\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 $0<T $ and
denote by $\mathbf{u}=(u_{1},\dots ,u_{N})$ the displacements vector, by
$\mathbf{v}=(v_{1},\dots , v_{N})$ the velocity
vector and the stress tensor by $\sigma =(\sigma _{ij})$, where here and
below $i,j=1,\dots, N$, a comma separates the components of a vector or
tensor from partial derivatives, and ``$\, '\,$'' denotes partial
time derivative, thus $\mathbf{v} =\mathbf{u} '$. The velocity of
the foundation is $\mathbf{v}_F$, and the setting is depicted in Fig.\,1.

The dynamic equations of motion, in dimensionless form, are
\begin{equation}
v_{i}'-\sigma _{ij,j}(\mathbf{u},\mathbf{v})=f_{Bi}\quad \text{ in }\Omega _{T},  \label{5.1}
\end{equation}
where $\mathbf{f}_{B}$ represents the volume force acting on the body, all
the variables are in dimensionless form, for the sake of simplicity we set
the material density to be $\rho =1$, and
\begin{equation}
\mathbf{u}(t) =\mathbf{u}_{0}+\int_{0}^{t}\mathbf{v}(s)\, ds.
\label{25febe5}
\end{equation}
The initial conditions are
\begin{equation}
\mathbf{u}(\mathbf{x},0)=\mathbf{u}_{0}(\mathbf{x}),\quad \mathbf{v}(
\mathbf{x},0)=\mathbf{v}_{0}(\mathbf{x})\;\;\text{ in }\Omega,  \label{5.2}
\end{equation}
where $\mathbf{u}_{0}$ is the initial displacement and $\mathbf{v}_{0}$ the
velocity, both prescribed functions.

The body is held fixed on $\Gamma _{D}$ and tractions $\mathbf{f}_{N}$ act
on $\Gamma _{N}$. Thus,
\begin{equation}
\mathbf{u}=0\quad \text{ on }\Gamma _{D},\quad \sigma \mathbf{n}=\mathbf{f}
_{N}\quad\text{on }\Gamma _{N}.  \label{5.3}
\end{equation}
The foundation is assumed completely rigid, so we use the Signorini
condition on the potential contact surface,
\begin{equation}
u_{n}-g\leq 0,\quad\sigma _{n}\leq 0,\quad( u_{n}-g) \sigma
_{n}=0\quad \text{on }\Gamma _{C}.  \label{25febe6}
\end{equation}
Here, $u_{n}=\mathbf{u}\cdot \mathbf{n}$ is the normal component of $\mathbf{
u}$ and $\sigma _{n}=\sigma _{ij} n_i n_j$ is the normal component of the
stress vector or the contact pressure on $\Gamma_C$.

The material is assumed to be linearly viscoelastic with constitutive
relation
\begin{equation}
\sigma ( \mathbf{u,v}) =A\varepsilon ( \mathbf{u})
+B\varepsilon (\mathbf{v}),  \label{25febe2}
\end{equation}
where $\varepsilon ( \mathbf{u}) $ is the small strain tensor, $A$
is the elasticity tensor, and $B$ is the viscosity tensor, both symmetric
and linear operators satisfying
\[
( A\xi, \xi) \geq \delta ^{2}\left| \xi\right| ^{2}, \quad (
B\xi, \xi) \geq \delta ^{2}\left| \xi\right| ^{2},
\]
for some $\delta$ and all symmetric second order tensors
$\xi =\{\xi_{ij}\}$, i.e., both are coercive or elliptic. In components,
 the Kelvin-Voigt constitutive relation is
\[
\sigma_{ij}=a_{ijkl}u_{k,l} +b_{ijkl} u'_{k,l},
\]
where $a_{ijkl}$ and $b_{ijkl}$ are the elastic moduli and viscosity
coefficients, respectively.

