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     {\LARGE \textsf{МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ}} \\[2mm]
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     {\flushleft\it Vol. 20, No. 2, pp. 144--145 (2023) \hspace{\fill}{\rm\small УДК \udcs \hspace{-4mm}}}
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\begin{document}
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\title[Equilibrium problem for a Kirchhoff-Love plate]{\large Equilibrium
problem for a Kirchhoff-Love plate contacting with the lateral
surface along a strip of a given width}
\author[N.P. LAZAREV]{{\bf N.P. Lazarev}\protect\orcidicon{0000-0001-8827-2630}}



\address{Nyurgun Petrovich Lazarev  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} North-Eastern Federal University,
\newline\hphantom{iii} Kulakovsky str., 48,
\newline\hphantom{iii} 677000, Yakutsk, Russia}%
\email{nyurgunlazarev@yandex.ru}%

\author{{\bf D.Y. Nikiforov}}%
\address{Djulustan  Yakovlevich Nikiforov  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} North-Eastern Federal University,
\newline\hphantom{iii} Kulakovsky str., 48,
\newline\hphantom{iii} 677000, Yakutsk, Russia}%
\email{dju92@mail.ru}%

\author{{\bf G.M. Semenova}}%
\address{Galina Mikhailovna Semenova  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} North-Eastern Federal University,
\newline\hphantom{iii} Kulakovsky str., 48,
\newline\hphantom{iii} 677000, Yakutsk, Russia}%
\email{sgm.08@yandex.ru}%


\thanks{\sc Lazarev, N.P.,
Equilibrium problem for a Kirchhoff-Love plate}
\thanks{\copyright \ 2023 Lazarev N.P.}
\thanks{\rm The results of the first and third sections were obtained
N. P.~Lazarev with the support of the Russian Science Foundation
(project 24-21-00081)}
\thanks{\rm The results of the second section were obtained
by D. Y.~Nikiforov and G. M.~Semenova with the support of the
Ministry of Education and Science of the Russian Federation
(agreement No. 075-02-2024-1441, February 28, 2024)}
\thanks{\it Received January, 1, 2023, Published December, 31, 2023}%





\semrtop \vspace{1cm}
\maketitle {\small

\vspace{-12pt}
%\centerline{{\it Communicated by} {\sc P.P. Petrov}
%({\it filled by Editorial board})
}

\bigskip
\bigskip

\begin{quote}
\noindent{\bf Abstract:} A new model of a Kirchhoff-Love plate is
justified, which may come into contact by its lateral surface with
a non-deformable obstacle along a strip of a given width. The
non-deformable obstacle restricts displacements of the plate along
the outer lateral surface. The obstacle is specified by a
cylindrical surface, the generatrices of which are perpendicular
to the midplane of the plate. A problem is formulated in
variational form. A set of admissible displacements is determined
in a suitable Sobolev space in the framework of a clamping
condition and a non-penetration condition of the Signorini type.
The non-penetration condition is given as a system of two
inequalities. The existence and uniqueness of a solution to the
problem is proven. An equivalent differential formulation and
optimality conditions are found under the assumption of additional
regularity of the solution to the variational problem. A
qualitative connection has been established between the proposed
model and a previously studied problem in which the plate is in
contact over the entire lateral surface.\medskip

\noindent{\bf Keywords:} contact problem, limit passage,
variational inequality, nonpenetration condition.
 \end{quote}


\bigskip

\section{Introduction}
Boundary value problems of the theory of elasticity with
inequality type conditions describing the equilibrium of bodies
are successfully studied on the basis of variational inequalities
\cite{Sign,Fichera2,DalMaso1988}.  Within the framework of
nonlinear problems with non-penetration conditions, mathematical
models are often studied in which elastic bodies may come into
mechanical contact interaction with non-deformable obstacles
\cite{Rademacher,KhludnevKovtunenko,Lazarev-Rudoy2022,LazarevLobach},
or with another deformable body
\cite{Furtsev2019JMS,FurtsevSEMR2020,Pyatkina,KhlHoffBot2006}. For
studies of the regularity of solutions to obstacle problems we
refer to \cite{Piersanti4th,Piersanti2nd,Grimaldi}. Asymptotic analysis for
problems of solid mechanics with inequality type constraints can
be found, for example, in
\cite{PiersantiAsymp1,LNP-KVA-Pmtf2024,PiersantiAsymp2}.
 In the case where a body has a crack (or cracks), the interaction of
opposite crack faces can also be described using models subject to
unilateral constraints
 \cite{Knees,Furtsev2024,Kovtunenko-Ef2022,Royal2022Kov-Rud,Shcherbakov2022,KovtunItou,KovtunRaj},
 etc.
 For problems of this type, numerical methods are proposed, for example, in \cite{RudoyFurtsev,RudoyShcherbakov}.
Within the framework of the theory of elasticity, a qualitative
connection between nonlinear problems describing contact
interaction with obstacles and problems of the crack theory has
been established for a number of mathematical models
\cite{StepanovKhlud2003,Nikolaeva,LazarevFict,LazarevSFU}. Note
that cases of simultaneous possible contact of the plate along the
front surface and lateral edge are also of interest
\cite{LazarevLobach1,LazarevChel2023}. We can mention the works
for pointwise contact problems \cite{laz-kov,laz-fed2022}, where
minimization problems  over nonconvex sets are investigated.



