# Tykhonov Well-posedness of a Heat Transfer Problem with Unilateral
Constraints

Mircea Sofonea, Perpignan, Domingo A. Tarzia, Rosario

Abstract. We consider an elliptic boundary value problem with unilateral
constraints and subdifferential boundary conditions. The problem describes the
heat transfer in a domain $D\subset{{\rm I}\mkern-3.0mu{\rm R}}^{d}$ and its
weak formulation is in the form of a hemivariational inequality for the
temperature field, denoted by ${\cal P}$. We associate to Problem ${\cal P}$
an optimal control problem, denoted by ${\cal Q}$. Then, using appropriate
Tykhonov triples, governed by a nonlinear operator $G$ and a convex
$\widetilde{K}$, we provide results concerning the well-posedness of problems
${\cal P}$ and ${\cal Q}$. Our main results are Theorems 14 and 18, together
with their corollaries. Their proofs are based on arguments of compactness,
lower semicontinuity and pseudomonotonicity. Moreover, we consider three
relevant perturbations of the heat transfer boundary valued problem which lead
to penalty versions of Problem ${\cal P}$, constructed with particular choices
of $G$ and $\widetilde{K}$. We prove that Theorems 14 and 18 as well as their
corollaries can be applied in the study of these problems, in order to obtain
various convergence results.

Keywords : heat transfer problem, unilateral constraint, subdifferential
boundary condition, hemivariational inequality, optimal control, Tykhonov
well-posedness, approximating sequence, convergence results.

MSC 2010 : 49J40, 49J20, 49J52, 49J45, 35A16, 35M86.

##  1 Introduction

A large number of mathematical models in Physics, Mechanics and Engineering
Sciences are expressed in terms of nonlinear boundary value problems which
involve inequalities. Their analysis and optimal control was made the object
of many books and papers and, therefore, the literature in the field is
extensive. Here we restrict ourselves to mention the books [1, 9, 11, 12, 13,
17, 18, 28, 30, 31] and, more recently, [5, 27, 37]. The results presented in
[1, 5, 9, 11, 13, 17, 18, 30] concern the analysis of various classes of
variational inequalities and are based on arguments of monotonicity and
convexity, including properties of the subdifferential of a convex function.
The results in [12, 27, 28, 31, 37] concern the analysis of hemivariational
inequalities and are based on properties of the subdifferential in the sense
of Clarke, defined for locally Lipschitz functions, which may be nonconvex.
Results in the study of optimal control for variational and hemivariational
inequalities have been discussed in several works, including [2, 3, 4, 8, 19,
23, 25, 26, 29, 32, 33, 34, 35, 36] and, more recently, in [36, 39].

The main task in the analysis and control of inequality problems is to provide
their well-posedness. The concepts of well-posedness vary from problem to
problem and from author to author. A few examples are the concept of well-
posedness in the sense of Hadamard for partial differential equations, the
concept of well-posedness in the sense of Tykhonov for a minimization problem,
the concept of well-posedness in the sense of Levitin-Polyak for a
constrainted optimization problem, among others. The concept of Tykhonov well-
posedness (well-posedness, for short) was introduced in [42] for a
minimization problem and then it has been generalized for different
optimization problems, see for instance [6, 7, 15, 16, 22, 45]. It has been
extended in the recent years to various mathematical problems like
inequalities, inclusions, fixed point and saddle point problems. Thus, the
well-posedness of variational inequalities was studied for the first time in
[20, 21] and the study of well-posedness of hemivariational inequalities was
initiated in [10]. References in the field include [40, 43], among others. A
general framework which unifies the view on the well-posedness for abstract
problems in metric spaces was recently introduced in [41]. Moreover, the
Tykhonov well-posedness of an antiplane shear problem was studied in our
recent paper [38].

In this paper we consider a stationary boundary value problem which describes
the heat transfer in a domain $D\subset{{\rm I}\mkern-3.0mu{\rm R}}^{d}$,
$d=1,2,3$. In particular cases, the problem was already considered in [3, 4].
There, the heat flux was assumed to be given on the part $\Gamma_{3}$ of the
boundary of $D$ and, therefore, the weak formulation of the problem was in a
form of the variational inequality for the temperature field. In contrast, in
this current paper we model the heat transfer on $\Gamma_{3}$ by using a
subdifferential boundary condition, governed by a locally Lipschitz potential
function. As a consequence, the weak formulation of the problem leads to an
elliptic hemivariational inequality, denoted by ${\cal P}$. Moreover, we
associate to this inequality an optimal control problem, denoted by ${\cal
Q}$.

Our aim in this paper is twofold. The first one is to study the well-posedness
of the hemivariational inequality ${\cal P}$. The second one is to study the
well-posedness of the associated optimal control problem ${\cal Q}$. To this
end, for both problems, we use specific Tykhonov triples, constructed with
approximating sets govened by a penalty-type operator and a set of
constraints. Proving the well-posedness of the corresponding hemivariational
inequality and the associated optimal control problem in this functional
setting is non standard and represents the main trait of novelty of our paper.
Moreover, the well-posedness of these problems implies general convergence
results which allows us to deduce the weak-strong dependence of the solution
to Problem ${\cal P}$ with respect to the data, its approch by a penalty-like
method as well as the weak compactness of the set of solution to Problem
${\cal Q}$. All these results represent a continuation of our previous results
obtained in [3, 4].

The rest of the manuscript is structured as follows. In Section 2 we recall
some preliminary material on hemivariational inequalities and well-posedness
in the sense of Tykhonov. In Section 3 we introduce the heat transfer problem,
list the assumptions on the data and state its variational formulation, ${\cal
P}$. Then, we introduce the associate optimal control problem ${\cal Q}$. In
Section 4 we prove our first result, Theorem 14, which state the well-
posedness of Problem ${\cal P}$. In Section 5 we prove our second result,
Theorem 18, which state the weakly generalized well-posedness of Problem
${\cal Q}$. Both theorems are completed by several corollaries. Their proofs
are based on arguments of pseudomonotonicity, compacteness, and lower
semicontinuity. Finally, in Section 6 we consider three relevant versions of
the heat transfer problem with penalty conditions. We prove that Theorems 14,
18 and their corollaries can be applied in the study of these problems in
order to deduce various convergence results.

##  2 Preliminaries

We start with some notation and preliminaries on hemivariational inequalities
and send the reader to [27, 28, 37] for more details on the material presented
below. Let $(X,\|\cdot\|_{X})$ be a reflexive Banach space and let $X^{*}$
denote its dual. We use the symbol $\langle\cdot,\cdot\rangle$ for the duality
pairing between $X^{*}$ and $X$. The limits, upper and lower limits below are
considered as $n\to\infty$, even if we do not mention it explicitely. The
symbols “$\rightharpoonup$” and “$\to$” denote the weak and the strong
convergence in $X$ or $X^{*}$.

For real valued functions defined on $X$ we recall the following definitions.

###### Definition 1

A function $j\colon X\to\mathbb{R}$ is said to be locally Lipschitz, if for
every $x\in X$, there exists $U_{x}$ a neighborhood of $x$ and a constant
$L_{x}>0$ such that $|j(y)-j(z)|\leq L_{x}\|y-z\|_{X}$ for all $y$, $z\in
U_{x}$. For such functions the generalized (Clarke) directional derivative of
$j$ at the point $x\in X$ in the direction $v\in X$ is defined by

$j^{0}(x;v)=\limsup_{y\to x,\ \lambda\downarrow 0}\frac{j(y+\lambda
v)-j(y)}{\lambda}.$

The generalized gradient (subdifferential) of $j$ at $x$ is a subset of the
dual space $X^{*}$ given by

$\partial j(x)=\\{\,\zeta\in X^{*}\mid
j^{0}(x;v)\geq{\langle\zeta,v\rangle}\quad\forall\,v\in X\,\\}.$

The function $j$ is said to be regular (in the sense of Clarke) at the point
$x\in X$ if for all $v\in X$ the one-sided directional derivative
$j^{\prime}(x;v)$ exists and $j^{0}(x;v)=j^{\prime}(x;v)$.

We shall use the following properties of the generalized directional
derivative and the generalized gradient.

###### Proposition 2

Assume that $j\colon X\to\mathbb{R}$ is a locally Lipschitz function. Then the
following hold:

(a) For every $x\in X$, the function $X\ni v\mapsto j^{0}(x;v)\in\mathbb{R}$
is positively homogeneous, i.e., $j^{0}(x;\lambda v)=\lambda j^{0}(x;v)$ for
all $\lambda\geq 0$, $v\in X$.

(b) For every $v\in X$, we have $j^{0}(x;v)=\max\,\\{\,\langle\xi,v\rangle\ :\
\xi\in\partial j(x)\,\\}$.

We shall use these definitions and properties both in the case when $X={{\rm
I}\mkern-3.0mu{\rm R}}$ and $X=V$ where $V$ is a Sobolev-type space which will
defined in Section 3. Next, we proceed with the definition of some classes of
nonlinear operators.

###### Definition 3

An operator $A\colon X\to X^{*}$ is said to be:

a) monotone, if for all $u$, $v\in X$, we have $\langle Au-Av,u-v\rangle\geq
0$;

b) strongly monotone, if there exists $m_{A}>0$ such that

$\langle Au-Av,u-v\rangle\geq m_{A}\|u-v\|_{X}^{2}\quad\mbox{for all}\
u,\,v\in X;$

c) bounded, if $A$ maps bounded sets of $X$ into bounded sets of $X^{*}$;

d) pseudomonotone, if it is bounded and $u_{n}\rightharpoonup u$ in $X$ with

$\displaystyle\limsup\,\langle Au_{n},u_{n}-u\rangle\leq 0$

imply $\displaystyle\liminf\,\langle Au_{n},u_{n}-v\rangle\geq\langle
Au,u-v\rangle$ for all $v\in X$;

e) demicontinuous, if $u_{n}\to u$ in $X$ implies $Au_{n}\rightharpoonup Au$
in $X^{*}$.

We shall use the following result related to the pseudomonotonicity of
operators.

###### Proposition 4

Assume that the operator $A\colon X\to X^{*}$ is bounded, demicontinuous and
monotone. Then $A$ is pseudomonotone.

We turn now to the study of hemivariational inequalities of the form

$u\in K,\quad\langle Au,v-u\rangle+j^{0}(u;v-u)\geq\langle
f,v-u\rangle\qquad\forall\,v\in K.$ (2.1)

and, to this end, we consider the following hypotheses on the data.

$\displaystyle K\ \mbox{is nonempty, closed and convex subset of}\ X.$ (2.2)
$\displaystyle\left\\{\begin{array}[]{l}A\colon X\to X^{*}\ \mbox{is
pseudomonotone and}\\\ \mbox{strongly monotone with constant}\
m_{A}>0.\end{array}\right.$ (2.5)
$\displaystyle\left\\{\begin{array}[]{l}j\colon X\to\mathbb{R}\ \mbox{is such
that}\\\\[5.69054pt] \ \ {\rm(a)}\ j\ \mbox{is locally
Lipschitz.}\\\\[5.69054pt] \ \ {\rm(b)}\
\|\xi\|_{X^{*}}\leq\widetilde{c}_{0}+\widetilde{c}_{1}\,\|v\|_{X}\ \mbox{for
all}\ v\in X,\ \xi\in\partial j(v),\ \mbox{with}\
\widetilde{c}_{0},\widetilde{c}_{1}\geq 0.\\\\[5.69054pt] \ \ {\rm(c)}\
j^{0}(v_{1};v_{2}-v_{1})+j^{0}(v_{2};v_{1}-v_{2})\leq\alpha_{j}\,\|v_{1}-v_{2}\|_{X}^{2}\\\\[2.84526pt]
\qquad\ \mbox{for all}\ v_{1},v_{2}\in X,\ \mbox{with}\
\alpha_{j}>0.\end{array}\right.$ (2.11) $\displaystyle\alpha_{j}<m_{A}.$
(2.12) $\displaystyle f\in X^{*}.$ (2.13)

The unique solvability of the variational-hemivariational inequality (2.1) is
provided by the following result proved in [37].

###### Theorem 5

Assume $(\ref{KK})$–$(\ref{ff})$. Then, there exists a unique solution to
inequality $(\ref{1})$.

