):=\pni\,,\quad
\Gamma_n(t):=\Kni\,,\quad\sigma_n(t):=\sigma^i_n\,,\quad\wn(t):=\wni\,,
\end{equation}
where $i$ is the largest integer such that $\tni\le t$.
Lemma~\ref{5.1} shows that for every $n$ and $t$ we have 
\begin{equation}\label{minkt}
\Q(\en(t))\le\Q(e^*)+{\mathcal{D}}((p^*\!,\Gamma^*),(\pn(t),\Gamma_n(t)))
\end{equation}
for every $(e^*\!,p^*\!,\Gamma^*)\in A(\wn(t))$.

We shall use the following notation
$$
\Om_n(t):=\Om\setmeno \Gamma_n(t)\,,\quad \hat\Om_n(t):=
\overline\Om\setmeno \Gamma_n(t)\,,\quad
\partial_D\Om_n(t):=\partial\Om\setmeno \Gamma_n(t)\,.
$$

We now prove a discrete energy estimate, which will be crucial in the proof 
of the compactness of the piecewise constant interpolations.
\begin{lemma}\label{energy estimate}
There exists a sequence $R_n\to 0$ such that  we have
\begin{eqnarray}\label{1999}
&\displaystyle 
\Q(e_n^k)+\sum_{i=1}^k\,{\mathcal{D}}((\pni,\Kni),(\pnim,\Knim))\le \Q(e_0)
+ \int_{0}^{t_n^k} \langle \sigma_n(s),
E\dot w(s) \rangle \,ds + R_n
\end{eqnarray}
for every $n$ and for every $k=1,\ldots,n$.
\end{lemma}
\begin{proof}
As $(\enim-E \wnim+E \wni,\pnim,\Knim)$ is admissible in problem~(\ref{2110}) 
which defines $(\eni,\pni,\Kni)$, we obtain
\begin{eqnarray*}%\label{ii-1}
&\displaystyle \Q(\eni)+{\mathcal{D}}((\pni,\Kni),(\pnim,\Knim))\le\\
&\displaystyle \le\Q(\enim)+\langle\sigma^{i-1}_n,E(\wni-\wnim )\rangle+\Q(E(\wni-\wnim))\,.
\nonumber
\end{eqnarray*}
Iterating this estimate and taking into account the regularity hypotheses on $w$ 
we get the thesis. We refer to \cite[Lemma 7.3]{DM-Toa} for the details.
\end{proof}

\begin{lemma}\label{globalbound}
There exists a constant $C>0$, depending only on $e_0$, $p_0$, $\Gamma_0$, 
$w$, $T$,  $\alpha_\C$, $\beta_\C$, and $r_\K$, such that for $n$ large enough
\begin{eqnarray}
&\displaystyle \|\en(t)\|_2\le C\,,\label{en}\\
&\displaystyle {\rm Diss}(p_n(\cdot), \Gamma_n(\cdot);0,T)\le C\,,\label{dissn}\\
&\displaystyle \|p_n(t)\|_1\le C\,,\label{varpn}\\
&\displaystyle \hone(\Gamma_n(t))\le C\,,\label{h1kn}
\end{eqnarray}
for every $t\in[0,T]$.
\end{lemma}

\begin{proof} 
From \eqref{lowerboundC},  \eqref{upperboundC}, and \eqref{1999} we deduce that 
$$
\alpha_\C\|\en(t)\|_2^2\le \beta_\C\|e_0\|^2_2+2\beta_\C\int_0^t\|\en(\tau)\|_2\|E\dot w(\tau)\|_2d\tau\,.
$$
By Gronwall's Lemma 
there exist a constant $c_1>0$, depending only on $\alpha_\C$, $\beta_\C$, $e_0$, $w$, and $T$, such that 
\eqref{en} holds if $C\ge c_1$.
Then inequality \eqref{1999} gives 
$$
\sum_{i=1}^n\,{\mathcal{D}}((\pni,\Kni),(\pnim,\Knim))\le c_2
$$
with $c_2>0$ depending only on $c_1$, $\beta_\C$, $e_0$,  $w$, and $T$.
This proves \eqref{dissn},
which gives \eqref{varpn} and \eqref{h1kn} thanks to \eqref{adm3k}, \eqref{varp}, and \eqref{misK}. 
\end{proof}

\begin{remark}\label{rem5.4}
Using \eqref{upperboundC} we deduce from Lemma \ref{energy estimate} and 
Lemma \ref{globalbound} 
that for $n$ large enough we have
\begin{equation}\label{512}
\Q(e_n(t))+{\rm Diss}(p_n(\cdot),\Gamma_n(\cdot);0,t)\le \Q(e_0)
+ \int_{0}^{t} \langle \sigma_n(s),
E\dot w(s) \rangle \,ds + \tilde R_n
\end{equation}
for every $t\in[0,T]$, where 
$$
\tilde R_n:=R_n+2\beta_\C C\sup_{1\le i\le n}\int_{\tnim}^{\tni}\|E\dot w(s)\|_2\,ds\to 0
$$
as $n\to\infty$.
\end{remark}

In order to pass to the limit we begin by proving the following result
which plays the role of Helly's Theorem in this setting.

\begin{lemma}\label{ppnKKn} 
For every~$n$ let $t\mapsto(p_n(t),\Gamma_n(t))$ be a function from $[0,T]$ into 
$  M_ b(\overline\Om;\Msym)\allowbreak{\times}\Kmf$ such that 
$t\mapsto \Gamma_n(t)$ is increasing and $|p_n(t)|(\Gamma_n(t))=0$ for every $t\in [0,T]$. 
Assume that there exists a constant $C$ such that 
\begin{equation}\label{dissn1}
\|p_n(0)\|_1+\hone(\Gamma_n(0))+{\rm Diss}(p_n(\cdot),\Gamma_n(\cdot);0,T)\le C\qquad\hbox{for every }n\,.
\end{equation}
Then there exist an increasing sequence of integers $n_k$  and a function 
$t\mapsto (p(t),\Gamma(t))$ from $[0,T]$ into 
$  M_ b(\overline\Om;\Msym){\times}\Kmf$, with $t\mapsto \Gamma(t)$
increasing and $|p(t)|(\Gamma(t))=0$, such that for every $t\in[0,T]$
\begin{eqnarray}
& \Gamma_{n_k}(t)\to \Gamma(t)\quad\hbox{in the Hausdorff metric}\,, \label{knkt}\\
& p_{n_k}(t)\wtos p(t)\quad\hbox{weakly}^*\hbox{ in }
  M_ b(\hat\Om(t);\Msym)\,,\label{pnpt}
\end{eqnarray}
where $\hat\Om(t)$ is defined by \eqref{omt}.
\end{lemma}
\begin{proof}
By \eqref{misK} inequality \eqref{dissn1} implies that $\hone(\Gamma_n(T))\le C$ for every $n$. 
By  Helly's Theorem in $\Kmf$ (see \cite[Theorem 6.3]{DM-Toa})  
we can pass to a subsequence, not relabelled, such that $\Gamma_{n}(t)\to \Gamma(t)$ 
in the Hausdorff metric for every $t\in[0,T]$, where $\Gamma\colon[0,T]\to\Kmf$ is increasing. 
As $|p_n(t)|(\Gamma_n(t))=0$, from \eqref{varp} and \eqref{dissn1} we obtain  
\be
|p_n(t)|(\overline\Om)\le (1+\tfrac{1}{r_\K})C\quad\hbox{  for every $n$ and every $t\in [0,T]$.}
\label{6000}
\ee