To describe the friction process we need additional notation. We denote the
tangential components of the displacements by $\mathbf{u}_{T}= \mathbf{u-}(
\mathbf{u\cdot n)n}$ and the tangential tractions by $\sigma _{Ti}=\sigma
_{ij}n_{j}-\sigma _{n}n_{i}$. The general law of dry friction, a version of
which we employ, is
\begin{gather}
|\sigma _{T}|\leq \mu ( | \mathbf{u}_{T}'-\mathbf{v} _{F}|
) \left| \sigma _{n}\right| , \\
\sigma _{T}=- \mu ( | \mathbf{u}_{T}'-\mathbf{v} _{F}| )
\left| \sigma _{n}\right|\frac{\mathbf{u}_{T}' -\mathbf{v} _{F}}{|
\mathbf{u}_{T}'- \mathbf{v}_{F}|} \quad \text{if}\quad \mathbf{u}
_{T}' \neq \mathbf{v}_{F}.
\end{gather}
This condition creates major mathematical difficulties in the weak
formulation of the problem, since the stress $\sigma$ does not have
sufficient regularity for its boundary values to be well-defined.
Nevertheless, some progress has been made in using this model in \cite{EJ99}
and \cite{Ja99} (see also the references therein). However, there the
contact condition was that of normal damped response, \textit{i.e.}, the
unilateral restriction was on the normal velocity rather than on the normal
displacement, as in (\ref{25febe6}). Such a condition implies that once
contact is lost, it is never regained, which in most applied situations is
not the case, and, moreover, the mathematical difficulties in dealing with
that model were considerable. These difficulties motivated \cite{MO87} and
many followers to model the normal contact between the body and the
foundation by a \textit{normal compliance} condition in which the normal
stress is given as a function of surface resistance to interpenetration.
This is usually justified by modeling the contact surface in terms of
``surface asperities.''

To overcome the difficulties in giving meaning to the trace of the stress on
the contact surface we employ an averaged stress in the friction model.
Thus, it is a locally averaged stress which controls the friction and the
onset of sliding on the surface, however, we make no particular assumptions
on the form of the averaging process. It may be of interest to investigate
and deduce them from homogenization or experimental results (see \cite{Oden}
for a step in this direction). This procedure of averaging the stress has
been employed earlier by the authors in \cite{KSbi01} (see also references
therein) and recently in \cite{Co02} in a very interesting paper on the
existence of weak solutions for a linear viscoelastic model with the
Signorini boundary conditions (\ref{25febe6}) and an averaged Coulomb
friction law. In this paper we consider a similar situation, but with a slip
rate dependent coefficient of friction, and without the compatibility
assumptions on the initial data which are assumed in \cite{Co02}. The
averaged form of the friction which we will employ involves replacing the
normal stress $\sigma _{n}$ with an averaged normal stress, $(\mathcal{R}
\sigma)_{n}$, where $\mathcal{R}$ is an averaging operator to be described
shortly. Thus, we employ the following law of dry friction,
\begin{gather}  \label{5.5}
|\sigma _{T}|\leq \mu (|\mathbf{u}_{T}'-\mathbf{v} _{F}|) 
|(\mathcal{R}\sigma )_{n}|, \\
\label{5.7}
\sigma _{T}=- \mu ( | \mathbf{u}_{T}'-\mathbf{v} _{F}| )| ( \mathcal{R}\sigma ) _{n}|
\frac{\mathbf{u}_{T} ' - \mathbf{v}_{F}}{| \mathbf{u}_{T}'- \mathbf{v}_{F}| } 
\quad \text{if}\quad\mathbf{u}_{T}' \neq \mathbf{v}_{F}.
\end{gather}
Here, $\mu $ is the coefficient of friction, a positive bounded function
assumed to depend on the relative slip $\mathbf{u}_{T}'- \mathbf{v}
_{F}$ between the body and foundation. We could have let $\mu $ depend on $
\mathbf{x}\in \Gamma _{C}$ as well, to model the local roughness of the
contact surface, but we will not consider it here to simplify the
presentation; however, all the results below hold when $\mu$ is a Lipschitz
function of $\mathbf{x}$.

We assume that $\mathbf{v}_{F}\in L^{\infty}(0,T;(L^{2}(\Gamma _{C}))^{N})$,
and refer to \cite{Ad75, Kuf77} for standard notation and concepts related
to function spaces. The regularization $( \mathcal{R}\sigma )
_{n} $ of the normal contact force in (\ref{5.5}) and (\ref{5.7}) is such
that $\mathcal{R}$ is \textit{linear} and
\begin{equation}
\begin{array}{c}
\mathbf{v}^{k}\to \mathbf{v}\text{ in }L^{2}( 0,T;L^{2}(
\Omega ) ^{N}) \text{ implies }( \mathcal{R}\sigma
^{k}) _{n}\to ( \mathcal{R}\sigma ) _{n}\text{ in }
L^{2}( 0,T;L^{2}( \Gamma _{c}) )
\end{array}
\label{25febe1}
\end{equation}