We end this section by recalling some preliminaries concerning the concept of
well-posedness in the sense of Tykhonov. For more details in the matter we
send the reader to [41].

Assume that ${\cal M}$ is an abstract mathematic object called generic
“problem”, associated to a metric space $X$. Problem ${\cal M}$ could be an
equation, a minimization problem, a fixed point problem, an optimal control
problem, an inclusion or an inequality problem. Its rigorous statement vary
from exemple to example. We associate to ${\cal M}$ the concept of “solution”
which follows from the context and which will be clearly defined in each
exemple we present below in this paper. We now introduce the following
definitions.

###### Definition 6

A Tykhonov triple for Problem ${\cal M}$ is a mathematical object of the form
$\mbox{{${\cal T}$}}=(I,\Omega,\mbox{{${\cal C}$}})$ where $I$ is a given
nonempty set, $\Omega:I\to 2^{X}-\\{\emptyset\\}$ and $\mbox{{${\cal
C}$}}\subset\mbox{{${\cal S}$}}(I)$, $\mbox{{${\cal C}$}}\neq\emptyset$.

Note that in this definition and below in this paper $\mbox{{${\cal S}$}}(I)$
represents the set of sequences of elements of $I$ and $2^{X}$ denotes the set
of parts of the space $X$. A typical element of $I$ will be denoted by
$\theta$ and a typical element of $\mbox{{${\cal S}$}}(I)$ will be denoted by
$\\{\theta_{n}\\}$. We refer to the set $I$ as the set of indices. Moreover,
for any $\theta\in I$ we refer to the set $\Omega(\theta)\subset X$ as an
approximating set and ${\cal C}$ will represent the so-called convergence
criterion.

###### Definition 7

Given a Tykhonov triple $\mbox{{${\cal T}$}}=(I,\Omega,\mbox{{${\cal C}$}})$,
a sequence $\\{u_{n}\\}\subset X$ is called an approximating sequence if there
exists a sequence $\\{\theta_{n}\\}\subset\mbox{{${\cal C}$}}$ such that
$u_{n}\in\Omega(\theta_{n})$ for each $n\in\mathbb{N}$.

Note that approximating sequences always exist since, by assumption,
$\mbox{{${\cal C}$}}\neq\emptyset$ and, moreover, for any sequence
$\\{\theta_{n}\\}\subset\mbox{{${\cal C}$}}$ and any $n\in\mathbb{N}$, the set
$\Omega(\theta_{n})$ is not empty. Therefore, Definition 7 above make sense.

###### Definition 8

Given a Tykhonov triple $\mbox{{${\cal T}$}}=(I,\Omega,\mbox{{${\cal C}$}})$,
Problem ${\cal M}$ is said to be :

a) (strongly) well-posedness if it has a unique solution and every
approximating sequence converges in $X$ to this solution;

b) weakly well-posedness if it has a unique solution and every approximating
sequence converges weakly in $X$ to this solution;

c) weakly generalized well-posedness if it has at least one solution and every
approximating sequence contains a subsequence which converges weakly in $X$ to
some point of the solution set.

We remark that the concept of approximating sequence above depends on the
Tykhonov triple ${\cal T}$ and, for this reason, we sometimes refer to
approximating sequences with respect ${\cal T}$. As a consequence, the
concepts of strongly, weakly and weakly generalized well-posedness depend on
the Tykhonov triple ${\cal T}$ and, therefore, we refer to them as strongly,
weakly and weakly generalized well-posedness with respect ${\cal T}$,
respectively.

##  3 The heat transfer problem

In this section we introduce the heat transfer problem, list the assumptions
on the data, derive its variational formulation and state our optimal control
problem. The problem under consideration is the following.

Problem ${\cal H}$. Find a temperature field $u:D\to{{\rm I}\mkern-3.0mu{\rm
R}}$ such that

$\displaystyle u\geq 0,\qquad-\Delta u-f\geq 0,\qquad u(\Delta
u+f)=0\qquad{\rm a.e.\ in\ }D,$ (3.1) $\displaystyle u=0\hskip 68.2866pt{\rm
a.e.\ on\ }\Gamma_{1},$ (3.2) $\displaystyle u=b\hskip 68.2866pt{\rm a.e.\ on\
}\Gamma_{2},$ (3.3) $\displaystyle-\frac{\partial u}{\partial\nu}\in\partial
j_{\nu}(u)\hskip 22.76219pt{\rm a.e.\ on\ }\Gamma_{3}.$ (3.4)

In (3.1)–(3.4) and below $D$ is a bounded domain in $\mathbb{R}^{d}$
($d=1,2,3$ in applications) with smooth boundary $\partial
D=\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}$ and outer normal unit $\nu$. We
assume that $\Gamma_{1}$, $\Gamma_{2}$, $\Gamma_{3}$ are measurable sets such
that $\Gamma_{i}\cap\Gamma_{j}=\emptyset$ for $i\neq j$ and, moreover,
$meas\,(\Gamma_{1})>0$. In addition, in (3.1)–(3.4) we do not mention the
dependence of the different functions on the spatial variable
$\mbox{\boldmath{$x$}}\in D\cup\partial D$. The functions $f$, $b$ and
$j_{\nu}$ are given and will be described below. Here we restrict ourselves to
mention that $f$ represents the internal energy, $b$ is a the prescribed
temperature field on $\Gamma_{2}$, $\frac{\partial u}{\partial\nu}$ denotes
the normal derivative of $u$ on $\Gamma_{3}$ and $\partial j_{\nu}$ denotes
the Clarke subdifferential of the potential function $j_{\nu}$, assumed to be
locally Lipschitz. Note that Problem ${\cal H}$ represents an extension of the
problem considered in [3], where the boundary condition (3.4) was of the form

$-\frac{\partial u}{\partial\nu}=q\hskip 19.91692pt{\rm a.e.\ \ on\ \
}\Gamma_{3},$ (3.5)

with a given function $q:\Gamma_{3}\to\mathbb{R}$. It is obvious to see that
condition (3.5) can be recovered by condition (3.4) with an appropriate choice
of $j_{\nu}$.

For the variational analysis of Problem ${\cal H}$ we use standard notation
for $C^{1}$, Lebesque and Sobolev spaces. We use the symbols $``\to"$ and
$"\rightharpoonup"$ to indicate the strong and weak convergence in various
spaces which will be indicated below. We also use $``\to"$ for the convergence
in $\mathbb{R}$. Moreover, we shall use the space

$V=\ \\{\,v\in H^{1}(D)\ :\ v=0\quad{\rm on}\quad\Gamma_{1}\,\\},$

endowed with the inner product of the space $H^{1}(D)$, denoted by
$(\cdot,\cdot)_{V}$, and the associated norm, denoted by $\|\cdot\|_{V}$.
Since $meas(\Gamma_{1})>0$, it is well known that $(V,(\cdot,\cdot)_{V})$ is a
real Hilbert space. In addition, by the Friedrichs-Poincaré inequality and the
Sobolev trace theorem we have

$\displaystyle\|v\|_{V}\leq c_{0}\,\|\nabla v\|_{L^{2}(D)^{d}},$ (3.6)
$\displaystyle\|v\|_{L^{2}(\Gamma_{3})}\leq c_{3}\,\|v\|_{V}$ (3.7)

for all $v\in V$, respectively. Here and below in this paper $c_{0}$ and
$c_{3}$ are positive constants which depend on $D$, $\Gamma_{1}$ and
$\Gamma_{3}$. We denote in what follows by $V^{*}$ the dual of $V$, by
$\langle\cdot,\cdot\rangle$ the duality paring between $V^{*}$ and $V$ and by
$0_{V}$ the zero element of $V$.

We now list the assumptions on the data of Problem ${\cal H}$. First, for the
functions $f$ and $b$ we assume that

$\displaystyle f\in L^{2}(D),$ (3.8)
$\displaystyle\left\\{\begin{array}[]{ll}b\in L^{2}(\Gamma_{2})\ \mbox{and
there exists $u_{b}\in V$ such that}\\\ u_{b}\geq 0\ \ {\rm in}\ \ D\ {\rm
and}\ u_{b}=b\ \ {\rm on}\ \ \Gamma_{2}.\end{array}\right.$ (3.11)

Moreover, for the potential function $j_{\nu}$ we assume the following.

$\displaystyle\left\\{\begin{array}[]{ll}j_{\nu}\colon\Gamma_{3}\times\mathbb{R}\to\mathbb{R}\
\mbox{is such that}\\\\[5.69054pt] \ \ {\rm(a)}\ j_{\nu}(\cdot,r)\ \mbox{is
measurable on}\ \Gamma_{3}\ \mbox{for all}\ r\in\mathbb{R}\ \mbox{and
there}\\\ \qquad\mbox{exists}\ {\overline{e}}\in L^{2}(\Gamma_{3})\ \mbox{such
that}\ j_{\nu}(\cdot,{\overline{e}}(\cdot))\in
L^{1}(\Gamma_{3}).\\\\[5.69054pt] \ \ {\rm(b)}\
j_{\nu}(\mbox{\boldmath{$x$}},\cdot)\ \mbox{is locally Lipschitz on}\
\mathbb{R}\ \mbox{for a.e.}\
\mbox{\boldmath{$x$}}\in\Gamma_{3}.\\\\[5.69054pt] \ \ {\rm(c)}\
|\xi|\leq{\overline{c}}_{0}+{\overline{c}}_{1}\,|r|\ \mbox{for a.e.}\
\mbox{\boldmath{$x$}}\in\Gamma_{3},\\\ \qquad\mbox{ for all}\ r\in\mathbb{R},\
\xi\in\partial j_{\nu}(\mbox{\boldmath{$x$}},r),\ \mbox{with}\
{\overline{c}}_{0},\,{\overline{c}}_{1}\geq 0.\\\\[5.69054pt] \ \ {\rm(d)}\
j_{\nu}^{0}(\mbox{\boldmath{$x$}},r_{1};r_{2}-r_{1})+j_{\nu}^{0}(\mbox{\boldmath{$x$}},r_{2};r_{1}-r_{2})\leq\alpha_{j_{\nu}}\,|r_{1}-r_{2}|^{2}\\\
\qquad\mbox{ for a.e.}\ \mbox{\boldmath{$x$}}\in\Gamma_{3},\ \mbox{all}\
r_{1},\,r_{2}\in\mathbb{R},\ \mbox{with}\ \alpha_{j_{\nu}}\geq
0,\\\\[5.69054pt] \ \ {\rm(e)\ either}\ j_{\nu}(\mbox{\boldmath{$x$}},\cdot)\
{\rm or}\ -j_{\nu}(\mbox{\boldmath{$x$}},\cdot)\ {\rm is\ regular\ on}\
\mathbb{R}\ \ {\rm for\ a.e.}\
\mbox{\boldmath{$x$}}\in\Gamma_{3}.\end{array}\right.$ (3.21)

Finally, we assume that the following smallness condition holds:

$\alpha_{j_{\nu}}c_{0}^{2}c_{3}^{2}<1.$ (3.22)

Examples of sets $D$, $\Gamma_{1}$, $\Gamma_{2}$ and functions $b$ for which
such assumption is satisfied can be easily constructed. Below, we restrict
ourselves to the following ones.

###### Example 9

Let $\alpha>0$, $\beta>0$, and let

$\displaystyle D=\\{\,\mbox{\boldmath{$x$}}=(x_{1},x_{2})\in\mathbb{R}^{2}\ :\
0<x_{1}<\alpha,\ \ 0<x_{2}<\beta\,\\},$
$\displaystyle\Gamma_{1}=\\{\,\mbox{\boldmath{$x$}}=(x_{1},x_{2})\in\mathbb{R}^{2}\
:\ x_{1}=0,\ \ 0\leq x_{2}\leq\beta\,\\},$
$\displaystyle\Gamma_{2}=\\{\,\mbox{\boldmath{$x$}}=(x_{1},x_{2})\in\mathbb{R}^{2}\
:\ x_{1}=\alpha,\ \ 0\leq x_{2}\leq\beta\,\\}.$

Assume that $\varphi$ is a positive function with regularity $\varphi\in
C^{1}([0,\beta])$ and consider the functions $b:\Gamma_{2}\to\mathbb{R}$,
$u_{b}:D\to\mathbb{R}$ defined by $b(x_{1},x_{2})=\varphi(x_{2})$,
$u_{b}(x_{1},x_{2})=\frac{x_{1}}{\alpha}\,\varphi(x_{2})$. Then, it is easy to
see that condition $(\ref{b})$ is satisfied.