Let $D$ be a countable dense set in $[0,T]$ containing $0$. Using a diagonal argument 
we can pass to a subsequence, not relabelled, such that for every $s\in D$ 
$$
p_n(s)\wtos p(s) \quad\hbox{weakly}^*\hbox{ in }
  M_ b(\hat\Om(s);\Msym)\,,
$$
for a suitable measure $p(s)\in   M_ b(\hat\Om(s);\Msym)$. 
We extend $p(s)$ to $\overline\Om$ by setting $p(s)=0$ on $\Gamma(s)$.
 
Let $V_n(t):={\rm Diss}(p_n(\cdot),\Gamma_n(\cdot);0,t)$. By Helly's Theorem we may 
assume that $V_n(t)\to V(t)$ for every $t\in[0,T]$, for some nondecreasing function $V$.
By \eqref{Hrk}, \eqref{defH}, \eqref{D}, and \eqref{dissd} for every $t_1,t_2\in[0,T]$ with $t_1<t_2$ we have 
\begin{equation}\label{vnt12}
r_\K |p_n(t_2)-p_n(t_1)|(\hat\Om_n(t_2))\le V_n(t_2)-V_n(t_1)\,.
\end{equation}
By \eqref{semicont1} we obtain 
\begin{equation}\label{vt12}
r_\K |p(s_2)-p(s_1)|(\hat\Om(s_2))\le V(s_2)-V(s_1)
\end{equation}
for every $s_1,s_2\in D$ with $s_1<s_2$.
Therefore we can apply the Cauchy criterion, and for every $t\in [0,T]\setmeno D$ 
we find $p(t)\in  M_ b(\hat\Om(t);\Msym)$ such that 
$p(s)\to p(t)$ strongly in $  M_ b(\hat\Om(t);\Msym)$ as $s\to t$, with $s\in D$ and $s<t$. 
We extend $p(t)$ to $\overline\Om$ by setting $p(t)=0$ on $\Gamma(t)$.
From  \eqref{vt12} we obtain
\begin{equation}\label{vts}
r_\K |p(t)-p(s)|(\hat\Om(t))\le V(t)-V(s)
\end{equation}
for every $s\in D$ and $t\in[0,T]$ with $s<t$. 

Let us fix  a continuity point $t$ of $V$ in $]0,T]$. To prove that $p_n(t)\wtos p(t)$ 
weakly$^*$ in $  M_ b(\hat\Om(t);\Msym)$, by \eqref{6000} it is enough to show that
\be
\langle \varphi,p_n(t)\rangle\to\langle \varphi,p(t)\rangle \label{6001}
\ee
for every $\varphi\in C_c(\hat\Om(t);\R^2)$.
 Let us fix such a $\varphi$. Since 
 ${\rm supp}(\varphi)\subset\hat\Om(t)$ we have 
 ${\rm supp}(\varphi)\subset\hat\Om_n(t)$ for $n$ large enough.
Given $s\in D$  with $s<t$, from \eqref{vnt12} and \eqref{vts} we obtain
\begin{eqnarray*}
&|\langle \varphi,p_n(t)\rangle-\langle \varphi,p(t)\rangle|\le 
|\langle \varphi,p_n(t)-p_n(s)\rangle|+ |\langle \varphi,p_n(s)-p(s)\rangle|+
 |\langle \varphi,p(s)-p(t)\rangle|\le \\
& \le
\tfrac{ \|\varphi\|_\infty}{r_\K}(V_n(t)-V_n(s))+ |\langle \varphi,p_n(s)-p(s)\rangle|+
\tfrac{ \|\varphi\|_\infty}{r_\K}(V(t)-V(s))\,,
\end{eqnarray*}
hence
$$
\limsup_{n\to\infty}|\langle \varphi,p_n(t)\rangle-\langle \varphi,p(t)\rangle|\le 
2\tfrac{ \|\varphi\|_\infty}{r_\K}(V(t)-V(s))\,.
$$
Passing to the limit as $s\to t$ we obtain \eqref{6001}, which gives \eqref{pnpt} 
for every continuity point of $V$ in $]0,T]$. Since the discontinuity points are at most countable,  
\eqref{pnpt} can be obtained for them by extracting a further subsequence, 
using a diagonal argument. 
\end{proof}

\end{section}

\begin{section}{Proof of Theorem \ref{main}}
%: Proof of Theorem \ref{main}

\begin{proof}[Proof of the global stability] Let $\en(t)$, $\pn(t)$, $\Gamma_n(t)$, 
$\sigma_n(t)$, and $\wn(t)$ be defined by \eqref{kt}. By Lemma \ref{globalbound} we can apply Lemma~\ref{ppnKKn} and we obtain a sequence 
$n_k$ of integers satisfying \eqref{knkt} and \eqref{pnpt}. Let us fix $t\in[0,T]$.
By \eqref{en} there exist a subsequence $n^t_k$, of $n_k$, depending on 
$t$, and a function $e(t)\in L^2(\Om;\Msym)$ such that $e_{n^t_k}(t)\wto e(t)$ 
weakly in $L^2(\Om;\Msym)$. We define $\sigma(t):=\C e(t)$. 
Applying Lemma~\ref{closure} we obtain that  $(e(t),p(t),\Gamma(t))\in A(w(t))$.
 As $(e_n(t),p_n(t),\Gamma_n(t))$ solves \eqref{minkt}, the global 
 stability for $(e(t),p(t),\Gamma(t))$ (see Definition \ref{defquasist}(a)) 
 follows from Lemma~\ref{approx1}.