There are a number of ways to construct such a regularization. For example,
if $\mathbf{v}\in H^{1}(\Omega) $, one may extend it to $\mathbb{R}
^{N}$ in such a way that the extended function $E\mathbf{v}$ satisfies
$\| E\mathbf{v}\|_{1,\mathbb{R}^{N}}\leq C\|
\mathbf{v}\|_{1,\Omega }$, where $C$ is a positive constant
that is independent of $\mathbf{v}$, and define
\begin{equation}
\mathcal{R}\sigma \equiv A\varepsilon ( E\mathbf{u}\ast \psi )
+B\varepsilon ( E\mathbf{v}\ast \psi ) ,  \label{25febe3}
\end{equation}
where $\psi $ is a smooth function with compact support, and ``\thinspace
*\thinspace '' denotes the convolution operation. Thus, $\mathcal{R}\sigma
\in C^{\infty }( \mathbb{R}^{N}) $ and $( \mathcal{R}\sigma
) _{n}\equiv ( \mathcal{R}\sigma ) \mathbf{n\cdot n}$ is
well defined on $\Gamma _{C}$. In this way we average the displacements and
the velocity and consider the stress determined by the averaged variables.

Another way to obtain an averaging operator satisfying (\ref{25febe1}) is as
follows. Let $\psi :\Gamma _{C}\times \mathbb{R}^{N}\to \mathbb{R}$ be
such that $\mathbf{y}\to \psi ( \mathbf{x,y}) \text{ is
in } C_{c}^{\infty }(\Omega)$, $\psi $ is uniformly bounded
and, for $\mathbf{x}\in \Gamma _{C}$,
\[
\mathcal{R} \sigma _{n}\equiv \mathcal{R}\sigma \mathbf{n\cdot n}\quad
\mbox{where}\quad \mathcal{R}\sigma (\mathbf{x}) \equiv
\int_{\Omega }\sigma (\mathbf{y}) \psi ( \mathbf{x,y}
) dy.
\]
Then, the operator is linear and it is routine to verify that (\ref{25febe1}
) holds. Physically, this means that the normal component of stress, which
controls the friction process, is averaged over a part of $\Omega$, and to
be meaningful, we assume that the support of $\psi ( \mathbf{x,\cdot }
)$ is centered at $\mathbf{x}\in \Gamma _{C}$ and is small. Conditions
(\ref{5.5}) and (\ref{5.7}) are the model for friction which we employ in
this work. The tangential part of the traction is bounded by the so-called
friction bound, $\mu (| \mathbf{u} _{T}'-\mathbf{v}_{F}| ) |(
\mathcal{R}\sigma ) _{n}|$, which depends on the sliding velocity via the
friction coefficient, and on the regularized contact stress. The surface
point sticks to the foundation and no sliding takes place until $|\sigma
_{T}|$ reaches the friction bound and then sliding commences and the
tangential force has a direction opposite to the relative tangential
velocity. The contact surface $\Gamma _{C}$ is divided at each time instant
into three parts: \textit{separation} zone, \textit{slip} zone and \textit{
stick} zone.

A new feature in the model is the dependence, which can be observed
experimentally, of the friction coefficient on the magnitude of the slip
rate $| \mathbf{u}_{T}'-\mathbf{v}_{F}| $.

We assume that the coefficient of friction is bounded, Lipschitz continuous
and satisfies
\begin{equation}
\left| \mu ( r_{1}) -\mu ( r_{2}) \right| \leq \mathrm{
\ \ Lip}_{\mu }\left| r_{1}-r_{2}\right|, \qquad \| \mu \|_{\infty }\leq K_\mu.
\label{5.10}
\end{equation}
In Section 5 this assumption will be relaxed and we shall consider $\mu $
which is set-valued and models the jump from a static value to a dynamic
value when sliding starts.

 The classical formulation of the problem of \textit{dynamic
contact between a viscoelastic body and a rigid foundation} is:

