∎

11institutetext:  École Nationale des Sciences Appliquées, Ibn Zohr.
Polydisciplinary faculty of Ouarzazate, Ibn Zohr

# Hemivariational Inequality for Navier-Stokes Equations: Existence,
Dependence and Optimal Control

H. Mahdioui  S. Ben Aadi  K. Akhlil

(Received: date / Accepted: date)

###### Abstract

In this paper we study existence, dependence and optimal control results
concerning solutions to a class of hemivariational inequalities for stationary
Navier-Stokes equations but without making use of the theory of pseudo-
monotone operators. To do so, we consider a classical assumption, due to J.
Rauch, which constrains us to make a slight change on the definition of a
solution. The Rauch assumption, although it insures the existence of a
solution, does not allow the conclusion that the non-convex functional is
locally Lipschitz. Moreover, two dependence results are proved, one with
respect to changes of the boundary condition and the other with respect to the
density of external forces. The later one will be used to prove the existence
of an optimal control to the distributed parameter optimal control problem
where the control is represented by the external forces.

###### Keywords:

Navier-Stokes equations Hemivariational inequalities Galerkin method Optimal
control Nonconvex Optimization Subdifferential.

###### MSC:

35Q30 47H10 49J20 49J52 49J53

##  1 Introduction

This paper is devoted to the study of Navier-Stokes equations involving
subdifferential boundary conditions but without making use of the theory of
pseudo-monotone operators. We assume the nonslip boundary condition together
with a Clarke subdifferential relation between the pressure and the normal
components of the velocity. Navier-Stokes equations together with this type of
boundary condition model, in practice, the motion of an incompressible viscous
fluid that, when pumped into the domain, can leave through the orifices on the
boundary and, by a mechanism allowing the adjustment of the orifice’s
dimensions, the normal velocity on the boundary of the fluid is regulated to
reduce the dynamic pressure.

Let $\mathscr{O}$ be a bounded simply connected domain in $\mathbb{R}^{d}$
with connected boundary $\partial\mathscr{O}$ of class $C^{2}$ ($d=2,\,3$).
The stationary Navier-Stokes equations are described by the following system:

$\displaystyle-\nu$
$\displaystyle\sum_{j=1}^{d}\frac{\partial^{2}u_{i}}{\partial
x_{j}^{2}}+\sum_{j=1}^{d}u_{j}\frac{\partial u_{i}}{\partial
x_{j}}+\frac{\partial p}{\partial x_{i}}=f_{i},\quad i=1,2,...,d\text{ in
}\mathscr{O},$ (1) $\displaystyle\sum_{j=1}^{d}\frac{u_{j}}{x_{j}}=0\text{ in
}\mathscr{O}.$ (2)

where $u=\\{u_{i}\\}_{i=1}^{d}$ and $p$ are respectively the velocity and the
pressure of the fluid. The external forces are represented by
$f=\\{f_{i}\\}_{i=1}^{d}$ and the kinematic viscosity by the constant $\nu$.
Using the standard Lamb formulation ((girault2012finite, , Chapter I)), one
can rewrite the equations (1)-(2) in an equivalent form involving the
rotational operator and the dynamic pressure
$\widetilde{p}:=p+\frac{1}{2}|u|^{2}$. The new formulation of the problem is
then considered with the following boundary conditions:

$\widetilde{p}(z)\in\partial j(z,u_{N}(z))\text{ and }u_{\tau}=0\text{ on
}\partial\mathscr{O}$ (3)

Here $u_{N}=u.n$ and $u_{\tau}=u-u_{N}\,n$ denote the normal and the
tangential components of $u$ on the boundary $\partial\mathscr{O}$, $n$ being
the unit outward normal vector on $\partial\mathscr{O}$. The multivalued
mapping $\partial j$ denotes the Clarke subdifferential of a locally Lipschitz
function $j(x,.)$.

In some important applications, but also in our present paper, the function
$j$ can be expressed as

$j(t)=\int_{0}^{t}\Theta(s)\,ds$ (4)

for a locally bounded function $\Theta$ in $\mathbb{R}$ such that $\Theta(t\pm
0)$ exists fo all $t\in\mathbb{R}$. In this situation, we consider the
following classical assumption introduced by J. Rauch rauch1977discontinuous
to study discontinous semilinear differential equations

$\displaystyle\operatorname*{ess\,sup}_{]-\infty,-t_{0}[}\Theta(t)\leq
0\leq\displaystyle\operatorname*{ess\,inf}_{]t_{0},+\infty[}\Theta(t)$

This assumption will refer to us as Rauch assumption. Geometrically, it
describes the ultimate increase of the graph of the function $\Theta$. One of
the most important advantage of this choice of $j$ is that it simplifies
tremendously the calculation of the subdifferential $\partial j$. In fact, due
to K. C. Chang chang1981variational , the subdifferential of $j$ can be
obtained by ”filling in the gaps” in the discontinuous graph of $\Theta$.
Under Rauch assumption, the resulting weak formulation of the problem (1)-(3)
is not a variational one but leads to the so-called hemivariational
inequality. When regularized with the help of Galerkin method the problem
becomes a semilinear differential equation as discussed, in its simplest form,
in the seminal work of J. Rauch rauch1977discontinuous . This simple remark
allows us to say that the hemivariational inequality can be seen, at least in
our context, as a limit of a sequence of semilinear equations involving
nonmonotone discontinuous functions.

It is fundamental to mention that, without any additional growth hypothesis on
$j$, the Rauch assumption is sufficient to establish the existence of
solutions to (1)-(3). Unfortunately, this condition does not make the
functional

$J(u)=\int_{\partial\mathscr{O}}j(u)\,d\sigma$

locally Lipschitz or even finite on the whole space. Because of this reason,
the Aubin-Clarke result giving the relation between the subdifferential of J
and j can not be used. One strategy to encounter this problem is to modify
slightly the definition of being a solution of (1)-(3).

It is worth to mention that the theory of hemivariational inequalities was
introduced the first time by Panagiotopoulos Pana81 ; Pana83 ; Pana89 ; Pana88
; Panagiotopoulos1 , who by applying the generalized gradient of Clarke-
Rockafellar Clar75 ; Clar83 ; Rock80  studied such variational-like
expressions to discuss solutions of a class of mechanical problems involving
nonconvex and nonsmooth energy functionals. In the case of functions $j$
expressed as in (4), hemivariational inequalities was extensively studied both
in a mathematical and mechanical point of view, see Pana89 ; Pana88 ; Pana91 ;
Pana882 ; Mopana88 ; Panasta88 ; Panakol90 ; Panaban84  for more details.

The hemivariational inequalities for stationary and non-stationary Navier-
Stokes equations were considered by many researchers in recent years. For
convex functions $j(x,.)$, the problem has been studied by Chebotarev chebo92
; chebo97 ; chebo03 . The boundary condition (3), in the convex case, has been
also considered for the Boussinesq equations in chebo01  and in kono00  for
its evolution counterpart. In all these papers the considered problems was
formulated as variational inequalities. In the nonconvex case, the formulation
of (1)-(3) is no longer a variational inequality but it leads to an
hemivariational inequality. In the stationary case, the problem (1)-(3) with
nonconvex superpotentials $j$ was considered by Migórski and Ochal M05HE
Migórski mig04 , for non-Newtonian case see dudkalmig15 . In Orcliz spaces,
hemivariational inequalities for Newtonian and Non-newtonian Navier-Stokes
equations has been recently studied in migpac2018 , migpac19 . Hemivariational
inequalities for generalized Newtonian fluids are recently extensively studied
see dudkalmig17  and references therein, see also migdud18  for evolutionary
Oseen model for generalized Newtonian fluid. For an equilibrium problem
approach to hemivariational inequalities for Navier-Stokes equations we refer
to homancha19  and aadi18 . For different aspects about nonsmooth optimization
in the context of Navier-Stokes system we refer to fang2019finite ; fang16 ;
kalita2015large ; kalita2014attractors ; M19F ; M04he ; MH ; M19F ;
migorski2018evolutionary ; SZ17Ev .

The goal of this paper is threefold. We aim to

  * (1)

show the existence of weak solutions to the hemivariational inequality
corresponding to the problem (1)-(3),

  * (2)

prove a dependence result of solutions with respect to the hemivariational
part and to the density of the external forces,

  * (3)

formulate and study the distributed parameter optimal control where the
control is represented by the density of the external forces.

The paper is organized as follows. In section 2 we give the formulation of the
stationary Navier-Stokes equations with a subdifferential boundary condition
as an hemivariational inequality. We give a slight different definition to
this problem to have a solution. This definition is so formulated to overcome
the problem of the integrability and the local Lipschitzianity of $J$. We give
in the end of this section an example illustrating the practicality of this
model. In section 3, we first regularize the problem and then use Galerkin
approximation. The existence of solutions to the regularized finite-
dimentional problem is proven by using the Brouwer’s fixed point theorem. In
addition a weak precompactness result is obtained by the Dunford-Pettis
theorem. In section 4, we prove the existence of a solution to our problem and
we discuss why, in our opinion the question of uniqueness is difficult to
answer even with a monotonicity assumption similar to the one in M05HE . In
section 5, we prove the dependence of the solution with respect to changes of
the boundary condition by using an Aubin-Frankowski theorem. Finally in
section 6, we first prove the dependence of solutions on external forces and
use the result to prove the existence of an optimal control to a distributed
parameter optimal control problem formulated by considering the external
forces as controls.

##  2 Problem statement

Let $\mathscr{O}$ a bounded simply connected domain in $\mathbb{R}^{d}$ with
connected boundary $\partial\mathscr{O}$ of class $C^{2}$ ($d=2,3$). We
consider the following Navier-Stokes system:

$-\nu\Delta u+(u.\nabla u)+\nabla p=f,\quad{\rm div}u=0\,\mbox{ in
}\mathscr{O}.$ (5)

This system describes the flow of incompressible viscous fluid in the domain
$\Omega$, subjected to the external forces $f=\\{f_{i}\\}_{i=1}^{d}$. The
unknown are the velocity $u=\\{u_{i}\\}_{i=1}^{d}$ and the pressure $p$ of the
fluid . The positive constant $\nu$ is the kinematic viscosity of the fluid
($\nu=\frac{1}{Re}$ where $Re$ stands for the Reynolds number). The nonlinear
term $(u.\nabla)u$, called the convective term, is the symbolic notation of
the vector $\sum_{j=1}^{d}u_{j}\frac{\partial u_{i}}{\partial x_{j}}$. The
second condition, i.e ${\rm div}\,u=0$, expresses the fact that the fluid is
incompressible.

In order to give a variational-like formulation of (5), we will use the
approach developed by Chebotarev chebo92 ; chebo97 ; chebo01 , Konovalova
kono00  and Alekseev and Smishliaev alekseev2001solvability . By means of
standard Lamb formulation (girault2012finite, , Chapter I), one obtain the
following identities

$-\Delta u=\rm rot\,\rm rot\,\ u-\nabla\,{\rm div}\,u,$ (6)

$(u.\nabla)u=\rm rot\,u\times u-\frac{1}{2}\nabla(u.u).$ (7)

Using the expressions (6)-(7) and the incompressibility condition, the
equation (5) can be reformulated as follows

$\nu\rm rot\,\rm rot\,u+\rm rot\,u\times u+\nabla\widetilde{p}=f,$ $\quad{\rm
div}u=0\,\mbox{ in }\mathscr{O}.$

where $\widetilde{p}=p+\frac{1}{2}\left|u\right|^{2}$ is the total head of the
fluid, or ”total pressure” .

