\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 83, pp. 1--21.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/83\hfil Dynamic contact with Signorini's condition] {Dynamic contact with Signorini's condition and slip rate dependent friction} \author[Kenneth Kuttler, Meir Shillor \hfil EJDE-2004/83\hfilneg] {Kenneth Kuttler, Meir Shillor} % in alphabetical order \address{Kenneth Kuttler \hfill\break Department of Mathematics \\ Brigham Young University\\ Provo, UT 84602, USA} \email{klkuttle@math.byu.edu} \address{Meir Shillor \hfill\break Department of Mathematics and Statistics\\ Oakland University\\ Rochester, MI 48309, USA} \email{shillor@oakland.edu} \date{} \thanks{Submitted October 15,2003. Published June 11, 2004.} \subjclass[2000]{74M10, 35Q80, 49J40, 74A55, 74H20, 74M15} \keywords{Dynamic contact, Signorini condition, slip rate dependent friction, \hfill\break\indent nonlocal friction, viscoelastic body, existence} \begin{abstract} Existence of a weak solution for the problem of dynamic frictional contact between a viscoelastic body and a rigid foundation is established. Contact is modelled with the Signorini condition. Friction is described by a slip rate dependent friction coefficient and a nonlocal and regularized contact stress. The existence in the case of a friction coefficient that is a graph, which describes the jump from static to dynamic friction, is established, too. The proofs employ the theory of set-valued pseudomonotone operators applied to approximate problems and a priori estimates. \end{abstract} \maketitle \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \numberwithin{equation}{section} \section{Introduction} This work considers frictional contact between a deformable body and a moving rigid foundation. The main interest lies in the dynamic process of friction at the contact area. We model contact with the Signorini condition and friction with a general nonlocal law in which the friction coefficient depends on the slip velocity between the surface and the foundation. We show that a weak solution to the problem exists either when the friction coefficient is a Lipschitz function of the slip rate, or when it is a graph with a jump from the static to the dynamic value at the onset of sliding. Dynamic contact problems have received considerable attention recently in the mathematical literature. The existence of the unique weak solution of the problem for a viscoelastic material with normal compliance was established in Martins and Oden \cite{MO87}. The existence of solutions for the frictional problem with normal compliance for a thermoviscoelastic material can be found in Figueiredo and Trabucho \cite{FT}; when the frictional heat generation is taken into account in Andrews \textit{et al.} \cite{AKS97}, and when the wear of the contacting surfaces is allowed in Andrews \textit{et al.} \cite{AKSW97}. A general normal compliance condition was dealt with in Kuttler \cite{kut97} where the usual restrictions on the normal compliance exponent were removed. The dynamic frictionless problem with adhesion was investigated in Chau \textit{et al.} \cite{CSS03} and in Fern\'andez \textit{et al.} \cite{FjSS03}. An important one-dimensional problem with slip rate dependent friction coefficient was investigated in Ionescu and Paumier \cite{IP}, and then in Paumier and Renard \cite{PR03}. Problems with normal compliance and slip rate dependent coefficient of friction were considered in Kuttler and Shillor \cite{KSpsm} and a problem with a discontinuous friction coefficient, which has a jump at the onset of sliding, in Kuttler and Shillor \cite{KSmunc}. The problem of bilateral frictional contact with discontinuous friction coefficient can be found in Kuttler and Shillor \cite{KSbi01}. A recent substantial regularity result for dynamic frictionless contact problems with normal compliance was obtained in Kuttler and Shillor \cite{KSreg04}, and a regularity result for the problem with adhesion can be found in Kuttler \textit{et al.} \cite {KSF03}. For additional publications we refer to the references in these papers, and also to the recent monographs Han and Sofonea \cite{HS} and Shillor \textit{et al.} \cite{SSTbook}. Dynamic contact problems with a unilateral contact condition for the normal velocity were investigated in Jarusek \cite{Ja99}, Eck and Jarusek \cite {EJ99} and the one with the Signorini contact condition in Cocu \cite{Co02}, (see also references therein). In \cite{Co02} the existence of a weak solution for the problem for a viscoelastic material with regularized contact stress and constant friction coefficient has been established, using the normal compliance as regularization. After obtaining the necessary a priori estimates, a solution was obtained by passing to the regularization limit. The normal compliance contact condition was introduced in \cite{MO87} to represent real engineering surfaces with asperities that may deform elastically or plastically. However, very often in mathematical and engineering publications it is used as a regularization or approximation of the Signorini contact condition which is an idealization and describes a perfectly rigid surface. Since, physically speaking, there are no perfectly rigid bodies and so the Signorini condition is necessarily an approximation, admittedly a very popular one. The Signorini condition is easy to write and mathematically elegant, but seems not to describe well real contact. Indeed, there is a low regularity ceiling on the solutions to models which include it and, generally, there are no uniqueness results, unlike the situation with normal compliance. Moreover, it usually leads to numerical difficulties, and most numerical algorithms use normal compliance anyway. Although there are some cases in quasistatic or static contact problems where using it seems to be reasonable, in dynamic situations it seems to be a poor approximation of the behavior of the contacting surfaces. We believe that in dynamic processes the Signorini condition is an approximation of the normal compliance, and not a very good one. On the other hand, there is no rigorous derivation of the normal compliance condition either, so the choice of which condition to use is, to a large extent, up to the researcher. This work extends the recent result in \cite{Co02} in a threefold way. We remove the compatibility condition for the initial data, since it is an artifact of the mathematical method and is unnecessary in dynamic problems. We allow for the dependence of the friction coefficient on the sliding velocity, and we take into account a possible jump from a static value, when the surfaces are in stick state, to a dynamic value when they are in relative sliding. Such a jump is often assumed in engineering publications. For the sake of mathematical completeness we employ the Signorini condition together with a regularized non-local contact stress. The rest of the paper is structured as follows. In Section 2 we describe the model, its variational formulation, and the regularization of the contact stress. In Section 3 we describe approximate problems, based on the normal compliance condition. The existence of solutions for these problems is obtained by using the theory of set-valued pseudomonotone maps developed in \cite{KSpsm}. A priori estimates on the approximate solutions are derived in Section 4 and by passing to the limit we establish Theorem \ref{maintheorem} . In Section 5 we approximate the discontinuous friction coefficient, assumed to be a graph at the origin, with a sequence of Lipschitz functions, obtain the necessary estimates and by passing to the limit prove Theorem \ref {setvaluedthm}. We conclude the paper in Section 6. \section{The model and variational formulation} First, we describe the classical model for the process and the assumptions on the problem data. We use the isothermal version of the problem that was considered in \cite{AKS97, AKSW97} (see also \cite{ KO, MO87, DL76, PAN}) and refer the reader there for a detailed description of the model. A similar setting, with constant friction coefficient, can be found in \cite {Co02}. \begin{figure}[ht] \begin{center} \setlength{\unitlength}{.7pt} \begin{picture}(300,121)(50,-6) \put(157,115){$\Gamma_D$} \put(282,60){$\Gamma_{N}$} \put(158,24){$\Gamma_C$} \put(213,16){\vector(0,-1){18}} \put(216,-1){${\mathbf{n}}$} \put(62,-6){Foundation} \put(176,-3){\vector(1,0){18}} \put(160,-6){$\mathbf{v}_F$} \put(306,13){$g$ - gap} \put(280,5){\line(0,1){14}} \put(120,70){$\Omega$ - Body} \put(50,8){\line(1,0){300}} \put(201,74){\oval(220,114)} \end{picture} \end{center} \caption{The physical setting; $\Gamma_C$ is the contact surface.