 \noindent Find the displacements $\mathbf{u}$ and the velocity
$\mathbf{v}=\mathbf{u}'$, such that
\begin{gather}
\mathbf{v}'-\mathrm{Div}(\sigma (\mathbf{u},\mathbf{v}))
=\mathbf{f}_{B}\quad \text{ in }\;\Omega _{T},  \label{2.14} \\
\sigma ( \mathbf{u,v}) =A\varepsilon ( \mathbf{u})
+B\varepsilon ( \mathbf{v}) ,  \label{2.15} \\
\mathbf{u}(t) =\mathbf{u}_{0}+\int_{0}^{t}\mathbf{v}(
s) ds,  \label{2.16} \\
\mathbf{u}(\mathbf{x},0) =\mathbf{u}_{0}(\mathbf{x}),\quad \mathbf{v} (
\mathbf{x},0)=\mathbf{v}_{0}(\mathbf{x})\;\;\text{ in }\Omega,  \label{2.17}
\\
\mathbf{u} =0\;\text{ on }\Gamma _{D},\quad \sigma \mathbf{n}=\mathbf{f}
_{N}\quad \text{ on }\Gamma _{N},  \label{2.18}
\\
u_{n}-g\leq 0,\quad \sigma _{n} \leq 0,\quad ( u_{n}-g) \sigma
_{n}=0\quad \text{ on }\Gamma _{C},  \label{2.19} \\
\quad | \sigma _{T}| \leq \mu ( | \mathbf{u}_{T}'- \mathbf{v
}_{F}| ) \left| (\mathcal{R}\sigma )_{n}\right| ,\quad \text{ on }
\Gamma _{C},  \label{2.20} \\
\mathbf{u}_{T}'\neq \mathbf{v}_{F}\;\; \text{implies }\;\; \sigma
_{T} =-\mu ( | \mathbf{u}_{T}'-\mathbf{v} _{F}| )
| (\mathcal{R}\sigma ) _{n}|\frac{\mathbf{u}_{T}' -
\mathbf{v}_{F}}{| \mathbf{u}_{T}'- \mathbf{v}_{F}| }.  \label{2.21}
\end{gather}

We turn to the weak formulation of the problem, and to that end we need
additional notation. $V$ denotes a closed subspace of ($H^{1}(\Omega ))^{N}$
containing the test functions $(C_{c}^{\infty }(\Omega ))^{N}$, and $\gamma $
is the trace map from $W^{1,p}(\Omega )$ into $L^{p}(\partial \Omega )$. We
let $H=$ $(L^{2}(\Omega ))^{N}$ and identify $H$ and $H'$, thus,
\[
V\subseteq H=H'\subseteq V'.
\]
Also, we let $\mathcal{V}=L^{2}(0,T;V)$ and $\mathcal{H}=L^{2}(
0,T;H) $.

Next, we choose the subspace $V$ as follows. If the body is clamped over
$\Gamma _{D}$, with meas$\,\Gamma _{D}>0$, 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
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
\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}
\end{equation}
where $C=M$ or $L$, for some $\delta >0,\;\lambda \geq 0.$
Next, we define $\mathbf{f}\in L^{2}(0,T;V')$ as
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
\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}
\end{equation}

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{equation}
\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}
\end{equation}
for all $\mathbf{w}\in \mathcal{K}_{\mathbf{u}}$, where
\begin{equation}
\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}
\end{equation}
Here, $\gamma _{T}^{\ast }\xi $ satisfies, for $\mathbf{w}\in \mathcal{K}_{
\mathbf{u}}$,
\begin{equation}
\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{equation}
\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
\[
S=\{\mathbf{u}\in L^{p}(0,T;W):\mathbf{u}'\in L^{q}(0,T;Y), \|
\mathbf{u}\|_{L^{p}(0,T;W)}+\|\mathbf{u}'\|_{L^{q}(0,T;Y)}<R\}.
\]
Then $S$ is precompact in $L^{p}(0,T;U).$
\end{theorem}

