# Elastic-brittle reinforcement
of flexural structures

Francesco Maddalena∗, Danilo Percivale∗∗ & Franco Tomarelli∗∗∗ ∗ Politecnico
di Bari, Dipartimento di Meccanica, Matematica e Management, Italy  ∗∗
Università di Genova, Dipartimento della Produzione Termoenergetica e Modelli
Matematici, Italy  ∗∗∗ Politecnico di Milano, Dipartimento di Matematica,
Italy
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Key words and phrases:

Calculus of variations, free discontinuity, variational inequality, adhesion,
elastic clamped plate, crack, plastic yielding, flexural structures, coating,
reinforcement.

###### 1991 Mathematics Subject Classification:

49J45, 74K30, 74K35, 74R10

This work is supported by PRIN (M.I.U.R) “Variational methods for stationary
and evolution problems with singularities and interfaces” 2017 and Progetto
G.N.A.M.P.A. (INDAM) “Problemi asintotici e fenomeni singolari in meccanica
dei continui (2020)” .

Dedicated to Claudio Baiocchi with sore heart for the loss of a teacher and
friend

Abstract. This note provides a variational description of the mechanical
effects of flexural stiffening of a 2D plate glued to an elastic-brittle or an
elastic-plastic reinforcement. The reinforcement is assumed to be linear
elastic outside possible free plastic yield lines or free crack. Explicit
Euler equations and a compliance identity are shown for the reinforcement of a
1D beam.

###### Contents

  1. 1 Introduction
  2. 2 Statement of the problem and main results
  3. 3 One-dimensional analysis: reinforcement of flexural beams
  4. 4 Hard-device reinforcement of flexural plate
  5. 5 Strengthening reinforcement of flexural plate
  6. 6 Elastic-plastic reinforcement of flexural plate

##  1\. Introduction

The main theme moving the present paper relies in studying the mechanisms
ruling stress transfer between material structures affected by strongly
different constitutive properties. We consider some functionals of the kind

(1.1) $\displaystyle\ \ \Lambda(K,v_{r},v_{p})\ =\
\mathcal{H}^{1}\\!\left(K\cap\overline{\Omega}\right)\\!\,+\,\sigma\\!\\!\int_{K}\\!|\,[Dv_{r}]\,|\,d\mathcal{H}^{1}+\\!\\!\int_{\Omega\setminus
K}\\!\\!\\!\big{(}\eta\,|D^{2}v_{r}|^{2}-f_{r}v_{r}\big{)}\,d{\bf x}+$
$\displaystyle\qquad\qquad\qquad\quad+\,\mu\int_{\Omega}|v_{r}-v_{p}|^{2}\,d{\bf
x}+\\!\int_{\Omega}\\!\left(\gamma\,|D^{2}v_{p}|^{2}\\!-f_{p}\,v_{p}\right)\,d{\bf
x}$

dependent on competing triplets $(K,v_{r},v_{p})$. Here
$\Omega\subset\mathbb{R}^{2}$ is given together with the loads $f_{r},f_{p}$,
the nonnegative constitutive parameters $\sigma,\,\eta,,\,\mu,\,\gamma$ and a
suitable Dirichlet boundary condition, shared by both displacements $v_{r}$
and $v_{p}$. $\mathcal{H}^{1}$ denotes the 1-dimensional Hausdorff measure.
Concerning competing triplets $(K,v_{r},v_{p})$, the set $K$ is closed, the
function $v_{r}$ is smooth outside $K$ while $v_{p}$ is smooth on $\Omega$.
Structural reinforcements are extensively employed in manufactured engineering
systems, ranging from the traditional field of composite structures to the
more recent applications in microelectronic devices and nano-reinforced
composites ([37]). Indeed, external bonding of plates is a method of
strenghtening which involves an additional adhering reinforcement to a
structural element ([38]). The adhesive is needed to transfer the stresses
between the two elements. This technique is aimed to reduce deflection, hence
confine crack and plastic yielding location in order to increase load carrying
capacity (see e.g. [21], [26], [27], [36], [41]), as well as predicting the
behavior of paint coating layers ([37], [39]).

In a typical reinforcement system, represented by a brittle structure ([24])
bonded to a more compliant substrate, cracking and debonding instabilities
(delamination) of the brittle element may appear under the action of external
data that may be ruled by external loading, temperature change or even
residual thermal stresses. The occurrence of plastic yielding, cracking and
loss of adhesion (or delamination) constitute the main failure modes of
reinforcements, so a great deal of work has been done over past decades to
apply fracture mechanics in description of behavior and the influence of
cracks nucleated on or near an interface between two dissimilar materials and
a number of papers have been published on the problem (see, for instance,
[22], [25], [27]). The crucial questions, when studying the possible failure
of bonded structures, rely in understanding crack or yielding nucleation and
crack propagation in presence of debonding or delamination of the constituent
materials. This issue, precisely the role played by bonding layer in the
formation of singularities, has not been yet investigated in spite of its
influence in the mechanical behavior of such structural systems. Therefore, in
our opinion, an appropriate description of such problems should incorporate
all these strongly nonlinear effects in a mathematical theory which is able to
detect qualitative and quantitative features of the underlying physics
([32],[33]).
To this aim we propose in the present paper a variational approach in which
debonding and possible singular states arise as minimizers of suitable energy
functionals.
In the previous works [28], [29], [30], [31], we have studied the adhesion
interaction of linear and nonlinear elastic structures by focusing on the
influence of different constitutive choices for the adhesive material, while
in [31] we have investigated the occurrence of global collapse and the
interplay of cracking and debonding for a couple of plane elastic sheets.
Here reinforcements are modelled in the framework of Kirchhoff-Love theory,
while the addition of a fracture energy term according to the Griffith theory
allows to capture crack formation. This program is achieved by exploiting the
techniques developed in the study of second order variational problems with
free discontinuity ([9]-[12], [14]-[16], [34],[35]). Precisely we aim to
describe the mechanical effects of bending a 2-dimensional stiff substrate
fixed at the boundary, shortly called clamped plate from now on, which is
glued to an elastic-plastic-brittle reinforcement. The reinforcement is
assumed to be linear elastic outside possible free plastic yield lines or free
1D crack. The adhesive interaction between the structures is modelled through
an energetic contribution whose density is the square of the modulus of
difference of the displacement; concerning this assumption we recall that, if
one assumes the $q$-power of the modulus of difference, with $0<q<1$, then the
only stable configurations of the system are the completely detached or
completely glued ones, even in the case of a flat plate as it was proved in
Section 2.2 of [28].

The phenomenon is modeled as a variational problem which allows free
discontinuity and free gradient discontinuity for the reinforcement. We assume
that both configurations of the plate and its reinforcement are described as
graphs referred to the coordinates in the horizontal plane, undergo vertical
displacements and are subject to Dirichlet boundary conditions. We describe
the details in these three main cases.

  1. (1)

hard-device reinforcement: the structure consists in the glueing of a plate
with a reinforcement; the plate undergoes a prescribed configuration
(described by a given displacement) and consequently acts on the reinforcement
through the adhesive layer; the reinforcement behaves as a piece-wise
Kirchhoff-Love plate since it can develop lower-dimensional singularities of
two kinds, plastic yielding (free gradient-discontinuity) and/or crack (free
discontinuity).

  2. (2)

strengthening reinforcement: the structure consists in the glueing of two
objects, which are still labeled as plate and reinforcement; the plate behaves
like a Kirchhoff-Love plate, whose unstressed flat reference configuration is
horizontal, under the action of a given transverse (vertical) load $f$, while
the plate displacement acts on the reinforcement through the adhesive layer;
the reinforcement behaves as a piece-wise Kirchhoff-Love plate since it can
develop lower-dimensional singularities of two kinds, plastic yielding and/or
crack. We denote the admissible vertical displacement of the reinforcement by
$v_{r}$ and the admissible vertical displacement of the plate by $v_{p}$.

  3. (3)

elastic-plastic reinforcement: as like as in case (2), but without crack and
with a refinement of the yielding energy along the a priori. unknown plastic
yield lines.

As far as we know in structural mechanics literature there are few studies
([8]) of the interplay of plastic-yielding or fracture with bending bulk
energy: here we aim to study this coupling in terms of integral functionals
with free discontinuity and free gradient discontinuity by methods introduced
in calculus of variations (see [3],[15],[17],[18],[20]).
In Section 3 we discuss the analogous one-dimensional case, say elastic-
brittle reinforcement of flexural beams (Theorems 3.1, 3.2, 3.3). In Sections
2, 4 and 5 we exhamine details of the clamped plate reinforcement (hard-device
and strengthening) showing the existence of energy minimizing solutions
(Theorems 2.1, 2.2, 2.3).
In the 1D case (beam reiforcement) we provide explicit Euler equations
(Proposition 3.9), transmission conditions at free-discontinuity set and
compliance identities fulfilled by minimizers (Proposition 3.10). Lack of
convexity in these functionals may lead to non uniqueness of minimizers
([6],[7]). However we show uniqueness and hence smoothness in case of small
loads: see Theorem 3.4, Remark 3.5. A more detailed analysis of non uniqueness
phenomenon is postponed in a forthcoming article.
In present paper we omit the consideration of a unilateral constraint forcing
non-interpenetration the plate and its reinforcement: such constraint leads to
technical difficulties and substantial problems for existence of strong
solutions in the 2D case of the plate, since Euler equations are replaced by
variational inequalities ([4]); this issue is postponed to a subsequent paper
(see Remark 5.2). Here we only start the analysis of solutions for the 1D
cases of the beam reinforced by a hard device, obtaining a variational
inequality coupled with free discontinuity (Propositions 3.7 and 3.11), and of
the strengthening reinforced beam, obtaining quasi-variational inequalities
coupled with free discontinuity (Propositions 3.8 and 3.12).

##  2\. Statement of the problem and main results

Assume

(2.1)
$\vspace{-0.3cm}\Omega\subset\\!\subset\Omega_{p}\subset\\!\subset\mathbb{R}^{2}\qquad\hbox{are
bounded and connected $C^{2}$ open sets}$ (2.2) $\vspace{-0.1cm}w\in
C^{2}(\overline{\Omega_{p}})\,,$

where $\Omega_{p}$ represents the horizontal reference configuration of the
plate and $\Omega$ represents the horizontal reference configuration of the
reinforcement and $w:\Omega_{p}\mapsto\mathbb{R}$ prescribes the Dirichlet
datum of the clamped plate.
In the case of hard-device reinforcement $v:\Omega_{p}\mapsto\mathbb{R}$
denotes the generic admissible vertical displacement of the reinforcement,
while the vertical displacement $w$ of the plate is prescribed: since the
reinforcing structure has to accomplish a prescribed configuration w, this
amounts to deal with a pre-strained state of the material competing with other
energetic terms. In the case of strengthening reinforcement,
$v_{r}:\Omega_{p}\mapsto\mathbb{R}$ denotes the generic admissible vertical
displacement of the reinforcement, while the generic admissible vertical
displacement of the plate is denoted by $v_{p}$ where
$v_{p}:\Omega_{p}\mapsto\mathbb{R}$, where $v_{p}$ is subject to a Dirichlet-
type boundary condition, prescribed on $\Omega_{p}\setminus\Omega$. We face
the three cases mentioned in the Introduction, by studying the minimization of
suitable energy functionals:

  * •

the energy $E$ associated to a hard-device reinforcement, which is dependent
on pairs $(K,v)$ where $K$ denotes the damaged region of the reinforcement and
$v$ its transversal displacement;

  * •

the energy $F$ associated to a strengthening reinforcement, which is dependent
on triplets $(K,v_{r},v_{p})$ where $K$ still denotes the damaged region of
the reinforcement (free discontinuity and free gradient-discontinuity) while
$v_{r}$ and $v_{p}$ denote respectively the transversal displacements of
reinforcement and plate;

  * •

the energy $G$ associated to an elastic-plastic reinforcement, which is
dependent on triplets $(K,v_{r},v_{p})$ as above, but $v_{r}$ may undergo only
free gradient-discontinuity on $K$.

We state some results related to minimization of these energies; their proofs
are postponed in Sections 4 and 5. All functions under exam are real-valued.