###### Example 10

Let $\alpha>0$, $\beta>0$, $\gamma>0$ and let

$\displaystyle
D=\\{\,\mbox{\boldmath{$x$}}=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\ :\
0<x_{1}<\alpha,\ \ 0<x_{2}<\beta,\ \ 0<x_{3}<\gamma\\},$
$\displaystyle\Gamma_{1}=\\{\,\mbox{\boldmath{$x$}}=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\
:\ x_{1}=0,\ \ 0\leq x_{2}\leq\beta,\ \ 0\leq x_{3}\leq\gamma\,\\},$
$\displaystyle\Gamma_{2}=\\{\,\mbox{\boldmath{$x$}}=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\
:\ x_{1}=\alpha,\ \ 0\leq x_{2}\leq\beta,\ \ 0\leq x_{3}\leq\gamma\,\\}.$

Assume that $\varphi$ is a positive function with regularity $\varphi\in
C^{1}([0,\beta]\times[0,\gamma])$ and consider the functions
$b:\Gamma_{2}\to\mathbb{R}$, $u_{b}:D\to\mathbb{R}$ defined by
$b(x_{1},x_{2},x_{3})=\varphi(x_{2},x_{3})$,
$u(x_{1},x_{2},x_{3})=\frac{x_{1}}{\alpha}\,\varphi(x_{2},x_{3})$. Then, it is
easy to see that condition $(\ref{b})$ is satisfied.

Note also that various examples of functions $j_{\nu}$ which satisfy condition
(3.21) can be found in [14, 27].

Next, we define the set $K\subset V$, the bilinear form $a:V\times V\to{{\rm
I}\mkern-3.0mu{\rm R}}$ and the function $j:V\to{{\rm I}\mkern-3.0mu{\rm R}}$
by equalities

$\displaystyle K=\ \\{\,v\in V\ :\ v\geq 0\ \ {\rm in}\ \ D,\quad v=b\ \ {\rm
on}\ \ \Gamma_{2}\,\\},$ (3.23) $\displaystyle a(u,v)=\int_{D}\,\nabla
u\cdot\nabla v\,dx\qquad\forall\,u,\,v\in V,$ (3.24) $\displaystyle
j(v)=\int_{\Gamma_{3}}j_{\nu}(v)\,da\qquad\forall\,v\in V.$ (3.25)

It follows from assumption (3.11) that the set $K$ is not empty. Moreover,
Lemma 8 in [37] provides the following result.

###### Proposition 11

Under assumption $(\ref{j})$ the function $j:V\to{{\rm I}\mkern-3.0mu{\rm R}}$
defined by (3.25) satisfies conditions $(\ref{5jj})$ on the space $X=V$ with
$\widetilde{c}_{0}=\sqrt{2\,{\it meas}(\Gamma_{3})}\,{\overline{c}}_{0}c_{3}$,
$\widetilde{c}_{1}=\sqrt{2}\,{\overline{c}}_{1}c_{3}^{2}$ and
$\alpha_{j}=\alpha_{j_{\nu}}c_{3}^{2}$. In addition,

$j^{0}(u;v)=\displaystyle\int_{\Gamma_{3}}j_{\nu}^{0}(u,v)\,da\qquad\forall\,u,\
v\in V.$ (3.26)

Next, we turn to the variational formulation of Problem ${\cal H}$. To this
end we assume that $u$ is a regular function which satisfies (3.1)–(3.4) and
we consider an arbitrary element $v\in K$. Then, using an integration by parts
if follows that

$\int_{D}\nabla u\cdot\nabla(v-u)\,dx+\int_{D}\Delta
u\cdot(v-u)\,dx=\int_{\partial D}\frac{\partial u}{\partial\nu}\,(v-u)\,da.$

Therefore, by (3.1)–(3.3) we find that

$\int_{D}\nabla
u\cdot\nabla(v-u)\,dx\geq\int_{D}f(v-u)\,dx+\int_{\Gamma_{3}}\frac{\partial
u}{\partial\nu}\,(v-u)\,da$ (3.27)

and, using the boundary condition (3.4) combined with definition of the Clarke
subdifferential, we obtain that

$\int_{\Gamma_{3}}\frac{\partial
u}{\partial\nu}\,(v-u)\,da\geq-\int_{\Gamma_{3}}j^{0}_{\nu}(u;v-u)\,da.$
(3.28)

We now combine inequalities (3.27) and (3.28), then use notation (3.24) and
equality (3.26) to see that

$a(u,v-u)+j^{0}(u;v-u)\geq(f,v-u)_{L^{2}(D)}.$

Finally, using this inequality and the regularity $u\in K$, guaranteed by
(3.1)–(3.3), we deduce the following variational formulation of Problem ${\cal
H}$.

Problem ${\cal P}$. Find $u\in V$ such that

$u\in K,\quad
a(u,v-u)+j^{0}(u;v-u)\geq(f,v-u)_{L^{2}(\Omega)}\quad\forall\,v\in K.$ (3.29)

We refer to inequality (3.29) as a hemivariational inequality with unilateral
constraint. A function $u\in V$ which satisfies (3.29) and is called a weak
solution to the heat transfer problem ${\cal H}$.

We now introduce the set of admissible pairs for inequality Problem ${\cal P}$
defined by

${\cal V}_{ad}=\\{\,(u,f)\in K\times L^{2}(D)\ \mbox{such that}\ (\ref{hv})\
\mbox{holds}\,\\}.$ (3.30)

We also assume that $\mbox{{${\cal L}$}}:V\times L^{2}(D)\to{{\rm
I}\mkern-3.0mu{\rm R}}$ is a given cost functional which satisfies the
following conditions.

$\displaystyle\left\\{\begin{array}[]{l}\mbox{For all sequences
}\\{u_{n}\\}\subset V\mbox{ and }\\{f_{n}\\}\subset L^{2}(D)\mbox{ such
that}\\\\[5.69054pt] u_{n}\rightarrow u\mbox{\ \ in\ \ }V,\ \
f_{n}\rightharpoonup f\mbox{\ \ in\ \ }L^{2}(D),\ \mbox{we
have}\\\\[8.53581pt] \displaystyle\liminf_{n\to\infty}\,{\cal
L}(u_{n},f_{n})\geq{\cal L}(u,f).\par\end{array}\right.$ (3.34)
$\displaystyle\left\\{\begin{array}[]{l}\mbox{ There exists}\
h:L^{2}(D)\to{{\rm I}\mkern-3.0mu{\rm R}}\ \mbox{such that}\\\\[5.69054pt]
{\rm(a)}\ \ {\cal L}(u,f)\geq h(f)\quad\forall\,u\in V,\ f\in
L^{2}(D).\\\\[5.69054pt] {\rm(b)}\ \ \|f_{n}\|_{L^{2}(D)}\to+\infty\
\Longrightarrow\ h(f_{n})\to\infty.\end{array}\right.$ (3.38)

###### Example 12

A typical example of function ${\cal L}$ which satisfies conditions
$(\ref{L1})$ and $(\ref{L2})$ is obtained by taking

${\cal L}(u,f)=g(u)+h(f)\qquad\forall\,u\in V,\ f\in L^{2}(D),$

where $g:V\to{{\rm I}\mkern-3.0mu{\rm R}}$ is a continuous positive function
and $h:L^{2}(D)\to{{\rm I}\mkern-3.0mu{\rm R}}$ is a weakly lower
semicontinuous coercive function, i.e., it satisfies condition
$(\ref{L2}){\rm(b)}$.

We also assume that $\mbox{{${\cal L}$}}:V\times L^{2}(D)\to{{\rm
I}\mkern-3.0mu{\rm R}}$ is a given cost function and we associate to Problem
${\cal P}$ the following optimal control problem.

Problem ${\cal Q}$. Find $(u^{*},f^{*})\in{\cal V}_{ad}$ such that

$\mbox{{${\cal L}$}}(u^{*},f^{*})=\min_{(u,f)\in{\cal V}_{ad}}\mbox{{${\cal
L}$}}(u,f).$ (3.39)

Under the previous assumptions, the unique solvability of Problem ${\cal P}$
as well as the solvability of Problem ${\cal Q}$ can be obtained by using
standard arguments, already used in [37] and [23, 24], respectively. Our aim
in what follows is to study the well-posedness of these problems and to derive
some consequences. To this end, as already mentionned in Section 2, for each
problem we need to prescribe a Tykhonov triple, based on three ingredients: a
set of indices, a familly of approximating sets and a convergence criterion
for the sequences of indices.

##  4 A well-posedness result

This section is devoted to the well-posedness of the Problem ${\cal P}$, in
the sense precised in Section 2, with $X=V$. In order to construct an
appropriate Tykhonov triple for this problem, we consider a set
$\widetilde{K}$, a penalty operator $G:V\to V^{*}$ and a penalty parameter
$\lambda$ such that the following conditions hold.

$\displaystyle\left\\{\begin{array}[]{ll}\mbox{\rm(a)}\quad\mbox{{$\widetilde{K}$}}\
\mbox{\rm is a closed convex subset of}\ V.\\\\[5.69054pt] \mbox{\rm(b)}\quad
K\subset\mbox{{$\widetilde{K}$}}.\end{array}\right.$ (4.3)
$\displaystyle\left\\{\begin{array}[]{ll}\mbox{\rm(a)}\quad G:V\to V^{*}\ \
\mbox{is a bounded, demicontinuous and monotone operator}.\\\\[5.69054pt]
\mbox{\rm(b)}\quad\langle Gu,v-u\rangle\leq
0\qquad\forall\,u\in\mbox{{$\widetilde{K}$}},\ v\in K.\\\\[5.69054pt]
\mbox{\rm(c)}\quad u\in\mbox{{$\widetilde{K}$}},\quad\langle
Gu,v-u\rangle=0\quad\forall\,v\in K\ \ \Longrightarrow\ \ u\in
K.\end{array}\right.$ (4.7)

We start with the following result.

###### Proposition 13

Assume that $(\ref{b})$–$(\ref{sm})$, $(\ref{wK})$ and $(\ref{G})(a)$ hold.
Then for each $\lambda>0$ and $f\in L^{2}(D)$ there exists a unique element
$u=u(\lambda,f)$ such that

$u\in\mbox{{$\widetilde{K}$}},\quad a(u,v-u)+\frac{1}{\lambda}\,\langle
Gu,v-u\rangle+j^{0}(u,;v-u)\geq({f},v-u)_{L^{2}(D)}\quad\forall\,v\in\mbox{{$\widetilde{K}$}}.$
(4.8)

Proof. Define the operator $A:V\to V^{*}$ by equality

$\langle Au,v\rangle=a(u,v)+\frac{1}{\lambda}\,\langle
Gu,v\rangle\qquad\forall\,u,\ v\in V.$

Then, using the definition (3.24) of the form $a$, inequality (3.6) and the
properties (4.7)(a) of the operator $G$, it is easy to see that the operator
$A$ is bounded, demicontinuous and, moreover,

$\langle Au-
Av,u-v\rangle\geq\frac{1}{c_{0}^{2}}\,\|u-v\|_{V}^{2}\qquad\forall\,u,\,v\in
V.$ (4.9)

Therefore, it follows from Proposition 4 that $A$ is pseudomonotone. In
addition, inequality (4.9) shows that $A$ is strongly monotone with constant
$m_{A}=\frac{1}{c_{0}^{2}}$. On the other hand, the functional
$v\mapsto({f},v)_{L^{2}(D)}$ is linear and continuous on $V$ and, therefore,
it defines an element in $V^{*}$. We note that assumption (3.11) implies that
$K\neq\emptyset$, hence inclusion (4.3)(b) guarantees that
$\mbox{{$\widetilde{K}$}}\neq\emptyset$. Moreover, recall assumption (4.3)(a),
Proposition 11 and the smallness assumption (3.22). All these ingradients
allow us to apply Theorem 5 in order to deduce the unique solvabilty of
inequality (4.8), which concludes the proof. $\Box$

We now take $\mbox{{${\cal T}$}}=(I,\Omega,\mbox{{${\cal C}$}})$ where

$\displaystyle I=\\{\,\theta=(\lambda,{f})\ :\ \lambda>0,\ {f}\in
L^{2}(\Omega)\,\\},$ (4.10)
$\displaystyle\Omega(\theta)=\\{\,u\in\mbox{{$\widetilde{K}$}}\ {\rm such\
that}\ (\ref{hvp})\ {\rm holds}\,\\}\qquad\forall\,\theta=(\lambda,f)\in I,$
(4.11) $\displaystyle\mbox{{${\cal C}$}}=\\{\,\\{\theta_{n}\\}\ :\
\theta_{n}=(\lambda_{n},f_{n})\in I\ \ \forall\,n\in\mathbb{N},\ \
\lambda_{n}\to 0,\ \ f_{n}\rightharpoonup f\ \ {\rm in}\ L^{2}(D)\ {\rm as}\
n\to\infty\,\\}.$ (4.12)

Then, using Proposition 13 we see that for each $\theta=(\lambda,f)$ the set
$\Omega(\theta)$ defined by (4.11) is not empty and, therefore, the triple
(4.10)–(4.12) is a Tykhonov triple in the sense of Definition 6.