Moreover, by Remark \ref{unique}  $e(t)$ is uniquely determined by $(p(t),\Gamma(t))$.  
Therefore the limit of $e_{n^t_k}(t)$ does not depend on the subsequence, 
and we conclude that the whole sequence $e_{n_k}(t)$ converges to $e(t)$ 
weakly in $L^2(\Om;\Msym)$. Summarizing, we have constructed a function
 $t\mapsto(e(t),p(t),\Gamma(t))$  such that condition (a) in Definition \ref{defquasist} 
 holds for every $t\in[0,T]$ and 
\begin{eqnarray}
& (e(t),p(t),\Gamma(t))\in A(w(t))\,,\label{adwt2}\\
& \Gamma_{n_k}(t)\to \Gamma(t)\quad\hbox{in the Hausdorff metric}\,, \label{knkt2}\\
& p_{n_k}(t)\wtos p(t)\quad\hbox{weakly}^*\hbox{ in }  M_ b(\hat\Om(t);\Msym)\,,\label{pnpt2}\\
& e_{n_k}(t)\wto e(t)\qquad\,\,\hbox{ weakly in }L^2(\Om;\Msym)\,,\label{enet2}
\end{eqnarray}
where $\hat\Om(t)$ is defined by \eqref{omt}.
 By \eqref{en} and \eqref{enet2} we have $e\in L^\infty({[0,T]}; L^2(\Om;\Msym))$.
\end{proof}

\begin{proof}[Proof of the energy balance] 
By \eqref{en} and \eqref{enet2} we can apply the Dominated 
Convergence Theorem and we obtain that 
$t\mapsto \langle \sigma(t), E\dot w(t)\rangle$ belongs to $L^1({[0,T]})$ and 
$$
\int_0^t\langle \sigma_n(s),E\dot w(s)\rangle\,ds\to \int_0^t\langle \sigma(s),E\dot w(s)\rangle\,ds
$$
for every $t\in[0,T]$.
As $\Q$ is lower semicontinuous with respect to the weak convergence in $L^2$, by 
Lemma~\ref{lsemicont2} we may pass to the limit in \eqref{512} and we obtain
\begin{equation}\label{5200}
\Q(e(t))+{\rm Diss}(p(\cdot),\Gamma(\cdot);0,t)\le \Q(e_0)+\int_0^t\langle \sigma(s),E\dot w(s)\rangle\, ds
\end{equation}
for every $t\in[0,T]$.

Let $V(t):={\rm Diss}(p(\cdot),\Gamma(\cdot);0,t)$. 
Since $t\mapsto V(t)$ and $t\mapsto \Gamma(t)$ are increasing, by \cite[Proposition 6.1]{DM-Toa} 
there exists a set $\Theta\subset[0,T]$ with $[0,T]\setmeno\Theta$ at most countable such that 
$V(t_n)\to V(t)$ and $\Gamma(t_n)\to \Gamma(t)$ in the Hausdorff metric for every $t\in\Theta$ 
and every sequence $t_n\to t$. 
By Lemma~\ref{approx1}, using Remark \ref{unique} as in the proof of the global stability, we obtain that
\begin{equation}\label{5000}
e(t_n)\wto e(t)\qquad\hbox{weakly in }L^2(\Om;\Msym)\hbox{ for every }t\in\Theta\hbox{ and every }t_n\to t\,.
\end{equation}




Fix $t\in[0,T]$ and consider a sequence of $s_n^i$ of subdivisions of the interval $[0,t]$ satisfying
\begin{eqnarray}
0=s_n^0<s_n^1<\dots<s_n^{n-1}<s_n^n=t\nonumber\\
\lim_{n\to\infty}\max_{1\le i\le n}(s_n^i-s_n^{i-1})=0\,.\label{5300}
\end{eqnarray}
For $i=1,\dots,n$, we have 
$$
(e(\sni)-Ew(\sni)+Ew(\snim),p(\sni),\Gamma(\sni))\in A(w(\snim))\,.
$$
By the global stability we obtain
\begin{eqnarray*}
 \Q(e(\snim))&\le&\Q(e(\sni)-Ew(\sni)+Ew(\snim))+
 {\mathcal{D}}((p(\sni),\Gamma(\sni)),(p(\snim),\Gamma(\snim)))=\\
&=&\Q(e(\sni))+\langle \sigma(\sni),Ew(\snim)-Ew(\sni)\rangle+\Q(Ew(\snim)-Ew(\sni))+\\
&&{}+{\mathcal{D}}((p(\sni),\Gamma(\sni)),(p(\snim),\Gamma(\snim)))\,.
\end{eqnarray*}
Summing for $i=1,\dots, n$ we obtain
\begin{eqnarray*}
\Q(e_0)\le \Q(e(t))+\sum_{i=1}^n{\mathcal{D}}((p(\sni),\Gamma(\sni)),(p(\snim),\Gamma(\snim)))+\\
+\sum_{i=1}^n \langle \sigma(\sni),Ew(\snim)-Ew(\sni)\rangle+S_n\,,
\end{eqnarray*}
where $S_n:=\sum_i \Q(Ew(\snim)-Ew(\sni))$,
so that
\begin{equation}\label{5100}
\Q(e(t))+{\rm Diss}(p(\cdot),\Gamma(\cdot);0,t)\ge \Q(e_0)+
\sum_{i=1}^n \langle \sigma(\sni),Ew(\sni)-Ew(\snim)\rangle-S_n\,.
\end{equation}
As $w\in AC([0,T];H^1(\Om_0;\R^2))$, it is easy to prove that $S_n\to0$ 
(see \cite[Proof of Lemma 7.3]{DM-Toa} for the details).

Let us define the piecewise constant function $\tilde \sigma_n$ on 
${]0,t]}$ by setting $\tilde \sigma_n(s)=\sigma(\sni)$ for 
$\snim<s\le \sni$. By \eqref{5000} and \eqref{5300} we have 
$\tilde \sigma_n(s)\wto \sigma(s)$ weakly in $L^2(\Om;\Msym)$ for every $s\in\Theta$.
By \eqref{en} we can apply the Dominated Convergence Theorem and we obtain 
$$
\sum_{i=1}^n \langle \sigma(\sni),Ew(\sni)-Ew(\snim)\rangle=
\int_0^t\langle \tilde \sigma_n(s),E\dot w(s)\rangle\, ds\to 
\int_0^t\langle \sigma(s),E\dot w(s)\rangle\,ds\,.
$$
Passing now to the limit in \eqref{5100} as $n\to\infty$ we get 
\begin{equation}\nonumber
\Q(e(t))+{\rm Diss}(p(\cdot),\Gamma(\cdot);0,t)\ge \Q(e_0)+
\int_0^t\langle \sigma(s),E\dot w(s)\rangle\, ds\,,
\end{equation}
which, together with \eqref{5200}, proves condition {\rm (b)} in 
Definition \ref{defquasist}.
\end{proof}