######  Theorem 2.1.

(hard-device reinforcement) Assume (2.1),(2.2) and

(2.3) $\eta>0\,,\qquad\mu>0\,,\qquad f\in L^{4}(\Omega)\,,$

then there exists a pair $(Z,u)$ minimizing

(2.4) $E(K,v)\,=\
\mathcal{H}^{1}\left(K\cap\overline{\Omega}\right)\,+\int_{\Omega\setminus
K}\\!\\!\Big{(}\,\eta\,|D^{2}v|^{2}-fv\,\Big{)}\,d{\bf
x}\,+\,\mu\int_{\Omega}|v-w|^{2}\,d{\bf x}$

over essential admissible pairs $(K,v)$, say pairs s.t.

(2.5) $\left\\{\begin{array}[]{l}\displaystyle K\ \hbox{is the smallest closed
subset of }\mathbb{R}^{2}\ \hbox{s.t.}\\\ \displaystyle v\in
C^{2}(\Omega_{p}\setminus K),\ \qquad v\equiv w\ \hbox{a.e in
}\Omega_{p}\setminus\overline{\Omega}\,.\end{array}\right.$

Moreover $Z\cap\Omega_{p}=Z\cap\overline{\Omega}$ is an $(\mathcal{H}^{1},1)$
rectifiable set and $E(Z,u)<+\infty\,.$

Here and in the sequel we denote by ${\mathcal{H}}^{1}$ the $1$-dimensional
Hausdorff measure.
If $(Z,u)\in\mathop{{\rm argmin}}\nolimits E$, say $(Z,u)$ is an optimal pair
among the ones fulfilling (2.5), then $Z$ represents the damaged zone of the
reinforcement $\Omega$, say the 1D set where either plastic yielding or
fracture occur, and $u$ is the related transverse displacement of the
reinforcement.

######  Theorem 2.2.

(strengthening reinforcement) \- Assume (2.1),(2.2),(2.3) and

(2.6) $f_{r}\in L^{4}(\Omega),\quad f_{p}\in
L^{2}(\Omega)\,,\quad\gamma>0\,.\vspace{-0.1cm}$

Then there exists a triplet $(Z,u_{r},u_{p}):=(Z,U)$ minimizing

(2.7)
$\begin{array}[]{ll}\vspace{0.2cm}F(K,v_{r},v_{p})\,\\!\\!&=\,F(K,V)\,:=\vspace{0.1cm}\\\
&\displaystyle\mathcal{H}^{1}\\!\left(K\cap\overline{\Omega}\right)+\eta\int_{\Omega\setminus
K}\\!\big{(}|D^{2}v_{r}|^{2}-f_{r}v_{r}\big{)}d{\bf x}+\vspace{0.1cm}\\\
&\displaystyle\qquad\qquad+\mu\int_{\Omega}\\!|v_{r}-v_{p}|^{2}d{\bf
x}\,+\int_{\Omega}\\!\left(\gamma\,|D^{2}v_{p}|^{2}\\!-f_{p}\,v_{p}\right)d{\bf
x}\end{array}$

over essential admissible triplets $(K,v_{r},v_{p})$, say triplets s.t.

(2.8) $\left\\{\begin{array}[]{l}v_{p}\in H^{2}(\Omega_{p})\,,\qquad
v_{p}\equiv w\hbox{ a.e in
}\Omega_{p}\setminus\overline{\Omega}\,,\vspace{0.1cm}\\\ \displaystyle K\
\hbox{is the smallest closed subset of }\mathbb{R}^{2}\ \hbox{s.t.}\\\
\displaystyle v_{r}\in C^{2}(\Omega_{p}\setminus K),\ \qquad v_{r}\equiv w\
\hbox{a.e in }\Omega_{p}\setminus\overline{\Omega}\,.\end{array}\right.$

Moreover $Z\cap\Omega_{p}=Z\cap\overline{\Omega}$ is an $(\mathcal{H}^{1},1)$
rectifiable set and $F(Z,u_{r},u_{p})<+\infty\,.$

If $(Z,U)=(Z,u_{r},u_{p})$ is an optimal triplet among the ones fulfilling
(2.8), say it is an essential admissible pair $(Z,U)\in\mathop{{\rm
argmin}}\nolimits F$, then $Z$ represents the damaged zone of the
reinforcement $\Omega$, say the 1D set where either plastic yielding or
fracture occur, and $u_{r},\ u_{p}$ respectively are the related displacement
of the reinforcement and the plate.

######  Theorem 2.3.

(elastic-plastic reinforcement of flexural plate) \-
Assume (2.1),(2.2),(2.3), $\sigma\\!>\\!0,\
f_{r}\\!\in\\!L^{s}(\Omega),\,s\\!>\\!2$, and $f_{p}\\!\in\\!L^{2}(\Omega)$.
Then there is a triplet $(Z,u_{r},u_{p})=(Z,U)$ minimizing

$\displaystyle G(K,v_{r},v_{p})\ =\ G(K,V)\ :=\ $ $\displaystyle\ =\
\mathcal{H}^{1}\left(K\cap\overline{\Omega}\right)\,+\,\sigma\int_{K\cap\overline{\Omega}}|\,[Dv_{r}]\,|\,d\mathcal{H}^{1}\,+\,\int_{\Omega\setminus
K}\\!\\!\big{(}\eta|D^{2}v_{r}|^{2}-f_{r}v_{r}\big{)}\,d{\bf x}\,+\,$
$\displaystyle\qquad\,+\,\mu\int_{\Omega}|v_{r}-v_{p}|^{2}\,d{\bf
x}+\,\int_{\Omega}\\!\big{(}\gamma|D^{2}v_{p}|^{2}-f_{p}v_{p}\big{)}\,d{\bf
x}$

over elastic-plastic essential admissible triplets $(K,v_{r},v_{p})$, say
triplets s.t.

(2.10) $\left\\{\begin{array}[]{l}\displaystyle K\ \hbox{is the smallest
closed subset of }\mathbb{R}^{2}\ \hbox{s.t.}\\\ \displaystyle v_{r}\in
C^{0}(\Omega_{p})\cap C^{2}(\Omega_{p}\setminus K),\ \qquad v_{r}=v_{p}\equiv
w\ \hbox{a.e in
}\Omega_{p}\\!\setminus\\!\overline{\Omega}\,.\end{array}\right.$

Moreover $Z\cap\Omega_{p}=Z\cap\overline{\Omega}$ is an $(\mathcal{H}^{1},1)$
rectifiable set and $G(Z,u_{r},u_{p})<+\infty\,.$

Theorem 2.3 describes a situation where crack is a priori excluded, while
elastic deformation is present together with possible damage due to plastic
yielding on the one-dimensional subset $K$: the free gradient-discontinuity
set.

In the subsequent analysis we shall use the following notation for the various
contribution to the total mechanical energy:

(2.11) $F_{r}(K,v_{r})\ :=\
\mathcal{H}^{1}\left(K\cap\overline{\Omega}\right)\,+\,\int_{\Omega\setminus
K}\\!\Big{(}\,\eta\,|D^{2}v_{r}|^{2}-f_{r}v_{r}\,\Big{)}\,d{\bf x}\ ,$ (2.12)
$M(v_{r}-v_{p})\ :=\ \mu\\!\int_{\Omega}\\!|v_{r}-v_{p}|^{2}\,d{\bf x}\ ,$
(2.13) $F_{p}(v_{p})\ :=\
\\!\int_{\Omega}\\!\\!\left(\gamma\,|D^{2}v_{p}|^{2}-f_{p}\,v_{p}\right)\,d{\bf
x}\ .$

Hence

(2.14) $\displaystyle\,E(K,v)=F_{r}(K,v)+M(v-w)\,,\qquad\quad\ f_{r}=f,$
$\displaystyle\ \,\hbox{with domain }\eqref{admissible K v},\vspace{0.2cm}$
(2.15) $\displaystyle\
F(K,v_{r},v_{p})=F_{r}(K,v_{r})+M(v_{r}-v_{p})+F_{p}(v_{p})\,,$
$\displaystyle\ \ \hbox{with domain }\eqref{admissible K vr vp}\vspace{0.2cm}$
(2.16) $\displaystyle\ \
G(K,v_{r},v_{p})=F(K,v_{r},v_{p})+\sigma\int_{K\cap\overline{\Omega}}|\,[Dv_{r}]\,|\,d\mathcal{H}^{1}\,,$
$\displaystyle\ \ \hbox{with domain }\eqref{admissible K vr vp G}$

where $F_{r}$ represents the potential energy of the reinforcement under the
Griffith assumption on the fracture energy, $M$ represents the adhesive
interaction energy (dependent on the slip $|v_{r}-v_{p}|$ between the plate
and the reinforcement) and $F_{p}$ represents the elastic energy of the
Kirchhoff-Love plate under the action of a transverse dead load $f$.

######  Remark 2.4.

We emphasize that, when minimizing (2.7), the Dirichlet datum turns out to be
forced on the plate ($v_{p}=w$ on $\partial\Omega$) since
$v_{p}-w\\!\in\\!H^{2}_{0}(\Omega)$ and
$H^{2}(\Omega_{p})\\!\subset\\!C^{0}(\Omega_{p}),$ while the Dirichlet datum
is prescribed by penalization on the reinforcement (through $v_{r}=w$ a.e.
$\Omega_{p}\setminus\overline{\Omega}$). Hence the damage of the reinforcement
may develop also at the boundary: if this is the case then
$\mathcal{H}^{1}(K\cap\partial\Omega)>0$.
In any case: $K\subset\overline{\Omega}$; $K$ is the closure of the set where
either $v_{r}$ or $\nabla v_{r}$ is not continuous; $w\in C^{2}$ and $v_{r}=w$
in $\Omega_{p}\setminus\overline{\Omega}$.

######  Remark 2.5.

The notions of essential pair or triplet in (2.5),(2.8),(2.10) select those
pairs or triplets which are cleansed of every artifact that does not affect
the functional value and are good representatives in equivalence classes of
admissible displacements. These classes allow highly irregular displacement
function $v$ for $v_{r}$ for the reinforcement: see Remarks 2.3-2.5 and Lemmas
2.6, 2.7 in [18] for comparison with Definition 2.1 in [17] of admissible
triplets in the context of image segmentation and/or image inpainting.
Minimization among admissible triplets (as defined in [16]) would be
equivalent to minimization among essential admissible triplets.

######  Remark 2.6.

The more general case where $|D^{2}v_{p}|^{2}$ is replaced by $Q(D^{2}v_{p})$,
with $Q$ positive definite quadratic form, leads to claims similar to the ones
we prove here (in Theorems 2.2 and 2.3) without any change in the proofs.

######  Remark 2.7.

The present paper deals with the Dirichlet boundary condition for both
reinforcement and plate: explicitly the reinforcement acts on the whole plate
$\Omega_{p}$ and sticks perfectly to it outside $\overline{\Omega}.$
Nevertheless the study of Neumann boundary condition for the reinforcement,
still keeping the Dirichlet condition $w$ on the plate (this boundary
conditions correspond to a structure where the reinforcement is present only
on the proper subset $\Omega$ of the plate $\Omega_{p}$), can be easily
recovered by the present analysis with minor changes: by considering
admissible displacements for the reinforcement defined only in the smaller
domain reference set $\Omega$ and replacing (2.5),(2.8),(2.10) respectively by

(2.17) $\begin{array}[]{l}\displaystyle K\ \hbox{is the smallest closed subset
of }\mathbb{R}^{2}\ s.t.\,\displaystyle v\in C^{2}(\Omega\setminus
K)\,;\end{array}$ (2.18)
$\begin{array}[]{l}v_{p}\\!-\\!w\\!\in\\!H^{2}_{0}(\Omega_{p})\,,\
\displaystyle K\,\hbox{ smallest closed subset of }\mathbb{R}^{2}\
s.t.\,\displaystyle v_{r}\\!\in\\!C^{2}(\Omega\\!\setminus\\!K)\,;\end{array}$
(2.19)
$\begin{array}[]{l}v_{p}\\!-\\!w\\!\in\\!H^{2}_{0}(\Omega_{p}),\,\\!\displaystyle
K\,\hbox{smallest closed subset of }\mathbb{R}^{2}s.t.\,\displaystyle
v_{r}\\!\in\\!C^{2}(\Omega\\!\setminus\\!K)\\!\cap\\!C^{0}(\Omega).\end{array}$

All the claims in Theorems 2.1 and Theorem 2 still hold true under these
different admissible classes of pairs and triplets. The only change to be made
in the proofs amounts to refer to [12] instead of [16], to perform the
analysis of partial regularity for weak minimizers.