Our main result in this section is the following.

###### Theorem 14

Assume that $(\ref{f})$–$(\ref{sm})$, $(\ref{wK})$ and $(\ref{G})$ hold. Then
Problem ${\cal P}$ is well-posed with respect to the Tykhonov triple
$(\ref{I})$–$(\ref{C})$.

Proof. Following Definition 8 a), the proof is carried out in two main steps.

i) Unique solvability of Problem ${\cal P}$. Note that assumptions (4.3) and
(4.7)(a) are satisfied if $\mbox{{$\widetilde{K}$}}=K$ and $Gv=0_{V}$ for all
$v\in V$, respectively. Moreover, with this particular choice inequality (4.8)
reduces to inequality (3.29), for any $\lambda>0$. Therefore, the existence of
a unique solution $u\in K$ to Problem ${\cal P}$ is a direct consequence of
Proposition 13.

ii) Convergence of approximating sequences. To proceed, we consider an
approximating sequence for the Problem ${\cal P}$, denoted by $\\{u_{n}\\}$.
Then, according to Definition 7 it follows that there exists a sequence
$\\{\theta_{n}\\}$ of elements of $I$, with $\theta_{n}=(\lambda_{n},f_{n})$,
such that $u_{n}\in\Omega(\theta_{n})$ for each $n\in\mathbb{N}$ and,
moreover,

$\displaystyle\lambda_{n}\to 0,$ (4.13) $\displaystyle f_{n}\rightharpoonup
f\quad{\rm in}\quad L^{2}(D).$ (4.14)

Note that the inclusion $u_{n}\in\Omega(\theta_{n})$ combined with definition
(4.11) imples that for each $n\in\mathbb{N}$ the following inequality holds:

$\displaystyle u_{n}\in\mbox{{$\widetilde{K}$}},\quad
a(u_{n},v-u_{n})+\frac{1}{\lambda_{n}}\,\langle
Gu_{n},v-u_{n}\rangle+j^{0}(u_{n},;v-u_{n})$ (4.15)
$\displaystyle\qquad\qquad\geq({f}_{n},v-u_{n})_{L^{2}(D)}\quad\forall\,v\in\mbox{{$\widetilde{K}$}}.$

Our aim in what follows is to prove the convergence

$u_{n}\to u\quad{\rm in}\quad V,\quad{\rm as}\quad n\to\infty.$ (4.16)

To this end we proceed in three intermediate steps that we present below.

ii-a) A first weak convergence result. We claim that there is an element
${\widetilde{u}}\in\mbox{{$\widetilde{K}$}}$ and a subsequence of
$\\{u_{n}\\}$, still denoted by $\\{u_{n}\\}$, such that
$u_{n}\rightharpoonup{\widetilde{u}}$ in $X$, as $n\to\infty$.

To prove the claim, we establish the boundedness of the sequence $\\{u_{n}\\}$
in $V$. Let $n\in\mathbb{N}$ and let $u$ be the solution of Problem ${\cal
P}$. We use assumption (4.3)(b) and take $v=u$ in (4.15) to see that

$\displaystyle a(u_{n},u_{n}-u)\leq\frac{1}{\lambda_{n}}\,\langle
Gu_{n},u-u_{n}\rangle+j^{0}(u_{n},u-u_{n})+(f_{n},u_{n}-u)_{L^{2}(D)}.$

Then, using inequalities (3.6) and (4.7)(b) we obtain that

$\displaystyle\frac{1}{c_{0}^{2}}\,\|u_{n}-u\|_{V}^{2}\leq
a(u,u-u_{n})+j^{0}(u_{n};u-u_{n})+(f_{n},u_{n}-u)_{L^{2}(D)}.$ (4.17)

On the other hand, by Proposition 2(b) we have

$\displaystyle
j^{0}(u_{n};u-u_{n})=j^{0}(u_{n};u-u_{n})+j^{0}(u;u_{n}-u)-j^{0}(u;u_{n}-u)$
$\displaystyle\qquad\leq
j^{0}(u_{n};u-u_{n})+j^{0}(u;u_{n}-u)+|j^{0}(u;u_{n}-u)|$
$\displaystyle\quad\qquad=j^{0}(u_{n};u-u_{n})+j^{0}(u;u_{n}-u)+|\max\,\\{\,\langle\xi,u_{n}-u\rangle\,:\,\xi\in\partial
j(u)\,\\}|$

and, using Proposition 11, we deduce that

$\displaystyle
j^{0}(u_{n};u-u_{n})\leq\alpha_{j_{\nu}}c_{3}^{2}\|u_{n}-u\|_{V}^{2}+(\widetilde{c}_{0}+\widetilde{c}_{1}\|u\|_{V})\|u_{n}-u\|_{V}.$
(4.18)

Finally, note that

$a(u,u-u_{n})+(f_{n},u_{n}-u)_{L^{2}(D)}\leq\big{(}\|u\|_{V}+\|f_{n}\|_{L^{2}(D)}\big{)}\|u_{n}-u\|_{V}.$
(4.19)

We now combine inequalities (4.17)–(4.19) to see that

$\displaystyle\frac{1}{c_{0}^{2}}\,\|u_{n}-u\|_{V}^{2}\leq\big{(}\|u\|_{V}+\|f_{n}\|_{L^{2}(D)}\big{)}\big{)}\|u_{n}-u\|_{V}$
(4.20)
$\displaystyle\quad\qquad+\alpha_{j_{\nu}}c_{3}^{2}\|u_{n}-u\|_{V}^{2}+(\widetilde{c}_{0}+\widetilde{c}_{1}\|u\|_{V})\|u_{n}-u\|_{V}.$

Note that by (4.14) we know that the sequence $\\{f_{n}\\}$ is bounded in
$L^{2}(D)$. Therefore, using inequality (4.20) and the smallness assumption
(3.22), we deduce that there exists a constant $C>0$ independent of $n$ such
that $\|u_{n}-u\|_{X}\leq C$. This implies that the sequence $\\{u_{n}\\}$ is
bounded in $V$. Thus, from the reflexivity of $V$, by passing to a
subsequence, if necessary, we deduce that

$u_{n}\rightharpoonup{\widetilde{u}}\ \ {\rm in}\ \ V,\ \ \mbox{as}\ \
n\to\infty,$ (4.21)

with some ${\widetilde{u}}\in V$. Moreover, assumption (4.3)(a) and the
convergence (4.21) imply that
$\mbox{{$\widetilde{u}$}}\in\mbox{{$\widetilde{K}$}}$, which completes the
proof of the claim.

ii-b) A property of the weak limit. Next, we show that ${\widetilde{u}}$ is a
solution to Problem ${\cal P}$.

Let $v$ be a given element in $\widetilde{K}$ and let $n\in\mathbb{N}$. We use
(4.15) to obtain that

$\displaystyle\frac{1}{\lambda_{n}}\,\langle Gu_{n},u_{n}-v\rangle\leq
a(u_{n},v-u_{n})+j^{0}(u_{n},v-u_{n})+(f_{n},u_{n}-v)_{L^{2}(D)}.$ (4.22)

Then, using arguments similar to those used in the proof of (4.18), (4.19) and
the boundedness of the sequence $\\{u_{n}\\}$, we deduce that each term in the
right hand side of inequality (4.22) is bounded. This implies that there
exists a constant $M_{0}>0$ which does not depend on $n$ such that

$\langle Gu_{n},u_{n}-v\rangle\leq\lambda_{n}M_{0}.$

We now pass to the upper limit in this inequality and use the convergence
(4.13) to deduce that

$\limsup\,\langle Gu_{n},u_{n}-v\rangle\leq 0.$ (4.23)

We now take $v=\mbox{{$\widetilde{u}$}}$ in (4.23) and find that

$\limsup\,\langle Gu_{n},u_{n}-\mbox{{$\widetilde{u}$}}\rangle\leq 0.$ (4.24)

Therefore, using the pseudomonotonicity of the operator $G$, guaranted by
assumption (4.7)(a) and Proposition 4, we obtain that

$\liminf\,\langle Gu_{n},u_{n}-v\rangle\geq\langle
G\mbox{{$\widetilde{u}$}},\mbox{{$\widetilde{u}$}}-v\rangle.$ (4.25)

We now combine inequalities (4.25) and (4.23) to find that $\langle
G{\widetilde{u}},{\widetilde{u}}-v\rangle\leq 0$. On the other hand, by
(4.7)(b) we deduce that $\langle G{\widetilde{u}},{\widetilde{u}}-v\rangle\geq
0$. We conclude from above that

$\langle G{\widetilde{u}},{\widetilde{u}}-v\rangle=0$

and we recall that this equality holds for all $v\in\widetilde{K}$. Then,
using assumptions (4.3)(b), (4.7)(c), we find that ${\widetilde{u}}\in K$.

Next, we use (4.15), again, to obtain that

$\displaystyle a(u_{n},u_{n}-v)\leq\frac{1}{\lambda_{n}}\,\langle
Gu_{n},v-u_{n}\rangle+j^{0}(u_{n},v-u_{n})+(f_{n},u_{n}-v)_{L^{2}(D)}.$

Therefore, using assumption (4.7)(b) we find that

$a(u_{n},u_{n}-v)\leq j^{0}(u_{n},v-u_{n})+(f_{n},u_{n}-v)_{L^{2}(D)}.$ (4.26)

Moreover, from (4.21), (3.26), the compactness of the trace and the properties
of the integral it follows that

$\limsup j^{0}(u_{n};v-u_{n})\leq j^{0}(\widetilde{u},v-\widetilde{u})$ (4.27)

and, in addition,

$(f_{n},u_{n}-v)_{L^{2}(D)}\to(f,\mbox{{$\widetilde{u}$}}-v)_{L^{2}(D)}.$
(4.28)

We now gather relations (4.26)–(4.28) to see that

$\limsup\,a(u_{n},u_{n}-v)\leq
j^{0}({\widetilde{u}},v-{\widetilde{u}})+(f,\widetilde{u}-v)_{L^{2}(D)}.$
(4.29)

Now, taking $v={\widetilde{u}}\in K$ in (4.29) and using Proposition 2(a) we
obtain that

$\limsup\,a(u_{n},u_{n}-{\widetilde{u}})\leq 0.$ (4.30)

On the other hand, using the properties of the form $a$ we have that

$a(u_{n},v)\to a(\widetilde{u},v)\quad{\rm as}\quad n\to\infty$ (4.31)

and, since $a(u_{n}-\widetilde{u},u_{n}-\widetilde{u})\geq 0$, we deduce that

$a(u_{n},u_{n})\geq
a(\widetilde{u},u_{n})+a(u_{n},\widetilde{u})-a(\widetilde{u},\widetilde{u}),$

for each $n\in\mathbb{N}$. Using now (4.31) and the convergence (4.21) we find
that

$\liminf_{n\to\infty}\,a(u_{n},u_{n})\geq a(\widetilde{u},\widetilde{u}).$
(4.32)

Therefore, combining (4.31), (4.32) we obtain that

$a({\widetilde{u}},{\widetilde{u}}-v)\leq\liminf\,a(u_{n},u_{n}-v)$ (4.33)

and, using (4.33) and (4.29) we have

$a({\widetilde{u}},{\widetilde{u}}-v)\\\ \leq
j^{0}(\widetilde{u},v-\widetilde{u})+(f,\widetilde{u}-v)_{L^{2}(D)}.$

It follows from here that ${\widetilde{u}}\in K$ is a solution to Problem
${\cal P}$, as claimed.

ii-c) A second weak convergence result. We now prove the weak convergence of
the whole sequence $\\{u_{n}\\}$.