We suppose that, on the boundary $\partial\mathscr{O}$, the tangential
components of the velocity vector are known and without loss of generality we
put them equal to zero (the nonslip condition):

$u_{\tau}:=u-u_{N}\,n=0\text{ on }\partial\mathscr{O},$ (8)

where $n$ is the unit outward normal on the boundary $\partial\mathscr{O}$ and
$u_{N}=u.n$ denotes the normal component of the vector $u$. Moreover, we
assume the following subdifferential boundary condition:

$\widetilde{p}(z)\in\partial j(z,u_{N}(z))\text{ for }z\in\partial\mathscr{O}$
(9)

where $\partial j(\xi)$ is the generalized gradient of $j$ at $\xi$ and is
given by

$\partial j(\xi)=\\{\xi^{*}\in
V^{*}:j^{0}(\xi;h)\geq\langle\xi^{*},h\rangle_{V^{*}\times V}\text{ for all
}h\in V\\}$,

$j^{0}(\xi;h)$ is the generalized derivative of a locally Lipschitz function
$j$ at $\xi\in V$ in the direction $h\in V$ defined by:

$j^{0}(\xi;h)=\displaystyle\limsup_{\nu\rightarrow\xi,\lambda\downarrow
0}\frac{j(\nu+\lambda h)-j(\nu)}{\lambda}.$

In order to give the weak formulation of the problem (8)-(9), we introduce the
following functional spaces:

$\mathcal{W}=\\{u\in\mathcal{C}^{\infty}(\mathscr{O};\mathbb{R}^{d}):{\rm
div}u=0\text{ in }\mathscr{O},u_{\tau}=0\text{ on }\partial\mathscr{O}\\}.$
(10)

Let us denote by $\mathcal{V}$ and $\mathcal{H}$ the closure of $\mathcal{W}$
in the norms of $H^{1}(\mathscr{O};\mathbb{R}^{d})$ and
$L^{2}(\mathscr{O};\mathbb{R}^{d})$, respectively. We define the operators
$\mathscr{A}:\mathcal{V}\to\mathcal{V}^{*}$ and
$\mathscr{B}[.]:\mathcal{V}\times\mathcal{V}\to\mathcal{V}^{*}$ with
$\mathscr{B}[u]=\mathscr{B}(u,u)$ by

$\displaystyle\langle\mathscr{A}u,v\rangle=$
$\displaystyle\nu\int_{\mathscr{O}}\rm rot\,u.\rm rot\,v\,{\rm d}\lambda(x)$
$\displaystyle\langle\mathscr{B}(u,v),w\rangle=$
$\displaystyle\int_{\mathscr{O}}(\rm rot\,u\times v).w\,{\rm d}\lambda(x)$

for $u,v,w\in\mathcal{V}$.

We multiply the equation of motion (8) by $v\in\mathcal{V}$ and apply the
Gauss divergence theorem, we have:

$\langle\mathscr{A}u+\mathscr{B}[u],v\rangle+\displaystyle\int_{\partial\mathscr{O}}\widetilde{p}(z)\,v_{N}(z)\,d\sigma(z)=\langle
f,v\rangle,$ (11)

From the relation (9), by using the definition of the Clarke subdifferential,
we have

$\displaystyle\int_{\partial\mathscr{O}}\widetilde{p}(z)\,v_{N}(z)\,d\sigma(z)\leq\int_{\partial\mathscr{O}}j^{0}(z,u_{N}(z);v_{N}(z))\,d\sigma(z).$
(12)

The two relation (11)-(12) yield to the following weak formulation

$\text{(HVI)}\left\\{\begin{array}[]{ll}\text{ Find }u\in\mathcal{V}\text{
such that }\\\
\langle\mathscr{A}u+\mathscr{B}[u],v\rangle+\displaystyle\int_{\partial\mathscr{O}}j^{0}(z,u_{N}(z);v_{N}(z))\,d\sigma(z)\geq\langle
f,v\rangle,\text{ for every }v\in\mathcal{V},\par\end{array}\right.$

the equation above is called an hemivariational inequality.

We have already mentioned in the introduction that the Rauch assumption is not
sufficient to make the functional $J$ locally Lipchitz or even finite in the
whole space $\mathcal{V}$. Because of this reason, a slight modified
definition of being a solution should be adopted. Define the space
$\widetilde{\mathcal{V}}$ as follows:

$\widetilde{\mathcal{V}}=\\{u\in\mathcal{V}:\,\,u_{N}=\gamma(u).n\in
L^{\infty}(\partial\mathscr{O};\mathbb{R})\\}$

where $\gamma$ is the trace operator from $\mathcal{V}$ in
$L^{2}(\partial\mathscr{O};\mathbb{R}^{d})$. It is easy to prove that
$\widetilde{\mathcal{V}}$ is dense in $\mathcal{V}$ for the weak topology.
Now, we are able to give what we mean by a solution of the problem $(HVI)$.

###### Definition 1

A pair $(u,\xi)\in\mathcal{V}\times L^{1}(\partial\mathscr{O},\,\mathbb{R})$
is said to be solution of (HVI) if the following two relations are satisfied

$\left\\{\begin{array}[]{ll}\langle\mathscr{A}u+\mathscr{B}[u],v\rangle+\displaystyle\int_{\partial\mathscr{O}}\xi(z)\,v_{N}(z)\,d\sigma(z)=\langle
f,v\rangle,\text{ for every }v\in\widetilde{\mathcal{V}}\\\ {}\\\
\xi(z)\in\partial j(z;u_{N}(z)),\quad\text{for a.e.
}z\in\partial\mathscr{O}.\par\end{array}\right.$ (13)

Let us introduce the following operator
$\mathscr{E}:L^{1}(\partial\mathscr{O},\,\mathbb{R})\rightarrow\widetilde{\mathcal{V}}^{*}$
defined by

$\langle\mathscr{E}(\xi),\,v\rangle_{\mathcal{V}}=\int_{\partial\mathscr{O}}\xi(z)\,v_{N}(z)\,d\sigma(z),\quad\forall
v\in\widetilde{\mathcal{V}}.$

In order to justify this definition, let us observe that for any $\xi\in
L^{1}(\partial\mathscr{O},\,\mathbb{R}^{d})$ there may correspond no more that
one linear continuous functional $\mathscr{E}(\xi)\in\mathcal{V}^{*}$ with the
property that

$\langle\mathscr{E}(\xi),\,v\rangle_{\mathcal{V}}=\int_{\partial\mathscr{O}}\xi(z)\,v_{N}(z)\,d\sigma(z),\quad\forall
v\in\widetilde{\mathcal{V}}$

This fact is a consequence of the density of the space
$\widetilde{\mathcal{V}}$ in $\mathcal{V}$. Accordingly, if
$\mathscr{E}(\xi)\in\mathcal{V}^{*}$, then for each $v\in\mathcal{V}$ the
value $\langle\mathscr{E}(\xi),\,v\rangle_{\mathcal{V}}$ is determined
uniquely by $\xi$. The equation (13) can be written then as

$\langle\mathscr{A}u+\mathscr{B}[u]-f,v\rangle+\langle\mathscr{E}(\xi),\,v\rangle_{\mathcal{V}}=0,$

or more compactly as

$\Lambda(u,\xi)=f\text{ for all }v\in\mathcal{V}.$

For simplicity and if no ambiguity occurs, we write always our problem as

$\langle\mathscr{A}u+\mathscr{B}[u],v\rangle+\int_{\partial\mathscr{O}}\xi(z)\,v_{N}(z)\,d\sigma(z)=\langle
f,v\rangle$

for every $v\in\widetilde{\mathcal{V}}$.

The above procedure has been extensively used by Naniewicz nani94 ; nani95  to
study hemivariational inequalities with directional growth conditions. Such
non standard growth conditions give arise to problems involving functionals
$J$ which are not locally Liptschitz in the whole space.

In the following remark we will highlight the fact that there is an
equivalence, in some sense, between the Navier-Stokes system (8)-(9) and the
hemivariational inequality (13).

###### Remark 1

It’s clear that the hemivariational inequality (13) can be derived from
(8)-(9). Now we show that, in some sense, the converse also holds true. Let
$(u,\xi)\in\mathcal{V}\times L^{1}(\partial\mathscr{O};\mathbb{R})$ be a
solution to the problem (13), then by construction of $\mathcal{V}$, we have
${\rm div}u=0$ and $u_{\tau}=0$ on $\partial\mathscr{O}$. Now, let us take an
arbitrary element $w$ in $\mathcal{V}\cap
C^{\infty}_{0}(\mathscr{O};\mathbb{R}^{d})$, then also $w_{N}=0$ and one
obtains that $\langle\mathscr{A}u+\mathscr{B}[u],w\rangle=\langle f,w\rangle$.
Note $\tilde{f}_{u}=f-\mathscr{A}u-\mathscr{B}[u]$, and by the density of
$\mathcal{V}\cap C^{\infty}_{0}(\mathscr{O};\mathbb{R}^{d})$ in $\mathcal{V}$
we can write

$\langle\tilde{f}_{u},w\rangle=0,\quad\text{for all }w\in\mathcal{V}$

From Proposition 1.1 in Chapter I of Temam Temam  it follows that there exist
a distribution $h$ such that $\tilde{f}_{u}=\nabla h$. As a consequence we
have

$\mathscr{A}u+\mathscr{B}[u]+\nabla h=f$

which, by multiplying by $v$ and integrating by parts over $\mathscr{O}$,
implies

$\langle\mathscr{A}u+\mathscr{B}[u],v\rangle+\int_{\partial\mathscr{O}}h(z)v_{N}(z)d\sigma(z)=\langle
f,v\rangle$

Comparing this equality with the one in (13) entails

$\int_{\partial\mathscr{O}}[h(z)-\xi(z)]v_{N}(z)]d\sigma(z)=0,\quad\,\forall\,v\in
V.$

As $v$ is arbitrary, one can conclude that $h\in
L^{1}(\partial\mathscr{O},\mathbb{R})$ and $h=\xi\in\partial j(z,u(z))$ a.e.
This shows the subdifferential condition (9).

The following example shows the practicality of our framework

###### Example 1

The subdifferential condition appearing in the problem (8)-(9) refers, in
practice, to an artificial behaviour of the flow of the fluid through the
boundary $\partial\mathscr{O}$. The fluid pumped into $\mathscr{O}$ can leave
the domain through the orifices on the boundary. By a mechanism allowing the
adjustment of the orifices dimensions, the normal velocity on the boundary of
the fluid is regulated to reduce the dynamic pressure on
$\partial\mathscr{O}$.

We consider the boundary condition (9) by given real numbers $a$ and $b$ such
that $0\leq a\leq b$. The locally Lipschitz function
$j:\partial\mathscr{O}\times\mathbb{R}\rightarrow\mathbb{R}$ is defined by :

$j(x,s)=\left\\{\begin{array}[]{lll}\frac{\widetilde{p}}{2(b-a)}(s-a)^{2}&\text{
if }0\leq s<b;\\\ \frac{\widetilde{p}}{2}(b-a)&\text{ if }s\geq
b.\end{array}\right.$

For $x\in\partial\mathscr{O}$, we have:

$\partial
j(x,s)=\left\\{\begin{array}[]{lll}\frac{\widetilde{p}}{b-a}(s-a)&\text{ if
}0\leq s<b;\\\ \left[0,\widetilde{p}\right]&\text{ if }s=b;\\\ 0&\text{ if
}s>b.\end{array}\right.$

The condition $u_{N}>0$ refers to the fact that there is a flow through
$\partial\mathscr{O}$. The boundary condition $u_{N}=0$ means that there is no
flow across the boundary. If $u_{N}\in(0,b)$, the orifices allow the fluid to
infiltrate outside the tube. When the velocity of the fluid increases, the
total pressure is a linear function which takes its values between $0$ and
$\widetilde{p}$. If $u_{N}$ reaches the value $b$, a mechanism opens the holes
more widely and allows the fluid to pass to the outside. As a result, the
pressure drops to $0$. Finally, its worth to mention that the dependence of
$j$ on the space variable traduces the fact that the subdifferential boundary
condition can possibly take different values on the parts of
$\partial\mathscr{O}$. For other examples, see (M05HE, , Example 18) and
(M07HE, , Remark 2).