##  3\. One-dimensional analysis: reinforcement of flexural beams

We study the 1D case, namely the hard-device reinforcement and strengthening
reinforcement of a clamped beam, in order to make explicit some properties of
minimizers like compliance identity, Euler equations, issues related to
uniqueness and possible addition of the unilateral constraint describing the
non-interpenetration of beam and reinforcement. The displacement of the
clamped beam is modeled by a function of one variable which is free in the
interval $[-1,+1]$ while it must coincide with a given function outside
$[-1,+1]$ to take into account of boundary conditions. We consider possibly
different weights for energy dissipation when crack or crease do appear: the
constants $\alpha$ and $\beta$ introduced below.
We consider real valued functions defined on bounded intervals, and set

(3.1) $\Omega\\!=\\!(-1,1),\quad\Omega_{p}\\!=\\!(-2,2),\quad w\in
C^{2}(-2,+2),$

moreover, concerning the notation, $\dot{v}$ denotes the absolutely continuous
part of the distributional derivative $v^{\prime}$ of $v$, $\ddot{v}$ denotes
the absolutely continuous part of $(\dot{v})^{\prime}\,,$ $S_{v}$ denotes the
set of discontinuity points of $v$, $S_{\dot{v}}$ denotes the set of
discontinuity points of $\dot{v}$ and $v^{-}$, $v^{+}$ denote respectively the
left and right limit of $v$. Since we will consider admissible only piece-wise
$H^{2}$ functions $v:(-2,2)\to\mathbb{R}$ fulfilling $v=w$ in
$(-2,-1)\cup(1,2)$, we have $\left(S_{v}\cup S_{\dot{v}}\right)\subset[-1,1]$
for them all. Here $H^{2}(a,b)$ denotes the usual Sobolev space of real-valued
functions $v\in L^{2}(a,b)$ s.t. $v^{\prime},v^{\prime\prime}\in L^{2}(a,b)$.
We emphasize that the beam may develop singularities also at both clamped
endpoints $\pm 1$: namely, it may undergo crack discontinuity (if
$S_{v}\cap\\{\pm 1\\}$ is nonempty) or plastic-yield bending (if
$S_{\dot{v}}\cap\\{\pm 1\\}$ is nonempty).
After labeling by $\sharp$ the counting measure, we denote by

(3.2) $J(v)\,=\,\alpha\sharp\,(S_{v})+\beta\sharp\,(S_{\dot{v}}\setminus
S_{v})$

the whole energy associated to damage of the reinforcement: in this one-
dimensional setting we allow different release energy for crack and crease,
respectively $\alpha$ and $\beta$. In the one-dimensional setting the
functionals $E,\,F$ and $G$ are replaced respectively by $E_{1},\,F_{1}$ and
$G_{1}$ defined below: we emphasize that for them all the strong and weak
formulation of related free discontinuity problems coincide in the one-
dimensional case, since finite energy entails that only a finite number of
discontinuity points is allowed by finite energy, hence only piece-wise
regular functions have finite energy.
The total energy for hard-device reinforcement of a clamped beam is given by
functional $E_{1}$:

(3.3) $\begin{array}[]{l}\displaystyle E_{1}(v)\ =\
J(v)\,+\,\int_{-1}^{1}\big{(}\,\eta\,|\ddot{v}|^{2}-fv\,\big{)}\,dx\,+\,\mu\,\int_{-1}^{1}|v-w|^{2}\,dx\
;\end{array}$

functional $E_{1}$ has to be minimized among the admissible functions $v$ such
that

(3.4) $v\in\mathbf{H}^{2}(-2,2):=\\{v:(-2,2)\to\mathbb{R},\hbox{ s.t. }v\hbox{
is piece-wise }H^{2}\\}\,,$ (3.5) $v=w\qquad\hbox{a.e }(-2,-1)\cup(1,2)\,$

The total energy for strengthening reinforcement of a clamped beam is given by
functional $F_{1}$:

(3.6) $\begin{array}[]{l}\\\ F_{1}(v_{r},v_{p})\ =\ F_{1}(V)\ :=\\\
\displaystyle
J(v_{r})+\\!\int_{-1}^{1}\\!\Big{(}\eta|\ddot{v}_{r}|^{2}-f_{r}v_{r}\Big{)}dx+\mu\\!\int_{-1}^{1}\\!|v_{r}-v_{p}|^{2}\,dx+\\!\int_{-1}^{1}\\!\\!\Big{(}\gamma\,|v_{p}^{\prime\prime}|^{2}-f_{p}\,v_{p}\Big{)}dx\,.\end{array}$

Functional $F_{1}$ has to be minimized among the admissible pairs $V$ such
that

(3.7) $V=(v_{r},v_{p})\in\mathbf{H}^{2}(-2,2)\times H^{2}(-2,2)\hbox{ with }$
(3.8) $v_{r}=v_{p}=w\quad\hbox{in}\quad(-2,-1)\cup(1,2)$

The total energy for strengthening reinforcement of an elastic-plastic clamped
beam is given by functional $G_{1}$:

(3.9) $\begin{array}[]{l}\\\ G_{1}(v_{r},v_{p})\ =\ G_{1}(V)\ :=\\\ \\\
\displaystyle\beta\
\sharp\big{(}S_{\dot{v}_{r}}\big{)}\,+\,\sigma\,\sum_{S_{\dot{v}_{r}}}\,|\,[\dot{v}_{r}]\,|\,+\\\
\displaystyle\int_{-1}^{1}\\!\Big{(}\eta|\ddot{v}_{r}|^{2}-f_{r}v_{r}\Big{)}dx+\mu\\!\int_{-1}^{1}\\!|v_{r}-v_{p}|^{2}\,dx+\\!\int_{-1}^{1}\\!\\!\Big{(}\gamma\,|v_{p}^{\prime\prime}|^{2}-f_{p}\,v_{p}\Big{)}dx\,.\end{array}$

Functional $G_{1}$ has to be minimized among the admissible pairs $V$
fulfilling

(3.10)
$V=(v_{r},v_{p})\in\Big{(}C^{0}([-2,2])\cap\mathbf{H}^{2}(-2,2)\Big{)}\times
H^{2}(-2,2)\,,$ (3.11) $v_{r}=v_{p}=w\quad\hbox{in}\quad(-2,-1)\cup(1,2)\,.$

Concerning respectively (3.4), (3.7), (3.10), we recall that in all cases the
finiteness of total energy implies respectively
$\sharp\big{(}S_{u}\big{)}<+\infty$,
$\sharp\big{(}S_{u_{r}}\cup(S_{\dot{u}_{r}})\big{)}<+\infty$ and
$S_{u_{r}}=\emptyset$ with $\sharp\big{(}S_{\dot{u}_{r}}\big{)}<+\infty$,
hence $u$ and $u_{r}$, are made by finitely many $H^{2}$ pieces.

######  Theorem 3.1.

Assume (3.1), (3.2), (3.3), $\eta>0,$ $\mu>0$, $f\in L^{2}(-1,1)$ and

(3.12) $0<\beta\leq\alpha\leq 2\beta,$

then the functional $E_{1}$ defined by (3.3) achieves a finite minimum over
functions $v$ fulfilling conditions (3.4),(3.5).

Proof - After noticing that

(3.13)
$\int_{-1}^{1}\\!\\!\\!\big{(}\,\mu\,|v-w|^{2}\,-fv\,\big{)}\,dx=\int_{-1}^{1}\\!\\!\mu\Big{(}v-\big{(}w+f/(2\mu)\big{)}\Big{)}^{\\!2}dx\,-\\!\int_{-1}^{1}\\!\\!\Big{(}fw+f^{2}/(4\mu)\Big{)}dx$

where the last summand on right-hand side is a constant, we deduce that the
functional $E_{1}$ is bounded from below since all terms are nonnegative,
except such constant. Thus the claim follows by choosing $g=w+f/(2\mu)\in
L^{2}(-1,1)$ in the result of [23]. $\square$ constraint n=1.

######  Theorem 3.2.

Assume (3.1), (3.2), (3.6), (3.12), $\eta\\!>\\!0,$ $\mu\\!>\\!0$,
$\gamma\\!>\\!0$, $f_{r},f_{p}\in L^{2}(-1,1)$.

Then the functional $F_{1}$ defined by (3.6) achieves a finite minimum over
functions $v$ fulfilling conditions (3.7),(3.8).

Proof -

The only novelty with respect to Theorem 3.1 consists in the addition of the
functional
$\int_{-1}^{1}\\!\\!\left(\gamma\,|v_{p}^{\prime\prime}|^{2}-f_{p}\,v_{p}-f_{r}\,v_{r}\right)dx$
and adhesive interaction $\mu\int_{-1}^{1}|v_{r}-v_{p}|^{2}dx$ coupling
$v_{r}$ and $v_{p}$ .
In case of functional $F_{1}$ the identity (3.13) reads as follows

(3.14)
$\int_{-1}^{1}\\!\\!\big{(}\,\mu\,|v_{r}-v_{p}|^{2}\,-f_{r}v_{r}\,\big{)}\,dx=\int_{-1}^{1}\\!\\!\mu\Big{(}v_{r}-\big{(}v_{p}+f_{r}/(2\mu)\big{)}\Big{)}^{2}dx\,-\int_{-1}^{1}\\!\\!\Big{(}f_{r}v_{p}+f_{r}^{2}/(4\mu)\Big{)}dx$

where the last summand is not a priori bounded from below, unless we show an a
priori bound on $\|v_{p}\|_{L^{2}(-1,1)}$, moreover we have to check that
minimizing sequences are not made by pair sequences
$\big{(}(v_{r})_{n},(v_{p})_{n}\big{)}$ balancing
$\mu\|(v_{p})_{n}\|_{L^{2}(-1,1)}^{2}\to+\infty$ together with
$\int_{-1}^{1}\left(\gamma\,|v_{p}^{\prime\prime}|^{2}-f\,v_{p}\right)dx\to-\infty$.

This is prevented by the subsequent estimate from below (3.16) due to
$\,v_{p}-w\in H^{2}_{0}(-1,1)$, where $C_{P}$ denotes the best Poincaré
constant fulfilling

(3.15)
$\|v\|_{L^{2}(-1,1)}^{2}\,\leq\,C_{P}\,\|v^{\prime\prime}\|_{L^{2}(-1,1)}^{2}\qquad\forall\,v\\!\in\\!H^{2}_{0}(-1,1).$

and we denote shortly $\|\cdot\|^{2}$ in place of
$\|\cdot\|^{2}_{L^{2}(-1,1)}$ :

(3.16) $\begin{array}[]{l}\\\ F_{1}(v_{r},v_{p})\ =\\\
\displaystyle=J(v_{r})+\\!\int_{-1}^{1}\\!\\!\Big{(}\eta|\ddot{v}_{r}|^{2}+\mu|v_{r}-v_{p}|^{2}(\gamma\,|v_{p}^{\prime\prime}|^{2}\Big{)}\,dx\,-\\!\int_{-1}^{1}\\!\left(f_{r}v_{r}+f_{p}\,v_{p}\right)dx\\\
\displaystyle=J(v_{r})+\\!\int_{-1}^{1}\\!\\!\Big{(}\eta|\ddot{v}_{r}|^{2}+\mu|v_{r}-v_{p}|^{2}(\gamma\,|v_{p}^{\prime\prime}|^{2}\Big{)}\,dx\,+\\\
\displaystyle\qquad-\int_{-1}^{1}\\!\\!f_{r}(v_{r}-v_{p})\,dx-\int_{-1}^{1}\\!\\!(f_{r}+f_{p})(v_{p}-w)\,dx-\int_{-1}^{1}\\!\\!(f_{r}+f_{p})w\,dx\\\
\displaystyle\geq\\!J(v_{r})+\int_{-1}^{1}\\!\\!\Big{(}\eta|\ddot{v}_{r}|^{2}+\mu|v_{r}-v_{p}|^{2}+\gamma\,|v_{p}^{\prime\prime}|^{2}\Big{)}\,dx\,+\\\
\displaystyle\ \
-\frac{1}{4\mu}\,\|f_{r}\|^{2}\\!-\mu\,\|v_{r}-v_{p}\|^{2}-\sqrt{C_{P}}\,\|f_{r}+f_{p}\|\,\|(v_{p}-w)^{\prime\prime}\|-\|f_{r}+f_{p}\|\,\|w\|\\\
\displaystyle\geq\\!J(v_{r})+\int_{-1}^{1}\\!\\!\Big{(}\eta|\ddot{v}_{r}|^{2}+\gamma\,|v_{p}^{\prime\prime}|^{2}\Big{)}\,dx\,+\\\
\displaystyle\quad-\frac{1}{4\mu}\,\|f_{r}\|^{2}-\sqrt{C_{P}}\,\|f_{r}+f_{p}\|\,\|(v_{p}-w)^{\prime\prime}\|-\|f_{r}+f_{p}\|\,\|w\|\\\
\displaystyle\geq\\!J(v_{r})+\int_{-1}^{1}\\!\\!\Big{(}\eta|\ddot{v}_{r}|^{2}+\gamma\,|v_{p}^{\prime\prime}|^{2}\Big{)}\,dx\,-\sqrt{C_{P}}\,\|f_{r}+f_{p}\|\,\|v_{p}^{\prime\prime}\|\\\
\displaystyle\quad-\frac{1}{4\mu}\,\|f_{r}\|^{2}-\sqrt{C_{P}}\,\|f_{r}+f_{p}\|\,\|w^{\prime\prime}\|-\|f_{r}+f_{p}\|\,\|w\|\\\
\displaystyle\geq\\!J(v_{r})+\int_{-1}^{1}\\!\\!\Big{(}\eta|\ddot{v}_{r}|^{2}+\gamma\,|v_{p}^{\prime\prime}|^{2}\Big{)}\,dx\,-\frac{\gamma}{2}\|v_{p}^{\prime\prime}\|^{2}+\\\
\displaystyle\
-\frac{C_{P}}{2\,\gamma}\|f_{r}+f_{p}\|^{2}-\frac{1}{4\mu}\,\|f_{r}\|^{2}-\sqrt{C_{P}}\,\|f_{r}+f_{p}\|\,\|w^{\prime\prime}\|-\|f_{r}+f_{p}\|\,\|w\|\\\
\displaystyle\geq\\!J(v_{r})+\int_{-1}^{1}\\!\\!\Big{(}\eta|\ddot{v}_{r}|^{2}+(\gamma/2)\,|v_{p}^{\prime\prime}|^{2}\Big{)}\,dx\,-C(\mu,\gamma,f_{r},f_{p},C_{P})\
.\end{array}$