} \end{figure} We consider a viscoelastic body which occupies the reference configuration $\Omega \subset \mathbb{R}^{N}$ ($N=2$ or $3$ in applications) which may come in contact with a rigid foundation on the part $\Gamma _{C}$ of its boundary $\Gamma =\partial \Omega$. We assume that $\Gamma$ is Lipschitz, and is partitioned into three mutually disjoint parts $\Gamma _{D},\Gamma_{N}$ and $\Gamma _{C}$ and has outward unit normal $\mathbf{n} =(n_{1},\dots , n_{N})$. The part $\Gamma _{C}$ is the potential contact surface, Dirichlet boundary conditions are prescribed on $\Gamma _{D}$ and Neumann's on $\Gamma_{N}$. We set $\Omega _{T}=\Omega \times (0,T)$ for $00$, then we set $V=\{\mathbf{u}\in H^{1}(\Omega ))^{N}:\mathbf{u}=0\;\;\text{on }\; \Gamma _{D}\}$. If the body is not held fixed, so that meas$\,\Gamma _{D}=0$, then it is free to move in space, and we set $V=(H^{1}(\Omega ))^{N}$. We note that the latter leads to a noncoercive problem for the quasistatic model, the so-called punch problem, which in that context needs a separate treatment. We let $U$ be a Banach space in which $V$ is compactly embedded, $V$ is also dense in $U$ and the trace map from $U$ to $(L^{2}( \partial \Omega) ) ^{N}$ is continuous. We seek the solutions in the convex set $$\mathcal{K}\equiv \left\{\mathbf{w}\in \mathcal{V}: \mathbf{w}'\in \mathcal{V}', \; ( w_{n}-g) _{+}=0\text{ in }L^{2}( 0,T;L^{2}( \Gamma _{C}) ) \right\} . \label{4mare11}$$ Here, $(f)_{+}=\max \{0,f\}$ is the positive part of $f$. We shall need the following viscosity and elasticity operators, $M$ and $L$, respectively, defined as: $M,L:V\to V'$, \begin{gather} \langle M\mathbf{u},\mathbf{v}\rangle =\int_{\Omega }B\varepsilon ( \mathbf{u}) \varepsilon ( \mathbf{v}) dx, \label{6.6} \\ \langle L\mathbf{u},\mathbf{v}\rangle =\int_{\Omega }A\varepsilon ( \mathbf{u}) \varepsilon ( \mathbf{v}) dx. \label{6.7} \end{gather} It follows from the assumptions and Korn's inequality (\cite{nec81, Ole92}) that $M$ and $L$ satisfy $$\langle C\mathbf{u},\mathbf{u}\rangle \geq \delta ^{2}\|\mathbf{u} \|_{V}^{2}-\lambda |\mathbf{u}|_{H}^{2},\quad \langle C\mathbf{u},\mathbf{u} \rangle \geq 0,\quad \langle C\mathbf{u},\mathbf{v}\rangle =\langle C \mathbf{v},\mathbf{u}\rangle , \label{28febe1}$$ where $C=M$ or $L$, for some $\delta >0,\;\lambda \geq 0.$ Next, we define $\mathbf{f}\in L^{2}(0,T;V')$ as $$\langle \mathbf{f},\mathbf{z}\rangle _{\mathcal{V}',\mathcal{V} }=\int_{0}^{T}\int_{\Omega }\mathbf{f}_{B}\mathbf{z}\,dx\,dt+\int_{0}^{T}\int_{ \Gamma _{N}}\mathbf{f}_{N}\mathbf{z}d\Gamma dt, \label{6.9}$$ for $\mathbf{z}\in \mathcal{V}$. Here $\mathbf{f}_{B}\in L^{2} ( 0,T;H)$ is the body force and $\mathbf{f}_{N}\in L^{2}( 0,T;L^{2} ( \Gamma _{N}) ^{N})$ is the surface traction. Finally, we let $\gamma _{T}^{\ast }:L^{2}( 0,T;L^{2}( \Gamma _{C}) ^{N}) \to \mathcal{V}'$ be defined as $$\langle \gamma _{T}^{\ast }\xi ,\mathbf{w}\rangle \equiv \int_{0}^{T}\int_{\Gamma _{C}}\xi \cdot \mathbf{w}_{T}d\Gamma dt. \label{28febe2}$$ The first of our two main results in this work is the existence of weak solutions to the problem, under the above assumptions. \begin{theorem} \label{maintheorem} Assume, in addition, that $u_{0}\in V$, $u_{0n}-g\leq 0$ a.e. on $\Gamma _{C}$, $\mathbf{v}(0) =\mathbf{v}_{0}\in H$, and let $\mathbf{u}(t) =\mathbf{u}_{0}+\int_{0}^{t}\mathbf{v} (s) ds$. Then, there exist $\mathbf{u}\in C([ 0,T] ;U) \cap L^{\infty }(0,T;V)$, $\mathbf{u}\in \mathcal{K}$, $\mathbf{v}\in L^{2}(0,T;V) \cap L^{\infty }(0,T;H)$, $\mathbf{v}'\in L^{2}(0,T;H^{-1}(\Omega ) ^{N})$, and $\xi \in L^{2}( 0,T;L^{2}( \Gamma _{C})^{N})$ such that \begin{aligned} &-( \mathbf{v}_{0},\mathbf{u}_{0}-\mathbf{w}(0) ) _{H}+\int_{0}^{T}\langle M\mathbf{v},\mathbf{u-w}\rangle dt+\int_{0}^{T}\langle L\mathbf{u},\mathbf{u-w}\rangle dt \\ &-\int_{0}^{T}( \mathbf{v},\mathbf{v-w}') _{H}dt+\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi ,\mathbf{u}-\mathbf{w} \rangle dt\\ &\leq \int_{0}^{T}\langle \mathbf{f,u}-\mathbf{w}\rangle dt,\quad \end{aligned}\label{4mare9} for all $\mathbf{w}\in \mathcal{K}_{\mathbf{u}}$, where $$\mathcal{K}_{\mathbf{u}}\equiv \{\mathbf{w}\in \mathcal{V} :\mathbf{w}'\in \mathcal{V}, \;(w_{n}-g)_{+}=0\mbox{ in }\; L^{2}(0,T;L^{2}(\Gamma _{C})),\;\;\mathbf{u} (T)=\mathbf{w}(T)\}. \label{31julye1}$$ Here, $\gamma _{T}^{\ast }\xi$ satisfies, for $\mathbf{w}\in \mathcal{K}_{ \mathbf{u}}$, $$\langle \gamma _{T}^{\ast }\xi ,\mathbf{w}\rangle \leq \int_{0}^{T}\int_{\Gamma _{C}}\mu ( \left| \mathbf{v}_{T}-\mathbf{v} _{F}\right| ) \left| ( \mathcal{R}\sigma ) _{n}\right| ( \left| \mathbf{v}_{T}-\mathbf{v}_{F}+\mathbf{w}_{T}\right| -\left| \mathbf{v}_{T}-\mathbf{v}_{F}\right| ) \,d\Gamma dt. \label{4mare10}$$ \end{theorem} The proof of this theorem will be given in Section 4. It is obtained by considering a sequence of approximate problems, based on the normal compliance condition studied in Section 3, where the relevant a priori estimates are derived. \section{Approximate Problems} The Signorini condition leads to considerable difficulties in the analysis of the problem. Therefore, we first consider approximate problems based on the normal compliance condition, which we believe is a more realistic model. We establish the unique solvability of these problems and obtain the necessary a priori estimates which will allow us to pass to the Signorini limit. These problems have merit in and of themselves. We shall use the following two well known results, the first one can be found in Lions \cite{lio69} and the other one in Simon \cite{sim87} or Seidman \cite{sei89} see also \cite{kut98}). \begin{theorem}[\cite{lio69}] \label{t6.1} Let $p\geq 1$, $q>1$, $W\subseteq U\subseteq Y$, with compact inclusion map $W\to U$ and continuous inclusion map $U\to Y$, and let $S=\{\mathbf{u}\in L^{p}(0,T;W):\mathbf{u}'\in L^{q}(0,T;Y), \| \mathbf{u}\|_{L^{p}(0,T;W)}+\|\mathbf{u}'\|_{L^{q}(0,T;Y)}1 let \[ S_{T}=\{\mathbf{u}:\|\mathbf{u}(t)\|_{W}+\|\mathbf{u}^{\prime }\|_{L^{q}(0,T;Y)}\leq R,\quad t\in \lbrack 0,T]\}.$ Then $S_{T}$ is precompact in $C(0,T;U).