##  3 Regularized Problem

In the forthcoming study of the problem (13) we restrict ourselves to
superpotentials $j$ which are independent of $z$.

Let $\Theta\in L^{\infty}_{loc}(\mathbb{R})$. For $\mu>0$ and
$t\in\mathbb{R}$, we define:

$\underline{\Theta}_{\mu}(t)=\displaystyle\operatorname*{ess\,inf}_{\left|t-s\right|\leq\mu}\Theta(s),\quad\quad\overline{\Theta}_{\mu}(t)=\displaystyle\operatorname*{ess\,sup}_{\left|t-s\right|\leq\mu}\Theta(s).$

For a fixed $t\in\mathbb{R}$, the functions $\underline{\Theta}_{\mu}$ and
$\overline{\Theta}_{\mu}$ are decreasing and increasing in $\mu$,
respectively. Let

$\displaystyle\underline{\Theta}(t)=\lim_{\mu\to
0^{+}}\underline{\Theta}_{\mu}(t),\quad\quad\overline{\Theta}(t)=\lim_{\mu\to
0^{+}}\overline{\Theta}_{\mu}(t),$

and let $\widehat{\Theta}(t):\mathbb{R}\to 2^{\mathbb{R}}$ be a multifunction
defined by

$\widehat{\Theta}(t)=\left[\underline{\Theta}(t),\overline{\Theta}(t)\right]$

From Chang chang1981variational  we know that a locally Lipschitz function
$j:\mathbb{R}\to\mathbb{R}$ can be determined up to an additive constant by
the relation

$j(t)=\int_{0}^{t}\Theta(s)\,d\lambda(s)$

such that $\partial j(t)\subset\hat{\Theta}(t)$ for all $t\in\mathbb{R}$. If
moreover, the limits $\Theta(t\pm 0)$ exist for every $t\in\mathbb{R}$, then
$\partial j(t)=\widehat{\Theta}(t)$.

###### Remark 2

Here for the existence theory an abstract regularization procedure by
convolution is used. Such a regularization procedure can be modified in order
to get approximations of locally Lipschitz function that can be treated
numerically; see ovg14 .

Now, we consider the mollifier

$\mathfrak{h}\in C_{0}^{\infty}(-1,1),\mathfrak{h}\geq 0\text{ with
}\displaystyle\int_{-\infty}^{+\infty}\mathfrak{h}(s)\,{\rm d}\lambda(s)=1$

and let

$\Theta_{\varepsilon}=\mathfrak{h}_{\varepsilon}*\Theta\text{ with
}\mathfrak{h}_{\varepsilon}(s)=\frac{1}{\varepsilon}\mathfrak{h}(\frac{s}{\varepsilon})$

where $*$ denotes the convolution product and $0<\varepsilon<\varepsilon_{0}$.
Thus the regularized problem becomes:

$(\mathscr{P}_{\varepsilon})\left\\{\begin{array}[]{ll}\text{ Find
}u\in\mathcal{V}\text{ such that: for all }v\in\widetilde{\mathcal{V}}\\\
\displaystyle\langle\mathscr{A}u+\mathscr{B}[u],v\rangle+\int_{\partial{\mathscr{O}}}\Theta_{\varepsilon}(u_{N})\,v_{N}\,d\sigma=\langle
f,v\rangle.\end{array}\right.$

Now and in order to define the corresponding finite dimensional problem
$(\mathscr{P}_{\varepsilon})$, we consider a Galerkin basis of
$\widetilde{\mathcal{V}}$ and let $\mathcal{V}_{n}$ be the resulting
n-dimensional subspace. This problem reads:

$(\mathscr{P}^{n}_{\varepsilon})\left\\{\begin{array}[]{ll}\text{ Find
}u^{\varepsilon_{n}}\in\mathcal{V}_{n}\text{ such that: for all
}v\in\mathcal{V}_{n}\\\
\displaystyle\langle\mathscr{A}u^{\varepsilon_{n}}+\mathscr{B}[u^{\varepsilon_{n}}],v\rangle+\int_{\partial{\mathscr{O}}}\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})\,v_{N}d\sigma=\langle
f,v\rangle.\end{array}\right.$

For the existence of solutions we will need the following hypothesis
$H(\Theta)$:

  * (1)

(Chang assumption) $\Theta\in L_{loc}^{\infty}(\mathbb{R}),\,\Theta(t\pm
0)\text{ exists for any }t\in\mathbb{R}.$

  * (2)

(Rauch assumption) there is $t_{0}\in\mathbb{R}$ such that:

$\displaystyle\operatorname*{ess\,sup}_{]-\infty,-t_{0}[}\Theta(t)\leq
0\leq\displaystyle\operatorname*{ess\,inf}_{]t_{0},+\infty[}\Theta(t)$

###### Remark 3

If one assume more generally that

$\displaystyle\operatorname*{ess\,sup}_{]-\infty,-t_{0}[}\Theta(t)\leq\alpha\leq\displaystyle\operatorname*{ess\,inf}_{]t_{0},+\infty[}\Theta(t)$
(14)

for some real number $\alpha$, it is possible to come back to the situation
where the Rauch assumption is imposed by simply replacing $\Theta$ by
$\Theta-\alpha$ and $f$ by $f-\alpha$. In fact if we assume (14) the problem
(13) is equivalent to

$\left\\{\begin{array}[]{ll}\langle\mathscr{A}u+\mathscr{B}[u],v\rangle+\displaystyle\int_{\partial\mathscr{O}}(\xi(z)-\alpha)\,v_{N}(z)\,{\rm
d}\sigma(z)=\langle f-\alpha,v\rangle,\text{ for every
}v\in\widetilde{\mathcal{V}}\\\ {}\\\ \xi(z)-\alpha\in\partial
j(z;u_{N}(z))-\alpha=\partial\left[j(z;u_{N}(z))-\alpha
u_{N}(z)\right],\quad\text{for a.e.
}z\in\partial\mathscr{O}\par\end{array}\right.$

Let us note $\tilde{f}=f-\alpha$ and $\widetilde{\Theta}=\Theta-\alpha$. Thus,
$\tilde{j}(.)=\int_{0}^{.}\widetilde{\Theta}(s)\,{\rm d}\lambda(s)$ and the
problem (13) under (14) become to find $(u,\eta)$ such that

$\left\\{\begin{array}[]{ll}\langle\mathscr{A}u+\mathscr{B}[u],v\rangle+\displaystyle\int_{\partial\mathscr{O}}\eta\,v_{N}(z)\,d\sigma(z)=\langle\tilde{f},v\rangle,\text{
for every }v\in\widetilde{\mathcal{V}}\\\ {}\\\
\eta\in\partial\tilde{j}(z;u_{N}(z)),\quad\text{for a.e.
}z\in\partial\mathscr{O}\par\end{array}\right.$

where $\widetilde{\Theta}$ fulfill the Rauch assumption. This means that,
without loss of generality, we can always consider the initial Rauch
assumption (with $\alpha=0$).

###### Lemma 1

Suppose that $H(\Theta)$ holds. Then we can determine $a,b>0$ such that for
every $u\in\mathcal{V}$

$\int_{\partial{\mathscr{O}}}\Theta_{\varepsilon}(u_{N}(z))u_{N}(z)\,{\rm
d}\sigma(z)\geq-ab\,\sigma(\partial{\mathscr{O}}).$ (15)

###### Proof

From the hypothesis $H(\Theta)$ we obtain that

$\Theta_{\varepsilon}(\xi)=(\mathfrak{h}\star\Theta)(\xi)=\int_{-\varepsilon}^{+\varepsilon}\Theta(\xi-t)\mathfrak{h}_{\varepsilon}(t){\rm
d}\lambda(t)\leq\displaystyle\operatorname*{ess\,sup}_{\left|t\right|\leq\varepsilon}\Theta(\xi-t)$

and analogously

$\displaystyle\operatorname*{ess\,inf}_{\left|t\right|\leq\varepsilon}\Theta(\xi-t)\leq\Theta_{\varepsilon}(\xi)$

In the above two inequalities we set
$x=\xi-t,\left|x-\xi\right|\leq\varepsilon$ and enlarge the bounds for
$-\infty<x\leq\varepsilon+\xi$ and $\xi-\varepsilon\leq x<\infty$,
respectively. Then the supremum and the infimum for $\xi\in(-\infty,-\xi_{1})$
and $\xi\in(+\xi_{1},+\infty)$, respectively are formed and the bounds are
enlarged by replacing $\varepsilon+\xi$ by $1-\xi_{1}$ and $\xi-\varepsilon$
by $\xi_{1}-1(\varepsilon<1)$; we obtain from $\text{H}(\Theta)$ that there
exists $\xi\in\mathbb{R}$ such that

$\sup_{(-\infty,-\xi)}\Theta_{\varepsilon}(\xi_{1})\leq
0\leq\inf_{(+\xi,+\infty)}\Theta_{\varepsilon}(\xi_{1}).$

Thus we can determine $a>0$ and $b>0$ such that

$\left\\{\begin{array}[]{ll}\Theta_{\varepsilon}(\xi)\geq 0,&\text{ if
}\xi>a,\\\ \Theta_{\varepsilon}(\xi)\leq 0,&\text{ if }\xi<-a,\\\
\left|\Theta_{\varepsilon}(\xi)\right|\leq b,&\text{ if }\left|\xi\right|\leq
a,\end{array}\right.$

and may write

$\displaystyle\displaystyle\int_{\partial{\mathscr{O}}}\Theta_{\varepsilon_{n}}(u_{N}(z))u_{N}(z)\,{\rm
d}\sigma(z)=$
$\displaystyle\displaystyle\int_{\left|u_{N}(z)\right|>a}\Theta_{\varepsilon_{n}}(u_{N}(z))u_{N}(z)\,{\rm
d}\,\sigma(z)$ $\displaystyle\qquad+\int_{\left|u_{N}(z)\right|\leq
a}\Theta_{\varepsilon_{n}}(u_{N}(z))u_{N}(z)\,{\rm d}\,\sigma$
$\displaystyle\geq\int_{\left|u_{N}(z)\right|\leq
a}\Theta_{\varepsilon_{n}}(u_{N}(z))u_{N}(z)\,{\rm d}\,\sigma$
$\displaystyle\geq-ab\,\sigma(\partial{\mathscr{O}}).$

###### Lemma 2

Suppose $\Theta\in L^{\infty}_{\mathrm{loc}}(\mathbb{R})$. Then for every
$t,t^{\prime}\in\mathbb{R}$ we have

$|\Theta_{\varepsilon}(t)-\Theta_{\varepsilon}(t^{\prime})|\leq\operatorname*{ess\,sup}_{s\in]t\wedge
t^{\prime}-\varepsilon,t\vee
t^{\prime}+\varepsilon[}|\Theta(s)|\,\|\mathfrak{h}^{\prime}\|_{\infty}\,|t-t^{\prime}|\,\left(t\vee
t^{\prime}-t\wedge t^{\prime}+2\varepsilon\right)$

where
$\|\mathfrak{h}^{\prime}\|_{\infty}:=\|\mathfrak{h}^{\prime}\|_{C_{c}^{\infty}(]-\varepsilon,\varepsilon[)}$.