Then we can fix a minimizing sequence $\big{(}(v_{r})_{h},(v_{p})_{h}\big{)}$
for $F_{1}$, and get boundedness of $\|(v_{p})_{h}\|_{H^{2}(-1,1)}$, thanks to
$F_{1}(w,w)<+\infty$, (3.16) and the Poincaré inequality (3.15). There is
$u_{p}\in L^{2}$ such that we can extract a subsequence, without relabeling,
fulfilling $(v_{p})_{h}\to u_{p}$ weakly in $H^{2}$ and strongly in $L^{2}$,
with
$\|(v_{p})^{\prime\prime}_{h}\|_{L^{2}}\to\|u_{p}^{\prime\prime}\|_{L^{2}}$.
By (3.14) and (3.16) also $\|(v_{r})_{h}-(v_{p})_{h}\|_{L^{2}}$ and
$\|(v_{r})_{h}\|_{L^{2}}$ are bounded: by extracting again, without
relabeling, $(v_{r})_{h}\\!\to\\!u_{r}$ weakly in $L^{2}$. We write

(3.17)
$\begin{array}[]{l}F_{1}\big{(}(v_{r})_{h},u_{p}\big{)}=F_{1}\big{(}(v_{r})_{h},(v_{p})_{h}\big{)}+\Big{(}F_{1}\big{(}(v_{r})_{h},(v_{p})_{k}\big{)}-F_{1}\big{(}(v_{r})_{h},(v_{p})_{h}\big{)}\Big{)}+\vspace{-0.1cm}\\\
+\Big{(}F_{1}\big{(}(v_{r})_{h},u_{p}\big{)}\\!-\\!F_{1}\big{(}(v_{r})_{h},(v_{p})_{k}\big{)}\\!\Big{)}=F\big{(}(v_{r})_{h},(v_{p})_{h}\big{)}+A(h,k)+B(h,k).\end{array}$

By lower semicontinuity of the functional
$v\mapsto\mu\\!\int_{-1}^{1}\\!|v-v_{p}|^{2}\,dx+\\!\int_{-1}^{1}\\!\left(\gamma\,|v^{\prime\prime}|^{2}\\!-\\!f_{p}\,v\right)dx$
we get $\liminf_{k}B(h,k)\leq 0$. Moreover, $\forall\,\varepsilon>0$
$\exists\,h_{\varepsilon}:$ for every $h,k>h_{\varepsilon}$

$\displaystyle|A(h,k)|$ $\displaystyle\leq$
$\displaystyle\mu\\!\int_{-1}^{1}\\!|(v_{p})_{k}-(v_{p})_{h}||2(v_{r})_{h}-((v_{p})_{k}+(v_{p})_{h})|\,dx+$
$\displaystyle\
+\eta\big{(}\|(v_{p})^{\prime\prime}_{k}\|^{2}-\|(v_{p})^{\prime\prime}_{h}\|^{2}\big{)}+\|f_{p}\|_{L^{2}}\big{(}\|(v_{p})_{k}-(v_{p})_{h}\|\big{)}\
<\ \varepsilon\ .$

By evaluating on both sides of (3.17) first $\liminf_{k}$, then $\liminf_{h}$,
we obtain that also $\big{(}(v_{r})_{h},u_{p}\big{)}$ is a minimizing sequence
for the functional $F_{1}$ which is lower semicontinuous, or equivalently
$(v_{r})_{h}$ is a minimizing sequence for functional $E_{1}$ with datum
$u_{p}$. Then $(u_{r},u_{p})$ belongs to $\mathop{{\rm argmin}}\nolimits
F_{1}$. $\square$

######  Theorem 3.3.

Assume (3.1), (3.2), (3.9),
$\beta\\!\geq\\!0,\,\sigma\\!>\\!0,\,\eta\\!>\\!0,\,\mu\\!>\\!0,\,\gamma\\!>\\!0,$
and $\,f_{r},\,f_{p}$ belong to $L^{2}(-1,1).$ Then the functional $G_{1}$
defined by (3.9) achieves a finite minimum over pairs $(v_{r},v_{p})$
fulfilling Dirichlet condition (3.8) and

(3.18)
$V=(v_{r},v_{p})\in\Big{(}C^{0}(-2,2)\cap\mathbf{H}^{2}(-2,2)\Big{)}\times
H^{2}(-2,2)\,.$

Proof - Notice that $v_{r}\in C^{0}$ entails $S_{v_{r}}=\emptyset.$ We set
$I(v_{r})\,=\,\beta\,\sharp\big{(}S_{\dot{v}_{r}}\big{)}+\sigma\sum_{S_{\dot{v}_{r}}}\,|\,[\dot{v}_{r}]\,|$.
By arguing as like as in the derivation of (3.16) (the only difference
consists in replacing $J(v_{r})$ with $I(v_{r})$), we get

(3.19) $G_{1}(v_{r},v_{p})\geq
I(v_{r})+\int_{-1}^{1}\\!\\!\Big{(}\eta|\ddot{v}_{r}|^{2}+(\gamma/2)\,|v_{p}^{\prime\prime}|^{2}\Big{)}\,dx\,-C(\mu,\gamma,f_{r},f_{p},C_{P})\,.$

If $\beta>0$, we conclude by arguing as in the proof of Theorem 3.2 that a
minimizing sequence has a subsequence converging to a minimum. When $\beta=0$,
after finding an optimal $u_{p}$ again by the argument in the proof of Theorem
3.2, we exploit Theorem 2.1 of [35] to find the related optimal $u_{r}$.
We emphasize that the safe load condition assumed in [35] is unnecessary here
thanks to adhesion term $\int_{-1}^{1}\\!|v_{r}-v_{p}|^{2}dx$, providing
boundedness from below by (3.19). $\square$ Next result shows that, provided
the load and Dirichlet datum are suitably small, the strengthening
reinforcement of the clamped beam ($\mathop{{\rm argmin}}\nolimits F_{1}$) has
a unique solution $(u_{r},u_{p})$ where $v_{r}$ has neither crack nor hinges,
say $S_{u_{r}}\cup S_{\dot{u}_{r}}=\emptyset$.

######  Theorem 3.4.

In addition to assumptions of Theorem 3.2 we assume

(3.20)
$\frac{1}{4\mu}\|f_{r}\|^{2}_{L^{2}(-1,1)}+\frac{C_{\Omega}}{2\gamma}\|f_{r}+f_{p}\|^{2}_{L^{2}(-1,1)}+\gamma\|w^{\prime\prime}\|_{L^{2}(-1,1)}^{2}+\\!\int_{-1}^{1}\\!\\!\\!(f_{r}+f_{p})w\,dx<\beta-M,$

where $\ -\infty<M=\min\big{\\{}F_{1}(v_{r},v_{p})\\!:v_{r}-w\in
H^{2}_{0}(-1,1),v_{p}-w\in H^{2}_{0}(-1,1)\big{\\}}<+\infty.$
Then there is a unique minimizer $(u_{r},u_{p})$ of $F_{1}$ over
$\mathbf{H}^{2}\times H^{2}$ and such minimizer fulfils $S_{u_{r}}\cup
S_{\dot{u}_{r}}=\emptyset$, thus $u_{r}-w\in H^{2}_{0}(-2,2)$.

Proof -  We denote shortly $\|\cdot\|^{2}$ in place of
$\|\cdot\|^{2}_{L^{2}(-1,1)}$ . First we note that
$\min\big{\\{}F_{1}(v_{r},v_{p})\\!:v_{r}-w\\!\in\\!H^{2}_{0}(-1,1),v_{p}-w\\!\in\\!H^{2}_{0}(-1,1)\big{\\}}\leq
F_{1}(w,w)=(\eta+\gamma)\|w^{\prime\prime}\|^{2}-\int_{-1}^{1}(f_{r}+f_{p})w\,dx$;

moreover, thanks to (3.16), $F_{1}$ is bounded from below and coercive hence
its infimum $M$ is attained and is a finite minimum.
Assume by contradiction that a minimizer $(u_{r},u_{p})$ of $F_{1}$ on
$\mathbf{H}^{2}\times H^{2}$ has $S_{u_{r}}\\!\cup
S_{\dot{u}_{r}}\\!\not=\\!\emptyset$, we deucee
$\alpha\,\sharp\big{(}S_{u_{r}}\big{)}+\beta\,\sharp\big{(}S_{u_{r}}\\!\setminus\\!S_{\dot{u}_{r}}\big{)}\geq\beta$,
hence, exploiting Poincaré inequality (3.15) and assumption (3.20) we get

$\begin{array}[]{l}\\!\\!\\!\displaystyle\int_{-1}^{1}\\!f_{r}u_{r}+\int_{-1}^{1}\\!f_{p}u_{p}-\int_{-1}^{1}(f_{r}+f_{p})\,w=\int_{-1}^{1}f_{r}\,(u_{r}-u_{p})\,+\int_{-1}^{1}(f_{r}+f_{p})\,(u_{p}-w)\,\leq\\\
\vspace{0.1cm}\leq\displaystyle\frac{1}{4\mu}\|f_{r}\|^{2}+\mu\|u_{r}-u_{p}\|^{2}+\frac{C_{\Omega}}{2\gamma}\|f_{r}+f_{p}\|^{2}+\frac{\gamma}{2C_{\Omega}}\|u_{p}-w\|^{2}\leq\\\
\vspace{0.1cm}\leq\displaystyle\frac{1}{4\mu}\|f_{r}\|^{2}+\mu\|u_{r}-u_{p}\|^{2}+\frac{C_{\Omega}}{2\gamma}\|f_{r}+f_{p}\|^{2}+\frac{\gamma}{2}\,\|u_{p}^{\prime\prime}-w^{\prime\prime}\|^{2}\leq\\\
\leq\displaystyle\frac{1}{4\mu}\|f_{r}\|^{2}\\!+\frac{C_{\Omega}}{2\gamma}\|f_{r}+f_{p}\|^{2}\\!+\gamma\|w^{\prime\prime}\|^{2}+\eta\|\ddot{u}_{r}\|^{2}\\!+\mu\|u_{r}-u_{p}\|^{2}\\!+\gamma\,\|u_{p}^{\prime\prime}\|^{2}\\\
<\displaystyle\beta-M-\int_{-1}^{1}(f_{r}+f_{p})\,w\,+\,\eta\|\ddot{u}_{r}\|^{2}+\mu\|u_{r}-u_{p}\|^{2}+\gamma\,\|u_{p}^{\prime\prime}\|^{2}\vspace{-0.2cm}\\\
\leq\displaystyle\alpha\,\sharp\big{(}S_{u_{r}}\big{)}+\beta\,\sharp\big{(}S_{u_{r}}\\!\setminus\\!S_{\dot{u}_{r}}\big{)}\,+\eta\|\ddot{u}_{r}\|^{2}\\!+\mu\|u_{r}-u_{p}\|^{2}\\!+\gamma\,\|u_{p}^{\prime\prime}\|^{2}\\!-M-\\!\int_{-1}^{1}\\!\\!\\!(f_{r}+f_{p})\,w\,.\end{array}$

say, an inequality contradicting minimality of $(u_{r},u_{p})$ :
$M\,<\,F_{1}(u_{r},u_{p})\,.$ Uniqueness of minimizer over
$\mathbf{H}^{2}\times H^{2}$ with Dirichlet datum $w$ follows by uniqueness
over $H^{2}\times H^{2}$. $\square$