In this work, we consider a special configuration of a
non-deformable obstacle in contact with a strip on the lateral
cylindrical surface of a plate. In this case, the obstacle in the
initial state does not come into contact with the points of the
plate along the entire width, as, for example, in the works
\cite{LazarevFict,LazarevSFU,LazarevFict2014}, but along a strip
of a given width. It is shown that when the parameter of the width
of the contact zone tends to the value of the plate thickness, we get
as a limiting problem the previously known problem studied in
\cite{Nikolaeva}. Thus, the presented mathematical model
generalizes the previously known problem describing the contact of
a Kirchhoff-Love plate with a non-deformable obstacle.


\section{The Variational Problem}



Let $\Omega \subset \mathbf{R}^2$ be a bounded with a smooth
boundary $\Gamma$, which consists of two continuous curves
$\Gamma=\overline\Gamma_0\cup\overline\Gamma_1$, $\Gamma_0\cap\Gamma_1=\emptyset$,
$\mbox{mes}(\Gamma_0)> 0$, $\mbox{mes}(\Gamma_1)> 0$. Denote by
$\nu=(\nu_1,\nu_2)$ the external unit normal vector to $\Gamma$.
For simplicity, suppose the plate has a uniform thickness $2h$.
Let us assign a three-dimensional Cartesian space $\{x_1,x_2,z\}$
with the set $\{\Omega\}\times\{0\}\subset \mathbf{R}^3$
corresponding to the middle plane of the plate.

%%%%%
 Denote by
$\chi=\chi({x})=({W},w)$ the displacement vector of the
mid-surface points (${x}\in\Omega$), by ${W}=(w_1,w_2)$ the
displacements in the plane $\{x_1,x_2\}$, and by $w$ the
displacements along the axis  $z$ (deflections). The strain and
integrated stress tensors are denoted by
$\varepsilon_{ij}=\varepsilon_{ij}(W)$,
$\sigma_{ij}=\sigma_{ij}(W)$, respectively
\cite{KhludnevKovtunenko}:
$$
\varepsilon_{ij}(W)=\frac{1}{2}(\frac{\partial w_j}{\partial
x_i}+\frac{\partial w_i}{\partial x_j}), \quad
\sigma_{ij}(W)=a_{ijkl}\varepsilon_{kl}(W),\quad i,j=1,2,
$$
where $\{a_{ijkl}\}$ is the given elasticity tensor, assumed to be
symmetric and positive definite:
\begin{eqnarray}\nonumber
a_{ijkl}=a_{klij}=a_{jikl}, \quad i,j,k,l=1,2, \quad a_{ijkl}\in
L^{\infty}(\Omega),
\\
a_{ijkl}\xi_{ij}\xi_{kl}\geq c_0|\xi|^2, \quad \forall\xi, \quad
\xi_{ij}=\xi_{ji},\quad i,j=1,2,\quad c_0=const>0.\nonumber
\end{eqnarray}
A summation convention over repeated indices is used in the
sequel. Next we denote the bending moments by formulas
\cite{KhludnevKovtunenko}
$$
m_{ij}(w)=-d_{ijkl}w,_{kl},\quad i,j=1,2,\quad
(w,_{kl}=\frac{\partial^2 w}{\partial x_k\partial x_l})
$$
where tensor $\{d_{ijkl}\}$ has the same symmetry, boundedness,
and positive definiteness characteristics
 as tensor $\{a_{ijkl}\}$. Let
$B(\cdot\,,\cdot)$ be a bilinear form defined by the equality
\begin{equation}\label{Forma Bil_0}
B({\chi},{\bar\chi})=\int\limits_{\Omega}\!\big\{\sigma_{ij}(
W)\,\varepsilon_{ij}(\bar W)- m_{ij}(w)\bar w,_{ij}\big\}dx,
\end{equation}
where $\chi=(W,w)$,  $\bar{\chi}=(\bar{W},\bar w)$.