###### Proof

Let $t,\,t^{\prime}\in\mathbb{R}$, we have the following estimates

$\displaystyle|\Theta_{\varepsilon}(t)-\Theta_{\varepsilon}(t^{\prime})|\leq$
$\displaystyle\displaystyle\int_{\mathbb{R}}|\Theta(s)|\,|\mathfrak{h}_{\varepsilon}(t-s)-\mathfrak{h}_{\varepsilon}(t^{\prime}-s)|{\rm
d}\lambda(s)$ $\displaystyle\leq\int_{t\wedge t^{\prime}-\varepsilon}^{t\vee
t^{\prime}+\varepsilon}|\Theta(s)|\,|\mathfrak{h}_{\varepsilon}(t-s)-\mathfrak{h}_{\varepsilon}(t^{\prime}-s)|\,{\rm
d}\lambda(s)$ $\displaystyle\leq\operatorname*{ess\,sup}_{s\in]t\wedge
t^{\prime}-\varepsilon,t\vee
t^{\prime}+\varepsilon[}|\Theta(s)|\,\int_{t\wedge
t^{\prime}-\varepsilon}^{t\vee
t^{\prime}+\varepsilon}|\mathfrak{h}_{\varepsilon}(t-s)-\mathfrak{h}_{\varepsilon}(t^{\prime}-s)|\,{\rm
d}\lambda(s).$

By mean value theorem, there exists $c\in]t\wedge t^{\prime}-\varepsilon,t\vee
t^{\prime}+\varepsilon[$ such that

$\mathfrak{h}_{\varepsilon}(t-s)-\mathfrak{h}_{\varepsilon}(t^{\prime}-s)=\mathfrak{h}^{\prime}_{\varepsilon}(c)(t-t^{\prime})$

which give

$\displaystyle|\Theta_{\varepsilon}(t)-\Theta_{\varepsilon}(t^{\prime})|$
$\displaystyle\leq\operatorname*{ess\,sup}_{s\in]t\wedge
t^{\prime}-\varepsilon,t\vee
t^{\prime}+\varepsilon[}|\Theta(s)|\,\int_{t\wedge
t^{\prime}-\varepsilon}^{t\vee
t^{\prime}+\varepsilon}|\mathfrak{h}^{\prime}_{\varepsilon}(c)|\,|t-t^{\prime}|\,{\rm
d}\lambda(s)$ $\displaystyle\leq\operatorname*{ess\,sup}_{s\in]t\wedge
t^{\prime}-\varepsilon,t\vee
t^{\prime}+\varepsilon[}|\Theta(s)|\,\|\mathfrak{h}^{\prime}_{\varepsilon}\|_{\infty}\,|t-t^{\prime}|\,\left(t\vee
t^{\prime}-t\wedge t^{\prime}+2\varepsilon\right)$

###### Lemma 3

The operator
$\mathscr{K}_{\varepsilon}:\mathcal{V}_{n}\rightarrow\mathcal{V}_{n}^{\star}$
given by

$\langle\mathscr{K}_{\varepsilon}(v),w\rangle=\int_{\partial\mathscr{O}}\Theta_{\varepsilon}(v_{N})\,w_{N}\,{\rm
d}\,\sigma,\quad u,\,w\in\mathcal{V}_{n}$

is weakly continuous.

###### Proof

Let $(u^{k})_{k}$ be a sequence converging weakly to $u$ in $\mathcal{V}$. By
Rellich’s compactness criterion we may pass to a subsequence, which we still
denote as $u^{k}$, so that $u_{N}^{k}=u^{k}.n$ converges to $u_{N}=u.n$ in
$L^{2}(\partial\mathscr{O})$ and then almost everywhere on
$\partial\mathscr{O}$. It then follows that also $u_{N}^{k}\vee u_{N}$ and
$u_{N}^{k}\wedge u_{N}$ converge to $u_{N}$. Thus by applying Egoroff’s
theorem we can find that for any $\alpha>0$ we can determine
$\Gamma\subset\partial\mathscr{O}$ with $\sigma(\Gamma)<\alpha$ such that
$u^{k}_{N}$, $u_{N}^{k}\vee u_{N}$ and $u_{N}^{k}\wedge u_{N}$ converge to
$u_{N}$ uniformly on $\partial\mathscr{O}\setminus\Gamma$.

Let $\mu>0$, there exists $n_{0}$ such that for all $k\geq n_{0}$

$|u_{N}^{k}\vee u_{N}-u_{N}|<\frac{\mu}{2},$

and

$|u_{N}^{k}\wedge u_{N}-u_{N}|<\frac{\mu}{2}.$

If $s\in\mathbb{R}$ satisfies $u_{N}^{k}\wedge
u_{N}-\varepsilon<s<u_{N}^{k}\vee u_{N}+\varepsilon$, then it satisfies
$-\frac{\mu}{2}+u_{N}-\varepsilon<s<\frac{\mu}{2}+u_{N}+\varepsilon$. We
choose $\mu$ such that $\frac{\mu}{2}>\varepsilon_{0}>\varepsilon$. Then
$\varepsilon+\frac{\mu}{2}<\mu$ and $s$ satisfies $|s-u_{N}|<\mu$ which
implies that $|s-\|u_{N}\||<\mu$ with
$\|u_{N}\|=\|u_{N}\|_{L^{\infty}(\partial\mathscr{O}\setminus\Gamma)}$

By Lemma 2, we have for every $z\in\partial\mathscr{O}\setminus\Gamma$:

$\displaystyle|\Theta_{\varepsilon}(u_{N}^{k})-\Theta_{\varepsilon}(u_{N})|$
$\displaystyle\leq\operatorname*{ess\,sup}_{s\in]u_{N}^{k}\wedge
u_{N}-\varepsilon,u_{N}^{k}\vee
u_{N}+\varepsilon[}|\Theta(s)|\,\|\mathfrak{h}^{\prime}\|_{\infty}\,|u_{N}^{k}-u_{N}|\,$
$\displaystyle\qquad\qquad\times\left(u_{N}^{k}\vee u_{N}-u_{N}^{k}\wedge
u_{N}+2\varepsilon\right)$
$\displaystyle\leq\left(\mu+2\varepsilon\right)\operatorname*{ess\,sup}_{|s-\|u_{N}\||<\mu}|\Theta(s)|\,\|\mathfrak{h}^{\prime}\|_{\infty}\,|u_{N}^{k}-u_{N}|$
$\displaystyle\leq
2\mu\|\Theta\|_{L^{\infty}(]\|u_{N}\|-\mu,\|u_{N}\|+\mu[)}\,\|\mathfrak{h}^{\prime}\|_{\infty}\,|u_{N}^{k}-u_{N}|$
$\displaystyle\leq K_{0}\,|u_{N}^{k}-u_{N}|$

where
$K_{0}=4k_{0}\varepsilon_{0}\|\Theta\|_{L^{\infty}(]\|u_{N}\|-2k_{0}\varepsilon_{0},\|u_{N}\|+2k_{0}\varepsilon_{0}[)}\,\|\mathfrak{h}^{\prime}\|_{C_{c}^{\infty}(]-\varepsilon_{0},\varepsilon_{0}[)}$
for some $k_{0}\in]0,1[$.

It follows that for all $v\in\mathcal{V}_{n}$

$\displaystyle\int_{\partial\mathscr{O}\setminus\Gamma}\left|\Theta_{\varepsilon}(u_{N}^{k})-\Theta_{\varepsilon}(u_{N})\right|\,|v_{N}|\,{\rm
d}\sigma(z)$ $\displaystyle\leq
K_{0}\,\int_{\partial\mathscr{O}\setminus\Gamma}\left|u_{N}^{k}-u_{N}\right|\,|v_{N}|\,{\rm
d}\sigma(z)$ $\displaystyle\leq
K_{0}\,\sigma(\partial\mathscr{O}\setminus\Gamma)\|v_{N}\|_{L^{\infty}(\partial\mathscr{O}\setminus\Gamma)}\|u_{N}^{k}-u_{N}\|_{L^{\infty}(\partial\mathscr{O}\setminus\Gamma)}$

As $\alpha$ is arbitrary we then conclude

$\int_{\partial\mathscr{O}}\left|\Theta_{\varepsilon}(u_{N}^{k})-\Theta_{\varepsilon}(u_{N})\right|\,|v_{N}|\,{\rm
d}\sigma(z)\leq
K_{1}\,\|u_{N}^{k}-u_{N}\|_{L^{\infty}(\partial\mathscr{O})}\|v_{N}\|_{L^{\infty}(\partial\mathscr{O})}$

where $K_{1}:=K_{0}\,\sigma(\partial\mathscr{O})$. It then follows that

$\left|\langle\mathscr{K}_{\varepsilon}(u^{k})-\mathscr{K}_{\varepsilon}(u),v\rangle\right|\leq
K_{1}\,\|u_{N}^{k}-u_{N}\|_{L^{\infty}(\partial\mathscr{O})}\|v_{N}\|_{L^{\infty}(\partial\mathscr{O})}$

Which complete the proof.

###### Proposition 1

Suppose that $H(\Theta)$ is satisfied. Then the regularized problem
$(\mathscr{P}^{n}_{\varepsilon})$ has at least one solution
$u^{\varepsilon_{n}}\in\mathcal{V}_{n}$. Moreover the sequence
$(u^{\varepsilon_{n}})_{n}$ is uniformly bounded on $\mathcal{V}$.

###### Proof

Let $\mathfrak{i}_{n}:\mathcal{V}_{n}\rightarrow\widetilde{\mathcal{V}}$ be
the inclusion mapping of $\mathcal{V}_{n}$ into $\widetilde{\mathcal{V}}$ and
$\mathfrak{i}_{n}^{\star}$ the dual projection mapping of
$\widetilde{\mathcal{V}}^{\star}$ into $\mathcal{V}_{n}^{\star}$. Define
$\mathscr{A}_{n}=\mathfrak{i}_{n}^{\star}\mathscr{A}\,\mathfrak{i}_{n}$,
$\mathscr{B}_{n}[.]=\mathfrak{i}_{n}^{\star}\mathscr{B}[.]\,\mathfrak{i}_{n}$
and $f_{n}=\mathfrak{i}_{n}^{\star}f\in\mathcal{V}_{n}^{\star}$.