######  Remark 3.5.

By analogous computations to the ones in the last proof, we obtain that the
inequality

$\frac{1}{4\mu}\,\|f\|^{2}_{L^{2}(-1,1)}+\int_{-1}^{1}\\!f\,w\,dx\ <\
\beta-\widetilde{M}$

entails uniqueness and $H^{2}$ regularity for minimizer of $E_{1}$ in
$\mathbf{H}^{2}$ with Dirichlet boundary condition $w$, where

$-\infty\,<\,\widetilde{M}\,=\,\min\big{\\{}E_{1}(v)\\!:v-w\in
H^{2}_{0}(-1,1)\big{\\}}\,\leq\,E_{1}(w)<+\infty\,.$

We show the analysis of $E_{1}$ under the addition of the unilateral
constraint

(3.21) $\qquad v\geq w\qquad\hbox{ on }[-1,+1]\,.$

Concerning notation, from now on we set $v^{+}(x)=\lim_{t\to x_{+}}v(t)$,
$v^{-}(x)=\lim_{t\to x_{-}}v(t)$.

######  Remark 3.6.

Actually, the constraint (3.21) has to be understood as a pointwise everywhere
weak inequality, since it refers to functions $v\in\mathbf{H}^{2}(-2,2)$:
explicitly, $v(x)\geq w(x)$ at $x\in[-1,+1]\\!\setminus\\!S_{v}$;
$v^{+}(x)\geq w(x)$ at $x\in S_{v}\cup\\{-1\\}$; $v^{-}(x)\geq w(x)$ at $x\in
S_{v}\cup\\{+1\\}$.
Thus, the contact set $\\{x\\!\in\\![-2,2]\\!:v^{+}(x)\\!=\\!w(x)\hbox{ or
}v^{-}(x)\\!=\\!w(x)\\}$ is a closed set for every $v$ fulfilling (3.21); the
complement in $(-1,1)$ of the contact set is an open set.
Actually, the inequality (3.21) prevents interpenetration and refers to a
reinforcement placed above: this conventional choice is made here in order to
have agreement with the usual formulation of variational inequalities ([4]).

######  Theorem 3.7.

(hard device with unilateral constraint)
Assume (3.1),(3.2),(3.3),(3.12) $\eta>0,$ $\mu>0$ and $f\\!\in\\!L^{2}(-1,1)$.
Then the functional $E_{1}$ achieves a finite minimum over pairs
$(v_{r},v_{p})$ fulfilling conditions (3.4),(3.5) together with the unilateral
constraint (3.21).

Proof -  The proof can be achieved by exact repetition of the argument in the
proof of Theorem 3.1 for the unconstrained case: both unilateral constraint
$v\geq w$ on $[-1,+1]$ and Dirichlet condition $v=w$ a.e. on
$(-2,-1)\cup(1,2)$ affect neither the compactness, nor the lower
semicontinuity properties of $E_{1}$; moreover the a.e. convergence preserves
the constraint in the limit of minimizing sequences. $\square$

######  Theorem 3.8.

(reinforcement with unilateral constraint)
Assume (3.1), (3.2), (3.6), (3.12), $\eta\\!>\\!0,$ $\mu\\!>\\!0$,
$\gamma\\!>\\!0$, $f_{r},f_{p}\in L^{2}(-1,1)$.
Then the functional $F_{1}$ achieves a finite minimum over pairs
$(v_{r},v_{p})$ fulfilling the conditions (3.7) and (3.8) together with the
unilateral constraint (corresponding to a reinforcement placed above the
plate)

(3.22) $\qquad v_{r}\geq v_{p}\qquad\hbox{ on }[-1,+1]\,.$

Also the constraint (3.22) has to be understood as a pointwise everywhere
everywhere weak inequality, in the sense of Remark 3.6, as like as (3.21) but
here with $v_{p}$ replacing $w$: thus, the admissible pairs belong to the
convex set

$\mathbf{K}\\!:=\\!\left\\{(v_{r},v_{p})\\!\in\\!\mathbf{H}^{2}(-2,2)\\!\times\\!H^{2}(-2,2)\\!:\,v_{r}\\!\geq\\!v_{p}\,\hbox{on}\,[-1,1],\,v_{r}\\!=\\!v_{p}\\!=\\!w\,\,\hbox{on}\,(-2,-1)\\!\cup\\!(1,2)\right\\}$

Proof of Thm 3.8 - The proof can be achieved by exact repetition of the
argument in the proof of Theorem 3.2 for the unconstrained case: both
unilateral constraint $v_{r}\geq v_{p}$ a.e. on $(-1,+1)$ and Dirichlet
condition $v_{r}\\!=\\!v_{p}\\!=\\!w$ a.e. on $(-2,-1)\cup(1,2)$ affect
neither the compactness, nor the lower semicontinuity properties of $F_{1}$;
moreover a.e. convergence preserves the constraint in the limit of minimizing
sequences. $\square$ By performing all the admissible variations of minimizers
for $E_{1}$ and $F_{1}$ without the unilateral constraint, we can deduce the
necessary conditions for minimality listed below in Propositions 3.9 and 3.10.

######  Proposition 3.9.

. (Euler equations for functional $E_{1}$) Every $u\in\mathop{{\rm
argmin}}\nolimits E_{1}$ fulfils

(3.23) $\eta\,u^{\prime\prime\prime\prime}+\mu(u-w)=f/2\qquad\hbox{in
}(-1,1)\setminus\left(S_{u}\cup S_{\dot{u}}\right)\ ,$ (3.24)
$\ddot{u}^{+}=\ddot{u}^{-}=\dddot{u}^{+}=\dddot{u}^{-}=0\qquad\qquad\hbox{in
}S_{u}\setminus\\{\pm 1\\}\ ,$ (3.25)
$\ddot{u}^{+}=\ddot{u}^{-}=[\dddot{u}]=0\qquad\qquad\ \hbox{on
}S_{\dot{u}}\setminus\left(S_{u}\cup\\{\pm 1\\}\right)\ ,$ (3.26) $\ddot{u}\in
H^{2}(-1,1)\qquad\hbox{and}\quad\eta\,(\ddot{u})^{\prime\prime}+\mu(u-w)=f/2\quad\hbox{on
}\mathcal{D}^{\prime}(-1,1)\ .$ (3.27) $\left\\{\
\begin{array}[]{l}\ddot{u}^{+}(-1)=\dddot{u}^{+}(-1)=0\qquad\hbox{if }-1\in
S_{u}\setminus S_{\dot{u}}\ ,\vspace{0.2cm}\\\
\ddot{u}^{-}(+1)=\dddot{u}^{-}(+1)=0\qquad\hbox{if }+1\in S_{u}\setminus
S_{\dot{u}}\ ,\end{array}\right.$ (3.28) $\left\\{\
\begin{array}[]{l}\ddot{u}^{+}(-1)=0\qquad\hbox{if }-1\in S_{\dot{u}}\setminus
S_{u}\ ,\vspace{0.2cm}\\\ \ddot{u}^{-}(+1)=0\qquad\hbox{if }+1\in
S_{\dot{u}}\setminus S_{u}\ ,\end{array}\right.\qquad\phantom{.}$

When $\alpha\\!=\\!\beta$ the conditions (3.24),(3.25),(3.27),(3.28)
altogether are improved as follows:

(3.29) $\hbox{if }\alpha=\beta\quad\hbox{ then
}\quad\ddot{u}^{+}=\ddot{u}^{-}=\dddot{u}^{+}=\dddot{u}^{-}=0\quad\hbox{ on
}\left(S_{u}\cup S_{\dot{u}}\right).$

(Euler equations for functional $F_{1}$) Every $(u_{r},u_{p})\in\mathop{{\rm
argmin}}\nolimits F_{1}$ fulfils

(3.30)
$\eta\,u_{r}^{\prime\prime\prime\prime}+\mu(u_{r}-u_{p})=f_{r}/2\qquad(-1,1)\\!\setminus\\!\left(S_{u_{r}}\cup
S_{\dot{u}_{r}}\right)\ ,$ (3.31)
$\gamma\,u_{p}^{\prime\prime\prime\prime}+\mu(u_{p}-u_{r})=f_{p}/2\qquad\hbox{
in }\mathcal{D}^{\prime}(-1,1)\ ,$ (3.32)
$\eta\,u_{r}^{\prime\prime\prime\prime}+\gamma
u_{p}^{\prime\prime\prime\prime}=(f_{r}+f_{p})/2\qquad(-1,1)\\!\setminus\\!\left(S_{u_{r}}\cup
S_{\dot{u}_{r}}\right)\ ,$ (3.33)
$\ddot{u}_{r}^{+}=\ddot{u}_{r}^{-}=\dddot{u}_{r}^{+}=\dddot{u}_{r}^{-}=0\qquad\qquad\
S_{u_{r}}\setminus\\{\pm 1\\}\ ,$ (3.34)
$\ddot{u}_{r}^{+}=\ddot{u}_{r}^{-}=[\dddot{u}_{r}]=0\quad\qquad
S_{\dot{u}_{r}}\\!\setminus\\!\left(S_{u_{r}}\cup\\{\pm 1\\}\right)\ ,$

hence $\ddot{u}_{r}\in H^{2}(-1,1)$ and

(3.35)
$\eta\,(\ddot{u}_{r})^{\prime\prime}+\mu(u_{r}-u_{p})=f_{r}/2\qquad\hbox{ in
}\mathcal{D}^{\prime}(-1,1)\ ,$ (3.36)
$\eta\,(\ddot{u}_{r})^{\prime\prime}+\gamma
u_{p}^{\prime\prime\prime\prime}=(f_{r}+f_{p})/2\qquad\hbox{ in
}\mathcal{D}^{\prime}(-1,1)\ .$ (3.37) $\left\\{\
\begin{array}[]{l}\ddot{u}_{r}^{+}(-1)=\dddot{u}_{r}^{+}(-1)=0\qquad\hbox{if
}-1\in S_{u_{r}}\\!\setminus\\!S_{\dot{u}_{r}}\ ,\vspace{0.2cm}\\\
\ddot{u}_{r}^{-}(+1)=\dddot{u}_{r}^{-}(+1)=0\qquad\hbox{if }+1\in
S_{u_{r}}\\!\setminus\\!S_{\dot{u}_{r}}\ ,\end{array}\right.$ (3.38)
$\left\\{\ \begin{array}[]{l}\ddot{u}_{r}^{+}(-1)=0\qquad\hbox{if }-1\in
S_{\dot{u}_{r}}\\!\setminus\\!S_{u_{r}}\ ,\vspace{0.2cm}\\\
\ddot{u}_{r}^{-}(+1)=0\qquad\hbox{if }+1\in
S_{\dot{u}_{r}}\\!\setminus\\!S_{u_{r}}\ ,\end{array}\right.\qquad\phantom{.}$

So, (3.35) and (3.36), together give

(3.39) $2\,\eta\,(\ddot{u}_{r})^{\prime\prime}+\mu(u_{r}-u_{p})+\gamma
u_{p}^{\prime\prime\prime\prime}=f_{r}+(1/2)f_{p}\qquad\hbox{ in
}\mathcal{D}^{\prime}(-1,1)\ .$

When $\alpha\\!=\\!\beta$ the conditions (3.33),(3.34),(3.37),(3.38)
altogether are improved as follows:

(3.40) $\hbox{if }\alpha=\beta\quad\hbox{ then
}\quad\ddot{u}_{r}^{+}=\ddot{u}_{r}^{-}=\dddot{u}_{r}^{+}=\dddot{u}_{r}^{-}=0\quad\hbox{
on }\left(S_{u_{r}}\cup S_{\dot{u}_{r}}\right).$

Eventually we deduce the following compliance identities.