$ \end{theorem} We turn to an abstract formulation of problem (\ref{5.1})--(\ref{5.7}). We define the \textit{normal compliance} operator $\mathbf{u}\to P(\mathbf{u})$, which maps $\mathcal{V}$ to $\mathcal{V}'$, by $$\langle P(\mathbf{u}),\mathbf{w}\rangle =\int_{0}^{T}\int_{\Gamma _{C}}(u_{n}-g)_{+}w_{n}\,d\Gamma dt. \label{6.8}$$ It will be used to approximate the Signorini condition by penalizing it. The abstract form of the approximate problem, with $0<\varepsilon$, is as follows. \noindent Problem $\mathcal{P}_{\varepsilon }$: Find \thinspace\ $\mathbf{ u,v }\in \mathcal{V}$ \thinspace\ such that \begin{gather} \mathbf{v}'+M\mathbf{v}+L\mathbf{u}+\frac{1}{\varepsilon } P( \mathbf{u})+\gamma _{T}^{\ast }\xi =\mathbf{f}\quad \text{ in }\;\, \mathcal{V}', \label{6.10} \$2pt] \mathbf{v}(0)=\mathbf{v}_{0}\in H, \label{6.11} \\ \mathbf{u}(t)=\mathbf{u}_{0}+\int_{0}^{t}\mathbf{v}(s)ds,\quad \mathbf{u} _{0}\in V \label{6.12} \end{gather} and for all \mathbf{w}\in \mathcal{V}, $$\langle \gamma _{T}^{\ast }\xi ,\mathbf{w}\rangle \leq \int_{0}^{T}\int_{\Gamma _{C}} \mu ( \left| \mathbf{v}_{T}-\mathbf{v} _{F}\right| ) \left| ( \mathcal{R}\sigma )_{n}\right| ( \left| \mathbf{v}_{T}-\mathbf{v}_{F}+\mathbf{w}_{T}\right| -\left| \mathbf{v}_{T}-\mathbf{v}_{F}\right| ) d\Gamma dt. \label{28febe4}$$ The existence of solutions of the problem follows from a straightforward application of the existence theorem in \cite{KSpsm}, therefore, \begin{theorem} \label{25febt1} There exists a solution to problem \mathcal{P} _{\varepsilon}. \end{theorem} We assume, as in Theorem \ref{maintheorem}, that initially $$u_{0n}-g \leq 0\quad \mbox{on } \Gamma_{C}. \label{25febe7}$$ \vskip4pt We turn to obtain estimates on the solutions of Problem \mathcal{P} _{\varepsilon }. In what follows C will denote a generic constant which depends on the data but is independent of t\in [0,T]  or \varepsilon, and whose value may change from line to line. We multiply (\ref{6.10}) by \mathbf{v}\chi _{\left[ 0,t\right] }, where \[ \chi _{\left[ 0,t\right] }(s) = \begin{cases} 1&\text{if } s\in \left[ 0, t\right], \\ 0&\text{if } s\notin \left[ 0, t\right], \end{cases}$ and integrate over $(0,t)$. From (\ref{28febe1}) and (\ref{25febe7}) we obtain \begin{aligned} &\frac{1}{2}\left| \mathbf{v}(t) \right| _{H}^{2}-\frac{1}{2} \left| \mathbf{v}_{0}\right| _{H}^{2}+\int_{0}^{t}( \delta ^{2}\| \mathbf{v}\|_{V}^{2}-\lambda |\mathbf{v}|_{H}^{2}) ds+\frac{1}{2} \langle L\mathbf{u}(t) ,\mathbf{u}(t) \rangle \\ &-\frac{1}{2}\langle L\mathbf{u}_{0},\mathbf{u}_{0}\rangle +\frac{1}{2\varepsilon }\int_{\Gamma _{C}}( ( u_{n}(t) -g) _{+}) ^{2}\,d\Gamma +\int_{0}^{t}\int_{\Gamma _{C}}\xi \cdot \mathbf{v}_{T}\,d\Gamma ds\\ &\leq \int_{0}^{t}\langle \mathbf{f},\mathbf{v}\rangle ds. \end{aligned} \label{28febe3} Since $\mu$ is assumed to be bounded, it follows from (\ref{28febe4}) that $$\xi \in \left[ -K_{\mu }\left| ( \mathcal{R}\sigma ) _{n}\right| ,\,K_{\mu }\left| ( \mathcal{R}\sigma ) _{n}\right| \,\right], \label{28febe7}$$ a.e. on $\Gamma _{C}$. Now, assumption (\ref{25febe1}) implies that there exists a constant $C$ such that $$\| ( \mathcal{R}\sigma ) _{n} \|_{L^{2}( 0,t;L^{2}( \Gamma _{c}) ) }\leq C \| \mathbf{v}\|_{L^{2}( 0,t;H)}. \label{28febe5}$$ Therefore, we find from (\ref{28febe7}) that \begin{aligned} \left| \int_{0}^{t}\int_{\Gamma _{C}}\xi \cdot \mathbf{v}_{T}d\Gamma ds\right| &\leq \| \xi \|_{L^{2}( 0,t;L^{2}( \Gamma _{C}) ) }\| \mathbf{v} _{T}\|_{L^{2}( 0,t;( L^{2}( \Gamma _{C}) ) ^{N}) } \\ &\leq C\| \mathbf{v}\|_{L^{2}( 0,t;H) }\| \mathbf{v}\|_{L^{2}( 0,t;U)}, \end{aligned}\label{28febe6} where $U$ is the Banach space described above, such that $V\subset U$ compactly, and the trace into $(L^{2}(\partial \Omega )^{N})$ is continuous. Now, the compactness of the embedding implies that for each $0<\eta$ $$\| \mathbf{v}\|_{L^{2}( 0,t;U) }\leq \eta \| \mathbf{v}\|_{L^{2}( 0,t;V) }+C_{\eta }\| \mathbf{v}\|_{L^{2}( 0,t; H)}. \label{28febe9}$$ Therefore, (\ref{28febe3}) yields \begin{aligned} & \left| \mathbf{v}(t) \right| _{H}^{2}-\left| \mathbf{v} _{0}\right| _{H}^{2} + \int_{0}^{t}( \delta ^{2}\|\mathbf{v} \|_{V}^{2}-\lambda |\mathbf{v}|_{H}^{2}) ds+( \delta ^{2}\| \mathbf{u}(t) \|_{V}^{2}-\lambda |\mathbf{u}(t) |_{H}^{2}) \\ & -\langle L\mathbf{u}_{0},\mathbf{u}_{0}\rangle +\frac{1}{ \varepsilon }\int_{\Gamma _{C}}( ( u_{n}(t) -g) _{+}) ^{2}d\Gamma -C\| \mathbf{v}\| _{L^{2}( 0,t;H) }\| \mathbf{v} \|_{L^{2}( 0,t;U) } \\ & \leq C_{\eta }\int_{0}^{t}\| \mathbf{f}\|_{ \mathcal{V}'}^{2}ds+\eta \int_{0}^{t}\| \mathbf{v} \|_{V}^{2}ds. \end{aligned}\label{28febe10} We obtain from (\ref{28febe9}) and (\ref{6.9}), \begin{aligned} & \left| \mathbf{v}(t) \right| _{H}^{2}+\delta ^{2}\int_{0}^{t}\| \mathbf{v}\|_{V}^{2}ds+\delta ^{2}\|\mathbf{u}( t) \|_{V}^{2}+ \frac{1}{\varepsilon }\int_{\Gamma _{C}}( ( u_{n}(t) -g) _{+}) ^{2}d\Gamma \\ & \leq C\int_{0}^{t}\left| \mathbf{v}(s) \right| _{H}^{2}ds +\eta \int_{0}^{t}\|\mathbf{v}\|_{V}^{2}ds+\lambda |\mathbf{u} (t) |_{H}^{2}+\langle L\mathbf{u}_{0},\mathbf{u}_{0}\rangle \\ &+\left| \mathbf{v}_{0}\right| _{H}^{2}+C_{\eta }\int_{0}^{t}\| \mathbf{f} \|_{\mathcal{V}^{\prime }}^{2}ds+\eta \int_{0}^{t}\| \mathbf{v}\|_{V}^{2}ds. \end{aligned}\label{28febe11} Choosing $\eta$ small enough and using the inequality $|\mathbf{u}(t) |_{H}^{2}\leq \left| \mathbf{u}_{0}\right| _{H}+\int_{0}^{t}\left| \mathbf{v}(s) \right| _{H}^{2}ds,$ yields \begin{align*} &\left| \mathbf{v}(t) \right| _{H}^{2}+\frac{\delta ^{2}}{2} \int_{0}^{t}\|\mathbf{v}\|_{V}^{2}ds+\delta ^{2}\|\mathbf{u}(t) \|_{V}^{2}+\frac{1}{\varepsilon }\int_{\Gamma _{C}}( ( u_{n}( t) -g) _{+}) ^{2}d\Gamma \\ &\leq C( \mathbf{u}_{0},\mathbf{v}_{0}) +C\int_{0}^{t} \| \mathbf{f}\|_{\mathcal{V}'}^{2}ds+C\int_{0}^{t}\left| \mathbf{v}(s) \right| _{H}^{2}ds. \end{align*} An application of Gronwall's inequality gives \begin{aligned} &\left| \mathbf{v}(t) \right| _{H}^{2}+\frac{\delta ^{2}}{2} \int_{0}^{t}\|\mathbf{v}\|_{V}^{2}ds +\delta ^{2}\|\mathbf{u}( t) \|_{V}^{2}+\frac{1}{\varepsilon }\int_{\Gamma _{C}}( ( u_{n}(t) -g) _{+}) ^{2}d\Gamma \\ &\leq C( \mathbf{u}_{0},\mathbf{v}_{0}) +C\int_{0}^{T}\| \mathbf{f}\|_{\mathcal{V}'}^{2}ds=C. \end{aligned}\label{28febe12} Now, let $\mathbf{w}\in L^{2} ( 0,T;H_{0}^{1} ( \Omega ) ^{N})$ and apply ( \ref{6.10}) to it, thus, $$\langle \mathbf{v}',\mathbf{w\rangle }+\langle M\mathbf{v,w}\rangle +\langle L\mathbf{u,w}\rangle =( \mathbf{f}_{B}\mathbf{,w}) _{ \mathcal{H}}. \label{4mare1}$$ It follows from estimate (\ref{28febe12}) that $\mathbf{v}'$ is bounded in $L^{2} ( 0,T;H^{-1}( \Omega ) ^{N})$, independently of $\varepsilon$. We conclude that there exists a constant $C$ , which is independent of $\varepsilon$, such that $$| \mathbf{v}(t)| _{H}^{2}+\int_{0}^{t}\|\mathbf{v} \|_{V}^{2}ds +\|\mathbf{u}(t) \|_{V}^{2}+\frac{1}{\varepsilon } \int_{\Gamma _{C}}(( u_{n}(t) -g) _{+})^{2}d\Gamma +\| \mathbf{v}'\| _{L^{2}(0,T;H^{-1}(\Omega) ^{N}) }\leq C. \label{4mare2}$$ Next, recall that $\Omega _{T}\equiv [0,T] \times \Omega$, and we have the following result. \begin{lemma} \label{28febl1} Let $(\mathbf{u,v})$ be a solution of Problem $\mathcal{P} _{\varepsilon }$. Then, there exists a constant $C$, independent of $\varepsilon$, such that $$\| \mathbf{v}'-\mathrm{Div}( A\varepsilon ( \mathbf{u}) +B\varepsilon ( \mathbf{v}) ) \| _{L^{2}( \Omega _{T}) }\leq C. \label{28febe13}$$ \end{lemma} In $(\ref{28febe13})$ measurable representatives are used whenever appropriate. \vskip3pt\noindent \textit{Proof}. Let $\mathbf{\phi }\in C_{c}^{\infty }( \Omega _{T};\mathbb{R}^{N})$, then by (\ref{6.10} ), $\int_{0}^{T}\int_{\Omega }-\mathbf{v\cdot \phi }_{t}+( A\varepsilon ( \mathbf{u}) +B\varepsilon ( \mathbf{v}) ) \cdot \varepsilon ( \mathbf{\phi })\, dx\,dt=\int_{0}^{T}\int_{\Omega }\mathbf{f}_{B}\cdot \mathbf{\phi }\,dx\,dt.$ Therefore, $\left| ( \mathbf{v}'-\mathrm{Div}( A\varepsilon ( \mathbf{u}) +B\varepsilon (\mathbf{v}) ) ) ( \mathbf{\phi }) \right| \leq \| \mathbf{f} _{B}\|_{L^{2}(0,T;H) }\| \mathbf{\phi } \|_{L^{2}(0,T;H) }$ holds in the sense of distributions, which establishes (\ref{28febe13}). Note that nothing is being said about $\mathbf{v} '$ or $\mathrm{Div } ( A\varepsilon ( \mathbf{u}) +B\varepsilon ( \mathbf{v } ) )$ separately. This estimate holds because the term that involves $\varepsilon$ relates to the boundary and so is irrelevant when we deal with $\mathbf{\phi }\in C_{c}^{\infty }( \Omega _{T};\mathbb{R} ^{N})$ which vanishes near the boundary, and such functions are dense in $L^{2}( \Omega _{T};\mathbb{R}^{N}).