The regularized problem $(\mathscr{P}^{n}_{\varepsilon})$ can be written
equivalently in the form

$\Lambda_{\varepsilon}(u^{\varepsilon_{n}})=0$ (16)

where
$\Lambda_{\varepsilon}=\mathscr{A}_{n}+\mathscr{B}_{n}[.]+\mathscr{K}_{\varepsilon}-f_{n}$
from $\mathcal{V}_{n}$ into $\mathcal{V}_{n}^{\star}$. As the domain
$\mathscr{O}$ is simply connected, it follows from bykhovskii1960orthogonal ,
that the bilinear form

$\langle\langle u,v\rangle\rangle_{\mathcal{V}}=\displaystyle\int_{\Omega}\rm
rot\,u.\rm rot\,v\,d\lambda$

generates a norm in $\mathcal{V}$ which is equivalent to the
$H^{1}(\Omega;\mathbb{R}^{d})$-norm. From this and from the Cauchy-Schwartz
inequality one can deduce that there exist some $c>0$ such that

$\langle\mathscr{A}u,v\rangle\leq
c\,\|u\|_{\mathcal{V}}\|v\|_{\mathcal{V}},\quad\text{for all
}u,\,v\in\mathcal{V},$

which means that $\mathscr{A}$ is continuous. Moreover, from (Temam, , Chapter
II) $\mathscr{B}[.]$ is continuous. This obviously implies that
$\mathscr{A}_{n}$ and $\mathscr{B}_{n}[.]$ and consequently
$\mathscr{A}_{n}+\mathscr{B}_{n}[.]$ are continuous. Finally by using Lemma 3,
the continuity of $\Lambda_{\varepsilon}$ follows. Because of the coercivity
of $\mathscr{N}:=\mathscr{A}+\mathscr{B}[.]$ (see aadi18 ) and Lemma 1 we have
the estimate

$\langle\Lambda(u^{\varepsilon_{n}}),u^{\varepsilon_{n}}\rangle\geq
M\,\left\|u^{\varepsilon_{n}}\right\|^{2}-ab\sigma(\partial{\mathscr{O}})-\|f\|\left\|u^{\varepsilon_{n}}\right\|,$
(17)

where $M$ is the coerciveness constant of $\mathscr{N}$. By applying Brouwer’s
fixed point theorem (cf. lions1969quelques  p.53) we obtain that (16) admits a
bounded solution $u^{\varepsilon_{n}}$.

###### Proposition 2

The sequence $(\Theta_{\varepsilon_{n}}(u^{\varepsilon_{n}}))_{n}$ is weakly
precompact in $L^{1}(\partial{\mathscr{O}})$.

###### Proof

The Dunford-Pettis theorem (cf. ekeland1976convex , p.239) implies that it
suffices to show that for each $\mu>0$ a $\delta(\mu)>0$ can be determined
such that for $\Gamma\subset\partial{\mathscr{O}}$ with
$\sigma(\Gamma)<\delta$

$\int_{\Gamma}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})\right|{\rm
d}\sigma<\mu$ (18)

The inequality

$s_{0}\left|\Theta_{\varepsilon}(s)\right|\leq\left|\Theta_{\varepsilon}(s)s\right|+s_{0}\sup_{\left|s\right|<s_{0}}\left|\Theta_{\varepsilon}(s)\right|$

implies that

$\int_{\Gamma}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})\right|\,d\,\sigma\leq\frac{1}{s_{0}}\int_{\partial\mathscr{O}}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\right|\,{\rm
d}\,\sigma+\int_{\Gamma}\displaystyle\sup_{\left|u_{N}^{\varepsilon_{n}}(x)\right|\leq
s_{0}}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})\right|\,{\rm
d}\,\sigma$ (19)

But

$\displaystyle\int_{\partial\mathscr{O}}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\right|\,{\rm
d}\,\sigma=$
$\displaystyle\int_{\left|u_{N}^{\varepsilon_{n}}\right|>\rho_{1}}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\right|\,{\rm
d}\,\sigma+\int_{\left|u_{N}^{\varepsilon_{n}}\right|\leq\rho_{1}}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\right|\,{\rm
d}\,\sigma$ $\displaystyle=$
$\displaystyle\int_{\left|u_{N}^{\varepsilon_{n}}\right|>\rho_{1}}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\right|\,{\rm
d}\,\sigma-\int_{\left|u_{N}^{\varepsilon_{n}}\right|\leq\rho_{1}}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\right|\,{\rm
d}\,\sigma$
$\displaystyle\qquad+2\int_{\left|u_{N}^{\varepsilon_{n}}\right|\leq\rho_{1}}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\right|\,{\rm
d}\,\sigma$ $\displaystyle\leq$
$\displaystyle\int_{\left|u_{N}^{\varepsilon_{n}}\right|>\rho_{1}}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\right|\,{\rm
d}\,\sigma+\int_{\left|u_{N}^{\varepsilon_{n}}\right|\leq\rho_{1}}\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\,{\rm
d}\,\sigma$
$\displaystyle\qquad+2\int_{\left|u_{N}^{\varepsilon_{n}}\right|\leq\rho_{1}}\left|\Theta_{\varepsilon}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\right|\,{\rm
d}\,\sigma$ $\displaystyle=$
$\displaystyle\int_{\partial\mathscr{O}}\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\,{\rm
d}\,\sigma+2\int_{\left|u_{N}^{\varepsilon_{n}}\right|\leq\rho_{1}}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\right|\,{\rm
d}\,\sigma$ $\displaystyle=$ $\displaystyle\langle
f,u^{\varepsilon_{n}}\rangle-\langle\mathscr{A}u^{\varepsilon_{n}},u^{\varepsilon_{n}}\rangle-\langle\mathscr{B}[u^{\varepsilon_{n}}],u^{\varepsilon_{n}}\rangle$
$\displaystyle\qquad+2\int_{\left|u_{N}^{\varepsilon_{n}}\right|\leq\rho_{1}}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\right|\,{\rm
d}\,\sigma$ $\displaystyle=$ $\displaystyle\langle
f,u^{\varepsilon_{n}}\rangle-\langle\mathscr{A}u^{\varepsilon_{n}},u^{\varepsilon_{n}}\rangle+2\int_{\left|u_{\varepsilon_{n}}\right|\leq\rho_{1}}\left|\Theta_{\varepsilon}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\right|\,{\rm
d}\,\sigma$ $\displaystyle\leq$ $\displaystyle
c+2ab\,\sigma(\partial{\mathscr{O}}),\quad\text{ for some constant }c.$

In the last two inequalities we have used the boundedness of the solutions
$(u^{\varepsilon_{n}})_{n}$, the estimate (15) and the relation

$\sup_{\left|s\right|\leq
s_{0}}\left|\Theta_{\varepsilon}(s)\right|\leq\displaystyle\operatorname*{ess\,sup}_{\left|s\right|\leq
s_{0}+1}\left|\Theta(s)\right|,$ (20)

Now choose $s_{0}$ such that for all $\varepsilon$ and $n$

$\frac{1}{s_{0}}\int_{\Gamma}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})u_{N}^{\varepsilon_{n}}\right|{\rm
d}\sigma\leq\frac{1}{s_{0}}(c+2\rho_{1}\rho_{2}\sigma(\partial{\mathscr{O}}))\leq\frac{\mu}{2}$
(21)

and $\delta$ such that

$\displaystyle\operatorname*{ess\,sup}_{\left|s\right|\leq
s_{0}+1}\left|\Theta(s)\right|\leq\frac{\mu}{2\delta}$ (22)

Relation (20) implies with (21) that for $\sigma(\Gamma)<\delta$

$\int_{\Gamma}\sup_{\left|u_{N}^{\varepsilon_{n}}\right|\leq
s_{0}}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})\right|\,{\rm
d}\,\sigma\leq\displaystyle\operatorname*{ess\,sup}_{\left|u_{N}^{\varepsilon_{n}}\right|\leq
s_{0}+1}\left|\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})\right|\sigma(\Gamma)\leq\frac{\mu}{2\delta}.\delta\leq\frac{\mu}{2}.$

From the relations (19), (21) and (22), the relation (18) results, i.e. that
$\\{\Theta_{\varepsilon_{n}}(u_{N}^{\varepsilon_{n}})\\}$ is weakly precompact
in $L^{1}(\partial{\mathscr{O}})$.

##  4 Existence

In this section we present an existence result corresponding to the
hemivariational inequality for Navier-Stokes systems under Rauch-Chang
assumption $H(\Theta)$. The uniqueness question is discussed in Remark 4.

###### Theorem 4.1

Under assumption $H(\Theta)$, the problem (13) has at least one solution.

###### Proof

From Proposition 1, we have that $\|u^{\varepsilon_{n}}\|<c$, where $c$ is
independent of $\varepsilon$ and $n$. Thus as $\varepsilon\rightarrow 0$,
$n\rightarrow\infty$ and by considering subsequences if necessary, we may
write that

$u^{\varepsilon_{n}}\rightarrow u,\quad\text{ weakly in }\mathcal{V},$

with $u\in\mathcal{V}$. Then, by the compactness of $\gamma$ the trace of
$\mathcal{V}$ into $\mathrm{L}^{2}(\partial\mathscr{O},\mathbb{R}^{d})$, it
follows that

$\gamma\,u^{\varepsilon_{n}}\rightarrow\gamma\,u,\quad\text{ in
}\mathrm{L}^{2}(\partial\mathscr{O},\mathbb{R}^{d}).$

This implies that $u^{\varepsilon_{n}}_{N}=\gamma
u^{\varepsilon_{n}}.\,n\rightarrow\gamma u.\,n=u_{N}$ in
$L^{2}(\partial\mathscr{O},\mathbb{R}^{d})$ and thus
$u^{\varepsilon_{n}}_{N}(z)\rightarrow u_{N}(z)$ a.e.
$z\in\partial\mathscr{O}$. Moreover due to Proposition 2 we can write that

$\Theta_{\varepsilon}(u^{\varepsilon_{n}}_{N})\rightarrow\xi,\quad\text{weakly
in }L^{1}(\partial{\mathscr{O}})$ (23)

By applying the properties of the Galerkin basis and a simple passage à la
limite we obtain

$\langle\mathscr{A}u+\mathscr{B}[u],v\rangle+\displaystyle\int_{\partial\mathscr{O}}\xi
v_{N}{\rm d}\sigma(x)=\langle f,v\rangle,\quad\forall
v\in\widetilde{\mathcal{V}}$ (24)

from which it follows that

$\left|\int_{\partial\mathscr{O}}\xi v_{N}{\rm d}\sigma(x)\right|\leq
k\|v\|_{\mathcal{V}},$

and that the linear functional $\mathscr{E}$ can be uniquely extended to the
whole space with $\mathscr{E}(\xi)\in\mathcal{V}^{*}$. Thus the expression
(24) can be written in the form

$\langle\mathscr{A}u+\mathscr{B}[u],v\rangle+\displaystyle(\mathscr{E}(\xi),v)=\langle
f,v\rangle,\quad\forall v\in\mathcal{V}.$

In order to complete the proof it will be shown that

$\xi\in\hat{\Theta}(u_{N}(z))=\displaystyle\partial j(u_{N}(z)),\quad\text{
for a.e }z\in\partial{\mathscr{O}}.$

As $u_{N}^{\varepsilon_{n}}\rightarrow u_{N}$ a.e., then by applying Egoroff’s
theorem we can find that for any $\alpha>0$ we can determine
$\Gamma\subset\partial{\mathscr{O}}$ with $\sigma(\Gamma)<\alpha$ such that

$u_{N}^{\varepsilon_{n}}\rightarrow u_{N},\quad\text{ uniformly on
}\partial{\mathscr{O}}\setminus\Gamma,$

with $u_{N}\in\mathrm{L}^{\infty}(\partial{\mathscr{O}}\setminus\Gamma)$. Thus
for any $\alpha>0$ we can find $\Gamma\subset\partial{\mathscr{O}}$ with
$\sigma(\Gamma)<\alpha$ such that for any $\mu>0$ and for
$\varepsilon<\varepsilon_{0}<\mu/2$ and $n>n_{0}>2/\mu$ we have

$|u_{N}^{\varepsilon_{n}}-u_{N}|<\frac{\mu}{2},\quad\text{ on
}\partial{\mathscr{O}}\setminus\Gamma.$