######  Proposition 3.10.

Compliance identity for functional $E_{1}$:
Assume $w(-1)=w(1)=w^{\prime}(-1)=w^{\prime}(1)=0$. Then any $u\in\mathop{{\rm
argmin}}\nolimits E_{1}$ fulfils

(3.41) $E_{1}(u)\,=\,J(u)\,+\,\mu\int_{-1}^{1}(\,w^{2}\,-\,wu\,)\,dx\ -\
\frac{1}{2}\int_{-1}^{1}fu\,dx\ .$

If boundary conditions are nonhomogeneous then the right-and side of
compliance (3.41) has to be added with the correction
$+\,\eta\,\left[\,\ddot{u}\,\dot{u}-\dddot{u}\,u\,\right]_{-1}^{+1}$ , where
$[z]_{-1}^{1}=z^{-}(1)-z^{+}(-1)$. Notice that (due to (3.27),(3.28),(3.29))
some of the four terms in the correction may be null if one endpoint or the
other belongs to $S_{\dot{u}}\cup S_{u}$.

Compliance identity for functional $F_{1}$:
Assume $w(-1)=w(1)=w^{\prime}(-1)=w^{\prime}(1)=0$, then any
$(u_{r},u_{p})\in\mathop{{\rm argmin}}\nolimits F_{1}$ fulfils

(3.42)
$F_{1}(u_{r},u_{p})\,=\,J(u_{r})\,-\,\mu\int_{-1}^{1}(u_{r}-u_{p})^{2}dx\,\,-\,\frac{1}{2}\,\int_{-1}^{1}f_{r}\,u_{r}\,dx\,\,-\,\frac{1}{2}\,\int_{-1}^{1}f_{p}\,u_{p}\,dx\
.$

If boundary conditions are nonhomogeneous then the right-and side of
compliance (3.42) has to be added with the correction

$+\left[\,\gamma\,(u_{p}^{\prime\prime}w^{\prime}\\!-u_{p}^{\prime\prime\prime}w)+\eta\,(\,\ddot{u}_{r}\,\dot{u}-\dddot{u}_{r}\,u)\,\right]_{-1}^{+1}$
, where $[z]_{-1}^{1}=z^{-}(1)-z^{+}(-1)$.

Proof of Proposition 3.9 (Euler equations for $E_{1}$ and $F_{1}$) \-
Let $u$ be a minimizer of $E_{1}$ among
$v\in\mathbf{H}^{2}:=\mathbf{H}^{2}(-2,2)$ fulfilling (3.4) and (3.5). For any
$v\in\mathbf{H}^{2}$ we set
$\mathbf{\big{[}\kern-5.0pt\big{[}}v\mathbf{\big{]}\kern-5.0pt\big{]}}=v^{+}-v^{-}$
where $v^{-},\ v^{+}$ denote respectively the left and right values of $v$ on
$S_{v}$.
We introduce the localized version of functional $E_{1}$: given $w$, $\alpha$,
$\beta$, we set, for any $v$ in $\mathbf{H}^{2}$ and any Borel set
$A\subset[-1,1]$,

(3.43) $E_{1}(v,A)\ =\
\int_{A}\big{(}\eta|\ddot{v}^{2}|+\mu|v-w|^{2}\big{)}\,dx+\alpha\,\sharp(S_{v}\cap
A)+\beta\,\sharp\left((S_{\dot{v}}\setminus S_{v})\cap A\right)\,.$

Step 1 - (Green formula) Assume: $u\in\mathop{{\rm argmin}}\nolimits E_{1}$.
Since $J(u)\leq E_{1}(u)<+\infty$, the set $S_{u}\cup S_{\dot{u}}$ is finite
and contained in $[-1,1]$; $u\in H^{4}(I)$ for every interval
$I\subset(-2,2)\setminus\\{S_{u}\cup S_{\dot{u}}\\}$.
From now on we label $t_{0}=-1$ and $t_{T+1}=1$ and $t_{j}$, for
$j=1,...,\mathsf{T}$, the (possibly empty) finite ordered set $\left(S_{u}\cup
S_{\dot{u}}\right)\cap(-1,+1)$. Then, integrating by parts, the next identity
is achieved for every $\varphi\in\mathbf{H}^{2}$

(3.44)
$\sum_{l=0}^{\mathsf{T}}\int_{t_{l}}^{t_{l+1}}\\!\ddot{u}\,\ddot{\varphi}\ dx\
=\
\sum_{l=0}^{\mathsf{T}}\int_{t_{l}}^{t_{l+1}}\\!u^{\prime\prime}\varphi^{\prime\prime}\,dx\
=\
\sum_{l=0}^{\mathsf{T}}\int_{t_{l}}^{t_{l+1}}u^{\prime\prime\prime\prime}\varphi\,dx\
+\\\
\sum_{l=1}^{\mathsf{T}}\Big{(}\big{(}-\dddot{u}^{-}(t_{l+1})\varphi^{-}(t_{l+1})+\dddot{u}^{+}(t_{l})\varphi^{+}(t_{l})\big{)}+\big{(}\ddot{u}^{-}(t_{l+1})\dot{\varphi}^{-}(t_{l+1})-\ddot{u}^{+}(t_{l})\dot{\varphi}^{+}(t_{l})\big{)}\Big{)}\,+\\\
+\Big{(}-\dddot{u}^{-}(t_{1})\varphi^{-}(t_{1})\,+\,\ddot{u}^{-}(t_{1})\dot{\varphi}^{-}(t_{1})\,+\,\dddot{u}^{+}(t_{T})\varphi^{+}(t_{T})\,-\,\ddot{u}^{+}(t_{T})\dot{\varphi}^{+}(t_{T})\Big{)}\,+\\\
+\Big{(}\dddot{u}^{+}(-1)\varphi^{+}(-1)\,-\,\ddot{u}^{+}(-1)\dot{\varphi}^{+}(-1)\,-\,\dddot{u}^{-}(1)\varphi^{-}(1)\,+\,\ddot{u}^{-}(1)\dot{\varphi}^{-}(1)\Big{)}\,.$

Step 2 - At first we show that each minimizer $u$ solves the fourth order
elliptic equation (3.23) on the interior of $(-1,1)\setminus(S_{u}\cup
S_{\dot{u}})$, by performing smooth variations. For every open set
$A\subset\subset(-1,1)\setminus(S_{u}\cup S_{\dot{u}})$, for every
$\varepsilon\in\mathbb{R}$ and for every $\varphi\in C_{0}^{\infty}(A)$ we
have

$0\leq
E_{1}(u+\varepsilon\varphi,A)-E_{1}(u,A)=2\varepsilon\bigg{(}\eta\\!\int_{A}u^{\prime\prime}\varphi^{\prime\prime}\,dx+\mu\\!\int_{A}(u-w)\varphi\,dx\,-\\!\int_{A}\frac{f}{2}\varphi\,dx\bigg{)}+o(\varepsilon)$

where $o(\varepsilon)$ is an infinitesimal of higher order than $\varepsilon$.
Hence

$\eta\,\int_{A}u^{\prime\prime}\varphi^{\prime\prime}\,dx\
=\int_{A}\Big{(}f/2-\,\mu\,(u-w)\Big{)}\varphi\,dx\qquad\forall\varphi\in
C_{0}^{\infty}(A)\ .$

Then (3.23) follows integrating by parts with Green formula (3.44).
Now we seek the Euler conditions at inner discontinuity points and at clamped
endpoints. Step 3 - We prove necessary conditions (3.24) for extremality on
$S_{u}\,$ and necessary conditions (3.27) for extremality at endpoints when
they do belong to $S_{u}\\!\setminus\\!S_{\dot{u}}$.

Choose $\varphi\in\mathbf{H}^{2}\cap C^{2}([t_{l},t_{l+1}])$,
$l=0,...,\mathsf{T}$, $\mbox{{spt}}(\varphi)\subset A$, where $A$ is a Borel
subset of $[-1,1]$ with $(S_{\dot{u}}\setminus S_{u})\cap A=\emptyset$. Then
for every $\varepsilon\in\mathbb{R}$ we have

$\big{(}S_{u+\varepsilon\varphi}\cup
S_{\dot{u}+\varepsilon\dot{\varphi}}\big{)}\cap A\ \subset\ {S_{u}}\cap A$

By (3.44) we have:

$\displaystyle\vspace{-0.2cm}0$ $\displaystyle\leq$ $\displaystyle
E_{1}(u+\varepsilon\varphi,A)-E_{1}(u,A)\ =$ $\displaystyle=$
$\displaystyle\alpha\left(\sharp(S_{u+\varepsilon\varphi}\cap
A)-\sharp(S_{u}\cap A)\right)+\beta\sharp\left((S_{\dot{\varphi}}\setminus
S_{u+\varepsilon\varphi})\cap A\right)\,+$ $\displaystyle\qquad+\
2\varepsilon\left(\sum_{l=0}^{\mathsf{T}}\int_{t_{l}}^{t_{l+1}}\big{(}\eta\,u^{\prime\prime}\varphi^{\prime\prime}+\mu(u-w)\varphi-\frac{f}{2}\varphi\big{)}\,dx\right)\,+\,o(\varepsilon)\,=$
$\displaystyle=$ $\displaystyle\alpha\left(\sharp(S_{u+\varepsilon\varphi}\cap
A)-\sharp(S_{u}\cap A)\right)+\beta\sharp\left((S_{\dot{\varphi}}\setminus
S_{u+\varepsilon\varphi})\cap A\right)\,+$ $\displaystyle\qquad+\
2\varepsilon\bigg{(}\sum_{l=0}^{\mathsf{T}}\int_{t_{l}}^{t_{l+1}}\big{(}\eta\,u^{\prime\prime\prime\prime}\varphi+\mu(u-w)\varphi-\frac{f}{2}\varphi\big{)}\,dx\,+$
$\displaystyle\qquad+\
{\dddot{u}}^{+}(-1)\varphi^{+}(-1)-\ddot{u}^{+}(-1)\dot{\varphi}^{+}(-1)-{\dddot{u}}^{-}(1)\varphi^{-}(1)+\ddot{u}^{-}(1)\dot{\varphi}^{-}(1)\,+$
$\displaystyle\qquad+\ \,\eta\\!\sum_{\left(S_{u}\cap A\right)\setminus\\{\pm
1\\}}\\!\Big{(}\mathbf{\big{[}\kern-5.0pt\big{[}}\dddot{u}\varphi\mathbf{\big{]}\kern-5.0pt\big{]}}-\mathbf{\big{[}\kern-5.0pt\big{[}}\ddot{u}\dot{\varphi}\mathbf{\big{]}\kern-5.0pt\big{]}}\Big{)}\bigg{)}\
+\ o(\varepsilon)\,.$

Up to a finite set of possible values of $\varepsilon$ entailing cancellation
of discontinuity, we have $S_{u+\varepsilon\varphi}\cap A=S_{u}\cap A$. Then
by discarding such values we can choose arbitrarily small $\varepsilon$
satisfying

$\vspace{-0.1cm}\sharp\,((S_{\dot{\varphi}}\setminus
S_{u+\varepsilon\varphi})\cap A)=\sharp\,((S_{\dot{\varphi}}\setminus
S_{u})\cap A)=0$

By taking into account (3.23) and the arbitrariness of the two traces of
$\varphi$ and $\dot{\varphi}$ on the two sides of points in $S_{u}$, for small
$\varepsilon$, we can choose $\varphi$ with $\varphi^{\pm}=0$, and
$\dot{\varphi}^{+}=0$ together with $\dot{\varphi}^{-}$ arbitrary, or
viceversa to get $\,\ddot{u}^{\pm}=0$ on $S_{u}\setminus\\{\pm 1\\}$.
Similarly, we obtain $\,\dddot{u}^{\pm}=0$ on $S_{u}\setminus\\{\pm 1\\}$ by
choosing $\dot{\varphi}^{\pm}=0$, and $\varphi^{+}=0$ together with
$\varphi^{-}$ arbitrary or vice-versa. So (3.24) is proved.
If some clamped endpoint ($-1$ and/or $+1$) belong to $S_{u}$, then (3.27) is
obtained as above, but taking into account that $\varphi\equiv 0$ outside
$[-1,1]$. Step 4 - We prove the necessary condition (3.25) for extremality on
$S_{\dot{u}}\setminus\left(S_{u}\cup\\{\pm 1\\}\right)$:

