\documentclass[a4paper,reqno]{amsart} \usepackage{amsfonts} \textwidth14cm \textheight22cm \flushbottom % \usepackage{marvosym} %for \EUR % \usepackage[notref,notcite]{showkeys} \catcode`@=11 % we will access private macros of plain TeX (carefully) \def\blfootnote{\xdef\@thefnmark{}\@footnotetext} \catcode`@=12 % at signs are no longer letters \theoremstyle{plain} \begingroup \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \endgroup \theoremstyle{definition} \begingroup \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \endgroup \theoremstyle{remark} \begingroup \endgroup \mathsurround=1pt \mathchardef\emptyset="001F \numberwithin{equation}{section} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\e}{\varepsilon} \newcommand{\Om}{\Omega} \newcommand{\R}{{\mathbb R}} \newcommand{\C}{{\mathbb C}} \newcommand{\Rtwo}{{\R}^2} \newcommand{\Msym}{{\mathbb M}^{2{\times}2}_{sym}} \newcommand{\wto}{\rightharpoonup} \newcommand{\wtos}{\mathrel{\mathop{\rightharpoonup}\limits^*}} \newcommand{\setmeno}{\!\setminus\!} \newcommand{\hone}{{\mathcal H}^{1}} \newcommand{\Ltwo}{{\mathcal{L}}^2} \newcommand{\Q}{{\mathcal Q}} \newcommand{\HH}{{\mathcal H}} \newcommand{\gdot}{{\dot{g}}} \newcommand{\K}{{\mathbb K}} \newcommand{\Kmf}{{\mathcal C}_m} \newcommand{\Ku}{{\mathcal K}(\overline U)} \newcommand{\nii}{_n^i} \newcommand{\nj}{_n^j} \newcommand{\nim}{_n^{i-1}} \newcommand{\njm}{_n^{j-1}} \newcommand{\sni}{s\nii} \newcommand\snim{s\nim} \newcommand\tni{t\nii} \newcommand\tnim{t\nim} \newcommand\tnj{t\nj} \newcommand\tnjm{t\njm} \newcommand\un{u_n} \newcommand\wn{w_n} \newcommand\en{e_n} \newcommand\pn{p_n} \newcommand\uni{u\nii} \newcommand\eni{e\nii} \newcommand\pni{p\nii} \newcommand\wni{w\nii} \newcommand\wnim{w\nim} \newcommand\enim{e\nim} \newcommand\pnim{p\nim} \newcommand\unim{u\nim} \newcommand\unj{u\nj} \newcommand\unjm{u\njm} \newcommand\pnj{p\nj} \newcommand\pnjm{p\njm} \newcommand\vn{v_n} \newcommand\Kni{\Gamma\nii} \newcommand\Knj{\Gamma\nj} \newcommand\Knjm{\Gamma\njm} \newcommand\Knim{\Gamma\nim} \newcommand{\LM}[2]{\hbox{\vrule width.4pt \vbox to#1pt{\vfill \hrule width#2pt height.4pt}}} \newcommand{\LLL}{{\mathchoice {\>\LM{7}{5}\>}{\>\LM{7}{5}\>}{\,\LM{5}{3.5}\,}{\,\LM{3.35}{2.5}\,}}} % L symbol \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 $0t$ 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 $0\rho\,,\\ && \Gamma^*\cap (\overline Q{}^-_{\rho,\delta}(x)\cup \overline Q{}^+_{\rho,\delta}(x))= \emptyset\,,\\ && (\Gamma^*\cap Q_\rho(x))\cup L_{\rho,\delta}(x)\cup A_{\rho,\delta}(x)\quad \hbox{is connected}\,,\\ && \hone(A_{\rho,\delta}(x))0$ 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\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})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)\,. $$ Therefore \eqref{K*K} and \eqref{pp*1} give \begin{eqnarray*} && \displaystyle \Q(e^*)-\Q(e)+H(p^*-p)(\hat\Om^*)+ \hone(\Gamma^*\setmeno \Gamma)=\\ &&\displaystyle =\lim_{n\to\infty}(\Q(e^*_n)-\Q(e_n)+ H(p^*_n-p_n)(\hat\Om^*_n)+\hone(\Gamma^*_n\setmeno \Gamma_n))\ge0\,, \end{eqnarray*} where $\hat\Om^*:=\overline\Om\setmeno \Gamma^*$. This inequality implies \eqref{111}. \end{proof} \end{section} \begin{section}{The discrete time problems}\label{discrete} %:The discrete time problems Let us fix $T>0$ and a sequence of subdivisions $(\tni)_{0\le i\le n}$ of the interval $[0,T]$, with \begin{eqnarray} & 0=t_n^00$, depending only on $e_0$, $p_0$, $\Gamma_0$, $w$, $T$, $\alpha_\C$, $\beta_\C$, and $r_\K$, such that for $n$ large enough \begin{eqnarray} &\displaystyle \|\en(t)\|_2\le C\,,\label{en}\\ &\displaystyle {\rm Diss}(p_n(\cdot), \Gamma_n(\cdot);0,T)\le C\,,\label{dissn}\\ &\displaystyle \|p_n(t)\|_1\le C\,,\label{varpn}\\ &\displaystyle \hone(\Gamma_n(t))\le C\,,\label{h1kn} \end{eqnarray} for every $t\in[0,T]$. \end{lemma} \begin{proof} From \eqref{lowerboundC}, \eqref{upperboundC}, and \eqref{1999} we deduce that $$ \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