Introduce the Sobolev spaces
\begin{eqnarray}
&H^{1,0}_{\Gamma_0}(\Omega)=\left\{v\in H^1(\Omega)\ \Bigm|\
v=0\,\ \mbox{on}\,\,\Gamma_0\,\right\}, \nonumber\\
& H^{2,0}_{\Gamma_0}(\Omega)=\left\{v\in H^2(\Omega)\ \Bigm|\
v=\frac{\partial v}{\partial \nu}=0\,\,\
\mbox{on}\,\,\Gamma_0\,\right\},
\nonumber\\
&\nonumber H(\Omega)=H^{1,0}_{\Gamma_0}(\Omega)^2\times
H^{2,0}_{\Gamma_0}(\Omega).
\end{eqnarray}
It is well known that the standard expression for a potential
energy functional of a Kirchhoff--Love plate has the following
representation
$$
\Pi({\chi})=\frac{1}{2}B({\chi},{\chi})-\int\limits_{\Omega}
\!{F}\,{{\chi}}d{x},\,\,\,\quad{\chi}=(W,w),
$$
where vector ${F}=(f_1,f_2,f_3)\in L_2(\Omega)^3$ describes the
body forces \cite{KhludnevKovtunenko}.
 Note that
the following inequality providing coercivity of functional
$\Pi({\chi})$
\begin{equation}\label{ozen}
B(\chi,\chi) \geq c\|\chi\|^2\quad\forall\; \chi\in
H(\Omega),\quad (\|\chi\|=\|\chi\|_{H(\Omega)})
\end{equation}
with a constant $c>0$ independent of $\chi$,  holds for the
bilinear form $B( \cdot, \cdot)$ \cite{KhludnevKovtunenko}.


Let us start with the description of a non-deformable
obstacle. The obstacle has a special shape such that the plate in
the initial state is in contact along a strip of the width  $l$,
where $l\in \mathbf{R}$ is a fixed number such that $0<l\leq 2h$.
Namely, we specify the obstacle by the following set:
$$\{(x_1,x_2,z)\,|\,\,(x_1,x_2)\in\Gamma_1,\quad z\in(-\infty,-h+l]\}.$$
Obviously, for $l=2h$ we obtain full contact along the lateral
surface of the plate, studied in \cite{Nikolaeva}.


In order to introduce boundary conditions of the Signorini type,
we recall the well-known relations of the Kirchhoff-Love model for
displacements of points $(x,z)\in\Omega\times[-h,h]$
\begin{equation}\label{Kirch}
W^z(x,z)=W(x)-z\nabla w,\quad  |z|\leq h,\quad  w^z(x,z)=w(x).
\end{equation}
Taking into account \eqref{Kirch} and arguing as in
\cite{KhludnevKovtunenko,Nikolaeva}, we impose the following
condition for displacements on $\Gamma_1$ describing the
non-penetration of plate points into a non-deformable obstacle. We
require the following relations to be satisfied
\begin{equation}\label{np-cond}
W\nu-z\frac{\partial w}{\partial\nu}\leq 0 \quad\mbox{on}\quad
\Gamma_1,\quad z\in[-h,-h+l],
\end{equation}
where $W\nu=w_i\nu_i$. The inequality \eqref{np-cond}, due to
linearity, can be equivalently represented as a system of two
inequalities
\begin{equation}\label{np-cond2}
W\nu+h\frac{\partial w}{\partial\nu}\leq 0, \quad
W\nu+(h-l)\frac{\partial w}{\partial\nu}\leq 0 \quad\mbox{on}\quad
\Gamma_1.
\end{equation}
Now we can introduce the following set of admissible functions
$$
K_l=\{\chi=(W,w)\in
H(\Omega)\,|\,\chi\,\,\mbox{satisying}\,\,\eqref{np-cond}\}.
$$
Let us formulate a variational statement of an equilibrium
problem. It is required to find a function $\xi=(U,u)\in K_l$,
such that
\begin{equation}\label{minZad2}
\Pi(\xi)=\inf_{\chi\in K_l}\Pi(\chi).
\end{equation}
\begin{theorem}
\label{VT_thm:1} The problem \eqref{minZad2} has a unique
solution.
\end{theorem}
\begin{proof}
We will apply the well known Weierstrass theorem in order to show
solution existence of the minimization problem \cite{Bayoki2}. The
energy functional is coercive and weakly lower semicontinuous on
$H(\Omega)$ \cite{KhludnevKovtunenko}.  It is easy to see that the
set $K_l$ has the convexity and closedness properties. These
properties of the set of admissible displacements ensure that the
set $K_l$ is weakly closed. Consequently, for the minimization
problem \eqref{minZad2} all conditions of the Weierstrass theorem
are satisfied both for the functional $\Pi(\chi)$ and for the set
of admissible functions $K_l$. This means that problem
\eqref{minZad2} has at least one solution. The functional is
convex and differentiable, and as a consequence the problem
\eqref{minZad2} is equivalent to the following variational
inequality
\begin{equation}\label{varnertonk123}
\quad \xi\in K_l, \quad B(\xi,{\chi}-{\xi})\geq
\int\limits_{\Omega}F(\chi-{\xi})dx\quad \forall\,\chi\in K_l.
\end{equation}
Assuming that there are two different solutions ${\xi}_1$ and
${\xi}_2$, we extract two inequalities from the variational
inequality
\begin{equation}\nonumber
 \quad B(\xi_1,{\xi}_2-{\xi}_1)\geq
\int\limits_{\Omega}F(\xi_2-{\xi_1})dx,
\end{equation}
\begin{equation}\nonumber
 \quad B(\xi_2,{\xi}_1-{\xi}_2)\geq
\int\limits_{\Omega}F(\xi_1-{\xi}_2)dx.
\end{equation}
Adding the last two inequalities we get that
\begin{equation}\nonumber
 \quad B({\xi}_2-{\xi}_1,{\xi}_2-{\xi}_1) \leq 0.
\end{equation}
This means, in view of \eqref{ozen}, that $\xi_1={\xi}_2$, and
also entails the uniqueness of the solution to the problem
\eqref{minZad2}.
\end{proof}