\end{section}

\begin{section}{Flow rule}
%:Flow rule


In this section we prove that every quasistatic evolution satisfies a weak formulation 
of the flow rule $\dot p(t,x)\in N_\K(\sigma(t,x))$, where $N_\K(\sigma)$ denotes 
the normal cone of $\K$ at $\sigma$ in the sense of convex analysis. 
We define
 $$
 \lambda(t):=\hone(\Gamma(t))\,.
 $$

\begin{theorem}\label{flowrule} Let $w\in AC([0,T];H^1(\Om_0;\R^2))$ and let $t\mapsto(e(t),p(t),\Gamma(t))$ be a quasistatic evolution with boundary condition $w$.  
Let us fix a Lebesgue point $t_0\in{]0,T]}$  of $t\mapsto\|E\dot w(t)\|_2$  such that 
$\dot p(t_0)$ and $\dot \lambda(t_0)$ exist. Assume also that
$\sigma(t_0)\in C(\hat\Om(t_0);\Msym)$. Then 
\be
\frac{\dot p(t_0)}{|\dot p(t_0)|}(x)\in N_\K(\sigma(t_0,x))\quad\hbox{for }|\dot p(t_0)|\hbox{-a.e.\ }x\in\hat\Om(t_0)\,.\label{flow}
\ee
\end{theorem}
\begin{proof}
Let us fix $0\le t_1<t_2\le T$. For $i=1,2$ we set $e_i:=e(t_i)$, $\sigma_i:=\sigma(t_i)$, 
$p_i:=p(t_i)$, $w_i:=w(t_i)$, $\Gamma_i:=\Gamma(t_i)$, $\Om_i:=\Om\setmeno\Gamma(t_i)$, 
$\hat\Om_i:=\overline\Om\setmeno\Gamma(t_i)$, and $\lambda_i=\lambda(t_i)$.
Taking into account definitions  \eqref{D} and \eqref{dissd}, from the energy equality 
\eqref{eneq} we obtain
\be\label{8500}
\Q(e_2)-\Q(e_1)+H(p_2-p_1)(\hat\Om_2)+\lambda_2-\lambda_1
\le \int_{t_1}^{t_2}\langle \sigma(t),E\dot w(t)\rangle\,dt\,.
\ee
Let us fix $\e\in{]0,1[}$ and let 
\begin{eqnarray}
&& e^*:=e_1+(1-\e)(e_2-e_1)-(1-\e)(Ew_2-Ew_1)\,,\nonumber\\
&& p^*:=p_1+(1-\e)(p_2-p_1)\ \hbox{on }\hat\Om_2\,,\quad p^*=0\ \hbox{on }\Gamma_2\,,\\
&& \Gamma^*:=\Gamma_2\,.\nonumber
\end{eqnarray}
It is easy to check that $(e^*,p^*,\Gamma^*)\in A(w_1)$.
From the global stability condition \eqref{glstab} we have 
\begin{eqnarray*}
&\Q(e_1)\le \Q\big(e_1+(1-\e)(e_2-e_1)-(1-\e)(Ew_2-Ew_1)\big)+\\
& + (1-\e) H(p_2-p_1)(\hat\Om_2 )+
 \lambda_2-\lambda_1\,,
\end{eqnarray*}
which gives
\begin{eqnarray*}
&0\le \langle \sigma_1,e_2-e_1\rangle -\langle \sigma_1,Ew_2-Ew_1\rangle +(1-\e)
\Q\big(e_2-e_1-(Ew_2-Ew_1)\big)+\\
& +H(p_2-p_1)(\hat\Om_2 )+\tfrac{1}{1-\e}(\lambda_2-\lambda_1)\,.
\end{eqnarray*}
Since by convexity  
$$
\Q\big(e_2-e_1-(Ew_2-Ew_1)\big)\le\tfrac{1}{1-\e^2}\Q(e_2-e_1)+\tfrac{1}{\e^2}\Q(Ew_2-Ew_1)\,,
$$
we obtain
\begin{eqnarray*}
& \langle \sigma_1,e_1-e_2\rangle\le -\langle \sigma_1,Ew_2-Ew_1\rangle+
\tfrac{1}{1+\e}\Q(e_2-e_1)+\tfrac{1-\e}{\e^2}\Q(Ew_2-Ew_1)+{}\\
&{}+H(p_2-p_1)(\hat\Om_2)+
 \lambda_2-\lambda_1+\tfrac{\e}{1-\e}(\lambda_2-\lambda_1)\,.
\end{eqnarray*}



By adding the terms of \eqref{8500} we get
\begin{eqnarray*}
& \Q(e_2-e_1)\le   -\langle \sigma_1,Ew_2-Ew_1\rangle+\tfrac{1}{1+\e}\Q(e_2-e_1)+
\tfrac{1-\e}{\e^2}\Q(Ew_2-Ew_1)+{}\\
& \displaystyle{}+\tfrac{\e}{1-\e}(\lambda_2-\lambda_1)+ \int_{t_1}^{t_2}\langle \sigma(t),E\dot w(t)\rangle\,dt\,,
\end{eqnarray*}
hence
\be\label{8502}
\Q(e_2-e_1)\le \tfrac{1+\e}{\e}\int_{t_1}^{t_2}\langle \sigma(t)-
\sigma_1,E\dot w(t)\rangle\,dt+\tfrac{1-\e^2}{\e^3}\Q(Ew_2-Ew_1)+\tfrac{1+\e}{1-\e}(\lambda_2-\lambda_1)\,.
\ee

Taking $\e=\frac 12$ and  using the version of Gronwall's Inequality proved in 
Lemma \ref{Gronwall} below, we deduce that  there exist two constants $C_1, C_2$ such that
\be\label{6501}
\|e_2-e_1\|_2\le C_1\int_{t_1}^{t_2}\|E\dot w(t)\|_2dt+C_2\sqrt{\lambda_2-\lambda_1}\,.
\ee
This shows, in particular, that the function $t\mapsto e(t)$ is continuous where the function 
$t\mapsto\lambda(t)$ is continuous.