$ The proof of the following lemma is straightforward. \begin{lemma} \label{1marl1} If $\mathbf{v,u}\in \mathcal{V}$, $\mathbf{v=u}'$ and $\mathbf{v}'-\mathrm{Div}( A\varepsilon ( \mathbf{u} ) +B\varepsilon ( \mathbf{v}) ) \in L^{2}( \Omega _{T})$, then for $\phi \in W_{0}^{1,\infty }(0,T)$ and $\psi \in W_{0}^{1,\infty }(\Omega)$, \begin{aligned} &\int_{\Omega _{T}}( v_{i}v_{i}-( A_{ijkl}\varepsilon ( \mathbf{u}) _{kl}+B_{ijkl}\varepsilon ( \mathbf{v}) _{kl}) \varepsilon ( \mathbf{u}) _{ij}) \psi \phi \,dx\,dt \\ &=\int_{\Omega _{T}}( -v_{i}+( A_{ijlk}\varepsilon ( \mathbf{u} ) _{kl}+B_{ijkl}\varepsilon ( \mathbf{v}) ) _{,j} ) u_{i}\psi \phi \,dx\,dt\\ &\quad -\int_{\Omega _{T}}( u_{i}v_{i}\psi \phi '-A_{ijkl}\varepsilon ( \mathbf{u}) _{kl}u_{i}\psi _{,j}\phi -B_{ijkl}\varepsilon ( \mathbf{v}) _{kl}u_{i}\psi _{,j}\phi )\,dx\,dt. \end{aligned}\label{1mare1} \end{lemma} We now denote the solution of Problem $\mathcal{P}_{\varepsilon }$ by $\mathbf{u}^{\varepsilon }$ and let ${\mathbf{u}^{\varepsilon }}'= \mathbf{v}^{\varepsilon }$. We deduce from the estimates (\ref{28febe12}) and (\ref{28febe13}) and from Theorems \ref{t6.1} and \ref{t6.2} that there exists a subsequence, still indexed by $\varepsilon$, such that as $\varepsilon \to 0$, the following convergences take place: \begin{gather} \mathbf{u}^{\varepsilon } \to \mathbf{u}\text{ weak}\ast \text{ in }L^{\infty }(0,T;V) , \label{1mare2} \\ \mathbf{u}^{\varepsilon } \to \mathbf{u}\text{ strongly in } C([0,T];U) , \label{1mare4} \\ \mathbf{v}^{\varepsilon } \to \mathbf{v}\text{ weakly in }\mathcal{ \ \ V}\text{,} \label{1mare5} \\ \mathbf{v}^{\varepsilon } \to \mathbf{v}\text{ weak}\ast \text{ in }L^{\infty }(0,T;H) , \label{1mare6} \\ \mathbf{u}^{\varepsilon }(T) \to \mathbf{u}( T) \text{ weakly in }V, \label{1mare8} \\ \mathbf{v}^{\varepsilon } \to \mathbf{v}\text{ strongly in } L^{2}( 0,T;U), \label{4mare3} \\ \mathbf{v}^{\varepsilon }( \mathbf{x},t) \to \mathbf{v} ( \mathbf{x},t) \text{ pointwise a.e. on }\Gamma _{C}\times \left[ 0, T\right] , \label{4mare4} \\ \mathbf{v}^{\varepsilon \prime } \to \mathbf{v}'\text{ weak}\ast \text{ in }L^{2}( 0,T;H^{-1}(\Omega) ^{N}), \label{4mare7} \\ \mathbf{v}^{\varepsilon \prime }-\mathrm{Div}( A\varepsilon ( \mathbf{u}^{\varepsilon }) +B\varepsilon ( \mathbf{v} ^{\varepsilon }) ) \to \mathbf{v}'-\mathrm{Div} ( A\varepsilon ( \mathbf{u}) +B\varepsilon ( \mathbf{v} ) ) \text{ weakly in }\mathcal{H}, \label{1mare9} \end{gather} where a measurable representative is being used in (\ref{4mare4}). Moreover, we have $$\left| \mathbf{v}^{\varepsilon }(T) \right| _{H}\leq C. \label{4mare13}$$ The following is a fundamental result which will, ultimately, make it possible to pass to the limit in Problem $\mathcal{P}_{\varepsilon }$. \begin{lemma} \label{1marl2} Let $\left\{ \mathbf{u}^{\varepsilon }, \mathbf{v} ^{\varepsilon }\right\}$ be the sequence, found above, of solutions of Problem $\mathcal{P}_{\varepsilon }$. Then, \begin{align*} &\lim \sup_{\varepsilon \to 0}\int_{\Omega _{T}}( v_{i}^{\varepsilon }v_{i}^{\varepsilon } -( A_{ijkl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{kl}+B_{ijkl}\varepsilon ( \mathbf{v} ^{\varepsilon }) _{kl}) \varepsilon ( \mathbf{u} ^{\varepsilon }) _{ij}) \,dx\,dt \\ & \leq \int_{\Omega _{T}}( v_{i}v_{i}-( A_{ijkl}\varepsilon ( \mathbf{u}) _{kl}+B_{ijkl}\varepsilon ( \mathbf{v}) _{kl}) \varepsilon ( \mathbf{u}) _{ij}) \,dx\,dt. \end{align*} In terms of the abstract operators, \begin{aligned} &\lim \sup_{\varepsilon \to 0}\int_{0}^{T}-\langle M\mathbf{v} ^{\varepsilon },\mathbf{u}^{\varepsilon }\rangle dt+\int_{0}^{T}( \mathbf{v}^{\varepsilon },\mathbf{v}^{\varepsilon }) _{H}dt-\int_{0}^{T}\langle L\mathbf{u}^{\varepsilon },\mathbf{u} ^{\varepsilon }\rangle dt \\ &\leq \int_{0}^{T}-\langle M\mathbf{v},\mathbf{u}\rangle dt+\int_{0}^{T}( \mathbf{v},\mathbf{v}) _{H}dt-\int_{0}^{T}\langle L\mathbf{u},\mathbf{u}\rangle dt. \end{aligned}\label{3mare12} \end{lemma} \begin{proof} Let $\eta >0$ be given. Let $\phi _{\delta }$ be a piecewise linear and continuous function such that for small $\delta >0$, $\phi _{\delta }(t) =1$ on $\left[ \delta ,T-\delta \right]$, $\phi _{\delta }(0) =0$ and $\phi _{\delta }(T) =0$. Also, let $\psi _{\delta }\in C_{c}^{\infty }( \Omega )$ be such that $\psi _{\delta }(\mathbf{x}) \in \left[ 0,1\right]$ for all $\mathbf{x,}$ and also, \begin{gather} \mathrm{meas}( \Omega \setminus \left[ \psi _{\delta }=1\right] ) \equiv m( Q_{\delta }) <\delta , \label{1mare10} \\ \frac{1}{2\delta }\int_{\Omega }B_{ijkl}\varepsilon ( \mathbf{u} _{0}) _{kl}\varepsilon ( \mathbf{u}_{0}) _{ij}( 1-\psi _{\delta }) <\eta , \label{1mare11} \\ \int_{0}^{T}\int_{Q_{\delta }}v_{i}v_{i}<\eta ,\quad \int_{0}^{\delta }\int_{\Omega }v_{i}v_{i}<\eta ,\quad \int_{T-\delta }^{T}\int_{\Omega }v_{i}v_{i}<\eta . \label{1mare11.5} \end{gather} Now by (\ref{1mare2})-(\ref{1mare9}) and formula (\ref{1mare1}), \begin{aligned} &\lim_{\varepsilon \to 0}\int_{\Omega _{T}}( v_{i}^{\varepsilon }v_{i}^{\varepsilon }-( A_{ijkl}\varepsilon ( \mathbf{u} ^{\varepsilon }) _{kl}+B_{ijkl}\varepsilon ( \mathbf{v} ^{\varepsilon }) _{kl}) \varepsilon ( \mathbf{u} ^{\varepsilon }) _{ij}) \phi _{\delta }\psi _{\delta }\,dx\,dt \\ &= \int_{\Omega _{T}}( -v_{i}+( A_{ijlk}\varepsilon ( \mathbf{ u }) _{kl}+B_{ijkl}\varepsilon ( \mathbf{v}) ) _{,j}) u_{i}\psi _{\delta }\phi _{\delta }\,dx\,dt \\ &\quad -\int_{\Omega _{T}}( u_{i}v_{i}\psi \phi '-A_{ijkl}\varepsilon ( \mathbf{u}) _{kl}u_{i}\psi _{,j}\phi -B_{ijkl}\varepsilon ( \mathbf{v}) _{kl}u_{i}\psi _{\delta ,j}\phi _{\delta }) \,dx\,dt, \end{aligned} \label{1mare13} which equals $\int_{\Omega _{T}}( v_{i}v_{i}-( A_{ijkl}\varepsilon ( \mathbf{u}) _{kl}+B_{ijkl}\varepsilon ( \mathbf{v}) _{kl}) \varepsilon ( \mathbf{u}) _{ij}) \psi _{\delta }\phi _{\delta }\,dx\,dt,$ by Lemma \ref{1marl1}. Since $\psi _{\delta }$ and $\phi _{\delta }$ are not identically equal to one, we have to consider the integrals $$\begin{gathered} I_{1} \equiv \int_{\Omega _{T}}v_{i}^{\varepsilon }v_{i}^{\varepsilon }( 1-\psi _{\delta }\phi _{\delta }) \,dx\,dt, \\ I_{2}\equiv -\int_{\Omega _{T}}B_{ijkl}\varepsilon ( \mathbf{v} ^{\varepsilon }) _{kl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{ij}( 1-\psi _{\delta }\phi _{\delta }) \,dx\,dt, \\ I_{3} \equiv -\int_{\Omega _{T}}A_{ijkl}\varepsilon ( \mathbf{u} ^{\varepsilon }) _{kl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{ij}( 1-\psi _{\delta }\phi _{\delta }) \,dx\,dt. \end{gathered}\label{1mare14}$$ It is clear that $I_{3}\leq 0$. Next, $0\leq I_{1}\leq \int_{0}^{T}\int_{Q_{\delta }}v_{i}^{\varepsilon }v_{i}^{\varepsilon }\,dx\,dt+\int_{0}^{\delta }\int_{\Omega }v_{i}^{\varepsilon }v_{i}^{\varepsilon }\,dx\,dt+\int_{T-\delta }^{T}\int_{\Omega }v_{i}^{\varepsilon }v_{i}^{\varepsilon }\,dx\,dt.$ Now, $v_{i}^{\varepsilon }v_{i}^{\varepsilon }\to v_{i}v_{i}$ in $L^{1}( \Omega _{T})$ by (\ref{1mare6}), and so it follows from ( \ref{1mare11.5}), for $\varepsilon$ small enough, $\int_{0}^{T}\int_{Q_{\delta }}v_{i}^{\varepsilon }v_{i}^{\varepsilon }\,dx\,dt<\eta ,\int_{0}^{\delta }\int_{\Omega }v_{i}^{\varepsilon }v_{i}^{\varepsilon }\,dx\,dt<\eta ,\quad \int_{T-\delta }^{T}\int_{\Omega }v_{i}^{\varepsilon }v_{i}^{\varepsilon }\,dx\,dt\quad <\eta .$ Thus, $I_{1}\leq 3\eta$ when $\varepsilon$ is small enough. It remains to consider $I_{2}$ for small $\varepsilon$. Integrating $I_{2}$ by parts one obtains \begin{aligned} I_{2} &=-\frac{1}{2}\int_{\Omega }B_{ijkl}\varepsilon ( \mathbf{u} ^{\varepsilon }(T) ) _{kl}\varepsilon ( \mathbf{u} ^{\varepsilon }(T) ) _{ij}\,dx +\frac{1}{2} \int_{\Omega}B_{ijkl}\varepsilon ( \mathbf{u}_{0}) _{kl}\varepsilon ( \mathbf{u}_{0}) _{ij}\,dx \\ &\quad -\frac{1}{2\delta }\int_{\Omega }\int_{0}^{\delta }B_{ijkl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{kl}\varepsilon ( \mathbf{u} ^{\varepsilon }) _{ij}\psi _{\delta }\,dtdx \\ &\quad +\frac{1}{2\delta } \int_{\Omega }\int_{T-\delta }^{T}B_{ijkl}\varepsilon ( \mathbf{u} ^{\varepsilon }) _{kl}\varepsilon ( \mathbf{u} ^{\varepsilon }) _{ij}\psi _{\delta }\,dtdx \\ &\leq \frac{1}{2\delta }\int_{\Omega }\int_{T-\delta }^{T}( B_{ijkl}( \varepsilon ( \mathbf{u}^{\varepsilon }) _{kl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{ij}-\varepsilon ( \mathbf{u}^{\varepsilon }(T) ) _{kl}\varepsilon ( \mathbf{u}^{\varepsilon }(T) ) _{ij}) ) dt\,dx \\ &\quad+\frac{1}{2\delta }\int_{\Omega }\int_{0}^{\delta }( B_{ijkl}( \varepsilon ( \mathbf{u}_{0}) _{kl}\varepsilon ( \mathbf{u} _{0}) _{ij}-\varepsilon ( \mathbf{u}^{\varepsilon }) _{kl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{ij}) ) \psi _{\delta }dtdx \\ &\quad +\frac{1}{2\delta }\int_{\Omega }B_{ijkl}\varepsilon ( \mathbf{u} _{0}) _{kl}\varepsilon ( \mathbf{u}_{0}) _{ij}( 1-\psi _{\delta }) \,dx. \end{aligned} \label{1mare17} %\label{1mare19} It follows from (\ref{1mare11}) that (\ref{1mare17}) is less than $\eta$. Consider now the second term on the right-hand side, we have \begin{align*} &\Big| \frac{1}{2\delta }\int_{\Omega }\int_{0}^{\delta }( B_{ijkl}( \varepsilon ( \mathbf{u}_{0}) _{kl}\varepsilon ( \mathbf{u}_{0}) _{ij}-\varepsilon ( \mathbf{u} ^{\varepsilon }) _{kl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{ij}) ) \psi _{\delta }dtdx\Big| \\ &\leq \frac{1}{2\delta }\int_{\Omega }\int_{0}^{\delta }\left| B_{ijkl}( \varepsilon ( \mathbf{u}_{0}) _{kl}\varepsilon ( \mathbf{u}_{0}) _{ij}-\varepsilon ( \mathbf{u} ^{\varepsilon }) _{kl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{ij}) \right| dtdx \\ & \leq \frac{1}{2\delta }\int_{\Omega }\int_{0}^{\delta }\int_{0}^{t}\left| \frac{d}{dt}( B_{ijkl}\varepsilon ( \mathbf{u} ^{\varepsilon }) _{kl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{ij}) \right| dsdtdx \\ & \leq \frac{1}{\delta }\int_{\Omega }\int_{0}^{\delta }\int_{0}^{t}\left| B_{ijkl}\varepsilon ( \mathbf{v}^{\varepsilon }) _{kl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{ij}\right| ds\,dt\,dx \\ & = \frac{1}{\delta }\int_{0}^{\delta }\int_{0}^{t}\int_{\Omega }\left| B_{ijkl}\varepsilon ( \mathbf{v}^{\varepsilon }) _{kl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{ij}\right| \,dx\,ds\,dt \\ & \leq \frac{C}{\delta }\int_{0}^{\delta }\int_{0}^{\delta } \| \mathbf{v}^{\varepsilon }\|_{V}\| \mathbf{u} ^{\varepsilon }\|_{V}dsdt\leq C\sqrt{\delta }<\eta , \end{align*} whenever $\delta$ is sufficiently small. Formula (\ref{1mare17}) is estimated similarly and this shows that, for the choice of a sufficiently small $\delta$, we have $I_{2}<3\eta$. Below, we choose such a $\delta$ and then \begin{align*} &\lim \sup_{\varepsilon \to 0}\int_{\Omega _{T}}( v_{i}^{\varepsilon }v_{i}^{\varepsilon }-( A_{ijkl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{kl}+B_{ijkl}\varepsilon ( \mathbf{v} ^{\varepsilon }) _{kl}) \varepsilon ( \mathbf{u} ^{\varepsilon }) _{ij}) \,dx\,dt \\ &\leq \lim \sup_{\varepsilon \to 0}\int_{\Omega _{T}}( v_{i}^{\varepsilon }v_{i}^{\varepsilon }-( A_{ijkl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{kl}+B_{ijkl}\varepsilon ( \mathbf{v} s ^{\varepsilon }) _{kl}) \varepsilon ( \mathbf{u} ^{\varepsilon }) _{ij}) ( 1-\psi _{\delta }\phi _{\delta }) \,dx\,dt \\ &\quad +\lim \sup_{\varepsilon \to 0}\int_{\Omega _{T}}( v_{i}^{\varepsilon }v_{i}^{\varepsilon }-( A_{ijkl}\varepsilon ( \mathbf{u}^{\varepsilon }) _{kl}+B_{ijkl}\varepsilon ( \mathbf{v} ^{\varepsilon }) _{kl}) \varepsilon ( \mathbf{u} ^{\varepsilon }) _{ij}) \psi _{\delta }\phi _{\delta }\,dx\,dt\\ &\leq 6\eta +\int_{\Omega _{T}}( v_{i}v_{i}-( A_{ijkl}\varepsilon ( \mathbf{u}) _{kl}+B_{ijkl}\varepsilon ( \mathbf{v}) _{kl}) \varepsilon ( \mathbf{u}) _{ij}) \,dx\,dt, \end{align*} and since $\eta$ was arbitrary, the conclusion of the lemma follows. \end{proof} \section{Existence} We prove our first main result, Theorem \ref{maintheorem}, which guarantees the existence of a weak solution for Problem (\ref{2.14}) - (\ref{2.21}). We recall Problem $\mathcal{P}_{\varepsilon }$ and restate it here for the sake of convenience. \noindent Problem $\mathcal{P}_{\varepsilon }$: Find \thinspace\ $\mathbf{u} ^{\varepsilon }\mathbf{,v}^{\varepsilon }\in \mathcal{V}$ \thinspace\ such that, for $\mathbf{u}_{0}\in V$ and $\mathbf{v}^{\varepsilon }(0)=\mathbf{v} _{0} \in H$, there hold $$\mathbf{v}^{\varepsilon \prime }+M\mathbf{v}^{\varepsilon }+L\mathbf{u} ^{\varepsilon }+\frac{1}{\varepsilon }P(\mathbf{u}^{\varepsilon })+\gamma _{T}^{\ast }\xi ^{\varepsilon }=\mathbf{f}\quad \text{ in }\;\,\mathcal{V}', \label{1mare20}$$ $\mathbf{u}^{\varepsilon }(t)=\mathbf{u}_{0}+\int_{0}^{t}\mathbf{v} ^{\varepsilon }(s)ds$, and for all $\mathbf{z}\in \mathcal{V}$, $$\label{1mare23} \langle \gamma _{T}^{\ast }\xi ^{\varepsilon}, \mathbf{z}\rangle \leq \int_{0}^{T}\int_{\Gamma _{C}}\left| ( \mathcal{R}\sigma ^{\varepsilon }) _{n}\right| \mu (|\mathbf{v}_{T}^{\varepsilon }-\mathbf{\ v}_{F}|) ( \left| \mathbf{v}_{T}^{\varepsilon }- \mathbf{v } _{F}+\mathbf{z} _{T}\right| -\left| \mathbf{v}_{T}^{\varepsilon }- \mathbf{v} _{F}\right| ) d\Gamma dt.$$ We note that the boundedness of $\mu$ implies $\xi ^{\varepsilon }\in \left[ -K\left| ( \mathcal{R}\sigma ^{\varepsilon }) _{n}\right| ,K\left| ( \mathcal{R}\sigma ^{\varepsilon }) _{n}\right| \right]$ a.e. on $\Gamma _{C}$, (\ref{28febe7}). Also, assumption (\ref{25febe1}) implies that there exists a constant $C$ such that $$\| ( \mathcal{R}\sigma ^{\varepsilon }) _{n} \| _{L^{2}( 0,t;L^{2}( \Gamma _{c}) ) }\leq C\| \mathbf{v}^{\varepsilon }\|_{L^{2}( 0,t;H) }. \label{1mare25}$$ Therefore, $\xi ^{\varepsilon }$ is bounded in $L^{2}(0,T;L^{2}( \Gamma _{C}) ^{N})$ and so we may take a further subsequence and assume, in addition to the above convergences, that $$\xi ^{\varepsilon }\to \xi \text{ weakly in }L^{2}( 0,T;L^{2}(\Gamma _{C}) ^{N}). \label{1mare26}$$ In addition, we may assume, after taking a suitable subsequence and using the fact that $L$ and $M$ are linear, that $$L\mathbf{u}^{\varepsilon }\to L\mathbf{u,\quad }M\mathbf{v} ^{\varepsilon }\to M\mathbf{v}\text{ in }\mathcal{V}'. \label{3mare.5}$$ It follows from (\ref{28febe12}) and (\ref{1mare4}) that $\int_{\Gamma _{C}}( ( u_{n}(t) -g) _{+}) ^{2}d\Gamma =0$ for each $t\in [0,T]$, and so $P( \mathbf{u}) =0$. Now, we recall that $\mathcal{K}$ and $\mathcal{K}_{\mathbf{u}}$ are given in (\ref{4mare11}) and (\ref{31julye1}), respectively. We multiply (\ref {1mare20}) by $\mathbf{u}^{\varepsilon }-\mathbf{w}$, with $\mathbf{w}\in \mathcal{K}_{\mathbf{u}}$ and integrate over $[0,T]$. Then, $\frac{1}{\varepsilon }\int_{0}^{T}\langle P(\mathbf{u}^{\varepsilon }), \mathbf{u}^{\varepsilon }-\mathbf{w}\rangle dt\geq 0,$ and thus \begin{aligned} &( \mathbf{v}^{\varepsilon }(T) ,\mathbf{u}^{\varepsilon }(T) -\mathbf{w}(T) ) _{H}-( \mathbf{v} _{0},\mathbf{u}_{0}-\mathbf{w}(0) ) _{H}+\int_{0}^{T}\langle M\mathbf{v}^{\varepsilon },\mathbf{u}^{\varepsilon }-\mathbf{w}\rangle dt \\ &\quad +\int_{0}^{T}\langle L\mathbf{u}^{\varepsilon },\mathbf{u} ^{\varepsilon }-\mathbf{w}\rangle dt+\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi ^{\varepsilon },\mathbf{u}^{\varepsilon }-\mathbf{w}\rangle dt \\ &\leq \int_{0}^{T}( \mathbf{v}^{\varepsilon },\mathbf{v} ^{\varepsilon }-\mathbf{w}') _{H}dt+\int_{0}^{T}\langle \mathbf{f,u}^{\varepsilon }-\mathbf{w}\rangle dt. \end{aligned} \label{3mare1} From (\ref{28febe12}), (\ref{4mare13}), and (\ref{1mare4}) we find, $$\lim_{\varepsilon \to 0}( \mathbf{v}^{\varepsilon }( T) ,\mathbf{u}^{\varepsilon }(T) -\mathbf{w}( T) ) _{H}=0, \label{3mare2}$$ and also $\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi ^{\varepsilon },\mathbf{u} ^{\varepsilon }-\mathbf{w}\rangle dt=\int_{0}^{T}\int_{\Gamma _{C}}\xi ^{\varepsilon }( \mathbf{u}_{T}^{\varepsilon }-\mathbf{w}_{T}) d\Gamma dt.