Since Problem ${\cal P}$ has a unique solution $u\in K$, we deduce from the
previous step that ${\widetilde{u}}=u$. Moreover, a careful analysis of the
proof in step ii-b) reveals that every subsequence of $\\{u_{n}\\}$ which
converges weakly in $V$ has the weak limit $u$. In addition, we recall that
the sequence $\\{u_{n}\\}$ is bounded in $V$. Therefore, using a standard
argument we deduce that the whole sequence $\\{u_{n}\\}$ converges weakly in
$V$ to $u$, as $n\to\infty$, i.e.,

${u}_{n}\rightharpoonup{u}\quad\mbox{ \rm{in} }\ V\quad\mbox{ \rm{as} }\
n\to\infty.$ (4.34)

ii-d) Strong convergence.  In the final step of the proof, we prove that
$u_{n}\to u$ in $V$, as $n\to\infty$.

We take $v={\widetilde{u}}\in K$ in (4.33) and use (4.30) to obtain

$0\leq\liminf\,a(u_{n},u_{n}-{\widetilde{u}})\leq\limsup\,a(u_{n},u_{n}-{\widetilde{u}})\leq
0,$

which shows that $a(u_{n},u_{n}-{\widetilde{u}})\to 0$, as $n\to\infty$.
Therefore, using equality $\widetilde{u}=u$, the coercivity of $a$ and the
convergence $u_{n}\rightharpoonup u$ in $V$, we have

$\frac{1}{c_{0}^{2}}\,\|u_{n}-u\|_{V}^{2}\leq
a(u_{n}-u,u_{n}-u)=a(u_{n},u_{n}-u)-a(u,u_{n}-u)\to 0,$

as $n\to\infty$. Hence, it follows that $u_{n}\to u$ in $V$, which completes
the proof. $\Box$

We now reformulate the convergence result obtained above. To this end, we
denote by $u(\lambda_{n},f_{n})$ and $u(f)$ the solution of the
hemivariational inequality (4.15) and (3.29), respectively. Then, if (4.13)
and (4.14) hold, the step ii) of the proof of Theorem 18 implies that

$u(\lambda_{n},f_{n})\to u(f)\quad{\rm in}\quad V.$ (4.35)

We shall use this convergence result in various places below in this paper.

We end this section with the following consequences of Theorem 14.

###### Corollary 15

Assume that $(\ref{b})$–$(\ref{sm})$, $(\ref{wK})$ and $(\ref{G})$ hold. Then
the operator $f\mapsto u$ which associates to every $f\in L^{2}(D)$ the
solution $u=u(f)\in V$ of inequality $(\ref{hv})$ is weakly-strongly
continuous, i.e.,

$f_{n}\rightharpoonup f\quad{\rm in}\quad L^{2}(D)\quad\Longrightarrow\quad
u(f_{n})\to u(f)\quad{\rm in}\quad V.$ (4.36)

Proof. Let $\\{f_{n}\\}\subset L^{2}(D)$ be a sequence such that
$f_{n}\rightharpoonup f\quad{\rm in}\quad L^{2}(D)$. We take
$\mbox{{$\widetilde{K}$}}=K$, $Gv=0_{V}$ for all $v\in V$ and let
$\\{\lambda_{n}\\}\subset{{\rm I}\mkern-3.0mu{\rm R}}$ be a sequence such that
$\lambda_{n}>0$ for each $n\in\mathbb{N}$ and $\lambda_{n}\to 0$. Then, it
follows that $u(\lambda_{n},f_{n})=u(f_{n})$ for each $n\in\mathbb{N}$ and,
therefore, the convergence (4.36) is a direct consequence of the convergence
(4.35). $\Box$

###### Corollary 16

Assume that $(\ref{b})$–$(\ref{sm})$, $(\ref{wK})$ and $(\ref{G})$ hold. Then,
for any $f\in L^{2}(D)$ the solution $u(\lambda,f)\in V$ of inequality
$(\ref{hvp})$ converges to the solution $u(f$) of inequality $(\ref{hv})$ as
$\lambda\to 0$, that is

$\lambda_{n}>0,\quad\lambda_{n}\to 0\quad\Longrightarrow\quad
u(\lambda_{n},f)\to u(f)\quad{\rm in}\quad V.$ (4.37)

Proof. The convergence (4.37) is a direct consequence of the convergence
(4.35). $\Box$

Note that in the case when $\mbox{{$\widetilde{K}$}}=V$ inequality (4.8)
represents a hemivariational inequality without constraints. In this
particular case Corollary 16 can be recovered by using convergence results for
penalty variational-hemivariational inequalities obtained in [37, 44].
Nevertheless, we stress that the functional framework used in the above
mentionned papers does not allow the use of these results in the case when
$\mbox{{$\widetilde{K}$}}\neq K$.

##  5 Weakly generalized well-posedness results

This section is devoted to the weakly generalized well-posedness of the
Problem ${\cal Q}$ in the sense precised in Section 2, with $X=V\times
L^{2}(D)$. We shall construct a relevant example of Tykhonov triple, which
will play a crucial role in the study of Problem ${\cal Q}$. To this end,
everywhere in this section we assume that $(\ref{b})$–$(\ref{sm})$,
$(\ref{L1})$, $(\ref{L2})$, $(\ref{wK})$ and $(\ref{G})$ hold. Then, for any
$\lambda>0$ and $f\in L^{2}(\Omega)$ we consider the hemivariatinal inequality
(4.8), which represents a perturbation of the hemivariational inequality
(3.29). Therefore, by analogy with (3.30), we introduce the set of perturbed
admissible pairs for inequality Problem ${\cal P}$, defined by

${\cal V}_{ad}^{\lambda}=\\{\,(u,f)\in\mbox{{$\widetilde{K}$}}\times L^{2}(D)\
\mbox{such that}\ (\ref{hvp})\ \mbox{holds}\,\\}.$ (5.1)

Moreover, for each $\mu\geq 0$ let $\mbox{{${\cal L}$}}_{\mu}:V\times
L^{2}(D)\to{{\rm I}\mkern-3.0mu{\rm R}}$ be a perturbation of the cost
functional, assumed to satisfy conditions (3.34) and (3.38), i.e.,

$\displaystyle\left\\{\begin{array}[]{l}\mbox{For all sequences
}\\{u_{n}\\}\subset V\mbox{ and }\\{f_{n}\\}\subset L^{2}(D)\mbox{ such
that}\hskip 68.2866pt\\\\[5.69054pt] u_{n}\rightarrow u\mbox{\ \ in\ \ }V,\ \
f_{n}\rightharpoonup f\mbox{\ \ in\ \ }L^{2}(D),\ \mbox{we
have}\\\\[8.53581pt] \displaystyle\liminf_{n\to\infty}\,{\cal
L}_{\mu}(u_{n},f_{n})\geq{\cal L}_{\mu}(u,f).\end{array}\right.$ (5.5)
$\displaystyle\left\\{\begin{array}[]{l}{\rm(a)}\ \ {\cal L}_{\mu}(u,f)\geq
h(f)\quad\forall\,u\in V,\ f\in L^{2}(D).\\\\[5.69054pt] {\rm(b)}\ \
\|f_{n}\|_{L^{2}(D)}\to+\infty\ \Longrightarrow\
h(f_{n})\to\infty.\end{array}\right.$ (5.8)

In addition, we assume that the following properties hold.

$\displaystyle\left\\{\begin{array}[]{l}\mbox{For all sequences
}\\{u_{n}\\}\subset V,\ \ \\{f_{n}\\}\subset L^{2}(D)\mbox{ and
}\\{\mu_{n}\\}\subset{{\rm I}\mkern-3.0mu{\rm R}}_{+}\mbox{ such
that}\\\\[5.69054pt] u_{n}\rightarrow u\mbox{\ \ in\ \ }V,\ \
f_{n}\rightharpoonup f\mbox{\ \ in\ \ }L^{2}(D),\ \mu_{n}\to 0\ \mbox{we
have}\\\\[8.53581pt] \displaystyle\liminf_{n\to\infty}\,{\cal
L}_{\mu_{n}}(u_{n},f_{n})\geq{\cal L}(u,f).\end{array}\right.$ (5.12)
$\displaystyle\left\\{\begin{array}[]{l}\mbox{For all sequences
}\\{u_{n}\\}\subset V\mbox{ and }\\{\mu_{n}\\}\subset{{\rm I}\mkern-3.0mu{\rm
R}}_{+}\mbox{ such that}\\\\[5.69054pt] u_{n}\rightarrow u\mbox{\ \ in\ \ }V,\
\mu_{n}\to 0\ \ \mbox{\rm and for any}\ f\in L^{2}(D)\ \mbox{we
have}\\\\[5.69054pt] \displaystyle\lim_{n\to\infty}\,{\cal
L}_{\mu_{n}}(u_{n},f)={\cal L}(u,f).\end{array}\right.$ (5.16)

An example of functionals ${\cal L}_{\mu}$ and ${\cal L}$ which satisfy
conditions (5.12) and (5.16) will be provided in the next section. Here we
restrict ourselves to remark that condition (5.16) implies that for $\mu=0$
the functionals ${\cal L}_{\mu}$ and ${\cal L}$ are the same, i.e.,

${\cal L}_{0}(u,f)={\cal L}(u,f)\qquad\forall\,u\in V,\ f\in L^{2}(D).$ (5.17)

It follows from here that if assuming conditions (5.16), (5.5) and (5.8) for
each $\mu\geq 0$ implies that conditions (3.34) and (3.38) hold, too.

We start with the following result.

###### Proposition 17

Assume $(\ref{b})$–$(\ref{sm})$, $(\ref{wK})$ and $(\ref{G})$. Then, for each
$\lambda>0$ and each $\mu\geq 0$ such that $(\ref{L1m})$ and $(\ref{L2m})$
hold, there exists a pair $(u^{*},f^{*})$ such that

$(u^{*},f^{*})\in{\cal V}^{\lambda}_{ad},\qquad\mbox{{${\cal
L}$}}_{\mu}(u^{*},f^{*})=\min_{(u,f)\in{\cal V}^{\lambda}_{ad}}\mbox{{${\cal
L}$}}_{\mu}(u,f).$ (5.18)

Proof. We use standard arguments that we resume in the following. Let
$\lambda>0$, $\mu\geq 0$, let

$\omega=\inf_{(u,f)\in{\cal V}^{\lambda}_{ad}}{\cal
L}_{\mu}(u,f)\in\mathbb{R}$ (5.19)

and let $\\{(u_{n},f_{n})\\}\subset{\cal V}^{\lambda}_{ad}$ be a minimizing
sequence for the functional ${\cal L}_{\mu}$, i.e.,

$\lim_{n\to\infty}\,{\cal L}_{\mu}(u_{n},f_{n})=\omega.$ (5.20)

We claim that the sequence $\\{f_{n}\\}$ is bounded in $L^{2}(D)$. Arguing by
contradiction, assume that $\\{f_{n}\\}$ is not bounded in $L^{2}(D)$. Then,
passing to a subsequence still denoted by $\\{f_{n}\\}$, we have

$\|f_{n}\|_{L^{2}(D)}\to+\infty\quad\text{as}\quad n\to+\infty$ (5.21)

and, using (5.8) it turns that

$\lim_{n\to+\infty}{\cal L}_{\mu}(u_{n},f_{n})=+\infty.$ (5.22)

Equalities (5.20) and (5.22) imply that $\omega=+\infty$ which is in
contradiction with (5.19).