From \eqref{8500} we obtain
\be\label{8503}
H(p_2-p_1)(\hat\Om_2 )+\lambda_2-\lambda_1
\le \langle \sigma_2,e_1-e_2\rangle+\int_{t_1}^{t_2}\langle \sigma(t),E\dot w(t)\rangle\,dt+\Q(e_2-e_1)\,.
\ee
By \eqref{8502}
\be\label{8504}
\begin{array}{c}
\displaystyle H(p_2-p_1)(\hat\Om_2 )\le \langle \sigma_2,e_1-e_2\rangle+\int_{t_1}^{t_2}\langle \sigma(t),E\dot w(t)\rangle\,dt+{}\\
\displaystyle{}+\tfrac{1+\e}{\e}\int_{t_1}^{t_2}\langle \sigma(t)-\sigma_1,E\dot w(t)\rangle\,dt+\tfrac{1-\e^2}{\e^3}\Q(Ew_2-Ew_1)+
\tfrac{2\e}{1-\e}(\lambda_2-\lambda_1)
\end{array}
\ee
For $i=1,2$ let $u_i\in BD_\ell(\Om_i,\partial_D\Om_i)$ be such that $u_i$, $e_i$, $p_i$, $\Gamma_i$, and $w_i$  
satisfy the weak kinematic admissibility conditions \eqref{adm1k}--\eqref{adm3k} on 
$\overline\Om$. We assume that $\sigma_2\in C(\hat\Om_2;\Msym)$. Therefore we have 
\be\label{8505}\begin{array}{c}
\displaystyle \langle \sigma_2,e_1-e_2\rangle_{\Om_2}=\langle\sigma_2,p_2-p_1\rangle_{\hat\Om_2}- 
\langle \sigma_2,Eu_2-Eu_1\rangle_{\Om_2}-{}\\
\displaystyle{}-\langle\sigma_2,(w_2-w_1)\odot\nu_\Om\rangle_{\partial_D\Om_2}+
\langle\sigma_2,(u_2-u_1)\odot\nu_\Om\rangle_{\partial_D\Om_2}\,,
\end{array}
\ee
where $\partial_D\Om_2=\partial\Om\setmeno\Gamma_2$.
Integrating by parts using Theorem \ref{2222}
we obtain
\begin{eqnarray*}
&\langle \sigma_2,Eu_2-Eu_1\rangle_{\Om_2}=\langle\sigma_2,(u_2-u_1)\odot\nu_\Om\rangle_{\partial_D\Om_2}\,,\\
& 
\langle \sigma_2,Ew_2-Ew_1\rangle_{\Om_2}=\langle\sigma_2,(w_2-w_1)\odot\nu_\Om\rangle_{\partial_D\Om_2}\,,
\end{eqnarray*}
so that \eqref{8505} gives
\be\label{8506}
\langle \sigma_2,e_1-e_2\rangle_{\Om_2}=\langle\sigma_2,p_2-p_1\rangle_{\hat\Om_2}- 
\langle \sigma_2,Ew_2-Ew_1\rangle_{\Om_2}\,.
\ee
By \eqref{8504} and \eqref{8506} we have
\be\label{8507}
\begin{array}{c}
\displaystyle H(p_2-p_1)(\hat\Om_2 )\le \langle \sigma_2,p_2-p_1\rangle_{\hat\Om_2}+
\int_{t_1}^{t_2}\langle \sigma(t)-\sigma_2,E\dot w(t)\rangle\,dt+{}\\
\displaystyle{}+\tfrac{1+\e}{\e}\int_{t_1}^{t_2}\langle \sigma(t)-
\sigma_1,E\dot w(t)\rangle\,dt+\tfrac{1-\e^2}{\e^3}\Q(Ew_2-Ew_1)+\tfrac{2\e}{1-\e}(\lambda_2-\lambda_1)\,.
\end{array}
\ee

Let us consider the function 
$F(x,\xi):=H(\xi)-\sigma_2(x)\,{:}\,\xi$, defined for $x\in\hat\Om_2$ and 
$\xi\in\Msym$, and let ${\mathcal{F}}\colon  M_ b(\hat\Om_2;\Msym)\to\R$ be defined by 
$$
{\mathcal{F}}(p):=\int_{\hat\Om_2}F\Big(x,\frac{p}{|p|}(x)\Big)\,d|p|(x)\,.
$$
Since, by \eqref{defH}, 
${\mathcal{F}}(p_2-p_1)=  H(p_2-p_1)(\hat\Om_2 )- 
\langle \sigma_2,p_2-p_1\rangle_{\hat\Om_2}$, 
 inequality \eqref{8507} gives
\be\label{8577}
\begin{array}{c}
\displaystyle {\mathcal{F}}(p_2-p_1)\le \int_{t_1}^{t_2}\langle \sigma(t)-\sigma_2,E\dot w(t)\rangle\,dt+ 
\tfrac{1+\e}{\e}\int_{t_1}^{t_2}\langle \sigma(t)-\sigma_1,E\dot w(t)\rangle\,dt+{}\\
\displaystyle{}+\tfrac{1-\e^2}{\e^3}\Q(Ew_2-Ew_1)+\tfrac{2\e}{1-\e}(\lambda_2-\lambda_1)\,.
\end{array}
\ee

We apply these results with $t_2=t_0$ and $t_1<t_0$.  By Theorem \ref{dotp} we have 
\be\label{88}
\tfrac {1}{t_2-t_1}(p_2-p_1)\wtos \dot p(t_0) \quad \hbox{weakly$^*$ in }  M_ b(\hat\Om(t_0);\Msym)
\ee
as $t_1\to t_2=t_0$, therefore, Reshetnyak's lower semicontinuity theorem \cite[Theorem 2]{Re} implies that
$$
{\mathcal{F}}(\dot p(t_0))\le \liminf_{t_1\to t_2-}{\mathcal{F}}(\tfrac {1}{t_2-t_1}(p_2-p_1))\,.
$$
Dividing each term of \eqref{8577} by $t_2-t_1$
and using \eqref{6501} 
we obtain
${\mathcal{F}}(\dot p(t_0))\le\tfrac{2\e}{1-\e}\dot\lambda(t_0)$. 
Taking the limit as $\e\to0$ we get ${\mathcal{F}}(\dot p(t_0))\le0$.