{\centering
\section{The Differential Statement }
}


Let $l$ be a fixed number such that $0<l\le2h$. Let us assume that
the solution $\xi=(U,u)\in K_l$ and elasticity tensors $\{a_{ijkl}\}$, $\{d_{ijkl}\}$ are sufficiently smooth. Namely, in addition to the prescribed properties of the solution, it is
sufficient to require that $\xi\in H^2(\Omega)^2\times
H^4(\Omega)$. Our aim is
to find from the variational inequality the equilibrium equations
fulfilled in $\Omega$ and optimality conditions satisfied on
$\Gamma_1$. We will apply the following Green's formulas
\eqref{Green1} for the functions $\chi=( W, w)\in K_l$
\cite{KhludnevKovtunenko},
\begin{eqnarray}\nonumber
\int\limits_{\Omega}\!\sigma_{ij}( U)\,\varepsilon_{ij}( W)dx=
-\int\limits_{\Omega}\sigma_{ij,j}(U)
w_i\,dx+\phantom{qqqqqqqqq}\\
\,\phantom{qqqqqqqqqaaaaaaaaaa}+\int\limits_{\Gamma}
\biggl(\sigma_\nu(U) W\nu+ \sigma_\tau(U) W\tau
\biggr)\,d\Gamma,\label{Green1}\end{eqnarray}
%\begin{equation}\label{Green1}
%\int\limits_{\Omega}\!\sigma_{ij}(
%U)\,\varepsilon_{ij}( W)dx=
%-\int\limits_{\Omega}\sigma_{ij,j}(U) w_i\,dx-
%\int\limits_{\gamma} \biggl[ \sigma_\nu(U) W\nu+
%\sigma_\tau(U) W\tau\biggr]\,d\gamma,
%\end{equation}
\begin{equation}\label{Green2}
\int\limits_{\Omega}m_{ij}(u) w,_{ij}dx=
\int\limits_{\Omega}m_{ij,ij}(u)\, w\,dx+ \int\limits_{\Gamma}
\biggl(t^\nu(u) w- m_\nu(u) \frac{\partial
w}{\partial\nu}\biggr)\,d\Gamma,
\end{equation}
where
$$
\sigma_\nu(U)=\sigma_{ij}(U)\nu_i\nu_j,\quad m_\nu(u)=-
m_{ij}\nu_i\nu_j,
$$
$$
\sigma_\tau(U)=(\sigma^1_\tau(U),\sigma^2_\tau(U))=(\sigma_{1j}(U)\nu_j,\sigma_{2j}(U)
\nu_j)-\sigma_\nu(U)\nu,
$$
$$
t^\nu(u)=-m_{ij,k}\tau_k\tau_j\nu_i - m_{ij,j}\nu_i,\quad
\tau=(-\nu_2,\nu_1),
$$
$$
{ W}\nu={ w}_i\nu_i,\quad { W}\tau=({ W}^1_\tau,{ W}^2_\tau),
\quad { w}_i=({ W}\nu) \nu_i+{ W}^i_\tau,\quad i=1,2.
$$