We conclude from above that the sequence $\\{f_{n}\\}$ is bounded in
$L^{2}(D)$, as clamed. Therefore, we deduce that there exists $f^{*}\in
L^{2}(D)$ such that, passing to a subsequence still denoted by $\\{f_{n}\\}$,
we have

$f_{n}\rightharpoonup f^{*}\quad\text{in}\quad L^{2}(D)\quad\text{as}\quad
n\to+\infty.$ (5.23)

Let $u^{*}$ be the solution of the hemivariational inequality (4.8) for
$f=f^{*}$. Then, by the definition (5.1) of the set ${\cal V}^{\lambda}_{ad}$
we have

$(u^{*},f^{*})\in{\cal V}^{\lambda}_{ad}.$ (5.24)

Moreover, using the convergence (5.23) and Corollary 15 it follows that

$u_{n}\to u^{*}\quad\text{in}\quad V\quad\text{as}\quad n\to+\infty.$ (5.25)

We now use the convergences (5.23), (5.25) and assumption (5.5), to deduce
that

$\liminf\limits_{n\to+\infty}{\cal L}_{\mu}(u_{n},f_{n})\geq{\cal
L}_{\mu}(u^{*},f^{*}).$ (5.26)

It follows from (5.20) and (5.26) that

$\omega\geq{\cal L}_{\mu}(u^{*},f^{*}).$ (5.27)

In addition, (5.24) and (5.19) yield

$\omega\leq{\cal L}_{\mu}(u^{*},f^{*}).$ (5.28)

We combine inequalities (5.27), (5.28) then we use (5.24) see that (5.18)
holds, which concludes the proof. $\Box$

We now take $\mbox{{${\cal T}$}}=(I,\Omega,\mbox{{${\cal C}$}})$ where

$\displaystyle I=\\{\,\theta=(\lambda,\mu),\ :\ \lambda>0,\ \mu\geq 0\,\\},$
(5.29) $\displaystyle\Omega(\theta)=\\{\,(u^{*},f^{*})\in V\times L^{2}(D)\
{\rm such\ that}\ (\ref{3d})\ {\rm
holds}\,\\}\qquad\forall\,\theta=(\lambda,\mu)\in I,$ (5.30)
$\displaystyle\mbox{{${\cal C}$}}=\\{\,\\{\theta_{n}\\}\ :\
\theta_{n}=(\lambda_{n},\mu_{n})\in I\ \ \forall\,n\in\mathbb{N},\ \
\lambda_{n}\to 0,\ \ \mu_{n}\to 0\ \ \ {\rm as}\ n\to\infty\,\\}.$ (5.31)

Then, using Proposition 17 we see that for each $\theta=(\lambda,\mu)$ the set
$\Omega(\theta)$ defined defined by (3.39) is not empty and, therefore, the
triple (5.29)–(5.31) is a Tykhonov triple in the sense of Definition 6.

Our main result in this section is the following.

###### Theorem 18

Assume $(\ref{b})$–$(\ref{sm})$, $(\ref{wK})$, $(\ref{G})$, $(\ref{L3})$,
$(\ref{L4})$ and, for all $\mu\geq 0$, assume that $(\ref{L1m})$,
$(\ref{L2m})$ hold. Then Problem ${\cal Q}$ is weakly generalized well-posed
with respect to the Tykhonov triple $(\ref{Ic})$–$(\ref{Cc})$.

Proof. Following Definition 8 the proof is carried out in two main steps.

i) Solvability of Problem ${\cal Q}$. Recall that for
$\mbox{{$\widetilde{K}$}}=K$ and $Gv=0_{V}$, for all $v\in V$ inequality (4.8)
reduces to inequality (3.29). Moreover, (5.17) shows that for $\mu=0$ we have
${\cal L}_{\mu}={\cal L}$. Therefore, the existence of at least one solution
to Problem ${\cal Q}$ is a direct consequence of Proposition 17.

ii) Convergence of approximating sequences. To proceed, we consider an
approximating sequence for the Problem ${\cal Q}$, denoted by
$\\{(u_{n}^{*},f_{n}^{*})\\}$. Then, according to Definition 7 it follows that
there exists a sequence $\\{\theta_{n}\\}$ of elements of $I$, with
$\theta_{n}=(\lambda_{n},\mu_{n})$, such that
$(u_{n},f_{n})\in\Omega(\theta_{n})$ for each $n\in\mathbb{N}$, (4.13) holds
and, moreover,

$\mu_{n}\to 0.$ (5.32)

In order to simplify the notation, for each $n\in\mathbb{N}$ we write ${\cal
V}^{n}_{ad}$ and ${\cal L}_{n}$ instead of ${\cal V}^{\lambda_{n}}_{ad}$ and
${\cal L}_{\mu_{n}}$, respectively. Then, exploiting the definition (5.30) we
deduce that $(u^{*}_{n},f_{n}^{*})\in{\cal V}_{ad}^{n}$ and

$\displaystyle{\cal L}_{n}(u^{*}_{n},f^{*}_{n})\leq{\cal L}_{n}(u_{n},f_{n})$
(5.33)

for each couple of functions $(u_{n},f_{n})\in{\cal V}_{ad}^{n}$, i.e., for
each couple of functions $(u_{n},f_{n})\in V\times L^{2}(D)$ which satisfies
inequality (4.8) in which $f$ is replaced by $f_{n}$, for each
$n\in\mathbb{N}$.

We shall prove that there exists a subsequence of the sequence
$\\{(u_{n}^{*},f_{n}^{*})\\}$, again denoted by $\\{(u_{n}^{*},f_{n}^{*})\\}$,
and an element $(u^{*},f^{*})\in V\times L^{2}(D)$ such that

$\displaystyle f_{n}^{*}\rightharpoonup f^{*}\quad\mbox{\rm in}\quad
L^{2}(D)\quad\mbox{\rm as}\quad n\to\infty,$ (5.34) $\displaystyle
u_{n}^{*}\rightarrow u^{*}\quad\mbox{\rm in}\quad V\quad\mbox{\rm as}\quad
n\to\infty,$ (5.35) $\displaystyle(u^{*},f^{*})\quad{\rm is\ a\ solution\ of\
Problem}\ {\cal Q}.$ (5.36)

To this end we proceed in four intermediate steps that we present below.

ii-a) A boundedness result. We claim that the sequence $\\{f_{n}^{*}\\}$ is
bounded in $L^{2}(D)$. Arguing by contradiction, assume that $\\{f_{n}^{*}\\}$
is not bounded in $L^{2}(D)$. Then, passing to a subsequence still denoted by
$\\{f_{n}^{*}\\}$, we have

$\|f_{n}^{*}\|_{L^{2}(D)}\to+\infty\quad\text{as}\quad n\to+\infty.$ (5.37)

We use assumption (3.38)μ of the cost function to deduce that

$\lim_{n\to\infty}{\cal L}_{n}(u_{n}^{*},f_{n}^{*})=+\infty.$ (5.38)

Next, let $f\in L^{2}(D)$ be given and let ${\mbox{{$\widetilde{u}$}}}_{n}$ be
the solution of the hemivariational inequality

$\widetilde{u}_{n}\in\mbox{{$\widetilde{K}$}}\ :\
a(\widetilde{u}_{n},v-\widetilde{u}_{n})+\frac{1}{\lambda_{n}}\,\langle
G\widetilde{u}_{n},v-\widetilde{u}_{n}\rangle+j^{0}(\widetilde{u}_{n},;v-\widetilde{u}_{n})\geq(f,v-\widetilde{u}_{n})_{V}\quad\forall\,v\in\mbox{{$\widetilde{K}$}},$
(5.39)

i.e., $\widetilde{u}_{n}=u(\lambda_{n},f)$. Note that Proposition 13
guarantees that this solution exists and is unique, for each $n\in\mathbb{N}$.
Moreover, using (4.13) and (4.35) it follows that

$\widetilde{u}_{n}\rightarrow u\quad\mbox{\rm in}\quad V\quad\mbox{\rm
as}\quad n\to\infty$ (5.40)

where $u=u(f)$ and, by (5.32) and assumption (5.16), we deduce that

${\cal L}_{n}(\widetilde{u}_{n},f)\to{\cal L}(u,f)\quad\mbox{\rm as}\quad
n\to\infty$ (5.41)

On the other hand (5.39) implies that the pair $(\widetilde{u}_{n},f)$
satisfies inequality (4.8), i.e., $(\widetilde{u}_{n},f)\in{\cal V}^{n}_{ad}$.
Therefore, (5.33) implies that

$\displaystyle{\cal L}_{n}(u^{*}_{n},f^{*}_{n})\leq{\cal
L}_{n}(\widetilde{u}_{n},f)\qquad\forall\,n\in\mathbb{N}.$ (5.42)

We now pass to the limit in (5.42) as $n\to\infty$ and use (5.38) and (5.41)
to obtain a contradiction, which proves the claim.

ii-b) Two convergence results. We conclude from step ii-a) that the sequence
$\\{f_{n}^{*}\\}$ is bounded in $L^{2}(D)$ and, therefore, we can find a
subsequence again denoted by $\\{f_{n}^{*}\\}$ and an element $f^{*}\in
L^{2}(D)$ such that (5.34) holds. Denote by $u^{*}$ the solution of inequality
(3.29) for $f=f^{*}$ and note that definition (3.30) implies that

$(u^{*},f^{*})\in{\cal V}_{ad}.$ (5.43)

Moreover, since $u_{n}^{*}=u(\lambda_{n},f_{n})$, $u^{*}=u(f)$, assumption
(4.13) and convergences (5.34), (4.36) imply that (5.35) holds, too.

ii-c) Optimality of the limit. We now prove that $(u^{*},f^{*})$ is a solution
to the optimal control problem ${\cal Q}$. To this end we use the convergences
(5.34), (5.35), (5.32) and assumption (5.12) to see that

${\cal L}(u^{*},f^{*})\leq\liminf_{n\rightarrow\infty}{\cal
L}_{n}(u_{n}^{*},f_{n}^{*}).$ (5.44)

Next, we fix a solution $({u}^{*}_{0},{f}^{*}_{0})$ of Problem ${\cal Q}$ and,
in addition, for each $n\in\mathbb{N}$ we denote by ${u}_{n}^{0}$ the unique
element of $\widetilde{K}$ which satisfies the inequality (4.8) with
$\lambda=\lambda_{n}$ and $f=f_{0}^{*}$, i.e.,
$u_{n}^{*}=u(\lambda_{n},f_{0}^{*})$. Therefore,
$({u}_{n}^{0},{f}^{*}_{0})\in{\cal V}_{ad}^{n}$ and, using the optimality of
the pair $(u_{n}^{*},f_{n}^{*})$, (5.33), we find that

${\cal L}_{n}(u_{n}^{*},f_{n}^{*})\leq{\cal
L}_{n}({u}_{n}^{0},{f}^{*}_{0})\qquad\forall\,n\in\mathbb{N}.$

We pass to the upper limit in this inequality to see that

$\limsup_{n\rightarrow\infty}{\cal
L}_{n}(u_{n}^{*},f_{n}^{*})\leq\limsup_{n\rightarrow\infty}{\cal
L}_{n}({u}_{n}^{0},{f}^{*}_{0}).$ (5.45)

Next, since $\lambda_{n}\to 0$ and $u_{0}^{*}=u(f_{0}^{*})$, it follows from
(4.35) that

${u}_{n}^{0}\to{u}^{*}_{0}\quad\text{in}\quad V\quad\text{as}\quad
n\to\infty.$

Using now this convergence and assumption (5.16) yields

$\lim_{n\rightarrow\infty}{\cal L}_{n}({u}_{n}^{0},{f}^{*}_{0})={\cal
L}({u}^{*}_{0},f^{*}_{0}).$ (5.46)

We now use (5.44)–(5.46) to see that

${\cal L}(u^{*},f^{*})\leq{\cal L}({u}^{*}_{0},{f}^{*}_{0}).$ (5.47)

On the other hand, since $({u}^{*}_{0},{f}^{*}_{0})$ is a solution of Problem
${\cal Q}$, we have

${\cal L}({u}^{*}_{0},{f}^{*}_{0})=\min_{(u,f)\in{\cal V}_{ad}}{\cal L}(u,f)$
(5.48)

and, therefore, inclusion (5.43) implies that

${\cal L}({u}^{*}_{0},{f}^{*}_{0})\leq{\cal L}(u^{*},f^{*}).$ (5.49)

We now combine the inequalities (5.47) and (5.49) to see that

${\cal L}(u^{*},f^{*})={\cal L}({u}^{*}_{0},{f}^{*}_{0}).$ (5.50)

Next, we use relations (5.43), (5.50) and (5.48) to see that (5.36) holds.

ii-d) End of proof. We remark that the convergences (5.34) and (5.35) imply
the weak convergence (in the product Hilbert space $V\times L^{2}(D)$) of the
sequence $\\{(u_{n}^{*},f_{n}^{*})\\}$ to the element $(u^{*},f^{*})$. Theorem
18 is now a direct consequence of Definition 8 c). $\Box$

We turn now to some direct consequence of Theorem 18.