(3.45) $\begin{array}[]{cc}\ddot{u}^{\pm}=0&\text{in
}S_{\dot{u}}\setminus\left(S_{u}\cup\\{\pm 1\\}\right)\ ,\end{array}$ (3.46)
$\begin{array}[]{cc}\mathbf{\big{[}\kern-5.0pt\big{[}}\dddot{u}\mathbf{\big{]}\kern-5.0pt\big{]}}=0&\text{in
}S_{\dot{u}}\setminus\left(S_{u}\cup\\{\pm 1\\}\right)\ .\end{array}$

Let $\varphi\in\mathbf{H}^{2}\cap C^{2}([t_{l},t_{l+1}])$,
$l=0,...,\mathsf{T}$, $\mbox{{spt}}(\varphi)\subset A$, with $A$ Borel subset
of $(-1,1)$ and $S_{\varphi}=\emptyset=(S_{u}\setminus S_{\dot{u}})\cap A$.
Then, up to a finite set of possible values of $\varepsilon$ entailing
cancelation of $\dot{u}$ discontinuity, we can choose $\varepsilon$
arbitrarily small such that

$(S_{u+\varepsilon\varphi}\cup S_{\dot{u}+\varepsilon\dot{\varphi}})\cap
A=S_{\dot{u}+\varepsilon\dot{\varphi}}\cap A=S_{\dot{u}}\ .$

Moreover, by Green formula (3.44):

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
E_{1}(u+\varepsilon\varphi,A)-E_{1}(u,A)\,\leq$ $\displaystyle\leq$
$\displaystyle\beta\big{(}\sharp\,(S_{\dot{u}+\varepsilon\dot{\varphi}}\cap
A)-\sharp\ (S_{\dot{u}}\cap A)\big{)}\,+$ $\displaystyle\qquad+\
2\varepsilon\,\bigg{(}\sum_{l=0}^{\mathsf{T}}\int_{t_{l}}^{t_{l+1}}\Big{(}\eta\,u^{\prime\prime}\varphi^{\prime\prime}+\mu(u-w)\varphi-\frac{f}{2}\varphi\Big{)}\,dx\bigg{)}\,+\,o(\varepsilon)\,=$
$\displaystyle=$ $\displaystyle
2\varepsilon\,\bigg{(}\sum_{l=0}^{\mathsf{T}}\int_{t_{l}}^{t_{l+1}}\Big{(}\eta\,u^{\prime\prime\prime\prime}\varphi\,dx+\mu(u-w)\varphi-\frac{f}{2}\varphi\Big{)}\,dx\,+$
$\displaystyle\qquad+\
{\dddot{u}}^{+}(-1)\varphi^{+}(-1)-\ddot{u}^{+}(-1)\dot{\varphi}^{+}(-1)-{\dddot{u}}^{-}(1)\varphi^{-}(1)+\ddot{u}^{-}(1)\dot{\varphi}^{-}(1)\,+$
$\displaystyle\qquad+\ \eta\sum_{\left(S_{\dot{u}}\cap A\right)\setminus\\{\pm
1\\}}\Big{(}\mathbf{\big{[}\kern-5.0pt\big{[}}+\dddot{u}\varphi\mathbf{\big{]}\kern-5.0pt\big{]}}-\mathbf{\big{[}\kern-5.0pt\big{[}}\ddot{u}\dot{\varphi}\mathbf{\big{]}\kern-5.0pt\big{]}}\Big{)}\bigg{)}\
+\ o(\varepsilon)\ .$

By taking into account (3.23), for small $\varepsilon$ and by the
arbitrariness of $\varphi$ and of the two traces of $\dot{\varphi}$ on the two
sides of $S_{\dot{u}}$, we can choose $\varphi$ with $\varphi^{\pm}=0$, and
arbitrary $\dot{\varphi}^{+}=\dot{\varphi}^{-}$, to get (3.45).
On the other hand, by choosing $\dot{\varphi}^{\pm}=0$ together with arbitrary
$\varphi$ and taking into account that
$\mathbf{\big{[}\kern-5.0pt\big{[}}{\varphi}\mathbf{\big{]}\kern-5.0pt\big{]}}=0$,
we obtain (3.46).
Then (3.25) follow from (3.45) and (3.46).
Step 5 - The analysis of minimizers at $\left(S_{\dot{u}}\setminus
S_{u}\right)\cap\\{\pm 1\\}$ can be done exactly in the same way as in Step 5,
but taking into account that $u=w$ and $\varphi=0$ on $[-2,-1]\cup[1,2]$, thus
obtaining (3.27) and (3.28). Step 6 - (3.26) is a straightforward consequence
of (3.23)-(3.25). Step 7 - Eventually, under the additional condition
$\alpha=\beta$, we prove the refinement (3.29) of (3.24),(3.25),(3.27),(3.28)
on $\left(S_{\dot{u}}\cup S_{u}\right)$ for every minimizer $u$.
We are left only to show that

(3.47) $\ \text{ if }\alpha=\beta\ \ \text{ then\,: }\ \ \ 1\in
S_{\dot{u}}\setminus S_{u}\Rightarrow\dddot{u}^{-}(1)=0\,;\ \ -1\in
S_{\dot{u}}\setminus S_{u}\Rightarrow\dddot{u}^{+}(-1)=0\,.$

Fix a Borel set $A$ s.t. $A\subset\subset(-2,2)$, $S_{u}\cap A=\emptyset\neq
S_{\dot{u}}\cap A$.
Let $\varphi\in\mathbf{H}^{2}\cap C^{2}([t_{l},t_{l+1}])$,
$l=0,...,\mathsf{T}$ and

$S_{\dot{u}}\cap A\,=\,S_{\varphi}\cap A\qquad\hbox{ and }\qquad S_{u}\cap
A\,=\,S_{\dot{\varphi}}\cap A\,=\,\,\emptyset\,.$

Then, for every value of $\varepsilon\in\mathbb{R}$ we have
$S_{u+\varepsilon\varphi}\cap A=S_{\varphi}\cap A$ and

$\left(S_{u+\varepsilon\varphi}\cup
S_{(\dot{u}+\varepsilon\dot{\varphi})}\right)\cap A=S_{\dot{u}}\cap A\ .$

By (3.44), (3.23), (3.24) and (3.25) we have

$\displaystyle 0\\!\\!$ $\displaystyle\leq$
$\displaystyle\\!\\!E_{1}(u+\varepsilon\varphi,A)-E_{1}(u,A)$ $\displaystyle=$
$\displaystyle\\!\\!\alpha\,\sharp\left(S_{u+\varepsilon\varphi}\cap
A\right)+\beta\left(\sharp\left((S_{(\dot{u}+\varepsilon\dot{\varphi})}\setminus
S_{u+\varepsilon\varphi})\cap A\right)-\beta\sharp\left(S_{\dot{u}}\cap
A\right)\right)$
$\displaystyle\\!\\!\quad+\,2\varepsilon\,\left(\sum_{l=0}^{\mathsf{T}}\int_{t_{l}}^{t_{l}+1}\Big{(}\eta\,u^{\prime\prime}\varphi^{\prime\prime}+\mu(u-w)\varphi-\frac{f}{2}\varphi\Big{)}\,dx\right)\,+\,o(\varepsilon)$
$\displaystyle=$ $\displaystyle\\!\\!\alpha\,\sharp(S_{\varphi}\cap
A)+\beta\,\sharp\left((S_{\dot{u}}\setminus S_{\varphi})\cap
A\right)-\beta\,\sharp(S_{\dot{u}}\cap A)$
$\displaystyle\\!\\!\quad+\,2\varepsilon\,\left(\sum_{l=0}^{\mathsf{T}}\int_{t_{l}}^{t_{l}+1}\Big{(}\eta\,u^{\prime\prime\prime\prime}\varphi+\mu(u-w)\varphi-\frac{f}{2}\varphi\Big{)}\,dx\
+\eta\\!\\!\\!\\!\sum_{\left(S_{\dot{u}}\cap A\right)\setminus\\{\pm
1\\}}\\!\\!\Big{(}\mathbf{\big{[}\kern-5.0pt\big{[}}\dddot{u}\varphi\mathbf{\big{]}\kern-5.0pt\big{]}}-\mathbf{\big{[}\kern-5.0pt\big{[}}\ddot{u}\dot{\varphi}\mathbf{\big{]}\kern-5.0pt\big{]}}\Big{)}\right.\vspace{-0.5cm}$
$\displaystyle\\!\\!\quad+\
\eta\,\Big{(}{\dddot{u}}^{+}(-1)\varphi^{+}(-1)-\ddot{u}^{+}(-1)\dot{\varphi}^{+}(-1)-{\dddot{u}}^{-}(1)\varphi^{-}(1)+\ddot{u}^{-}(1)\dot{\varphi}^{-}(1)\Big{)}\Bigg{)}\vspace{0.2cm}$
$\displaystyle\\!\\!\quad+\ o(\varepsilon)\vspace{0.1cm}$ $\displaystyle=$
$\displaystyle\\!\\!\alpha\,\sharp(S_{\varphi}\cap
A)-\beta\,\sharp(S_{\dot{u}}\cap
A)+2\eta\,\varepsilon\sum_{\left(S_{\dot{u}}\cap A\right)\setminus\\{\pm
1\\}}\mathbf{\big{[}\kern-5.0pt\big{[}}\dddot{u}\varphi\mathbf{\big{]}\kern-5.0pt\big{]}}+o(\varepsilon)\
.$

Since $S_{\varphi}\cap A=S_{\dot{u}}\cap A$, when $\alpha>\beta$ then the
inequality is fulfilled for $\varepsilon$ small enough, hence we do not obtain
further information (recall that the necessary condition for semicontinuity
$\alpha\geq\beta$ is always assumed). On the other hand, when $\alpha=\beta$,
we get

$0\leq E_{1}(u+\varepsilon\varphi,A)-E_{1}(u,A)\ =\\\ =\
2\eta\,\varepsilon\left(\sum_{S_{\dot{u}}\cap
A}\mathbf{\big{[}\kern-5.0pt\big{[}}\dddot{u}\varphi\mathbf{\big{]}\kern-5.0pt\big{]}}+{\dddot{u}}^{+}(-1)\varphi^{+}(-1)-{\dddot{u}}^{-}(1)\varphi^{-}(1)\right)+o(\varepsilon)\,.$

So the coefficient of $2\varepsilon$ must vanish, and by the arbitrariness of
the two traces of $\varphi$ at points in $S_{\dot{u}}\cap A$, of the right
trace at $-1$ and of the left trace at $+1$, taking into account that
$\varphi\equiv 0$ outside $[-1,1]$ we get (3.47). Step 8 - We make explicit
all the details for $E_{1}$ only, since the proof of Euler equations for
$F_{1}$ is identical. In fact
$F_{1}(v_{r},v_{p})-E_{1}(v_{r})=\int_{-1}^{1}(\gamma|\ddot{v}_{p}|^{2}-fv_{p})$
is a classical integral functional: so the analysis of any minimizer
$U=(u_{r},u_{p})$ of $F_{1}$ can be done by performing all the admissible
variations separately for $u_{r}$ and $u_{p}$. $\square$

Proof of Proposition 3.10 (compliance identities) - Assume
$u\\!\in\\!\mathop{{\rm argmin}}\nolimits E_{1}$ and label $t_{0}=-1$ and
$t_{T+1}=1$ and $t_{j}$, for $j=1,...,\mathsf{T}$, the (possibly empty) finite
ordered set $\left(S_{u}\cup S_{\dot{u}}\right)\cup(-1,+1)$.
Then, by (3.23)-(3.28), integrating by parts on the intervals
$[t_{j},t_{j+1}]$ we get

$\displaystyle\eta\int_{-1}^{1}|\ddot{u}|^{2}\,dx$ $\displaystyle=$
$\displaystyle\eta\sum_{j=0}^{T}\int_{t_{j}}^{t_{j+1}}|\ddot{u}|^{2}\,dx\,=\,$
$\displaystyle=$
$\displaystyle-\,\eta\int_{-1}^{1}(\ddot{u})^{\prime}\,\dot{u}\,dx\,+\,\eta\,\left[\,\ddot{u}\,\dot{u}\,\right]_{-1}^{+1}\,=$
$\displaystyle=$
$\displaystyle\eta\int_{-1}^{1}(\ddot{u})^{\prime\prime}\,u\,dx\,+\,\eta\,\left[\,\ddot{u}\,\dot{u}\,-\,\dddot{u}\,u\,\right]_{-1}^{+1}\,=$
$\displaystyle=$
$\displaystyle\mu\,\int_{-1}^{1}(w-u)\,u\,dx\,+\,\int_{-1}^{1}(f/2)\,u\,dx\,+\,\eta\left[\,\ddot{u}\,\dot{u}\,-\,\dddot{u}\,u\,\right]_{-1}^{+1}\
,$

here above and in the sequel, the notation $[z]_{-1}^{+1}$ stands for
$z^{-}(1)-z^{+}(-1)$. Hence