On the other hand, as $\sigma_2(x)=\sigma(t_0,x)\in \K=\partial H(0)$ (see \cite[Corollary 23.5.3]{Ro}) 
for every $x\in\hat\Om(t_0)$, we have 
$F(x,\xi)\ge0$ for  every $x\in\hat\Om(t_0)$ and every $\xi\in\Msym$. 
Therefore the inequality  ${\mathcal{F}}(\dot p(t_0))\le0$ 
 and the definition of $F$ give 
\be\label{8509}
 H\Big(\frac{\dot p(t_0)}{|\dot p(t_0)|}(x)\Big)= \sigma(t_0,x)\,{:}\,\frac{\dot p(t_0)}{|\dot p(t_0)|}(x)\quad
 \hbox{for $|\dot p(t_0)|$-a.e.\ }x\in \hat\Om(t_0)\,.
\ee
Since $\sigma(t_0,x)\in \K=\partial H(0)$  for every $x\in\hat\Om(t_0)$, we have  
$\sigma(t_0,x)\,{:}\,\xi\le H(\xi)$ for every $x\in\hat\Om(t_0)$ and every $\xi\in \Msym$.
Using \eqref{8509}, for $|\dot p(t_0)|$-a.e.\ $x\in\hat\Om(t_0)$ we obtain 
$$
\sigma(t_0,x)\,{:}\,\Big(\xi-\frac{\dot p(t_0)}{|\dot p(t_0)|}(x)\Big)\le H(\xi)-
H\Big(\frac{\dot p(t_0)}{|\dot p(t_0)|}(x)\Big)\quad\hbox{for every }\xi\in \Msym\,,
$$
which gives 
$\displaystyle\sigma (t_0,x)\in\partial H\Big(\frac{\dot p(t_0)}{|\dot p(t_0)|}(x)\Big)$. 
By \cite[Theorem 23.5]{Ro} this implies 
$$
\frac{\dot p(t_0)}{|\dot p(t_0)|}(x)\in N_\K(\sigma(t_0,x))\,\quad\hbox{for $|\dot p(t_0)|$-a.e.\ } x\in\hat\Om(t_0)\,,
$$
concluding the proof of \eqref{flow}.
\end{proof}


\begin{lemma}\label{Gronwall}
Let $\phi\colon[0,T]\to[0,+\infty[$ be a bounded measurable function, let $\psi\colon[0,T]\to[0,+\infty[$ 
be an integrable function, and let $\chi\colon[0,T]\to[0,+\infty[$ be a nondecreasing function.
Suppose that
\begin{equation}\label{hypothesis}
\phi(t)^2\le \int_0^t \phi(s)\, \psi(s)\,ds + \Big( \int_0^t\psi(s)\,ds \Big)^2+\chi(t)
\end{equation}
for every $t\in[0,T]$.
Then
\begin{equation}\label{thesis}
\phi(t)\le\tfrac32 \int_0^t\psi(s)\,ds+\tfrac32\sqrt{\chi(t)}
\end{equation}
for every $t\in[0,T]$.
\end{lemma}

\begin{proof}
Let us fix $t_0\in [0,T]$ and let 
$\gamma_0:= ( \int_0^{t_0}\psi(s)\,ds)^2+\chi(t_0)$.
For every $t\in[0,t_0]$ we define
$f(t):=  \int_0^t \phi(s)\, \psi(s)\,ds$.
Then $f$ is absolutely continuous on $[0,t_0]$,
$\phi(t)^2 \le f(t)+\gamma_0$ for every $t\in [0,t_0]$, and $\dot f(t)\le \psi(t) ({f(t)+\gamma_0})^{1/2}$ 
for a.e.\ $t\in [0,t_0]$. Integrating between $0$ and $t_0$ we get 
$2({f(t_0)+\gamma_0})^{1/2}\allowbreak\le 2\gamma_0^{1/2} +  \int_0^{t_0}\psi(s)\,ds\le   
3  \gamma_0^{1/2} $. By (\ref{hypothesis}) we have $\phi(t_0)\le ({f(t_0)+\gamma_0})^{1/2}$, 
so that the previous inequality gives $2\,\phi(t_0)\le 3  \gamma_0^{1/2} \le 3 \int_0^{t_0}\psi(s)\,ds+3\sqrt{\chi(t_0)}$.
\end{proof}
\end{section}

\begin{section}{Some properties of the solutions}\label{some}
%:Some properties of the solutions

In the first subsection we prove that the crack tips cannot be isolated from the plastic region. 
In the second subsection we show that, when the yield surface tends to infinity, 
 the quasistatic evolutions converge to a solution of the brittle crack evolution problem studied in \cite{Ch}.


\subsection{Plastic behaviour near the crack tip}\label{tip}

Let $t\mapsto(e(t),p(t),\Gamma(t))$ be a quasistatic evolution.  Given $t\in [0,T]$,  
a {\em regular crack tip} at time $t$ is a point $x(t)$ having an open neighbourhood $U$ 
such that $\Gamma(t)\cap U$ is a $C^2$ arc with an end-point at $x(t)$.

\begin{theorem}\label{thtip} Let $t\mapsto(e(t),p(t),\Gamma(t))$ be a quasistatic evolution,
 let  $t\in [0,T]$, and let $x(t)$ be a regular crack tip of $\Gamma(t)$. 
 Then $x(t)\in {\rm supp}\,p(t)$. In other words, $|p(t)|(U)\neq0$ for every open 
 neighbourhood $U$ of $x(t)$.
\end{theorem}
\begin{proof}
Assume by contradiction that there exists an open neighbourhood $U$ of  
$x(t)$ such that $|p(t)|(U)=0$. It is not restrictive to assume that $U$ is an open ball 
and that $\Gamma(t)$ meets $\partial U$ transversally. 
By the weak kinematic admissibility condition \eqref{adm1k} and by Korn's inequality  there exists 
$u(t)\in H^1(U\setmeno\Gamma(t))$ such that $e(t)=Eu(t)$ in $U\setmeno\Gamma(t)$. 
By the global stability condition we have 
$$
\int_{U\setmeno\Gamma(t)}Q(Eu(t))\,dx\le\int_{U\setmeno\Gamma(t)}Q(Eu^*)\,dx
$$ 
for every $u^*\in H^1(U\setmeno\Gamma(t))$ with $u^*=u(t)$ on $\partial U\setmeno\Gamma(t)$.
Classical results (see, e.g., \cite{Gris}) about  the solutions of the Lam\'e system with Neumann 
boundary conditions on a crack show that $|\sigma(t,x)|\to+\infty$ as $x$ tends to the crack tip $x(t)$. 
This contradicts the constraint $\sigma(t,x)\in\K$ for a.e.\ $x\in U\setmeno\Gamma(t)$, 
which follows from the global stability condition and from 
Theorem \ref{2222}.
\end{proof}


\subsection{Convergence to brittle crack growth}

In this subsection we study the dependence of the solutions on the elastic domain $\K$,  
when $\K$ tends to the whole space $\Msym$. We prove that the cracks of the corresponding 
quasistatic evolutions converge to a crack $\Gamma(t)$ such that $t\mapsto\Gamma(t)$ is a 
quasistatic evolution according to the model for brittle cracks studied in \cite{Ch}. 