Consequently, one obtain that

$\begin{array}[]{ll}\Theta_{\varepsilon}(u^{\varepsilon_{n}}_{N})&\leq\displaystyle\operatorname*{ess\,sup}_{|u_{N}^{\varepsilon_{n}}-\xi|\leq\varepsilon}\Theta(\xi)\\\
&\leq\displaystyle\operatorname*{ess\,sup}_{|u_{N}^{\varepsilon_{n}}-\xi|\leq\frac{\mu}{2}}\Theta(\xi)\\\
&\leq\displaystyle\operatorname*{ess\,sup}_{|u_{N}-\xi|\leq\mu}\Theta(\xi)\\\
&=\overline{\Theta}_{\mu}(u_{N})\end{array}$

Analogously we prove the inequality

$\underline{\Theta}_{\mu}(u_{N})=\displaystyle\operatorname*{ess\,inf}_{|u_{N}-\xi|\leq\mu}\Theta(\xi)\leq\Theta_{\varepsilon}(u_{N}^{\varepsilon_{n}})$

We take now $v\geq 0$ a.e. on $\partial{\mathscr{O}}\setminus\Gamma$ with
$v\in\mathrm{L}^{\infty}(\partial{\mathscr{O}}\setminus\Gamma)$. This implies

$\int_{\partial{\mathscr{O}}\setminus\Gamma}\underline{\Theta}_{\mu}(u_{N})\,v{\rm
d}\sigma\leq\int_{\partial{\mathscr{O}}\setminus\Gamma}\Theta_{\varepsilon}(u_{N}^{\varepsilon_{n}})\,v{\rm
d}\sigma\leq\int_{\partial{\mathscr{O}}\setminus\Gamma}\overline{\Theta}_{\mu}(u_{N})\,v{\rm
d}\sigma$

Taking the limit $\varepsilon\rightarrow 0$ as $n\rightarrow\infty$ we obtain
that

$\int_{\partial{\mathscr{O}}\setminus\Gamma}\underline{\Theta}_{\mu}(u_{N})\,v{\rm
d}\sigma\leq\int_{\partial{\mathscr{O}}\setminus\Gamma}\xi\,v{\rm
d}\sigma\leq\int_{\partial{\mathscr{O}}\setminus\Gamma}\overline{\Theta}_{\mu}(u_{N})\,v{\rm
d}\sigma$

and as $\mu\rightarrow 0$ that

$\int_{\partial{\mathscr{O}}\setminus\Gamma}\underline{\Theta}(u_{N})\,v{\rm
d}\sigma\leq\int_{\partial{\mathscr{O}}\setminus\Gamma}\xi\,v{\rm
d}\sigma\leq\int_{\partial{\mathscr{O}}\setminus\Gamma}\overline{\Theta}(u_{N})\,v{\rm
d}\sigma$

Since $v$ is arbitrary we have that

$\xi\in[\underline{\Theta}(u_{N}),\overline{\Theta}(u_{N})]=\widehat{\Theta}(u_{N})$

where $\sigma(\partial\mathscr{O})<\alpha$. For $\alpha$ as small as possible,
we obtain the result.

Several of the arguments applied in the proof of this theorem are borrowed
from the method developed in rauch1977discontinuous  for the existence proof
for semilinear differential equations.

###### Remark 4

The question of uniqueness is more delicate. In fact even if we suppose the
following monotonicity type assumption on $\Theta$ in the way did in M05HE

$\displaystyle\operatorname*{ess\,inf}_{\xi_{1}\neq\xi_{2}}\frac{\Theta(\xi_{1})-\Theta(\xi_{2})}{\xi_{1}-\xi_{2}}>-m$

a problem occurs when one needs to get estimates in
$L^{2}(\partial\mathscr{O})$ for $\xi\in L^{1}(\partial\mathscr{O})$, where
$(u,\xi)$ is a solution of (13). But as $L^{2}(\partial\mathscr{O})\subset
L^{1}(\partial\mathscr{O})$ it may occur that $\xi\in
L^{1}(\partial\mathscr{O})\setminus L^{2}(\partial\mathscr{O})$. As we have a
weak assumption and we don’t make use of any type of growth conditions we have
in fact enlarged the space where we are looking for a solution. By doing so we
loose any hope to prove a uniqueness result without a growth condition.

##  5 Dependence result

In this section, we characterize the dependence of solutions on the
hemivariational part, particularly on the functions $\Theta$. Consider a
sequence of functions $\Theta^{k}$ converging in some sense to a function
$\Theta^{\infty}$. Our aim is to prove that the solutions $u^{k}$ constructed
from $\Theta^{k}$ converge to $u^{\infty}$ corresponding to the function
$\Theta^{\infty}$. To do so we make the following hypothesis:

  1. $(H^{k})$

    1. (i)

$(\Theta^{k})_{k\in\mathbb{N}}\subset L^{\infty}_{loc}(\mathbb{R})$ and
$\Theta^{k}(t\pm 0)$ exists for any $t\in\mathbb{R}$ and $k\in\mathbb{N}$.

    2. (ii)

there is $t_{0}>0$ such that

$\sup_{]-\infty,-t_{0}[}\Theta^{k}(t)\leq
0\leq\inf_{]t_{0},+\infty[}\Theta^{k}(t),\quad\text{ for all }k\in\mathbb{N}$
(25)

  2. $(H^{\infty})$

    1. (iii)

$\Theta^{\infty}\in L^{\infty}_{loc}(\mathbb{R})$ and $\Theta^{\infty}(t\pm
0)$ exists for any $t\in\mathbb{R}$.

    2. (iv)

$\limsup_{k\to\infty}\mathrm{Graph}(\widehat{\Theta}^{k}(.))\subset\mathrm{Graph}(\widehat{\Theta}^{\infty}(.))$
(in the sense of Kuratowski, see AF90 ).

###### Theorem 5.1

Assume $(H^{k})$ and $(H^{\infty})$ hold and $f\in\mathcal{V}^{\star}$. Let
$(u^{k})_{k\in\mathbb{N}}$ denotes a sequence of solutions of the problem
(13), where $\Theta$ is replaced by $\Theta^{k}$. Then there exists a
subsequence of $(u^{k})_{k\in\mathbb{N}}$(denoted by the same symbol) such
that $u^{k}\rightarrow u^{\infty}$ weakly in $\mathcal{V}$, where
$u^{\infty}\in\mathcal{V}$ is a solution to (13) corresponding to
$\Theta^{\infty}$.

The assumption $(H^{k})(ii)$ is slightly stronger than the one needed usually
to ensure the existence of solutions but not too restrictive. As we will see
in the proof of the theorem one can always find a constant $\delta>0$ such
that $\Theta^{k}$ is positive in $]\delta,+\infty[$ including for the
discontinuities(which is not the case for the usual assumption). This Change
allow us to have similar lower bound for the integral part, i.e.
$\displaystyle\int_{\partial{\mathscr{O}}}\Theta^{k}(u_{N}^{k}(z)\,u_{N}^{k}(z)\,{\rm
d}\sigma(z)$ which make it possible to obtain the boundedness of $(u^{k})_{k}$
and the weak precompactness of $\\{\Theta^{k}(u_{N}^{k})\\}$ in
$\mathrm{L}^{1}(\partial{\mathscr{O}})$.

###### Proof

The sequence $(u^{k})_{k\in\mathbb{N}}$ is bounded. In fact, from $(H^{k})$
(ii) we can find $\delta_{1}>0$ and $\delta_{2}>0$ such that

$\left\\{\begin{array}[]{ll}\Theta^{k}(t)\geq 0&\text{ si }t>\delta_{1}\\\
\Theta^{k}(t)\leq 0&\text{ si }t<-\delta_{1}\\\
|\Theta^{k}(t)|\leq\delta_{2}&\text{ si }|t|\leq\delta_{1}\\\
\end{array}\right.$

This in hand, one can prove in the same way as Lemma 1 that

$\int_{\partial{\mathscr{O}}}\Theta^{k}(u_{N}^{k}(z)).\,u_{N}^{k}(z)\,{\rm
d}\sigma(z)\geq-\delta_{1}\delta_{2}\,\sigma(\partial{\mathscr{O}})$ (26)

Now by using the fact that $u^{k}$ is a solution of (13) by replacing $\Theta$
by $\Theta^{k}$, we have

$\langle\mathscr{A}u^{k},u^{k}\rangle+\langle\mathscr{\mathscr{missing}}B[u^{k}],u^{k}\rangle+\int_{\partial{\mathscr{O}}}\Theta^{k}(\gamma\,u_{N}^{k}(z))\gamma\,u_{N}^{k}(z)\,{\rm
d}\sigma(z)=\langle f,u^{k}\rangle$

which leads, from (26) and the coerciveness of $\mathscr{A}$, to
$\alpha\|u^{k}\|^{2}-\delta_{1}\delta_{2}\sigma(\partial{\mathscr{O}})\leq
c\|u^{k}\|$. If $\|u^{k}\|$ were unbounded(i.e. $\|u^{k}\|\rightarrow+\infty$
as $k\rightarrow+\infty$) it will leads to contradiction. Now, by considering
a subsequence if necessary, we may write

$u^{k}\rightarrow u^{\infty},\quad\text{ weakly in }\mathcal{V},$

with $u^{\infty}\in\mathcal{V}$. Then, by the compactness of $\gamma$ the
trace of $\mathcal{V}$ into
$\mathrm{L}^{2}(\partial\mathscr{O},\mathbb{R}^{d})$, it follows that

$\gamma\,u^{k}\rightarrow\gamma\,u^{\infty},\quad\text{ in
}\mathrm{L}^{2}(\partial\mathscr{O},\mathbb{R}^{d}).$

This implies that $u_{N}^{k}=u^{k}.\,n\rightarrow
u^{\infty}.\,n=u_{N}^{\infty}$ in
$\mathrm{L}^{2}(\partial\mathscr{O},\mathbb{R}^{d})$ and thus
$u_{N}^{k}(z)\rightarrow u_{N}^{\infty}(z)$ a.e. $z\in\partial\mathscr{O}$.

By Dunford-Pettis theorem, one can get, without much difficulties, the weak
compactness in $\mathrm{L}^{1}(\partial{\mathscr{O}})$ of
$\\{\Theta^{k}(u_{N}^{k})\\}$. It follows that there exists
$\Xi^{\infty}\in\mathrm{L}^{1}(\partial{\mathscr{O}})$ such that

$\Theta^{k}(u_{N}^{k})\rightarrow\Xi^{\infty}\text{ as
}k\to+\infty\quad\text{weakly in }\mathrm{L}^{1}(\partial{\mathscr{O}}).$

Moreover, as $k\to+\infty$, we have

$\langle\mathscr{A}\,u^{k}+\mathscr{B}[u^{k}],v\rangle\rightarrow\langle\mathscr{A}\,u^{\infty}+\mathscr{B}[u^{\infty}],v\rangle,\quad\text{
for all }v\in\tilde{\mathcal{V}}.$

It follows that

$\langle\mathscr{A}u^{\infty}+\mathscr{B}[u^{\infty}],v\rangle+\int_{\partial{\mathscr{O}}}\Xi.\,v_{N}(z)\,{\rm
d}\sigma(z)=\langle f,v\rangle,\quad\text{ for all
}\tilde{v}\in\tilde{\mathcal{V}}.$

On the other hand $\tilde{\mathcal{V}}$ is dense in $\mathcal{V}$, and from
the equality

$\int_{\partial{\mathscr{O}}}\Xi.\,v_{N}(z)\,{\rm d}\sigma(z)=\langle
f,v\rangle-\langle\mathscr{A}u^{\infty}+\mathscr{B}[u^{\infty}],v\rangle,\quad\text{
for all }\tilde{v}\in\tilde{\mathcal{V}},$

it follows that

$\mathscr{A}u^{\infty}+\mathscr{B}[u^{\infty}]+\mathscr{E}(\Xi)=f,\quad\text{
weakly in }\mathcal{V}^{*}$

It remains to show that $\Xi\in\widehat{\Theta}^{\infty}(u^{\infty}_{N})$ a.e.
in $\partial\mathscr{O}$. Recall that $u_{N}^{k}(z)\rightarrow
u_{N}^{\infty}(z)$ a.e. $z\in\partial\mathscr{O}$. By applying Egoroff’s
theorem we find that for any $\alpha>0$ we can determine
$\Gamma\in\partial\mathscr{O}$ with $\sigma(\Gamma)<\alpha$ such that

$u_{N}^{k}\rightarrow u_{N}^{\infty},\quad\text{uniformly on
}\partial\mathscr{O}\setminus\Gamma.$

By applying Theorem 7.2.1 of Aubin and Frankowska AF90  we deduce

$\Xi\in\overline{\mathrm{conv}}\left(\limsup_{t\to
u_{N}^{\infty}(z),\,k\to\infty}\widehat{\Theta}^{k}(t)\right)\subset\widehat{\Theta}^{\infty}(u_{N}^{\infty}(z)),\quad\text{for
all }z\in\partial\mathscr{O}\setminus\Gamma.$

The latter inclusion follows from the assumption $(H^{\infty})(iv)$, where
$\sigma(\Gamma)<\alpha$. For $\alpha$ as small as possible, we obtain the
result.