###### Corollary 19

Assume that $(\ref{b})$–$(\ref{sm})$, $(\ref{L1})$ and $(\ref{L2})$ hold.
Then, the set of solutions of Problem ${\cal Q}$ is weakly sequentially
compact.

Proof. Assume that $\\{(u_{n}^{*},f_{n}^{*})\\}$ is a sequence of solutions to
Problem ${\cal Q}$. Let $\\{\lambda_{n}\\}\subset{{\rm I}\mkern-3.0mu{\rm R}}$
be such that $\lambda_{n}>0$ for each $n\in\mathbb{N}$, $\lambda_{n}\to 0$,
and let $\mu_{n}=0$ for each $n\in\mathbb{N}$. Also, consider the particular
case when $\mbox{{$\widetilde{K}$}}=K$, $Gv=0_{V}$ for any $v\in V$, and note
that in this case we have ${\cal V}_{ad}^{n}={\cal V}_{ad}$, ${\cal
L}_{n}={\cal L}$, for each $n\in\mathbb{N}$. It follows from above that
$\\{(u_{n}^{*},f_{n}^{*})\\}$ is an approximating sequence for Problem ${\cal
Q}$. Therefore, the step (ii) in the proof of Theorem 18 implies that there
exists a subsequence of the sequence $\\{(u_{n}^{*},f_{n}^{*})\\}$, again
denoted by $\\{(u_{n}^{*},f_{n}^{*})\\}$, and an element $(u^{*},f^{*})\in
V\times L^{2}(D)$ such that (5.34–(5.36) hold, which concludes the proof.
$\Box$

###### Corollary 20

Assume $(\ref{b})$–$(\ref{sm})$, $(\ref{L3})$, $(\ref{L4})$ and $(\ref{L1m})$,
$(\ref{L2m})$, for all $\mu\geq 0$. Then, for each $\mu\geq 0$ there exists a
pair $(u^{*}_{\mu},f^{*}_{\mu})$ such that

$(u^{*}_{\mu},f^{*}_{\mu})\in{\cal V}_{ad},\qquad\mbox{{${\cal
L}$}}_{\mu}(u^{*},f^{*})=\min_{(u,f)\in{\cal V}_{ad}}\mbox{{${\cal
L}$}}_{\mu}(u,f).$ (5.51)

Moreover, if $\\{(u^{*}_{n},f^{*}_{n})\\}$ represents a sequence pairs such
that $(u^{*}_{n},f^{*}_{n})$ is a solution of Problem ${\cal Q}$ corresponding
to $\mu_{n}\geq 0$ for each $n\in\mathbb{N}$, then there exists a subsequence
of the sequence $\\{(u_{n}^{*},f_{n}^{*})\\}$, again denoted by
$\\{(u_{n}^{*},f_{n}^{*})\\}$, and an element $(u^{*},f^{*})\in V\times
L^{2}(D)$ such that $(\ref{se1})$–$(\ref{se3})$ hold.

The proof of Corollary 20 follows from arguments similar to those used in the
proof of Corollary 19 and, therefore, we skip it.

###### Corollary 21

Assume $(\ref{b})$–$(\ref{sm})$, $(\ref{wK})$, $(\ref{G})$, $(\ref{L3})$,
$(\ref{L4})$ and $(\ref{L1m})$, $(\ref{L2m})$, for all $\mu\geq 0$. Moreover,
assume that Problem ${\cal Q}$ has a unique solution. Then Problem ${\cal Q}$
is weakly well-posed with respect to the Tykhonov triple
$(\ref{Ic})$–$(\ref{Cc})$.

Proof. Let $(u^{*},f^{*})$ be the unique solution to Problem ${\cal Q}$ and
let $\\{u_{n}^{*},f_{n}^{*}\\}$ be an approximating sequence for the Problem
${\cal Q}$ with respect to the Tykhonov triple $(\ref{Ic})$–$(\ref{Cc})$
First, it follows from the proof of Theorem 18 that the sequence
$\\{f_{n}^{*}\\}$ is bounded in $L^{2}(D)$. Therefore, using arguments similar
to those used in step i) of the proof of Theorem 14 we deduce that the
sequence $\\{u_{n}^{*}\\}$ is bounded in $V$. We conclude from here that the
sequence $\\{(u_{n}^{*},f_{n}^{*})\\}$ is bounded in the product space
$V\times L^{2}(D)$. Second, a careful analysis of the proof of Theorem 18
reveals that $(u^{*},f^{*})$ is the weak limit (in $V\times L^{2}(D)$) of any
weakly convergent subsequence of the sequence $\\{(u_{n}^{*},f_{n}^{*})\\}$.
The two properties above allow us to use a standard argument in order to
deduce that the whole sequence $\\{(u_{n}^{*},f_{n}^{*})\\}$ converges weakly
in $V\times L^{2}(D)$ to $(u^{*},f^{*})$, as $n\to\infty$. Corollary 21 is now
a direct consequence of Definition 8 b). $\Box$

We end this section with the following remarks. First, Corollary 19 provides a
topological property of the set of solutions of Problem ${\cal Q}$. Moreover,
Corollary 20 provides a continuity result of the solutions of this problem
with respect to the parameter $\mu$ and, implicitely, with respect to the cost
functional. Finally, Corollary 21 provides a weakly well-posedness result for
Problem ${\cal Q}$, in the particular case when this problem has a unique
solution. Such situations arise for specific boundary condition and cost
functionals, as explained in [3, 4].

##  6 Some relevant examples

We start this section with examples of perturbed heat transfer problems which
lead to hemivariational inequalities of the form (4.8) for which conditions
$(\ref{wK})$ and $(\ref{G})$ are satisfied.

A unilateral problem with penalty conditions on $D$. The first problem we
introduce in this section is obtained by considering a penalty version of the
unilateral condition in (3.1). The problem is formulated as follows.

Problem $\mbox{{${\cal H}$}}_{0}$. Find a temperature field $u:D\to{{\rm
I}\mkern-3.0mu{\rm R}}$ such that

$\displaystyle\Delta u+f=\frac{1}{\lambda}\,p_{0}(u)\qquad{\rm a.e.\ in\ }D,$
(6.1) $\displaystyle u=0\hskip 85.35826pt{\rm a.e.\ on\ }\Gamma_{1},$ (6.2)
$\displaystyle u=b\hskip 85.35826pt{\rm a.e.\ on\ }\Gamma_{2},$ (6.3)
$\displaystyle-\frac{\partial u}{\partial\nu}\in\partial j_{\nu}(u)\hskip
39.83385pt{\rm a.e.\ on\ }\Gamma_{3}.$ (6.4)

Here $\lambda>0$ is a penalty parameter and $p_{0}$ is a function which is
assumed to satisfy the following condition.

$\left\\{\begin{array}[]{ll}p_{0}\colon D\times\mathbb{R}\to\mathbb{R}\
\mbox{is such that}\\\\[2.84526pt] {\rm(a)}\
|p_{0}(\mbox{\boldmath{$x$}},r)-p_{0}(\mbox{\boldmath{$x$}},s)|\leq
L_{0}|r-s|\\\ \qquad\quad\mbox{for all}\,r,\,s\in\mathbb{R},\ {\rm a.e.\
}\mbox{\boldmath{$x$}}\in D,\ {\rm with}\ L_{0}>0,\\\\[2.84526pt] {\rm(b)\
}(p_{0}(\mbox{\boldmath{$x$}},r)-p_{0}(\mbox{\boldmath{$x$}},s))\,(r-s)\geq
0\\\ \qquad\quad\mbox{for all}\ \ r,\,s\in\mathbb{R},\ {\rm a.e.\
}\mbox{\boldmath{$x$}}\in D,\\\\[2.84526pt] {\rm(c)}\
\mbox{\boldmath{$x$}}\mapsto p_{0}(\mbox{\boldmath{$x$}},r){\rm\ is\
measurable\ on}\ D\ {\rm for\ all\ }r\in\mathbb{R},\\\\[2.84526pt] {\rm(d)}\
p_{0}(\mbox{\boldmath{$x$}},r)=0\ \ \mbox{if and only if}\ \ r\geq 0,\ {\rm
a.e.}\ \mbox{\boldmath{$x$}}\in D.\end{array}\right.$ (6.5)

A typical example of such function is given by

$p_{0}(x,r)=-cr^{-}\qquad{\rm for\ all\ }r\in\mathbb{R},\
\mbox{\boldmath{$x$}}\in D$

where $c>0$ and $r^{-}$ represents the negative part of $r$, i.e.,
$r^{-}=max\,\\{-r,0\\}$. Note that in this case the term $p_{0}(u)$ in (6.1)
vanishes in the points of $D$ where $u\geq 0$ and equals
$c\,\frac{u}{\lambda}$ in the points of $D$ where $u<0$.

Using standard arguments it is easy to see that Problem $\mbox{{${\cal
H}$}}_{0}$ leads to a variational formulation of the form $(\ref{hvp})$ in
which

$\displaystyle\mbox{{$\widetilde{K}$}}=\ \\{\,v\in V\ :\ v=b\ \ {\rm on}\ \
\Gamma_{2}\,\\},$ (6.6) $\displaystyle\langle
Gu,v\rangle=\int_{D}p_{0}(u)v\,da\qquad\forall\,u,\ v\in V.$ (6.7)

We have the following result.

###### Proposition 22

Assume $(\ref{b})$ and $(\ref{p0})$. Then, the set $(\ref{wK0})$ and the
operator $(\ref{G0})$ satisfy conditions $(\ref{wK})$ and $(\ref{G})$,
respectively.

Proof. Conditions $(\ref{wK})$ and $(\ref{G})$(a) are clearly satisfied.
Assume now that $u\in\mbox{{$\widetilde{K}$}}$ and $v\in K$ where, recall, the
set $K$ is defined by (3.23). Then, using (6.5) we see that

$p_{0}(u)v\leq 0\quad{\rm and}\quad p_{0}(u)u\geq 0\quad{\rm a.e.\ in}\ D$
(6.8)

which imply that $p_{0}(u)(v-u)\leq 0$ a.e. in $D$. We conclude from here that

$\int_{D}p_{0}(u)(v-u)\,dx\leq 0$

and, therefore, condition $(\ref{G})$(b) holds.

Next, we assume that $u\in\mbox{{$\widetilde{K}$}}$ and $\langle
Gu,v-u\rangle=0$ for all $v\in K$, which implies that

$\int_{D}p_{0}(u)u\,dx=\int_{D}p_{0}(u)v\,dx\qquad\forall\,v\in K.$ (6.9)

We now use inequalities (6.8) to deduce that

$\int_{D}p_{0}(u)u\,dx=0.$ (6.10)

Therefore, (6.8), (6.9) combined with the implication

$h\geq 0,\quad\int_{\omega}h=0\quad\Longrightarrow\quad h=0\quad{\rm a.e.\ \
on}\ \ \omega$ (6.11)

show that $p_{0}(u)u=0$ ${\rm a.e.\ on\ }D$. This equality together with
condition (6.5)(d) implies that $u\geq 0$ ${\rm a.e.\ on\ }D$. Recall now that
$u\in\mbox{{$\widetilde{K}$}}$. We deduce from here that $u=b$ ${\rm a.e.\ on\
}\Gamma_{2}$. Therefore, $u\in K$, which concludes the proof. $\Box$

A unilateral problem with penalty conditions on $\Gamma_{2}$. The second
problem we introduce in this section is obtained by considering a penalty
version of the boundary condition (3.3) on $\Gamma_{2}$. The problem is
formulated as follows.