Along with the variational statement \eqref{minZad2}, one can deal
with the corresponding differential statement. Namely, the
following theorem holds.
\begin{theorem}\label{ekvth}
Supposing the solution $\xi=(U,u)$ as well as elasticity tensors
$\{a_{ijkl}\}$, $\{d_{ijkl}\}$ to be sufficiently smooth, the
variational problem \eqref{minZad2} is equivalent to the following
boundary value problem
\begin{equation}\label{e1}
-m_{ij,ij}(u)=f_3\quad\mbox{in}\quad\Omega,
\end{equation}
\begin{equation}\label{e2}
\quad-\sigma_{ij,j}(U)=f_i\quad\mbox{in}\quad\Omega,\quad i=1,2,
\end{equation}
\begin{equation}\label{e3}
\sigma_\nu(U)-\frac{1}{h}m_\nu(u)\leq 0,\quad -\sigma_\nu(U)(h-l)
+ m_\nu(u)\le 0\quad\mbox{on}\quad\Gamma_1,
\end{equation}
\begin{equation}\label{e4}
\sigma_\nu(U)\leq 0,\,\, U\nu+h\frac{\partial
u}{\partial\nu}\leq 0,\,\, U\nu+(h-l)\frac{\partial
u}{\partial\nu}\leq 0\quad\mbox{on}\quad \Gamma_1,
\end{equation}
\begin{equation}\label{e5}
\sigma_\tau(U)=(0,0),\quad t^\nu(u)= 0,\quad\sigma_\nu(U)U\nu + m_\nu(u) \frac{\partial u}{\partial\nu}=0
\quad\mbox{on}\quad \Gamma_1,
\end{equation}
\begin{equation}\label{e7}
U=(0,0),\quad u=\frac{\partial u}{\partial \nu}=0\quad \mbox{on}\quad\Gamma_0.
\end{equation}

\end{theorem}


\begin{proof}
Substituting $\bar\chi= \xi\pm \tilde\chi$, where $\tilde\chi\in
C^\infty_0(\Omega)^3$, as a test function in
\eqref{varnertonk123}, we obtain the following relation
$$
\int\limits_{\Omega}\!(\sigma_{ij}(U)\,\varepsilon_{ij}(\tilde
W)-m_{ij}(u)\tilde w,_{ij})dx =\int\limits_{\Omega}F\tilde\chi
dx,$$ that is, the equilibrium equations
\begin{equation}\label{urav-rav1}
-m_{ij,ij}(u)=f_3\quad\mbox{in}\quad\Omega,
\end{equation}
\begin{equation}\label{urav-rav2}
\quad-\sigma_{ij,j}(U)=f_i\quad\mbox{in}\quad\Omega,\quad i=1,2,
\end{equation}
hold in terms of distribution.

 Applying Green's formulas to
\eqref{varnertonk123} and using \eqref{urav-rav1},
\eqref{urav-rav2}, one can show that
\begin{equation}\label{var-ner3}
\int\limits_{\Gamma} \Biggl(\sigma_\nu(U)( W-U)\nu+
\sigma_\tau(U)( W-U)\tau-
$$
$$t^\nu(u)(w-u)+m_\nu(u)\left(\frac{\partial w}{\partial\nu}-
\frac{\partial u}{\partial\nu}\right)\biggr)\,d\Gamma\ge 0\quad
\forall\, \chi= ( W, w)\in K_l.
\end{equation}