##  6 Optimal Control

In this section, we provide a result on dependence of solutions with respect
to the density of the external forces and use it to study the distributed
parameter optimal control problem corresponding to it.

Before we start to discuss the optimal control problem, we first prove the
following auxiliary result.

###### Theorem 6.1

Under $H(\Theta)$ assume that $f_{n}$, $f\in
L^{2}(\mathscr{O};\mathbb{R}^{d})$ such that $f_{n}\rightarrow f$ weakly in
$L^{2}(\mathscr{O};\mathbb{R}^{d})$. Then for every $\\{u^{n}\\}_{n}$ solution
to the problem (13) corresponding to $f_{n}$, we can find a subsequence (still
denoted with the same symbol) such that $u^{n}\rightarrow u$ in $\mathcal{V}$
and $u$ is a solution to problem (13) corresponding to $f$.

###### Proof

Let $f_{n},\,f\in L^{2}(\mathscr{O};\mathbb{R}^{d})$ with $f_{n}\rightarrow f$
weakly in $L^{2}(\mathscr{O};\mathbb{R}^{d})$. Then by Theorem 4.1, there
exists $(u^{n},\xi_{n})\in\mathcal{V}\times
L^{1}(\partial\mathscr{O};\mathbb{R})$ such that $\xi_{n}\in\partial
j(u_{N}^{n})$ and

$\langle\mathscr{A}u^{n}+\mathscr{B}[u^{n}],v\rangle+\int_{\partial{\mathscr{O}}}\xi_{n}\,v_{N}(z)\,d\sigma(z)=\langle
f_{n},v\rangle,\quad\text{ for all }v\in\widetilde{\mathcal{V}}$

It is possible to prove a similar result as Lemma 1 with eventually different
constants. By the coerciveness of $\mathscr{A}$ and the continuity of the
injection $L^{2}(\mathscr{O};\mathbb{R}^{d})\subset\mathcal{V}^{*}$, we get

$M\,\|u^{n}\|_{\mathcal{V}}^{2}-ab\sigma(\partial\mathscr{O})-\|f_{n}\|_{L^{2}(\mathscr{O};\mathbb{R}^{d})}\,\|u^{n}\|_{\mathcal{V}}\leq
0,$

which simplifies to

$\|u^{n}\|_{\mathcal{V}}\leq\frac{ab\sigma(\partial\mathscr{O})}{\|u^{n}\|_{\mathcal{V}}}+\|f_{n}\|_{L^{2}(\mathcal{V})}.$

One can see immediately that if $\|u^{n}\|_{\mathcal{V}}$ converges to
$+\infty$, so will do $\|f_{n}\|_{L^{2}(\mathscr{O};\mathbb{R}^{d})}$, which
means that $\\{u^{n}\\}_{n}$ is in fact bounded(with bound independent of
$n$). As $\mathcal{V}$ is a reflexive Banach space, we may assume, by passing
to a subsequence if necessary, that there exists $u\in\mathcal{V}$ such that
$u^{n}$ converges to $u$. From the continuity of $\mathscr{A}$ and
$\mathscr{B}[.]$, we have $\mathscr{A}\,u^{n}\rightarrow\mathscr{A}\,u$ and
$\mathscr{B}[u^{n}]\rightarrow\mathscr{B}[u]$ weakly in $\mathcal{V}^{*}$.
Using the compactness of the trace operator $\gamma$, we may assume that
$\gamma\,u^{n}\rightarrow\gamma\,u$ in $L^{2}(\mathscr{O};\mathbb{R}^{d})$ and
then $\gamma\,u^{n}(z)\rightarrow\gamma\,u(z)$ for a.e.
$z\in\partial\mathscr{O}$. Consequently, $u_{N}^{n}(z)\rightarrow u_{N}(z)$
for a.e. $z\in\partial\mathscr{O}$. On the other hand one have

$\int_{\partial{\mathscr{O}}}\xi_{n}\,v_{N}(z)\,{\rm d}\sigma(z)=\langle
f_{n}-\mathscr{A}u^{n}-\mathscr{B}[u^{n}],v\rangle,\quad\text{ for all
}v\in\widetilde{\mathcal{V}}$

which means that $\xi_{n}\rightarrow\xi$ in $L^{1}(\partial\mathscr{O})$ and

$\langle\mathscr{A}u+\mathscr{B}[u],v\rangle+\int_{\partial{\mathscr{O}}}\xi\,v_{N}(z)\,d\sigma(z)=\langle
f,v\rangle,\quad\text{ for all }v\in\widetilde{\mathcal{V}},$

in order to complete the proof it will be shown that

$\xi\in\partial j(u_{N}(z))\text{ for a.e }z\in\partial\mathscr{O}.$

To do this, we first show that

$\partial j(u_{N}^{n})\subset\partial j(u_{N}),\quad\text{for all }n.$

As $u_{N}^{n}\rightarrow u_{N}$ for a.e. $z\in\partial\mathscr{O}$, we can
find, by Egoroff’s theorem, that for any $\alpha>0$ we can determine
$\Gamma\subset\partial{\mathscr{O}}$ with $\sigma(\Gamma)<\alpha$ such that

$u_{N}^{n}\rightarrow u_{N},\quad\text{ uniformly on
}\partial{\mathscr{O}}\setminus\Gamma,$ (27)

with $u_{N}\in\mathrm{L}^{\infty}(\partial{\mathscr{O}}\setminus\Gamma)$. Thus
for any $\mu>0$ there exists $n_{0}$ such that for all $n>n_{0}$ we have

$|u_{N}^{n}(z)-u_{N}(z)|<\frac{\mu}{2},\quad\forall
z\in\partial{\mathscr{O}}\setminus\Gamma.$

By using triangle inequality, we have that

$\begin{array}[]{ll}\overline{\Theta}_{\frac{\mu}{2}}(u^{{}_{n}}_{N})&=\displaystyle\operatorname*{ess\,sup}_{|u_{N}^{n}-\xi|\leq\frac{\mu}{2}}\Theta(\xi)\\\
&\leq\displaystyle\operatorname*{ess\,sup}_{|u_{N}-\xi|\leq\mu}\Theta(\xi)\\\
&=\overline{\Theta}_{\mu}(u_{N}).\end{array}$

Analogously we prove the inequality

$\underline{\Theta}_{\mu}(u_{N})\leq\underline{\Theta}_{\frac{\mu}{2}}(u^{n}_{N}).$

Taking the limit as $\mu\rightarrow 0^{+}$, we obtain that

$\xi_{n}\in\widehat{\Theta}(u_{N}^{n}(z))\subset\widehat{\Theta}(u_{N}(z)),\quad\forall
n\geq n_{0},\,z\in\partial\mathscr{O}\setminus\Gamma,$

from which we conclude that

$\xi(z)\in\overline{\mathrm{conv}}\,\,\widehat{\Theta}(u_{N}(z))=\widehat{\Theta}(u_{N}(z)),\quad\forall\,z\in\partial\mathscr{O}\setminus\Gamma,$

where $\sigma(\Gamma)<\alpha$. For $\alpha$ as small as possible, we obtain
the result.

We follow Migórski law2013note  and we let
$\mathcal{U}=L^{2}(\mathscr{O};\mathbb{R}^{d})$ be the space of controls. For
every $f\in\mathcal{U}$, we denote by $S(f)\subset\mathcal{V}\times
L^{1}(\partial\mathscr{O};\mathbb{R})$ the solution set corresponding to $f$
of the problem (13). It is then clear that, by definition, $S(f)$ is nonempty
for all $f\in\mathcal{U}$.

Let $\mathcal{U}_{ad}$ be a nonempty subset of $\mathcal{U}$ consisting of
admissible controls. Let $\mathscr{F}:\mathcal{U}\times\mathcal{V}\times
L^{1}(\partial\mathscr{O};\mathbb{R})\rightarrow\mathbb{R}$ be the objective
functional we want to minimize. The control problem reads as follows: Find a
control $\hat{f}\in\mathcal{U}_{ad}$ and a state $(\hat{u},\hat{\xi})\in
S(\hat{f})$ such that

$\mathscr{F}(\hat{f},\hat{u},\hat{\xi})=\inf\left\\{\mathscr{F}(f,u,\xi):\,f\in\mathcal{U}_{ad},\,(u,\xi)\in
S(f)\right\\}.$ (28)

A triple which solves (28) is called an optimal solution. The existence of
such optimal control can be proved by using Theorem 6.1. To do so, we need the
following additional hypothesis:

  * $H(\mathcal{U}_{ad})$

$\quad\mathcal{U}_{ad}$ is a bounded and weakly closed subset of
$\mathcal{U}$.

  * $H(\mathscr{F})$

$\quad\mathscr{F}$ is lower semicontinuous with respect to
$\mathcal{U}\times\mathcal{V}\times
L^{1}(\partial\mathscr{O};\mathbb{R})\rightarrow\mathbb{R}$ endowed with the
weak topology.

###### Theorem 6.2

Assume that $H(\Theta)$, $H(\mathcal{U}_{ad})$ and $H(\mathscr{F})$ are
fulfilled. Then the problem 28 has an optimal control.

###### Proof

Let $(f_{n},u_{n},\xi_{n})$ be a minimizing sequence for the problem (28), i.e
$f_{n}\in\mathcal{U}_{ad}$ and $(u_{n},\xi_{n})\in S(f_{n})$ such that

$\displaystyle\lim_{n\to\infty}\mathscr{F}(f_{n},u_{n},\xi_{n})=\inf\left\\{\mathscr{F}(f,u,\xi):\,f\in\mathcal{U}_{ad},\,(u,\xi)\in
S(f)\right\\}=:m$

It follows that the sequence $f_{n}$ belongs to a bounded subset of the
reflexive Banach space $\mathcal{V}$. We may then assume that
$f_{n}\rightarrow\hat{f}$ weakly in $\mathcal{V}$ (by passing to a subsequence
if necessary). By $H(\mathcal{U}_{ad})$, we have $\hat{f}\in\mathcal{U}_{ad}$.
From Theorem 6.1, we obtain, by again passing to a subsequence if necessary,
that $u_{n}\rightarrow\hat{u}$ weakly in $\mathcal{V}$ with
$(\hat{u},\hat{\xi})\in S(\hat{f})$. By $H(\mathscr{F})$, we have
$m\leq\mathscr{F}(\hat{f},\hat{u},\hat{\xi})\leq\displaystyle\liminf_{n\to\infty}\mathscr{F}(f_{n},u_{n},\xi_{n})=m$.
Which completes the proof.