\begin{theorem}[\cite{sei89,sim87}]
\label{t6.2}  Let $W,U$ and $Y$ be as above and for $q>1$
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
\begin{equation}
\langle P(\mathbf{u}),\mathbf{w}\rangle =\int_{0}^{T}\int_{\Gamma
_{C}}(u_{n}-g)_{+}w_{n}\,d\Gamma dt.  \label{6.8}
\end{equation}
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 $\mathbf{w}\in \mathcal{V}$,
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
u_{0n}-g \leq 0\quad \mbox{on } \Gamma_{C}.  \label{25febe7}
\end{equation}
\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{equation}
\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}
\end{equation}
Since $\mu $ is assumed to be bounded, it follows from (\ref{28febe4}) that
\begin{equation}
\xi \in \left[ -K_{\mu }\left| ( \mathcal{R}\sigma ) _{n}\right|
,\,K_{\mu }\left| ( \mathcal{R}\sigma ) _{n}\right| \,\right],
\label{28febe7}
\end{equation}
a.e. on $\Gamma _{C}$. Now, assumption (\ref{25febe1}) implies that there
exists a constant $C$ such that
\begin{equation}
\| ( \mathcal{R}\sigma ) _{n}
\|_{L^{2}( 0,t;L^{2}( \Gamma _{c}) ) }\leq C
\| \mathbf{v}\|_{L^{2}( 0,t;H)}.
\label{28febe5}
\end{equation}
Therefore, we find from (\ref{28febe7}) that
\begin{equation}
\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}
\end{equation}
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 $
\begin{equation}
\| \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}
\end{equation}
Therefore, (\ref{28febe3}) yields
\begin{equation}
\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}
\end{equation}
We obtain from (\ref{28febe9}) and (\ref{6.9}),
\begin{equation}
\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}
\end{equation}
 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{equation}
\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}
\end{equation}
Now, let $\mathbf{w}\in L^{2} ( 0,T;H_{0}^{1} ( \Omega ) ^{N}) $ and apply (
\ref{6.10}) to it, thus,
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
| \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}
\end{equation}
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
\begin{equation}
\| \mathbf{v}'-\mathrm{Div}( A\varepsilon (
\mathbf{u}) +B\varepsilon ( \mathbf{v}) )
\| _{L^{2}( \Omega _{T}) }\leq C.  \label{28febe13}
\end{equation}
\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{equation}\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{equation}
\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
\begin{equation}
\left| \mathbf{v}^{\varepsilon }(T) \right| _{H}\leq C.
\label{4mare13}
\end{equation}

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{equation}
\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{equation}
\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{equation} \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}
\end{equation}
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{equation}
\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}
\end{equation}
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{equation} \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}
\end{equation}
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
\begin{equation}
\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}
\end{equation}
$\mathbf{u}^{\varepsilon }(t)=\mathbf{u}_{0}+\int_{0}^{t}\mathbf{v}
^{\varepsilon }(s)ds$, and for all $\mathbf{z}\in \mathcal{V}$,
\begin{equation}  \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.
\end{equation}
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
\begin{equation}
\| ( \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}
\end{equation}
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
\begin{equation}
\xi ^{\varepsilon }\to \xi \text{ weakly in }L^{2}(
0,T;L^{2}(\Gamma _{C}) ^{N}).  \label{1mare26}
\end{equation}
In addition, we may assume, after taking a suitable subsequence and using
the fact that $L$ and $M$ are linear, that
\begin{equation}
L\mathbf{u}^{\varepsilon }\to L\mathbf{u,\quad }M\mathbf{v}
^{\varepsilon }\to M\mathbf{v}\text{ in }\mathcal{V}'.  \label{3mare.5}
\end{equation}
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{equation} 
\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}
\end{equation}
From (\ref{28febe12}), (\ref{4mare13}), and (\ref{1mare4}) we find,
\begin{equation}
\lim_{\varepsilon \to 0}( \mathbf{v}^{\varepsilon }(
T) ,\mathbf{u}^{\varepsilon }(T) -\mathbf{w}(
T) ) _{H}=0,  \label{3mare2}
\end{equation}
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
\begin{equation}
\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}
\end{equation}
Together with inequality (\ref{3mare1}) these imply
\begin{equation}
\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}
\end{equation}
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{equation} \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}
\end{equation}
This, in turn, implies
\begin{equation} \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}
\end{equation}
It only remains to verify that for all $\mathbf{z}\in \mathcal{V}$,
\begin{equation}
\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}
\end{equation}
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{equation}
\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}
\end{equation}
which holds for all $\mathbf{w}\in \mathcal{K}_{\mathbf{u}}$. Moreover, for
all $\mathbf{z}\in \mathcal{V}$,
\begin{equation}
\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{equation}
\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
\begin{equation}
\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}
\end{equation}
This estimate together with (\ref{1mare25}) and (\ref{28febe7}) yield
\begin{equation}
\| \xi _{\varepsilon }\|_{L^{2}
(0,T;L^{2}( \Gamma _{C}) ^{N}) }\leq C,  \label{10mare5}
\end{equation}
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
\begin{equation}
\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}
\end{equation}
and, thus, the conclusion of Lemma \ref{1marl2} also holds,
\begin{equation} \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}
\end{equation}
As above, (\ref{10mare11}) implies
\begin{equation}
\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}
\end{equation}
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{equation} \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}
\end{equation}
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{equation}
\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}
\end{equation}
which implies
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
\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}
\end{equation}
such that
\begin{equation} \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}
\end{equation}
where
\begin{equation}
\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}
\end{equation}
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}