$ Now, (\ref{1mare4}) and (\ref{1mare26}) show that this term converges to $$\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi ,\mathbf{u}-\mathbf{w}\rangle dt=\int_{0}^{T}\int_{\Gamma _{C}}\xi ( \mathbf{u}_{T}-\mathbf{w} _{T}) d\Gamma dt. \label{3mare9}$$ Together with inequality (\ref{3mare1}) these imply \begin{aligned} &( \mathbf{v}^{\varepsilon }(T) ,\mathbf{u}^{\varepsilon }(T) -\mathbf{w}(T) ) _{H}-( \mathbf{v} _{0},\mathbf{u}_{0}-\mathbf{w}(0) ) _{H}+\int_{0}^{T}\langle M\mathbf{v}^{\varepsilon },\mathbf{u}^{\varepsilon }\rangle dt\\ &-\int_{0}^{T}( \mathbf{v}^{\varepsilon },\mathbf{v}^{\varepsilon }) _{H}dt+\int_{0}^{T}\langle L\mathbf{u}^{\varepsilon },\mathbf{u} ^{\varepsilon }\rangle dt+\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi ^{\varepsilon },\mathbf{u}^{\varepsilon }-\mathbf{w}\rangle dt \\ &\leq \int_{0}^{T}\langle L\mathbf{u}^{\varepsilon },\mathbf{w}\rangle dt+\int_{0}^{T}\langle M\mathbf{v}^{\varepsilon },\mathbf{w}\rangle dt+\int_{0}^{T}( \mathbf{v}^{\varepsilon },\mathbf{w}') _{H}dt+\int_{0}^{T}\langle \mathbf{f,u}^{\varepsilon }-\mathbf{w}\rangle dt. \end{aligned}\label{3mare10} We take the $\lim \inf$ of both sides of (\ref{3mare10}) as $\varepsilon \to 0$ and use Lemma \ref{1marl2} with (\ref{3mare9}), (\ref{3mare.5}) and (\ref{3mare2}) to conclude \begin{aligned} &-( \mathbf{v}_{0},\mathbf{u}_{0}-\mathbf{w}(0) ) _{H}+\int_{0}^{T}\langle M\mathbf{v},\mathbf{u}\rangle dt-\int_{0}^{T}( \mathbf{v},\mathbf{v}) _{H}dt \\ &+\int_{0}^{T}\langle L\mathbf{u},\mathbf{u}\rangle dt+\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi ,\mathbf{u}-\mathbf{w}\rangle dt \\ &\leq \int_{0}^{T}\langle L\mathbf{u},\mathbf{w}\rangle dt+\int_{0}^{T}\langle M\mathbf{v},\mathbf{w}\rangle dt+\int_{0}^{T}( \mathbf{v},\mathbf{w}') _{H}dt+\int_{0}^{T}\langle \mathbf{f,u }-\mathbf{w}\rangle dt. \end{aligned}\label{3mare13} This, in turn, implies \begin{aligned} &-( \mathbf{v}_{0},\mathbf{u}_{0}-\mathbf{w}(0) ) _{H}+\int_{0}^{T}\langle M\mathbf{v},\mathbf{u-w}\rangle dt-\int_{0}^{T}( \mathbf{v},\mathbf{v-w}') _{H}dt\\ &+\int_{0}^{T}\langle L\mathbf{u},\mathbf{u-w}\rangle dt+\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi ,\mathbf{u}-\mathbf{w}\rangle dt\\ &\leq \int_{0}^{T}\langle \mathbf{f,u}-\mathbf{w}\rangle dt. \end{aligned} \label{3mare14} It only remains to verify that for all $\mathbf{z}\in \mathcal{V}$, $$\langle \gamma _{T}^{\ast }\xi ,\mathbf{z}\rangle \leq \int_{0}^{T}\int_{\Gamma _{C}}\left| ( \mathcal{R}\sigma ) _{n}\right| \mu ( \left| \mathbf{v}_{T}-\mathbf{v}_{F}\right| ) ( \left| \mathbf{v}_{T}-\mathbf{v}_{F}+\mathbf{z}_{T}\right| -\left| \mathbf{v}_{T}-\mathbf{v}_{F}\right| ) d\Gamma dt. \label{4mare5}$$ It follows from (\ref{4mare3}), (\ref{4mare4}) and (\ref{25febe1}) that $\left| ( \mathcal{R}\sigma ^{\varepsilon }) _{n}\right| \to \left| ( \mathcal{R}\sigma ) _{n}\right| \; \mbox{ in }\;L^{2}( 0,T;L^{2}( \Gamma _{C}) ) ,$ and also $\mathbf{v}_{T}^{\varepsilon }\to \mathbf{v}_{T}\text{ in } L^{2}(0,T;L^{2}(\Gamma _{C})^{N})$, while $\mu ( \left| \mathbf{v} _{T}^{\varepsilon }-\mathbf{v}_{F}\right| ) \to \mu ( \left| \mathbf{v}_{T}-\mathbf{v}_{F}\right| )$ pointwise in $\Gamma _{C}\times \lbrack 0,T]$ , and is bounded uniformly. This allows us to pass to the limit $\varepsilon \to 0$ in the inequality (\ref{1mare23}) and obtain (\ref{4mare5}). The proof of Theorem \ref{maintheorem} is now complete. \section{Discontinuous friction coefficient} In this section we consider a discontinuous, set-valued friction coefficient $\mu$, depicted in Fig.\,2, which represents a sharp drop from the static value $\mu_0$ to a dynamic value $\mu_s(0)$, when relative slip commences. \begin{figure}[ht] \begin{center} \begin{picture}(227,120)(-27,-20) \put(-20,0){\vector(1,0){220}} \put(0,-20){\line(0,1){120}} \put(0,100){\thicklines \line(0,-1){60}} \put(-15,100){$\mu_0$} \put(184,-18){$| \mathbf{v}_{*}|$} \qbezier(0,40)(100,5)(200,5) \put(0,100){\qbezier(-2,0)(0,0)(2,0)} \put(100,20){$\mu_s$} \put(0,0){\thicklines \qbezier(0,100)(10,65)(20,34)} \thicklines \qbezier[8](20,34)(20,17)(20,0) \put(20,0){\qbezier(0,-2)(0,0)(0,2)} \put(19,-10){$\varepsilon$} \put(18,50){$\mu_\varepsilon$} \thicklines \qbezier[6](20,34)(10,34)(0,34) \put(0,34){\qbezier(-2,0)(0,0)(2,0)} \put(0,40){\qbezier(-2,0)(0,0)(2,0)} \put(-27,42){$\mu_s(0)$} \end{picture} \end{center} \caption{The graph of $\mu$ vs. the slip rate $|\mathbf{v}_{*}|$, and its approximation $\mu _{\varepsilon }$.} \end{figure} The jump in the friction coefficient $\mu$ when slip begins is given by the vertical segment $[\mu_s(0), \mu_0]$. Thus, $\mu(v)=\begin{cases} [\mu_s(0), \mu_0] & v=0, \\ \mu_s(v) & v>0, \end{cases}$ where $\mu_s$ is a Lipschitz, bounded, and positive function which describes the dependence of the coefficient on the slip rate. The function $\mu _{\varepsilon }$, shown in Fig.\,2, is a Lipschitz continuous approximation of the set-valued function for $0\leq v\leq \varepsilon$. It follows from Theorem \ref{maintheorem} that the problem obtained from (\ref{2.14})--(\ref{2.21}) by replacing $\mu$ with $\mu _{\varepsilon }$ has a weak solution. For convenience we list the conclusion of this theorem with appropriate modifications. \begin{theorem} \label{10maret1} There exist $\mathbf{u}$ in $C( [0,T];U) \cap L^{\infty }(0,T;V)$, $\mathbf{u}$ in $\mathcal{K}$ and $\mathbf{v}$ in $L^{2}( 0,T;V) \cap L^{\infty }(0,T;H)$, $\mathbf{v}(0) =\mathbf{v}_{0}\in H$, such that $\mathbf{v} '\in L^{2}( 0,T;H^{-1}(\Omega) ^{N})$, $\mathbf{u}(t) = \mathbf{u}_{0}+\int_{0}^{t}\mathbf{\ v}(s) ds$, and \begin{aligned} &-( \mathbf{v}_{0},\mathbf{u}_{0}-\mathbf{w}(0) ) _{H}+\int_{0}^{T}\langle M\mathbf{v},\mathbf{u-w}\rangle dt-\int_{0}^{T}( \mathbf{v},\mathbf{v-w}') _{H}dt\\ &+\int_{0}^{T}\langle L\mathbf{u},\mathbf{u-w}\rangle dt+\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi ,\mathbf{u}-\mathbf{w}\rangle dt\\ &\leq \int_{0}^{T}\langle \mathbf{f,u}-\mathbf{w}\rangle dt, \end{aligned} \label{10mare1} which holds for all $\mathbf{w}\in \mathcal{K}_{\mathbf{u}}$. Moreover, for all $\mathbf{z}\in \mathcal{V}$, $$\langle \gamma _{T}^{\ast }\xi ,\mathbf{z}\rangle \leq \int_{0}^{T}\int_{\Gamma _{C}}\left| ( \mathcal{R}\sigma ) _{n}\right| \mu _{\varepsilon }( \left| \mathbf{v}_{T}-\mathbf{v} _{F}\right| ) ( \left| \mathbf{v}_{T}-\mathbf{v}_{F}+\mathbf{z} _{T}\right| -\left| \mathbf{v}_{T}-\mathbf{v}_{F}\right| ) d\Gamma dt. \label{10mare2}$$ \end{theorem} Here, $\mathcal{K}$ and $\mathcal{K}_{\mathbf{u}}$ are given in (\ref {4mare11}) and (\ref{31julye1}), respectively. We refer to $\{\mathbf{u} _{\varepsilon}, \mathbf{v}_{\varepsilon}\}$ as a solution which is guaranteed by the theorem. We point out that here $\varepsilon$ is not related to the penalization parameter used earlier, but indicates the extent to which $\mu _{\varepsilon }$ approximates $\mu$ (Fig.\,2). In the proof of Theorem \ref{maintheorem} the Lipschitz constant for $\mu$ was not used in the derivation of the estimates, therefore, there exists a constant $C$, which is independent of $\varepsilon$, such that $$\left| \mathbf{v}_{\varepsilon }(t) \right| _{H}^{2}+\int_{0}^{t}\|\mathbf{v}_{\varepsilon }\|_{V}^{2}ds+\delta ^{2}\| \mathbf{u}_{\varepsilon }(t) \|_{V}^{2}+\| \mathbf{v} _{\varepsilon }'\|_{L^{2}( 0,T;H^{-1}( \Omega ) ^{N}) }\leq C. \label{10mare4}$$ This estimate together with (\ref{1mare25}) and (\ref{28febe7}) yield $$\| \xi _{\varepsilon }\|_{L^{2} (0,T;L^{2}( \Gamma _{C}) ^{N}) }\leq C, \label{10mare5}$$ where here and below $C$ is a generic positive constant independent of $\varepsilon$. Therefore, there exists a subsequence $\varepsilon \to 0$ for which \begin{gather} \mathbf{u}_{\varepsilon } \to \mathbf{u}\text{ weak}\ast \text{ in }L^{\infty }(0,T;V) , \label{10mare6} \\ \mathbf{u}_{\varepsilon } \to \mathbf{u}\text{ strongly in } C([0,T];U) , \label{10mare7} \\ \mathbf{v}_{\varepsilon } \to \mathbf{v}\text{ weakly in }\mathcal{ \ V}\text{,} \label{10mare8} \\ \mathbf{v}_{\varepsilon } \to \mathbf{v}\text{ weak}\ast \text{ in }L^{\infty }(0,T;H) , \label{10mare9} \\ \mathbf{u}_{\varepsilon }(T) \to \mathbf{u}( T) \text{ weakly in }V, \label{10mare10} \\ \mathbf{v}_{\varepsilon } \to \mathbf{v}\text{ strongly in } L^{2}( 0,T;U), \label{10mare11} \\ \mathbf{v}_{\varepsilon }( \mathbf{x},t) \to \mathbf{v} ( \mathbf{x},t) \text{ pointwise a.e. on }\Gamma _{C}\times \left[ 0,T\right] , \label{10mare12} \\ \mathbf{v}_{\varepsilon }' \to \mathbf{v}'\text{ weak}\ast \text{ in }L^{2}(0,T;H^{-1}(\Omega) ^{N}), \label{10mare13} \\ \left| \mathbf{v}_{\varepsilon }(T) \right| _{H} \leq C. \label{10mare14} \end{gather} The result of Lemma \ref{28febl1} still holds, and