###### Acknowledgements.

We thank Prof. S. Migórski for pointing out that the Rauch and the growth
conditions are completely independent.

## Conflict of interest

The authors declare that they have no conflict of interest.

## References

  * (1) Al-Homidan, S., Ansari, Q. H., Chadli, O. : Noncoercive Stationary Navier-Stokes Equations of Heat-Conducting Fluids Modeled by Hemivariational Inequalities: An Equilibrium Problem Approach. Results in Mathematics, 74(4), (2019) 132.
  * (2) Alekseev, G. V., Smishliaev, A. B. : Solvability of the boundary-value problems for the Boussinesq equations with inhomogeneous boundary conditions. Journal of Mathematical Fluid Mechanics, 3(1),(2001), 18-39.
  * (3) Aubin, J.P. : Applied functional analysis, John Wiley and Sons Inc., New York, USA, 1979.
  * (4) Aubin, J.P., Frankowska, H. : Set-Valued Analysis, Birkhäuser, Berlin, 1990.
  * (5) Ben Aadi, S., Chadli, O., Koukkous, A.: ”Evolution Hemivariational Inequalities for Non-stationary Navier-Stokes Equations: Existence of Periodic Solutions by an Equilibrium Problem Approach, Minimax Theory and its Applications, Volume 3 (2018), No. 1, 107-130.
  * (6) Bykhovskii, É. B., Smirnov, N. V. E.: Orthogonal decomposition of the space of vector functions square-summable on a given domain, and the operators of vector analysis. Trudy Matematicheskogo Instituta imeni VA Steklova, 59, (1960), 5-36.
  * (7) Chang, K. C. : Variational methods for non-differentiable functionals and their applications to partial differential equations. Journal of Mathematical Analysis and Applications, 80(1), (1981), 102-129.
  * (8) Chebotarev, A.Y. : Subdifferential boundary value problems for stationary Navier-Stokes equations. Differentsial’nye Uravneniya, 28(8), (1992), 1443-1450.
  * (9) Chebotarev, A.Y. : Stationary variational inequalities in the model of inhomogeneous incompressible fluids, Sibirsk. Math. Zh. (Siberian Math. J.) 38 (1997), 1184-1193.
  * (10) Chebotarev, A.Y. : Variational Inequalities for Navier-Stokes Type Operators and One-Sided Problems for Equations of Viscous Heat-Conducting Fluids. Mathematical Notes, 70(1-2), (2001), 264-274.
  * (11) Chebotarev, A.Y. : Modeling of steady flows in a channel by Navier-Stokes variational inequalities, J. Appl. Mech. Tech. Phys. 44 (2003) 852-857.
  * (12) Clarke, F. H. : Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, New York, 1983.
  * (13) Clarke, F. H. : Generalized Gradient and Applications, Transactions of American Mathematical Society. , Vol. 205, pp. 247-262, 1975.
  * (14) Dudek, S., Kalita, P., Migórski, S. : Steady flow of generalized Newtonian fluid with multivalued rheology and nonmonotone friction law. Computers and Mathematics with Applications, 74(8), 1813-1825, (2017).
  * (15) Dudek, S., Kalita, P., Migórski, S. : Stationary flow of non-Newtonian fluid with nonmonotone frictional boundary conditions. Z. Angew. Math. Phys. 66(5), 2625-2646, (2015).
  * (16) Ekeland, I., Temam, R. : Convex Analysis and Variational Problems (Studies in Mathematics and its Applications, Vol. 1 , (1976), North-Holland, Amsterdam.
  * (17) Fang, C., Czuprynski, K., Han, W., Cheng, X., Dai, X. : Finite element method for a stationary Stokes hemivariational inequality with slip boundary condition. IMA Journal of Numerical Analysis (2019).
  * (18) Fang, C., Han, W., Migórski, S. : A class of hemivariational inequalities for nonstationary Navier-Stokes equations, Nonlinear Analysis Series B: Real World Applications 31 (2016), 257-276
  * (19) Girault, V., Raviart, P. A. : Finite element methods for Navier-Stokes equations: theory and algorithms (Vol. 5), (2012), Springer Science and Business Media.
  * (20) Kalita, P., Łukaszewicz, G. On large time asymptotics for two classes of contact problems. In Advances in variational and hemivariational inequalities (pp. 299-332), (2015) Springer, Cham.
  * (21) Kalita, P., Łukaszewicz, G. : Attractors for Navier-Stokes flows with multivalued and nonmonotone subdifferential boundary conditions. Nonlinear Analysis: Real World Applications, 19 (2014), 75-88.
  * (22) Konovalova, D. S. : Subdifferential boundary value problems for evolution Navier-Stokes equations. Differ. Uravn.(Differential Equations), 36, , (2000), 792-798.
  * (23) Lions, J. L. : Quelques méthodes de résolution des problemes aux limites non linéaires (1969), Dunod.
  * (24) Migórski, S., Dudek, S. : Evolutionary Oseen Model for Generalized Newtonian Fluid with Multivalued Nonmonotone Friction Law. Journal of Mathematical Fluid Mechanics, 20(3), 1317-1333 (2018).
  * (25) Migórski, S., Paczka, D. : Frictional Contact Problems for Steady Flow of Incompressible Fluids in Orlicz Spaces, in ”Current Trends in Mathematical Analysis and Its Interdisciplinary Applications” Hemen Dutta • Ljubisa D. R. Kocinac Hari M. Srivastava (Editors), Springer Nature Switzerland AG 2019.
  * (26) Migórski, S., Paczka, D. : On steady flow of non-Newtonian fluids with frictional boundary conditions in reflexive Orlicz spaces, Nonlinear Analysis: Real World Applications Volume 39, February 2018, Pages 337-361
  * (27) Migórski, S. : Hemivariational inequalities modeling viscous incompressible fluids. J. Nonlinear Convex Anal. 5, 217-227 (2004)
  * (28) Migórski, S., Ochal, A. : Hemivariational inequalities for stationary Navier-Stokes equations, Journal of Mathematical Analysis and Applications 306 (2005), 197-217.
  * (29) Migórski, S., Ochal, A. : Navier-Stokes problems modelled by evolution hemivariational inequalities, Discrete and Continuous Dynamical Systems, Supplement Volume (2007), 731-740.
  * (30) Migórski, S. : A note on optimal control problem for a hemivariational inequality modeling fluid flow. Dynamical systems, (2013) 545-554.
  * (31) Migórski, S., Szafraniec, P. : Nonmonotone slip problem for miscible liquids. Journal of Mathematical Analysis and Applications, 471(1-2) (2019), 342-357.
  * (32) Migórski, S. : Hemivariational inequalities modeling viscous incompressible fluids. Journal of Nonlinear and Convex Analysis, 5(2), (2004), 217-228.
  * (33) Migórski, S. : Hemivariational inequality for a planar flow of incompressible generalized Newtonian fluid.
  * (34) Migórski, S., Pczka, D. : Frictional Contact Problems for Steady Flow of Incompressible Fluids in Orlicz Spaces. In Current Trends in Mathematical Analysis and Its Interdisciplinary Applications (pp. 1-53), (2019), Birkhauser, Cham.
  * (35) Migórski, S., Dudek, S. : Evolutionary Oseen model for generalized Newtonian fluid with multivalued nonmonotone friction law. Journal of Mathematical Fluid Mechanics, 20(3),(2018), 1317-1333.
  * (36) Moreau, J. J., Panagiotopoulos, P. D. : Nonsmooth Mechanics and Applications, CISM Courses and Lectures, Springer Verlag, Wien, Austria, Vol. 302, pp. 81-176, 1988.
  * (37) Naniewicz, Z. : Hemivariational inequalities with functions fulfilling directional growth condition, Applicable Analysis, 55 (3-4), 259-285, 1994.
  * (38) Naniewicz, Z. : Hemivariational inequalities with functionals which are not locally Lipschitz. Nonlinear Analysis: Theory, Methods and Applications, 25(12), 1307-1320 (1995).
  * (39) Ovcharova, N., Gwinner J.: A study of regularization techniques of nondifferentiable optimization in view of application to hemivariational inequalities, J. Optim. Theory Appl. 162 (2014), no. 3, 754-778.
  * (40) Panagiotopoulos, P. D. : Coercive and Semicoercive Hemivariational Inequalities, Nonlinear Analysis, Theory, Methods and Applications, Vol. 16, pp. 209-231, 1991.
  * (41) Panagiotopoulos, P. D. : Variational-Hemivariational Inequalities in Nonlinear Elasticity, Aplikace Matematiky, Vol. 33, pp. 249-268, 1988.
  * (42) Panagiotopoulos ,P. D., Stavroulakis, G. E. : A Variational-Hemivariational Inequality Approach to the Laminated Plate Theory under Subdifferential Boundary Conditions, Quarterly of Applied Mathematics, Vol. 46, pp. 409-430, 1988.
  * (43) Panagiotopoulos, P. D., Koltsakis, E. : The Nonmonotone Skin Effect in Plane Elasticity. Problems Obeying to Subdifferential Materials Laws, ZAMM, Vol. 70, pp. 13-21, 1990.
  * (44) Panagiotopoulos, P. D., Baniotopoulos, C. C. : A Hemivariational Inequality and Substationary Approach to the Interface Problem: Theory and Prospects of Applications, Engineering Analysis, Vol. 1, pp. 20-31, 1984.
  * (45) Panagiotopoulos, P.D. : Nonconvex Superpotentials in the Sense of F. H. Clarke and Applications, Mechanics Research Communications, Vol. 8, pp. 335-340, 1981.
  * (46) Panagiotopoulos, P.D. : Nonconvex Energy Functions. Hemivariational Inequalities and Substationary Principles, Acta Mechanica, Vol. 42, pp. 160-183, 1983.
  * (47) Panagiotopoulos, P.D. : Inequality Problems in Mechanics and Applications, Birhhauser Verlag, Basel, Switzerland, 1985 (Russian translation, MIH Publisher, Moscow, 1989).
  * (48) Panagiotopoulos, P.D. : Hemivariational Inequalities and their Applications, In J. J. Moreau, and P.D. Panagiotopoulos and G. Strang (eds), Topics in Nonsmooth Mechanics, Birkhauser Verlag, Basel, Switzerland, 1988.
  * (49) Panagiotopoulos, P. D. : Inequality Problems in Mechanics and Applications: Convex and nonconvex energy functions. Springer Science and Business Media, (2012).
  * (50) Rauch, J. : Discontinuous semilinear differential equations and multiple valued maps. Proceedings of the American Mathematical Society, 64(2), (1977), 277-282.
  * (51) Rockafellar, R. T. : Generalized Derivatives and Subgradients of Nonconvex Functions, Canadian Journal of Mathematics, Vol. 32, pp. 257-280, 1980.
  * (52) Szafraniec, P. : Evolutionary Boussinesq model with nonmonotone friction and heat flux boundary conditions. Nonlinear Analysis: Real World Applications, 34, (2017) 403-415.
  * (53) Temam, R. : Navier-Stokes equations, volume 2 of Studies in Mathematics and its Applications. Mathematical Surveys and Monographs. North-Holland Publishing Co., Amsterdam ,(1979).