For every $\Gamma\in\Kmf$ we introduce the space 
$$
LD_\ell^2(\Om_\Gamma):=\{u\in L^2_{loc}(\Om_\Gamma;\R^2): Eu\in L^2(\Om_\Gamma;\Msym)\}\,,
$$
which in \cite{Ch} is denoted by $LD(\Om_\Gamma)$. As in Section \ref{prelim}, we continue to use  the notation 
$LD(\Om_\Gamma)$ for the space studied in \cite[Chapters I and II]{Tem}.
Since $LD(\Om_\Gamma)$ is contained in $L^2_{loc}(\Om_\Gamma;\R^2)$ by the Sobolev imbedding theorem in dimension two, we have
\be\label{korn}
LD_\ell^2(\Om_\Gamma)=\{u\in L^1_{loc}(\Om_\Gamma;\R^2): Eu\in L^2(\Om_\Gamma;\Msym)\}\,.
\ee
We shall use the following approximation result.
\begin{lemma}\label{approx2} Let $w\in H^1(\Om_0;\R^2)$.
Let $\Gamma_n,\Gamma\in\Kmf$ and assume that $\Gamma_n\to\Gamma$ in the Hausdorff metric. 
For every $v\in LD_\ell^2(\Om_{\Gamma})$, with $v=w$ $\hone$-a.e.\ on $\partial_D\Om_\Gamma$, 
there exists $v_n\in LD_\ell^2(\Om_{\Gamma_n})$, with $v_n=w$ $\hone$-a.e.\ on 
$\partial_D\Om_{\Gamma_n}$, such that $Ev_n\to Ev$ strongly in $L^2(\Om;\Msym)$.
\end{lemma}
\begin{proof} The proof of this lemma is contained in the proof of \cite[Theorem 3]{Ch}. 
 Bucur and Varchon \cite{B-V} proved the result when $v\in H^1(\Om_\Gamma;\R^2)$ and 
 $\partial_D\Om_{\Gamma_n}=\emptyset$.  If $v\notin H^1(\Om_\Gamma;\R^2)$ one has to 
 apply Chambolle's approximation result \cite[Theorem 1]{Ch}. 
 The Dirichlet boundary condition on $\partial_D\Om_{\Gamma_n}$ requires another 
 argument due to Chambolle, for which we refer also to \cite[Theorem 6.3]{DM-E-P}.
\end{proof}

We are now in a position to prove the main result of this section.
\begin{theorem}\label{convbrittle}  Let $w\in AC([0,T];H^1(\Om_0;\R^2))$, with $w(0)=0$, 
and let  $\K_n$ be an increasing sequence of compact convex sets containing $0$ as an interior point. 
Assume that $\K_n\nearrow \Msym$, in the sense that $\Msym$ is the  union of the sets $\K_n$. 
For every $n$ let $t\mapsto(e_n(t),p_n(t),\Gamma_n(t))$ be a 
quasistatic evolution corresponding to $\K=\K_n$, 
 with boundary condition $w$ and initial condition  $(0,0,\Gamma_0)$. 
 Then there exists a subsequence, not relabelled, and a quasistatic evolution 
 $t\mapsto \Gamma(t)$ of brittle cracks in the sense of \cite[Theorem 3]{Ch}, 
 such that for every $t\in[0,T]$
\begin{eqnarray}
&e_n(t)\to e(t)\qquad\hbox{strongly in }L^2(\Om;\Msym)\,,\label{ekbr}\\
&p_n(t)\to0\qquad\hbox{strongly in }   M_ b(\overline\Om;\Msym)\,,\label{pkbr}\\
&\Gamma_n(t)\to\Gamma(t)\quad\hbox{in the Hausdorff metric}\hbox{ and}\quad
\hone(\Gamma_n(t))\to\hone(\Gamma(t))\,,\label{gkbr}
\end{eqnarray}
where $e(t):=Eu(t)$ a.e.\ in $\Om(t):=\Om\setmeno\Gamma(t)$ and $u(t)$ is a minimizer of
$$
\min \{\Q(Ev): v\in LD_\ell^2(\Om(t))\,,\ v=w(t)\ \hone\hbox{-a.e.\ on }
\partial_D\Om(t):=\partial\Om\setmeno\Gamma(t)\}\,.
$$
\end{theorem}
\begin{proof}
Under our hypotheses there exists a sequence $r_n\to+\infty$ such that
\be\label{rK}
\{\xi\in\Msym:|\xi|\le r_n\}\subset \K_n\,.
\ee
Let $H_n$, $H_n(\mu)$, ${\mathcal{D}}_n$, and ${\rm Diss}_n$ be defined by 
\eqref{support}, \eqref{defH}, \eqref{D}, and \eqref{dissd},  with $\K=\K_n$. 
Applying Gronwall's Lemma to the energy balance for $(e_n(t),p_n(t),\Gamma_n(t))$ 
we deduce that there exists a positive constant $C$ such that 
\be\label{superstima}
\|e_n(t)\|_2\le C\,,\quad\hone(\Gamma_n(t))\le C\,,\quad H_n(p_n(t))(\hat\Om_n(t))\le C
\ee
for every $t\in[0,T]$ and every $n$, where $\hat\Om_n(t):=\overline\Om\setmeno\Gamma_n(t)$ . 
Using \eqref{Hrk} and \eqref{rK}, from the last inequality we infer
$$
r_n|p_n(t)|(\hat\Om_n(t))\le C\,,
$$ 
hence $|p_n(t)|(\overline\Om)\to 0$ by \eqref{adm3k}. This concludes the proof of 
\eqref{pkbr}. 

By \eqref{superstima} we can apply Helly's Theorem in $\Kmf$ 
(see \cite[Theorem 6.3]{DM-Toa})  and find a subsequence, not relabelled, and 
an increasing function $\Gamma\colon[0,T]\to\Kmf$ such that 
$\Gamma_n(t)\to \Gamma(t)$ in the Hausdorff metric for every $t\in[0,T]$. 

As in the proof of Theorem \ref{main}, we fix $t\in[0,T]$. By \eqref{superstima} 
there exists a sequence $n^t_k\to\infty$, depending on $t$, and a function 
$e(t)\in L^2(\Om;\Msym)$ such that $e_{n^t_k}\wto e(t)$ weakly in $L^2(\Om;\Msym)$. 
 By Lemma \ref{closure}, $(e(t),0)\in A(w(t),\Gamma(t))$. By 
\eqref{adm1k}--\eqref{adm3k}, and \eqref{korn} this implies that there exists 
$u(t)\in LD_\ell^2(\Om(t))$ such that $Eu(t)=e(t)$ a.e.\ in $\Om(t)$ and $u(t)=w(t)$ 
$\hone$-a.e.\ on $\partial_D\Om(t)$.