so there exists a further subsequence such that $$\mathbf{v}_{\varepsilon }'-\mathrm{Div}( A\varepsilon ( \mathbf{u}_{\varepsilon }) +B\varepsilon ( \mathbf{v} _{\varepsilon }) ) \to \mathbf{v}'-\mathop{\rm Div} ( A\varepsilon ( \mathbf{u}) +B\varepsilon ( \mathbf{v} ) ) \text{ weakly in }\mathcal{H}, \label{10mare14.5}$$ and, thus, the conclusion of Lemma \ref{1marl2} also holds, \begin{aligned} & \lim \sup_{\varepsilon \to 0}\int_{0}^{T}-\langle M\mathbf{v} ^{\varepsilon },\mathbf{u}^{\varepsilon }\rangle dt +\int_{0}^{T}( \mathbf{v}^{\varepsilon },\mathbf{v}^{\varepsilon }) _{H}dt-\int_{0}^{T}\langle L\mathbf{u}^{\varepsilon },\mathbf{u} ^{\varepsilon }\rangle dt \\ & \leq \int_{0}^{T}-\langle M\mathbf{v},\mathbf{u}\rangle dt+\int_{0}^{T}( \mathbf{v},\mathbf{v}) _{H}dt-\int_{0}^{T}\langle L\mathbf{u},\mathbf{u}\rangle dt. \end{aligned} \label{10mare14.6} As above, (\ref{10mare11}) implies $$\left| ( \mathcal{R}\sigma _{\varepsilon }) _{n}\right| \to \left| ( \mathcal{R}\sigma ) _{n}\right| \quad \text{ in }L^{2}( 0,T;L^{2}( \Gamma _{C}) ) , \label{10mare14.7}$$ and we may take a further subsequence, if necessary, such that$\;\xi _{\varepsilon }\to \xi$ weakly in $L^{2}( 0,T;L^{2}(\Gamma _{C}) ^{N})$. Passing now to the limit we obtain $\mathbf{u}(t) =\mathbf{u}_{0}+\int_{0}^{t}\mathbf{v}( s) ds,\quad \mathbf{u}_{0}\in V,\quad \mathbf{v}(0) = \mathbf{v}_{0}\in H.$ Letting $\mathbf{w}\in \mathcal{K}$, it follows from Theorem \ref{10maret1} that \begin{aligned} &( \mathbf{v}_{\varepsilon }(T) ,\mathbf{u}_{\varepsilon }(T) -\mathbf{w}(T) ) _{H}-( \mathbf{v} _{0},\mathbf{u}_{0}-\mathbf{w}(0) ) _{H}+\int_{0}^{T}\langle M\mathbf{v}_{\varepsilon },\mathbf{u}_{\varepsilon } \mathbf{-w}\rangle dt \\ &-\int_{0}^{T}( \mathbf{v}_{\varepsilon },\mathbf{v} _{\varepsilon }\mathbf{-w}') _{H}dt +\int_{0}^{T}\langle L \mathbf{u}_{\varepsilon },\mathbf{u}_{\varepsilon } \mathbf{-w}\rangle dt+\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi _{\varepsilon },\mathbf{u} _{\varepsilon }-\mathbf{w}\rangle dt \\ & \leq \int_{0}^{T}\langle \mathbf{f,u}_{\varepsilon }-\mathbf{w} \rangle dt. \end{aligned}\label{10mare15} The strong convergence in (\ref{10mare7}) allows us to pass to the limit $\lim_{\varepsilon \to 0}\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi _{\varepsilon },\mathbf{u}_{\varepsilon }-\mathbf{w}\rangle dt=\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi ,\mathbf{u}-\mathbf{w}\rangle dt.$ It follows now from the boundedness of $\left| \mathbf{v}_{\varepsilon }(T) \right| _{H}$, the weak convergence of $u_{\varepsilon }(T)$ to $u(T)$ in $V$ and the compactness of the embedding of $V$ into $H$ that $\lim_{\varepsilon \to 0}( \mathbf{v}_{\varepsilon }( T) ,\mathbf{u}_{\varepsilon }(T) -\mathbf{w}( T) ) _{H}=0.$ Rewriting (\ref{10mare15}), taking the $\lim \sup$ of both sides and using (\ref{10mare14.6}) yields \begin{aligned} &-( \mathbf{v}_{0},\mathbf{u}_{0}-\mathbf{w}(0) ) _{H}-\int_{0}^{T}\langle L\mathbf{u},\mathbf{w}\rangle dt+\int_{0}^{T}( \mathbf{v},\mathbf{w}') _{H}dt \\ &-\int_{0}^{T}\langle M\mathbf{v},\mathbf{w}\rangle dt+\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi ,\mathbf{u}-\mathbf{w}\rangle dt \\ &\leq \int_{0}^{T}-\langle M\mathbf{v},\mathbf{u}\rangle\, dt +\int_{0}^{T}( \mathbf{v},\mathbf{v}) _{H}dt-\int_{0}^{T}\langle L\mathbf{u},\mathbf{u}\rangle dt+\int_{0}^{T}\langle \mathbf{f,u}-\mathbf{w}\rangle dt, \end{aligned}\label{10mare17} which implies \begin{aligned} &-( \mathbf{v}_{0},\mathbf{u}_{0}-\mathbf{w}(0) ) _{H}+\int_{0}^{T}\langle L\mathbf{u},\mathbf{u-w}\rangle dt-\int_{0}^{T}( \mathbf{v},\mathbf{v-w}') _{H}dt \\ &+ \int_{0}^{T}\langle M\mathbf{v},\mathbf{u-w}\rangle dt+\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi ,\mathbf{u}-\mathbf{w}\rangle dt\\ &\leq \int_{0}^{T}\langle \mathbf{f,u}-\mathbf{w}\rangle dt. \end{aligned}\label{10mare18} It only remains to examine (\ref{10mare2}), without $\varepsilon$. It follows from (\ref{10mare11}) that $\mathbf{v}_{\varepsilon }\rightarrow \mathbf{v}$ strongly in $L^{2}( 0,T;U)$ and so $\mathbf{v} _{\varepsilon T}\to \mathbf{v}_{T}$ strongly in $L^{2}(0,T;L^{2} ( \Gamma _{C}) ^{N})$. Taking a measurable representative, we may assume $\mathbf{v}_{\varepsilon T}( \mathbf{x},t) \rightarrow \mathbf{v}_{T}( \mathbf{x},t)$ pointwise a.e. and in $L^{2}( \Gamma _{C}\times [0,T] )$. Taking a further subsequence we may assume, in addition, that $\mu _{\varepsilon }( \left| \mathbf{v}_{\varepsilon T}-\mathbf{v} _{F}\right| ) \rightarrow q$ weak* in $L^{\infty}( E)$, where $E=\left\{ ( \mathbf{x} ,t) :\left| \mathbf{v}_{T}( \mathbf{x},t) -\mathbf{v} _{F}( \mathbf{x},t) \right| =0\right\} .$ Let $\varepsilon _{0}>0$ be given. Then, for a.e $( \mathbf{x},t) \in E$, we have $\mu _{\varepsilon }( \left| \mathbf{v}_{\varepsilon T}-\mathbf{v} _{F}\right| ( \mathbf{x},t) ) \in \left[ \mu _{s}( 0+) -h( \varepsilon _{0}) ,\mu _{0}\right] ,$ whenever $\varepsilon$ is small enough. Therefore, $q( \mathbf{x}, t) \in \left[ \mu _{s}( 0+) -h( \varepsilon _{0}) ,\mu _{0}\right]$, a.e., where $h(\varepsilon )=\mu _{s}(0)-\mu _{s}(\varepsilon )$ is the function measuring the approach to the graph. Since $\varepsilon _{0}$ is arbitrary, we obtain $q( \mathbf{x} ,t) \in \left[ \mu _{s}( 0+) ,\mu _{0}\right]$, a.e. on $\Gamma _{C}$. If $\left| \mathbf{v}_{T}( \mathbf{x}, t) -\mathbf{v} _{F}( \mathbf{x},t) \right| >0$, we find that for a.e. $(\mathbf{x},t)$ and for $\varepsilon$ small enough, $\mu _{\varepsilon }( \left| \mathbf{v}_{\varepsilon T}-\mathbf{v} _{F}\right| ( \mathbf{x},t) ) =\mu _{s}( \left| \mathbf{v}_{\varepsilon T}-\mathbf{v}_{F}\right| ( \mathbf{x,t}) ) ,$ and this converges to $\mu _{s}( \left| \mathbf{v}_{T}-\mathbf{v} _{F}\right| ( \mathbf{x},t) )$. From (\ref{10mare2}) we obtain \begin{align*} \langle \gamma _{T}^{\ast }\xi _{\varepsilon },\mathbf{z}\rangle & \leq \int_{0}^{T}\int_{\Gamma _{C}}\left| ( \mathcal{R}\sigma _{\varepsilon }) _{n}\right| \mu _{\varepsilon }( \left| \mathbf{v } _{\varepsilon T}-\mathbf{v}_{F}\right| ) ( \left| \mathbf{v} _{\varepsilon T}-\mathbf{v}_{F}+\mathbf{z}_{T}\right| -\left| \mathbf{v} _{\varepsilon T}-\mathbf{v}_{F}\right| ) d\Gamma dt \\ & =\int_{E}\left| ( \mathcal{R}\sigma _{\varepsilon }) _{n}\right| \mu _{\varepsilon }( \left| \mathbf{v}_{\varepsilon T}- \mathbf{v} _{F}\right| ) ( \left| \mathbf{v}_{\varepsilon T}- \mathbf{v}_{F}+ \mathbf{z}_{T}\right| -\left| \mathbf{v}_{\varepsilon T}- \mathbf{v} _{F}\right| ) d\Gamma dt \\ &\quad +\int_{E^{C}}\left| ( \mathcal{R}\sigma _{\varepsilon }) _{n}\right| \mu _{\varepsilon }( \left| \mathbf{v}_{\varepsilon T}- \mathbf{v}_{F}\right| ) ( \left| \mathbf{v}_{\varepsilon T}- \mathbf{v}_{F}+\mathbf{z}_{T}\right| -\left| \mathbf{v}_{\varepsilon T}- \mathbf{v}_{F}\right| ) d\Gamma dt, \end{align*} and it follows from (\ref{10mare11}) and (\ref{10mare14.7}) that we may pass to the limit, thus, \begin{align*} \langle \gamma _{T}^{\ast }\xi ,\mathbf{z}\rangle &\leq \int_{E}\left| ( \mathcal{R}\sigma ) _{n}\right| q( \left| \mathbf{v}_{T}-\mathbf{ v }_{F}+\mathbf{z}_{T}\right| -\left| \mathbf{v}_{T}-\mathbf{v}_{F}\right| ) d\Gamma dt \\ &\quad +\int_{E^{C}}\left| ( \mathcal{R}\sigma ) _{n}\right| \mu _{s}( \left| \mathbf{v}_{T}-\mathbf{v}_{F}\right| ) ( \left| \mathbf{v}_{T}-\mathbf{v}_{F}+\mathbf{z}_{T}\right| -\left| \mathbf{v}_{T}-\mathbf{v}_{F}\right| ) d\Gamma dt. \end{align*} Since $q( \mathbf{x},t) \in \left[ \mu _{s}( 0+) ,\mu _{0}\right]$ a.e., it shows that there exists a measurable function $k_{\mu }$, with the property that $k_{\mu }( \mathbf{x},t) \in \mu ( \left| \mathbf{v}_{T}-\mathbf{v}_{F}\right| ( \mathbf{x} ,t) )$ a.e., such that $\langle \gamma _{T}^{\ast }\xi ,\mathbf{z}\rangle \leq \int_{0}^{T}\int_{\Gamma _{C}}\left| ( \mathcal{R}\sigma ) _{n}\right| k_{\mu }( \left| \mathbf{v}_{\varepsilon T}-\mathbf{v}_{F} \mathbf{z}_{T}\right| -\left| \mathbf{v}_{\varepsilon T}-\mathbf{v} _{F}\right| ) d\Gamma dt.