Since $K_l$ is a convex cone in $H(\Omega)$, one can substitute
$\chi= \lambda\xi$ with $\lambda\geq 0$ in (\ref{var-ner3}) and
deduce
\begin{equation}\label{toj1}
\int\limits_{\Gamma} \Biggl(\sigma_\nu(U)U\nu+ \sigma_\tau(U)
U\tau- t^\nu(u)u+m_\nu(u) \frac{\partial
u}{\partial\nu}\biggr)\,d\Gamma= 0,
\end{equation}
\begin{equation}\label{toj2}\int\limits_{\Gamma}
\Biggl(\sigma_\nu(U)W\nu+ \sigma_\tau(U)W\tau- t^\nu(u)w+m_\nu(u)
\frac{\partial w}{\partial\nu}\biggr)\,d\Gamma\ge 0,
\end{equation}
for all $\chi=(W,w)\in K_l$. Since the function $\chi=(W,w)\in
K_l$ satisfies zero boundary conditions on $\Gamma_0$, we can rewrite
\eqref{toj2} as follows
\begin{equation}\label{toj22}\int\limits_{\Gamma_1} \Biggl(\sigma_\nu(U)W\nu+
\sigma_\tau(U)W\tau - t^\nu(u)w + m_\nu(u) \frac{\partial
w}{\partial\nu}\biggr)\,d\Gamma\ge 0.
\end{equation}
Since the inequalities
\eqref{np-cond} does not depend on $W\tau$, due to arbitrariness of $W\tau$ on $\Gamma_1$, we
infer that
$$
\sigma_\tau(U)=(0,0)\quad\mbox{on}\quad \Gamma_1.
$$
Therefore, we can reduce \eqref{toj22} in the following form
\begin{equation}\label{toj23}\int\limits_{\Gamma_1} \Biggl(\sigma_\nu(U)W\nu
- t^\nu(u)w + m_\nu(u) \frac{\partial
w}{\partial\nu}\biggr)\,d\Gamma\ge 0\quad \forall\, \chi= ( W,
w)\in K_l.
\end{equation}
By choosing functions $\chi=(W,w)$ such that $W=(0,0)$,
$\frac{\partial w}{\partial\nu}=0$ on $\Gamma_1$ for
\eqref{toj23}, we get
$$
t^\nu(u)= 0\quad\mbox{on}\quad\Gamma_1.
$$
Now we can substitute test functions with the property $w=0$,
$W\nu+h\frac{\partial w}{\partial\nu}=0$ and $\frac{\partial
w}{\partial\nu}\geq 0$ on ${\Gamma_1}$

As a result, we have
\begin{equation}\label{toj24}
\int\limits_{\Gamma_1} \Biggl(\sigma_\nu(U)W\nu -
\frac{1}{h}m_\nu(u) W\nu\biggr)\,d\Gamma\ge 0.
\end{equation}
From here, since the value $W\nu\geq 0$ can be arbitrary, we get
$$
\sigma_\nu(U)-\frac{1}{h}m_\nu(u)\leq 0\quad\mbox{on}\quad
\Gamma_1.
$$
Now, substituting into \eqref{toj23} test functions satisfying
$W\nu+(h-l)\frac{\partial w}{\partial\nu}=0$, $\frac{\partial
w}{\partial\nu}\leq 0$ on ${\Gamma_1}$, we find
\begin{equation}\nonumber
\int\limits_{\Gamma_1} \Biggl(-\sigma_\nu(U)(h-l)\frac{\partial
w}{\partial\nu} + m_\nu(u) \frac{\partial
w}{\partial\nu}\biggr)\,d\Gamma\ge 0.
\end{equation}
Whence it follows that
\begin{equation}\label{toj2422}
-\sigma_\nu(U)(h-l) + m_\nu(u)\le 0\quad\mbox{on}\quad \Gamma_1.
\end{equation}
Substituting further into \eqref{toj23} $\eta=(W,w)$ such that
$w=0$, $W\nu\leq 0$, $\frac{\partial w}{\partial\nu} = 0$, it is
not difficult to establish the inequality
\begin{equation}\nonumber
\int\limits_{\Gamma_1} \sigma_\nu(U)W\nu\,d\Gamma\ge 0,
\end{equation}
which means that
\begin{equation}\label{toj2433}
\sigma_\nu(U)\le 0\quad\mbox{on}\quad \Gamma_1.
\end{equation}