Let us prove the minimality condition
\be\label{brittlemin}
\Q(Eu(t))\le\Q(Ev^*)+\hone(\Gamma^*\setmeno\Gamma(t))
\ee
 for every $\Gamma^*\in\Kmf$ with $\Gamma^*\supset\Gamma(t)$ and every 
 $v^*\in LD_\ell^2(\Om_{\Gamma^*})$ with $v^*=w(t)$ $\hone$-a.e.\ on 
 $\partial_D\Om_{\Gamma^*}$.

Let us fix $\Gamma^*$ and $v^*$. By \cite[Lemma 3.5]{DM-Toa} there exists a sequence 
$\Gamma^*_n\in\Kmf$ which converges to $\Gamma^*$ in the Hausdorff metric and 
such that $\Gamma^*_n\supset\Gamma_n(t)$ and 
$\hone(\Gamma^*_n\setmeno\Gamma_n(t))\to\hone(\Gamma^*\setmeno\Gamma(t))$.  
By Lemma \ref{approx2} there exist $v_n\in LD_\ell^2(\Om_{\Gamma_n(t)})$ and 
$v^*_n\in LD_\ell^2(\Om_{\Gamma_n^*})$, with  $v_n=w(t)$ $\hone$-a.e.\ on 
 $\partial_D\Om_{\Gamma_n(t)}$  and $v^*_n=w(t)$ $\hone$-a.e.\ on 
 $\partial_D\Om_{\Gamma_n^*}$, such that  $Ev_n\to Eu(t)=e(t)$ and 
 $Ev^*_n\to Ev^*$ strongly in $L^2(\Om;\Msym)$.

Since $(e_n(t),p_n(t))\in A(w(t),\Gamma_n(t))$, there exists 
$u_n(t)\in BD_\ell(\Om_{\Gamma_n(t)},\partial_D\Om_{\Gamma_n(t)})$ such that 
$u_n(t)$, $e_n(t)$, $p_n(t)$, $\Gamma_n(t)$, and $w(t)$ satisfy the weak kinematic admissibility conditions 
\eqref{adm1k}--\eqref{adm3k} on $\overline\Om$. 
We define
$$
u_n^*:=u_n(t)-v_n+v^*_n\,,\quad e^*_n:=e_n(t)-Ev_n+Ev^*_n\,,\quad
\hbox{and}\quad p^*_n:=p_n(t)\LLL\Om_{\Gamma_n^*}\,.
$$
It is easy to see that $u^*_n$, $e^*_n$, $p^*_n$, $\Gamma^*_n$, and $w(t)$ satisfy the weak 
kinematic admissibility conditions \eqref{adm1k}--\eqref{adm3k} on 
$\overline\Om$, i.e., $(e^*_n, p^*_n)\in A(w(t),\Gamma^*_n)$. 
By the global stability condition for $(e_n(t),p_n(t),\Gamma_n(t))$ we have
$$
\Q(e_n(t))\le \Q(e_n(t)-Ev_n+Ev^*_n)+\hone(\Gamma^*_n\setmeno\Gamma_n(t))\,,
$$
which gives 
$$
0\le\langle\sigma_n(t),Ev^*_n-Ev_n\rangle+\Q(Ev^*_n-Ev_n)+\hone(\Gamma^*_n\setmeno\Gamma_n(t))\,,
$$ 
where $\sigma_n(t):=\C e_n(t)$. Passing to the limit along the subsequence $n_k^t$ we obtain
$$
0\le\langle\sigma(t),Ev^*-Eu(t)\rangle+\Q(Ev^*-Eu(t))+\hone(\Gamma^*\setmeno\Gamma(t))\,,
$$
where $\sigma(t):=\C e(t)$. This inequality implies \eqref{brittlemin}.

In particular, we have $\Q(Eu(t))\le \Q(Ev^*)$ for every $v^*\in LD^2_\ell(\Om(t))$ with 
$v^*=w(t)$ $\hone$-a.e.\ on $\partial_D\Om(t)$. This implies that $Eu(t)=e(t)$ does 
not depend on the subsequence, hence the whole sequence $e_n(t)$ converges 
to $e(t)$ weakly in $L^2(\Om;\Msym)$.

The energy balance for  $(e_n(t),p_n(t),\Gamma_n(t))$ gives
\be\label{eneqk}
\Q(e_n(t))+{\rm Diss}_n(p_n(\cdot),\Gamma_n(\cdot);0,t)=
\int_0^t\langle \sigma_n(s),E\dot w(s)\rangle\,ds\,,
\ee
which implies
\be\label{disugk}
\Q(e_n(t))+\hone(\Gamma_n(t))\le \hone(\Gamma_0)+\int_0^t\langle \sigma_n(s),E\dot w(s)\rangle\,ds\,,
\ee
Passing to the limit and using the lower semicontinuity of $\Q$ and $\hone$ (see \cite[Corollary 3.4]{DM-Toa}), we obtain  
$$
\Q(e(t))+\hone(\Gamma(t))\le \hone(\Gamma_0)+\int_0^t\langle \sigma(s),E\dot w(s)\rangle\,ds\,,
$$
The opposite inequality follows from the global stability, as in the second part 
of the proof of Theorem \ref{main}, hence 
\be\label{enbr}
\Q(e(t))+\hone(\Gamma(t))=\hone(\Gamma_0)+ \int_0^t\langle \sigma(s),E\dot w(s)\rangle\,ds\,.
\ee

To prove \eqref{ekbr} and \eqref{gkbr} it remains to show that
\be\label{convbr}
\Q(e_n(t))\to \Q(e(t))\quad\hbox{and}\quad\hone(\Gamma_n(t))\to\hone(\Gamma(t))\,. 
\ee
From \eqref{disugk} and \eqref{enbr} we deduce that
\begin{eqnarray*}
&\displaystyle \limsup_{n\to\infty}\big(\Q(e_n(t))+\hone(\Gamma_n(t))\big)\le
\hone(\Gamma_0)+\lim_{n\to\infty}\int_0^t\langle \sigma_n(s),E\dot w(s)\rangle\,ds=\\
&\displaystyle
=\hone(\Gamma_0)+ \int_0^t\langle \sigma(s),E\dot w(s)\rangle\,ds=\Q(e(t))+\hone(\Gamma(t))\,,
\end{eqnarray*}
which gives \eqref{convbr} thanks to the lower semicontinuity of $\Q$ and 
$\hone$ (see \cite[Corollary 3.4]{DM-Toa}). \end{proof}



\end{section}


\bigskip
\noindent {\bf Acknowledgments.} {This work is part of the Project ``Variational problems with 
multiple scales" 2006, 
supported by the Italian Ministry of University and Research.}
\par

{\frenchspacing

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\end{thebibliography}
}


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