$ Thus, we have established the second main result in this work, an existence theorem which extends Theorem \ref{maintheorem} to the case when the friction coefficient is discontinuous. \begin{theorem} \label{setvaluedthm} There exist $\mathbf{u}\in C( (0,T] ;U) \cap L^{\infty }(0,T;V) \cap \mathcal{K}$ and $\mathbf{ v}\in L^{2}(0,T;V) \cap L^{\infty }(0,T;H)$, $\mathbf{v}'\in L^{2}( 0,T;H^{-1}(\Omega) ^{N})$, and $$\mathbf{u}(t) =\mathbf{u}_{0}+\int_{0}^{t}\mathbf{v}( s) ds,\quad \mathbf{u}_{0}\in V,\quad \mathbf{v}(0) = \mathbf{v}_{0}\in H, \label{10mare19}$$ such that \begin{aligned} &-( \mathbf{v}_{0},\mathbf{u}_{0}-\mathbf{w}(0) ) _{H}+\int_{0}^{T}\langle M\mathbf{v},\mathbf{u-w}\rangle dt-\int_{0}^{T}( \mathbf{v},\mathbf{v-w}') _{H}dt \\ &+\int_{0}^{T}\langle L\mathbf{u},\mathbf{u-w}\rangle dt+\int_{0}^{T}\langle \gamma _{T}^{\ast }\xi ,\mathbf{u}-\mathbf{w}\rangle dt\\ &\leq \int_{0}^{T}\langle \mathbf{f,u}-\mathbf{w}\rangle dt, \end{aligned}\label{10mare20} where $$\langle \gamma _{T}^{\ast }\xi ,\mathbf{z}\rangle \leq \int_{0}^{T}\int_{\Gamma _{C}}\left| ( \mathcal{R}\sigma ) _{n}\right| k_{\mu }( \left| \mathbf{v}_{T}-\mathbf{v}_{F}+\mathbf{z} _{T}\right| -\left| \mathbf{v}_{T}-\mathbf{v}_{F}\right| ) d\Gamma dt, \label{10mare21}$$ for all $\mathbf{z}\in \mathcal{V}$. Here, $k_{\mu }$ is a function in $L^{\infty }( \Gamma _{C}\times (0,T] )$ with the property that $k_{\mu }( \mathbf{x},t) \in \mu ( \left| \mathbf{v}_{T}-\mathbf{v}_{F}\right| )$ for a.e. $( \mathbf{x},t) \in \Gamma _{C}\times [0,T]$. \end{theorem} \section{Conclusions} We considered a model for dynamic frictional contact between a deformable body and a moving rigid foundation. The contact was modelled with the Signorini condition and friction with a general nonlocal law in which the friction coefficient depended on the slip velocity between the surface and the foundation. We have shown that there exists a weak solution to the problem when the friction coefficient is a Lipschitz function of the slip rate, or when it is a graph with a jump from the static to the dynamic value at the onset of sliding. The existence of weak solutions for these problems was obtained by using the theory of set-valued pseudomonotone maps of \cite{KSpsm}. The regularization of the contact stress was introduced in Section 2, and it remains an open problem either to justify it from the surface microstructure considerations, or to eliminate it. Whereas the uniqueness of the weak solutions for the problem with Lipschitz friction coefficient is unknown, and seems unlikely, there are uniqueness results for problems with the normal compliances condition. Moreover, the Signorini condition has a very low regularity ceiling, since once the surface comes into contact with the rigid foundation the velocity is discontinuous, which means that the acceleration is a measure or a distribution. On the other hand, the normal compliance condition lead to a much better regularity \cite{KSreg04}. The thermoviscoelastic contact problem with Signorini's contact condition is of some interest, and will be investigated in the future. \begin{thebibliography}{99} \bibitem{Ad75} R. Adams, Sobolev Spaces", Academic Press, New York, 1975. \bibitem{AKS97} K. T. Andrews, K. L. Kuttler and M. Shillor, \textit{On the dynamic behaviour of a thermoviscoelastic body in frictional contact with a rigid obstacle,} Euro. J. Appl. Math., \textbf{8} (1997), 417-436. \bibitem{AKSW97} K. T. Andrews, A. Klarbring, M. Shillor and S. Wright, \textit{A dynamic contact problem with friction and wear,} Int. J. Engng. Sci., \textbf{35}(14) (1997), 1291--1309. \bibitem{CSS03} O. Chau, M. Shillor and M. Sofonea, \textit{Dynamic frictionless contact with adhesion}, Z. angew Math. Phys. (ZAMP), \textbf{55} (2004), 32--47. \bibitem{Co02} M. Cocu, \textit{Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity}, Z. angew Math. Phys. (ZAMP), \textbf{53} (2002), 1099-1109. \bibitem{DL76} G. Duvaut and J .L. Lions,Inequalities in Mechanics and Physics", Springer, New York, 1976. \bibitem{EJ99} C. Eck and J. Jarusek, \textit{\ The solvability of a coupled thermoviscoelastic contact problem with small Coulomb friction and linearized growth of frictional heat}, Math. Meth. Appl. Sci., \textbf{22} (1999), 1221--1234. \bibitem{FjSS03} J. R. Fern\'andez, M. Shillor and M. Sofonea, \textit{\ Numerical analysis and simulations of a dynamic contact problem with adhesion}, Math. Comput. Modelling, \textbf{37} (2003), 1317--1333. \bibitem{FT} I. Figueiredo and L. Trabucho, \textit{A class of contact and friction dynamic problems in thermoelasticity and in thermoviscoelasticity,} Int. J. Engng. Sci., \textbf{33}(1) (1995), 45--66. \bibitem{HS} W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity," AMS and International Press, 2002. \bibitem{IP} I. R. Ionescu and J-C. Paumier, \textit{On the contact problem with slip rate dependent friction in elastodynamics}, Eur. J. Mech. A/Solids, \textbf{13}(4) (1994), 555--568. \bibitem{Ja99} J. Jarusek, \textit{\ Dynamic contact problems with small Coulomb friction for viscoelastic bodies, Existence of solutions}, Math. Models Methods Appl. Sci., \textbf{9}(1) (1999), 11-34. \bibitem{KO} N. Kikuchi and T. J. Oden, Contact Problems in Elasticity", SIAM, Philadelphia, 1988. \bibitem{Kuf77} A. Kufner, O. John and S. Fucik, Function Spaces," Noordhoff International, Leyden, 1977. \bibitem{kut97} K. L. Kuttler, \textit{Dynamic friction contact problems for general normal and friction laws,} Nonlinear Anal., \textbf{28} (1997), 559-575. \bibitem{kut98} K. L. Kuttler, Modern Analysis", CRC Press, 1998. \bibitem{KSpsm} K. L. Kuttler and M. Shillor, \emph{Set-valued pseudomonotone maps and degenerate evolution inclusions}, Commun. Contemp. Math., \textbf{1} (1999), 87--123. \bibitem{KSbi01} K. L. Kuttler and M. Shillor, \textit{Dynamic bilateral contact with discontinuous friction coefficient}, Nonlinear Anal., \textbf{ 45 } (2001), 309--327. \bibitem{KSmunc} K. L. Kuttler and M. Shillor, \textit{Dynamic contact with normal compliance wear and discontinuous friction coefficient}, SIAM J. Math. Anal., \textbf{34}(1) (2002), 1--27. \bibitem{KSreg04} K. L. Kuttler and M. Shillor, \textit{Regularity of solutions for dynamic frictionless contact problems with normal compliance}, preprint, 2004. \bibitem{KSF03} K. L. Kuttler, M. Shillor and J. R. Fern\'andez, \textit{ Existence and regularity for dynamic viscoelastic adhesivecontact with damage }, preprint, 2003. \bibitem{lio69} J. L. Lions, Quelques Methods de Resolution des Problemes aux Limites Non Lineaires", Dunod, Paris, 1969. \bibitem{MO87} J. A. C. Martins and J. T. Oden, \textit{Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws,} Nonlinear Anal., \textbf{II}(3) (1987), 407-428. \bibitem{nec81} Ne\v {c}as J. and Hlav\'{a}\v {c}ek, Mathematical Theory of Elasic and Elasto-Plastic Bodies: An introduction", Elsevier, 1981. \bibitem{Oden} J. T. Oden, \textit{New models of friction nonlinear elastodynamics problems}, J. Theoretical Appl. Mech., \textbf{7}(1), (1992), 47 -- 54. \bibitem{Ole92} O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization", North-Holland, New York 1992. \bibitem{PAN} P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications", Birkhauser, Basel, 1985. \bibitem{PR03} J.-C. Paumier and Y. Renard, \emph{Surface perturbation of an elastodynamic contact problem with friction}, Euro. Jnl. Appl. Math., \textbf{14}(2003), 465--483. \bibitem{sei89} Seidman T.I., The transient semiconductor problem with generation terms, II, in nonlinear semigroups, partial differential equations and attractors'', \textit{Springer Lecture Notes in Math.,} \textbf{1394}(1989), 185-198. \bibitem{SSTbook} M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact", Springer, to appear. \bibitem{sim87} J. Simon, \emph{Compact sets in the space $L^{p}( 0,T;B)$,} Ann. Mat. Pura. Appl., \textbf{146} (1987), 65-96. \end{thebibliography} \end{document}