Note that due to $\xi=(U,u)\in K_l$ and the following revealed
relations
$$
t^\nu(u)= 0,\quad\sigma_\nu(U)\leq 0\quad\mbox{on}\quad
\Gamma_1,
$$
$$
\sigma_\nu(U)-\frac{1}{h}m_\nu(u)\le
0,\quad-\sigma_\nu(U)(h-l) + m_\nu(u)\le 0\quad\mbox{on}\quad
\Gamma_1,
$$
the expression $\sigma_\nu(U)W\nu +m_\nu(u)\frac{\partial
w}{\partial\nu}$ is non-negative on $\Gamma_1$. Indeed, for the
subset  $\Gamma_1^+$ of  $\Gamma_1$, where $\frac{\partial w}{\partial\nu}\geq 0$ a.e. on $\Gamma_1^+$, we have
\begin{equation}\label{toj1533}
\sigma_\nu(U)W\nu +m_\nu(u)\frac{\partial
w}{\partial\nu}=$$$$=\sigma_\nu(U)\left(W\nu+h\frac{\partial
w}{\partial\nu}\right)
+\left(m_\nu(u)-h\sigma_\nu(U)\right)\frac{\partial
w}{\partial\nu}\ge 0,
\end{equation}
and for the
subset  $\Gamma_1^-$ of  $\Gamma_1$, where $\frac{\partial w}{\partial\nu}\leq 0$
a.e. on $\Gamma_1^-$, we get
\begin{equation}\label{toj15331}
\sigma_\nu(U)W\nu +m_\nu(u)\frac{\partial
w}{\partial\nu}=$$$$=\sigma_\nu(U)\left(W\nu+(h-l)\frac{\partial
w}{\partial\nu}\right) +\left(-\sigma_\nu(U)(h-l) +
m_\nu(u)\right)\frac{\partial w}{\partial\nu}\ge 0.
\end{equation}
Now we recall the relation \eqref{toj1}. Since the integrand of
\eqref{toj1} is non-negative a.e. on $\Gamma$. Therefore, we get
$$
\sigma_\nu(U)U\nu + m_\nu(u) \frac{\partial u}{\partial\nu}=0
\quad\mbox{on}\quad \Gamma_1.
$$

Conversely, in order to obtain from \eqref{e1}--\eqref{e7} the
variational inequality \eqref{varnertonk123} we multiply
\eqref{e1} by $(u-w)$ and each equality of \eqref{e2} by
corresponding $(u_i-w_i)$, $i=1,2,$ where $W=(w_1,w_2)$, $w$ such
that $\chi=(W,w)\in K_l$. Then after integrating over $\Omega$ and
summing, we get
$$
-\int\limits_{\Omega}(\sigma_{ij,j}(U)(U-W)+m_{ij,ij}(u)(w-u))dx=\int\limits_{\Omega}F(\chi-{\xi})dx.
$$
 At this point, recalling the Green formulas, we get
\begin{align}
& \int\limits_{\Omega}\Big(\sigma_{ij}( U)\,\varepsilon_{ij}(
W-U)-m_{ij}(u) (w-u),_{ij}\Big)dx-\nonumber
\\
& -\int\limits_{\Gamma} \biggl(\sigma_\nu(U) (W\nu-U\nu)+
\sigma_\tau(U) (W\tau-U\tau) \biggr)\,d\Gamma+ \nonumber
\\
& +\int\limits_{\Gamma} \biggl(t^\nu(u) (w-u)- m_\nu(u)
(\frac{\partial w}{\partial\nu}-\frac{\partial
u}{\partial\nu})\biggr)\,d\Gamma=\int\limits_{\Omega}F(\chi-{\xi})dx.\label{big1}
\end{align}
Taking into account that $\sigma_\tau(U)=(0,0)$ on $\Gamma_1$, and zero boundary conditions for
$\xi$, $\chi$ on $\Gamma_0$, we can represent the sum of
integrals over ${\Gamma}$ in the left side of \eqref{big1} as
follows
\begin{equation}\label{toj33}
I=\int\limits_{\Gamma_1} \biggl(t^\nu(u) (w-u)- m_\nu(u)
(\frac{\partial w}{\partial\nu}-\frac{\partial u}{\partial\nu}) -
\sigma_\nu(U) (W\nu-U\nu)\biggr)\,d\Gamma.
\end{equation}
Then bearing in mind the equalities $\sigma_\tau(U)=(0,0)$,
$t^\nu(u)=0$ on $\Gamma_1$, we can rewrite \eqref{toj33} as the
following sum
\begin{align}
&I= \int\limits_{\Gamma_1} \biggl(- m_\nu(u) (\frac{\partial
w}{\partial\nu}-\frac{\partial u}{\partial\nu}) - \sigma_\nu(U)
(W\nu-U\nu)\biggr)\,d\Gamma.\label{toj34}
\end{align}
 As we can see the integrand in the last integral is nonpositive because of $\chi\in K_l$ and relations \eqref{e3}--\eqref{e7}.
 It remains to note that since $I\leq 0$, the equality
 \eqref{big1} yields the variational inequality
 \eqref{varnertonk123}. The theorem is proved.
\end{proof}
\begin{remark}
We can note that in the framework of the theorem \ref{VT_thm:1}
instead of the fixed number $l$ the existence and uniqueness can
be proved for some function $l\in L^2(\Gamma_1)$ satisfying
$0<l<2h$ a.e. on $\Gamma_1$. Furthermore, bearing in mind
reasonings of the proof to the theorem \ref{ekvth} it can be seen
that same result is true for some continuous $l\in
C(\overline\Gamma_1)$ such that $0<l(x)<2h$ for all
$x\in\Gamma_1$.
\end{remark}