Problem $\mbox{{${\cal H}$}}_{2}$. Find a temperature field $u:D\to{{\rm
I}\mkern-3.0mu{\rm R}}$ such that

$\displaystyle u\geq 0,\qquad-\Delta u-f\geq 0,\qquad u(\Delta
u+f)=0\qquad{\rm a.e.\ in\ }D,$ (6.12) $\displaystyle u=0\hskip
102.42992pt{\rm a.e.\ on\ }\Gamma_{1},$ (6.13) $\displaystyle-\frac{\partial
u}{\partial\nu}=\frac{1}{\lambda}\,p_{2}(u-b)\hskip 31.29802pt{\rm a.e.\ on\
}\Gamma_{2},$ (6.14) $\displaystyle-\frac{\partial u}{\partial\nu}\in\partial
j_{\nu}(u)\hskip 55.48286pt{\rm a.e.\ on\ }\Gamma_{3}.$ (6.15)

Here, again, $\lambda>0$ is a penalty parameter and $p_{2}$ is a function
which is assumed to satisfy the following condition.

$\left\\{\begin{array}[]{ll}p_{2}\colon\Gamma_{2}\times\mathbb{R}\to\mathbb{R}\
\mbox{is such that}\\\\[2.84526pt] {\rm(a)}\
|p_{2}(\mbox{\boldmath{$x$}},r)-p_{2}(\mbox{\boldmath{$x$}},s)|\leq
L_{2}|r-s|\\\ \qquad\quad\mbox{for all}\,r,\,s\in\mathbb{R},\ {\rm a.e.\
}\mbox{\boldmath{$x$}}\in\Gamma_{2},\ {\rm with}\ L_{2}>0,\\\\[2.84526pt]
{\rm(b)\
}(p_{2}(\mbox{\boldmath{$x$}},r)-p_{2}(\mbox{\boldmath{$x$}},s))\,(r-s)\geq
0\\\ \qquad\quad\mbox{for all}\ \ r,\,s\in\mathbb{R},\ {\rm a.e.\
}\mbox{\boldmath{$x$}}\in\Gamma_{2},\\\\[2.84526pt] {\rm(c)}\
\mbox{\boldmath{$x$}}\mapsto p_{2}(\mbox{\boldmath{$x$}},r){\rm\ is\
measurable\ on}\ \Gamma_{2}\ {\rm for\ all\ }r\in\mathbb{R},\\\\[2.84526pt]
{\rm(d)}\ p_{2}(\mbox{\boldmath{$x$}},r)=0\ \ \mbox{if and only if}\ \ r=0,\
{\rm a.e.}\ \mbox{\boldmath{$x$}}\in\Gamma_{2}.\end{array}\right.$ (6.16)

A typical example of such function is given by

$p_{2}(\mbox{\boldmath{$x$}},r)=cr\qquad{\rm for\ all\ }r\in\mathbb{R},\
\mbox{\boldmath{$x$}}\in\Gamma_{2}$

where $c>0$. Using standard arguments it is easy to see that Problem
$\mbox{{${\cal H}$}}_{2}$ leads to a variational formulation of the form
$(\ref{hvp})$ in which

$\displaystyle\mbox{{$\widetilde{K}$}}=\ \\{\,v\in V\ :\ v\geq 0\ \ {\rm in}\
\ D\,\\},$ (6.17) $\displaystyle\langle
Gu,v\rangle=\int_{\Gamma_{2}}p_{2}(u-b)v\,da\qquad\forall\,u,\ v\in V.$ (6.18)

We have the following result.

###### Proposition 23

Assume $(\ref{b})$ and $(\ref{p2})$. Then, the set $(\ref{wK2})$ and the
operator $(\ref{G2})$ satisfy conditions $(\ref{wK})$ and $(\ref{G})$,
respectively.

Proof. Conditions $(\ref{wK})$ and $(\ref{G})$(a) are clearly satisfied.
Assume now that $u\in\mbox{{$\widetilde{K}$}}$ and $v\in K$. Then, using
(6.16) we see that $p_{2}(u-b)(v-u)=p_{2}(u-b)(b-u)$ a.e. on $\Gamma_{2}$ and,
using the properties (6.16) of the function $p_{2}$ we deduce that
$p_{2}(u-b)(v-u)\leq 0$ a.e. on $\Gamma_{3}$. We conclude from here that

$\int_{\Gamma_{2}}p_{2}(u-b)(v-u)\,da\leq 0$

and, therefore, condition $(\ref{G})$(b) holds.

Assume now that $u\in\mbox{{$\widetilde{K}$}}$ and $\langle Gu,v-u\rangle=0$
for all $v\in K$. This implies that

$\int_{\Gamma_{3}}p_{2}(u-b)(b-u)\,da=0.$

Now, since (6.16) guarantees that $p_{2}(u-b)(b-u)\leq 0$ a.e. on $\Gamma_{2}$
we use implication (6.11), again, to deduce that $p_{2}(u-b)(b-u)=0$ a.e. on
$\Gamma_{2}$. In now follows from condition (6.16)(d) that $u=b$ a.e. on
$\Gamma_{3}$ and, therefore, $u\in K$. $\Box$

A fully penalty problem. The third problem we introduce in this section is
obtained by considering a penalty version of both the conditions (3.1) and
(3.3). The problem is formulated as follows.

Problem $\mbox{{${\cal H}$}}_{02}$. Find a temperature field $u:D\to{{\rm
I}\mkern-3.0mu{\rm R}}$ such that

$\displaystyle\Delta u+f=\frac{1}{\lambda}\,p_{0}(u)\qquad\ {\rm a.e.\ in\
}D,$ (6.19) $\displaystyle u=0\hskip 91.04881pt{\rm a.e.\ on\ }\Gamma_{1},$
(6.20) $\displaystyle-\frac{\partial
u}{\partial\nu}=\frac{1}{\lambda}\,p_{2}(u-b)\hskip 19.91692pt{\rm a.e.\ on\
}\Gamma_{2},$ (6.21) $\displaystyle-\frac{\partial u}{\partial\nu}\in\partial
j_{\nu}(u)\hskip 42.67912pt{\rm a.e.\ on\ }\Gamma_{3}.$ (6.22)

Here $\lambda>0$ is a penalty parameter and $p_{0}$ and $p_{2}$ are functions
which satisfy conditions (6.5) and (6.16), respectively. Using standard
arguments it is easy to see that Problem $\mbox{{${\cal H}$}}_{2}$ leads to a
variational formulation of the form $(\ref{hvp})$ in which

$\displaystyle\mbox{{$\widetilde{K}$}}=V,$ (6.23) $\displaystyle\langle
Gu,v\rangle=\int_{D}p_{0}(u)v\,da+\int_{\Gamma_{2}}p_{2}(u-b)v\,da\qquad\forall\,u,\
v\in V.$ (6.24)

We have the following result.

###### Proposition 24

Assume $(\ref{b})$, $(\ref{p0})$ and $(\ref{p2})$. Then, the set
$(\ref{wK02})$ and the operator $(\ref{G02})$ satisfy conditions $(\ref{wK})$
and $(\ref{G})$, respectively.

Proof. Conditions $(\ref{wK})$ and $(\ref{G})$(a) are clearly satisfied.
Assume now that $u\in\mbox{{$\widetilde{K}$}}$ and $v\in K$. Then, using (6.5)
and (6.16) it is easy to see that $p_{0}(u)(v-u)\leq 0$ a.e. in $D$ and
$p_{2}(u-b)(v-u)\leq 0$ a.e. on $\Gamma_{3}$. We conclude from here that

$\int_{D}p_{0}(u)(v-u)\,dx+\int_{\Gamma_{2}}p_{2}(u-b)(v-u)\,da\leq 0$

and, therefore, condition $(\ref{G})$(b) holds.

Assume now that $u\in\mbox{{$\widetilde{K}$}}$ and $\langle Gu,v-u\rangle=0$
for all $v\in K$. This implies that

$\int_{D}p_{0}(u)v\,dx+\int_{\Gamma_{3}}p_{2}(u-b)(b-u)\,da=\int_{D}p_{0}(u)u\,dx.$
(6.25)

Now, recall that (6.5) and (6.16) guarantee that

$\displaystyle\mbox{$p_{0}(u)v\leq 0$\ \ a.e. \ in\
$D$,\quad$p_{2}(u-b)(b-u)\leq 0$\ \ a.e.\ on\ $\Gamma_{2}$},$ (6.26)
$\displaystyle\mbox{$p_{0}(u)u\geq 0$\ \ a.e.\ in\ $D$}.$ (6.27)

We now combine equality (6.25) with inequalities (6.26) and (6.27) to find
that

$\int_{D}p_{0}(u)u\,dx=0.$ (6.28)

Next, (6.27), (6.28) and (6.11) imply that $p_{0}(u)u=0$ a.e. in $D$ and,
using condition (6.5) we find that

$u\geq 0\qquad{\rm a.e.\ in}\ D.$ (6.29)

We conclude from here that $p_{0}(u)=0$ a.e. on $D$ and, therefore, (6.25)
yields

$\int_{\Gamma_{3}}p_{2}(u-b)(b-u)\,da=0.$ (6.30)

Next, (6.26), (6.30) and (6.11) imply that $p_{2}(u-b)(b-u)=0$ a.e. on
$\Gamma_{2}$. In now follows from condition (6.16)(d) that

$u=b\qquad{\rm a.e.\ on}\ \Gamma_{3}.$ (6.31)

We now use (6.29) and (6.31) de deduce that $u\in K$ which concludes the
proof. $\Box$

An example of cost functional. A large number of cost functionals for which
our results in Section 5 hold can be considered. Here we restrict ourselves to
provide the following example. Assume that

$\displaystyle a_{0}>0,\qquad a_{2}>0,\qquad\phi\in L^{2}(\Gamma_{2}),$ (6.32)
$\displaystyle\omega=[0,+\infty)\to{{\rm I}\mkern-3.0mu{\rm R}}\ \mbox{\rm is
a continuous function such that}\ \omega(0)=1.$ (6.33)

Consider the cost functional ${\cal L}:V\times L^{2}(D)\to{{\rm
I}\mkern-3.0mu{\rm R}}$ defined by

${\cal L}(u,f)=a_{0}\int_{D}f^{2}\,dx+a_{2}\int_{\Gamma_{2}}(u-\phi)^{2}\,da$
(6.34)

and, for each $\mu\geq 0$, let ${\cal L}_{\mu}:V\times L^{2}(D)\to{{\rm
I}\mkern-3.0mu{\rm R}}$ be defined by

${\cal
L}_{\mu}(u,f)=a_{0}\int_{D}f^{2}\,dx+a_{2}\int_{\Gamma_{2}}(u-\omega(\mu)\phi)^{2}\,da$
(6.35)

Then, it is easy to see that the following result holds.

###### Proposition 25

Under assumptions $(\ref{r1})$ and $(\ref{r2})$ the functionals ${\cal L}$ and
${\cal L}_{\mu}$ satisfy conditions $(\ref{L1})$, $(\ref{L2})$, $(\ref{L3})$,
$(\ref{L4})$ and $(\ref{L1m})$, $(\ref{L2m})$, for each $\mu\geq 0$.

Final remarks. We end this section with the remark that Propositions 22–24
allow us to apply Theorem 14 and and Corollaries 15, 16 in the study of the
corresponding boundary value problems. In this way we obtain the unique
solvability of Problems $\mbox{{${\cal H}$}}_{0}$, $\mbox{{${\cal H}$}}_{2}$
and $\mbox{{${\cal H}$}}_{02}$, the weak-strong continuous dependence of their
weak solutions with respect to $f\in L^{2}(D)$ and the convergence of these
solutions to the weak solution of Problem ${\cal H}$, as $\lambda\to 0$.
Moreover, Proposition 25 allows us to apply Theorem 18 and Corollaries 19–21
in the study of the corresponding optimal control problems. This allows to
obtain the solvability of these optimùal control problems, the sequential
compactness of the sets of their solutions and their continuous dependence
with respect to the parameter $\mu$.

## Acknowledgement

This project has received funding from the European Union’s Horizon 2020
Research and Innovation Programme under the Marie Sklodowska-Curie Grant
Agreement No 823731 CONMECH.

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Authors’address:

Mircea Sofonea, Laboratoire de Mathématiques et Physique University of
Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France, e-mail:
<EMAIL_ADDRESS>
Domingo A. Tarzia, Departamento de Matematica-CONICET, Universidad Austral,
Paraguay 1950, S2000FZF Rosario, Argentina, e-mail<EMAIL_ADDRESS>