{\centering
\section{Passage to the limit as $l\to 2h$}
}

Passages to the limit as parameters characterizing the sizes or
relative positions of structural elements, or the distances
between objects inside solid bodies have been studied, for
example, in
\cite{Rudoy2021JAIM,Rudoy-Saj2023,Fankina-Rud2024,Fankina-Sazh2023,LazarevMZ2023}.
The dependence of solutions on perturbation of the geometry of
objects within the framework of nonlinear models of solid
mechanics is also of scientific interest, see
\cite{KovKunisch2022,Khludnev2013AAM} etc. In this section we will
show that the previously studied problem in \cite{Nikolaeva},
which corresponds to the value $l=2h$, is a limit problem for a
family of problems with different values $l\in (0,2h]$. Namely, we
consider the sequence of functions $\{l_n\}\subset L_2(\Gamma_1)$,
satisfying the properties $0\leq l_n\leq 2h$ and $l_n\to 2h$ in
the space $L_2(\Gamma_1)$.

Let us consider a family of variational problems with different
sets of admissible displacements $K_{l_n}$, $n\in \mathbf{N}$:
\begin{equation}\label{varnertonk123l}
\quad \xi_n\in K_{l_n}, \quad B(\xi_n,{\chi}-{\xi}_n)\geq
\int\limits_{\Omega}F(\chi-{\xi_n})dx\quad \forall\,\chi\in
K_{l_n}.
\end{equation}
Substituting the test function ${\chi}=(0,0,0)$ into
\eqref{varnertonk123l} we obtain the inequality
\begin{equation}\nonumber
\quad B(\xi_n,{\xi}_n)\leq \int\limits_{\Omega}F{\xi_n}dx.
\end{equation}
Hence, we get the following uniform estimate
$$
\|\xi_n\|\leq C,
$$
where $C>0$ does not depend on $n\in\mathbf{N}$. The reflexivity
of the space allows us to extract a subsequence $\xi_{n_k}$ that
weakly converges in $H(\Omega)$ to some function $\tilde\xi$. As
the next step we show that $\tilde\xi=(\tilde{U},\tilde{u})\in
K_{2h}$. Since $l_n\to 2h$ converges strongly in $L_2(\Gamma_1)$,
we can exract a subsequence $l_{n_k}$ that converges almost
everywhere on $\Gamma_1$ to $2h$. Extracting a subsequence again
if necessary, we assume that $\{\xi_{n_k}\}$ converges on
$\Gamma_1$ almost everywhere. Based on the properties of these
convergent subsequences, we can pass to the limit as $k\to\infty$
in the following inequalities:
$$
U_{n_k}\nu+h\frac{\partial u_{n_k}}{\partial\nu}\leq 0, \quad
U_{n_k}\nu+(h-l_{n_k})\frac{\partial u_{n_k}}{\partial\nu}\leq 0
\quad\mbox{a.e. on}\quad \Gamma_1.
$$
As limiting relations we obtain
$$
\tilde U\nu+h\frac{\partial \tilde{u}}{\partial\nu}\leq 0, \quad
\tilde{U}\nu-h\frac{\partial\tilde{u}}{\partial\nu}\leq 0
\quad\mbox{a.e. on}\quad \Gamma_1.
$$
That is $\tilde\xi\in K_{2h}$. Let $\tilde\eta\in K_{2h}$ be an
arbitrary test function, it is obvious that in this case
$\tilde\eta\in K_{l_n}$, for all $n\in\mathbf{N}$. Therefore,
passing to the limit as $n\to\infty$ in the inequalities
 \begin{equation}\nonumber
\quad \xi_n\in K_{l_n}, \quad B(\xi_n,{\tilde\eta}-{\xi}_n)\geq
\int\limits_{\Omega}F(\tilde\eta-{\xi_n})dx
\end{equation}
with the fixed function $\tilde\eta$, we get
 \begin{equation}\label{varnertonk123l3}
B(\tilde\xi,{\tilde\eta}-{\tilde\xi})\geq
\int\limits_{\Omega}F(\tilde\eta-{\tilde\xi})dx.
\end{equation}
Thus, due to the uniqueness of the solution to the variational
inequality, it follows from \eqref{varnertonk123l3} that
$\tilde\xi$ is a solution to the problem \eqref{minZad2}
corresponding to the value $l=2h$ for $K_{2h}$.


\bigskip
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\end{document}