$\displaystyle E_{1}(u)$ $\displaystyle=$
$\displaystyle\eta\int_{-1}^{1}|\ddot{u}|^{2}dx-\int_{-1}^{1}fu\,dx\,+\,\mu\int_{-1}^{1}|u-w|^{2}dx\,+J(u)\,=$
$\displaystyle=$ $\displaystyle
J(u)+\mu\int_{-1}^{1}\\!\\!\big{(}(w-u)u+(u-w)^{2}\big{)}dx+\,(-1\\!+\\!1/2)\\!\int_{-1}^{1}fu\,dx\,+$
$\displaystyle\,+\,\eta\,\Big{(}-\ddot{u}^{+}(-1)\dot{u}(-1)\,+\,\ddot{u}^{-}(1)\dot{u}(1)+\dddot{u}^{+}(-1)u(-1)\,-\,\dddot{u}^{-}(1)u(1)\Big{)}\,.$

Now assume $(u_{r},u_{p})\in\mathop{{\rm argmin}}\nolimits F_{1}$ and label
$t_{j}$ as above.
By taking into account (3.30)-(3.38) and performing integrations by parts, we
get

$\eta\\!\int_{-1}^{1}\\!|\ddot{u}_{r}|^{2}dx\,=\eta\\!\int_{-1}^{1}\\!\\!(\ddot{u}_{r})^{\prime\prime}u_{r}\,dx\,=\,-\,\mu\\!\int_{-1}^{1}\\!\\!(u_{r}-u_{p})\,u_{r}\,dx\,+\int_{-1}^{1}\\!\\!(f_{r}/2)\,u_{r}\,dx\,+\,\eta\left[\ddot{u}_{r}\dot{u}_{r}\\!-\\!\dddot{u}_{r}\,u_{r}\right]_{-1}^{+1}.$

Performing two integrations by parts and taking into account (3.31), we get

$\displaystyle\gamma\\!\int_{-1}^{1}\\!|u_{p}^{\prime\prime}|^{2}dx$
$\displaystyle=$
$\displaystyle\gamma\\!\int_{-1}^{1}\\!\\!\\!u_{p}^{\prime\prime\prime\prime}u_{p}\,dx+\gamma\,\left[\,u_{p}^{\prime\prime}u_{p}^{\prime}\\!-u_{p}^{\prime\prime\prime}u_{p}\,\right]_{-1}^{1}$
$\displaystyle=$
$\displaystyle\mu\\!\int_{-1}^{1}\\!\\!(u_{r}-u_{p})\,u_{p}\,dx\,+\,\frac{1}{2}\int_{-1}^{1}\\!\\!f_{p}u_{p}\,dx+\gamma\,\\!\left[\,u_{p}^{\prime\prime}u_{p}^{\prime}\\!-u_{p}^{\prime\prime\prime}u_{p}\,\right]_{-1}^{+1}\
.$

Then for any $(u_{r},u_{p})\in\mathop{{\rm argmin}}\nolimits F_{1}$ we obtain

$\displaystyle F_{1}(u_{r},u_{p})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!J(u_{r})\,+\,(-1\\!+\\!1/2)\\!\left(\int_{-1}^{1}f_{r}u_{r}\,dx\,+\,\\!\int_{-1}^{1}f_{p}u_{p}\,dx\right)-\,\mu\int_{-1}^{1}(u_{r}-u_{p})^{2}dx$
$\displaystyle+\,\left[\,\eta\,(\,\ddot{u}_{r}\,\dot{u}_{r}-\dddot{u}_{r}\,u_{r})+\gamma\,(\,u_{p}^{\prime\prime}w^{\prime}-u_{p}^{\prime\prime\prime}w)\,\right]_{-1}^{1}\
.\quad\square$

######  Proposition 3.11.

. (Variational conditions for the minimizers of $E_{1}$ under unilateral
constraint)
Every minimizer $u$ of $E_{1}$ over the closed convex set

(3.48) $K:=\left\\{\,v\\!\in\\!H^{2}(-2,2):\,v\\!\geq\\!w\,\hbox{
on}\,[-1,1],\,v\\!=\\!w\,\hbox{ on}\,(-2,-1)\cup(1,2)\right\\}$

(see Remark 3.6 about the pointwise everywhere meaning the unilateral
constraint) fulfils the variational inequality

(3.49) $\left\\{\begin{array}[]{l}\qquad u\in K:\\\
\displaystyle\int_{(-1,1)\setminus\left(S_{u}\cup
S_{\dot{u}}\right)}\\!\\!\big{(}\eta\,u^{\prime\prime\prime\prime}+\mu(u-w)-f/2\big{)}\,\big{(}u-v\big{)}\
\leq\ 0\qquad\forall v\in K\,,\end{array}\right.$

together with the bilateral conditions at the free discontinuity and free-
gradient discontinuity set where the contact does not play a role:

(3.50) $\ddot{u}^{+}=\ddot{u}^{-}=0\qquad\,\ \hbox{on
}\big{(}S_{u}\\!\setminus\\!\\{\pm
1\\}\big{)}\cap\\{u^{+}\\!>\\!w\\}\cap\\{u^{-}\\!>\\!w\\},$ (3.51)
$\dddot{u}^{+}=\dddot{u}^{-}=0\qquad\,\hbox{on
}\big{(}S_{u}\\!\setminus\\!\\{\pm
1\\}\big{)}\cap\\{u^{+}\\!>\\!w\\}\cap\\{u^{-}\\!>\\!w\\},$ (3.52)
$\ddot{u}^{+}=\ddot{u}^{-}=[\dddot{u}]=0\qquad\hbox{on
}\big{(}S_{\dot{u}}\\!\setminus\\!\left(S_{u}\cup\\{\pm
1\\}\right)\big{)}\cap\\{u^{+}\\!>\\!w\\}\cap\\{u^{-}\\!>\\!w\\},$ (3.53)
$\begin{array}[]{l}\ddot{u}\in H^{2}\Big{(}(-1,1)\setminus\\{u^{+}=w\hbox{ or
}u^{-}=w\\}\Big{)}\quad\hbox{and}\vspace{0.1cm}\\\
\eta\,(\ddot{u})^{\prime\prime}+\mu(u-w)=f/2\quad\hbox{in }\
\mathcal{D}^{\prime}\big{(}(-1,1)\\!\setminus\\!\\{u^{+}=w\hbox{ or
}u^{-}=w\\}\big{)}\ ,\end{array}$ (3.54) $\left\\{\
\begin{array}[]{l}\ddot{u}^{+}(-1)=\dddot{u}^{+}(-1)=0\qquad\hbox{if
}-1\not\in S_{\dot{u}}\setminus S_{u}\hbox{ and
}u^{+}(-1)\\!>\\!w(-1),\vspace{0.2cm}\\\
\ddot{u}^{-}(+1)=\dddot{u}^{-}(+1)=0\qquad\hbox{if }+1\not\in
S_{\dot{u}}\setminus S_{u}\hbox{ and
}u^{-}(+1)\\!>\\!w(+1),\end{array}\right.$ (3.55) $\left\\{\
\begin{array}[]{l}\ddot{u}^{+}(-1)=0\qquad\hbox{if }-1\in
S_{\dot{u}}\\!\setminus\\!S_{u}\hbox{ and
}u^{+}(-1)\\!>\\!w(-1),\vspace{0.2cm}\\\ \ddot{u}^{-}(+1)=0\qquad\hbox{if
}+1\in S_{\dot{u}}\\!\setminus\\!S_{u}\hbox{ and
}u^{-}(+1)\\!>\\!w(+1),\end{array}\right.\qquad\phantom{,}$

jump condition $[\dddot{u}^{+}]=0$ in (3.52) can be improved when
$\alpha\\!=\\!\beta$, hence

(3.56) $\hbox{if }\alpha=\beta\hbox{ then }\left\\{\
\begin{array}[]{l}\dddot{u}^{+}=\dddot{u}^{-}=0\quad\hbox{ on
}\Big{(}\left(S_{u}\cup S_{\dot{u}}\right)\\!\setminus\\!\\{\pm
1\\}\Big{)}\cap\\{u^{+}\\!>\\!w^{+}\\}\cap\\{u^{-}\\!>\\!w\\},\\\
\dddot{u}^{+}(-1)=0\quad\hbox{if }-1\in S_{u}\cup S_{\dot{u}}\hbox{ and
}u^{+}(-1)\\!>\\!w(-1),\vspace{0.1cm}\\\ \dddot{u}^{-}(+1)=0\quad\hbox{if
}+1\in S_{u}\cup S_{\dot{u}}\hbox{ and
}u^{-}(+1)\\!>\\!w(+1),\end{array}\right.$

and in addition the unilateral conditions at the free discontinuity and free-
gradient discontinuity sets of $u$ where the contact with the obstacle plays a
role:

(3.57) $\ddot{u}^{+}\\!\geq 0\qquad\,\hbox{on }\big{(}(S_{u}\cup
S_{\dot{u}})\\!\setminus\\!\\{+1\\}\big{)}\cap\\{u^{+}\\!=\\!w\\}\,,$ (3.58)
$\ddot{u}^{-}\\!\geq 0\qquad\,\hbox{on }\big{(}(S_{u}\cup
S_{\dot{u}})\\!\setminus\\!\\{-1\\}\big{)}\cap\\{u^{-}\\!=\\!w\\}\,.$

No condition on $\dddot{u}^{\pm}$ is present on $S_{u}\cup S_{\dot{u}}$.

Proof -  The proof repeats the first 7 steps of Proposition 3.9 proof, but
achieves less information since there is a strictly smaller set of admissible
variations.
Step 1 is fully recovered thus, here we can exploit the Green formula (3.44).
We repeat Steps 2-7, by performing all the admissible variations of $u$ which
are of the kind $u+\varepsilon(v-u)$, with $\varepsilon\in[0,1]$ and $v\in K$:
for comparison, here $\varphi=v-u$.
As in Step 3 for the case of non constrained competitors, for $\varphi=v-u$
belonging to $\mathbf{H}^{2}\cap C^{2}([t_{l},t_{l+1}])$,
$l=0,...,\mathsf{T}$, with $\mbox{{spt}}(\varphi)\subset A$ Borel subset of
$[-1,1]$ and $(S_{\dot{u}}\setminus S_{u})\cap A=\emptyset$ we still get
$\big{(}S_{u+\varepsilon\varphi}\cup
S_{\dot{u}+\varepsilon\dot{\varphi}}\big{)}\cap A\ \subset\ {S_{u}}\cap A$ and

(3.59) $\displaystyle 0\\!\\!$ $\displaystyle\leq$
$\displaystyle\\!\\!E_{1}(u+\varepsilon\varphi,A)-E_{1}(u,A)\ =$
$\displaystyle=$
$\displaystyle\\!\\!\alpha\left(\sharp(S_{u+\varepsilon\varphi}\cap
A)-\sharp(S_{u}\cap A)\right)+\beta\sharp\left((S_{\dot{\varphi}}\setminus
S_{u+\varepsilon\varphi})\cap A\right)\,+$ $\displaystyle\qquad+\
2\varepsilon\left(\sum_{l=0}^{\mathsf{T}}\int_{t_{l}}^{t_{l+1}}\big{(}\eta\,u^{\prime\prime}\varphi^{\prime\prime}+\mu(u-w)\varphi-(f/2)\varphi\big{)}\,dx\right)\,+\,o(\varepsilon)\,=$
$\displaystyle=$
$\displaystyle\\!\\!\alpha\left(\sharp(S_{u+\varepsilon\varphi}\cap
A)-\sharp(S_{u}\cap A)\right)+\beta\sharp\left((S_{\dot{\varphi}}\setminus
S_{u+\varepsilon\varphi})\cap A\right)\,+$ $\displaystyle\qquad+\
2\varepsilon\bigg{(}\sum_{l=0}^{\mathsf{T}}\int_{t_{l}}^{t_{l+1}}\big{(}\eta\,u^{\prime\prime\prime\prime}\varphi+\mu(u-w)\varphi-(f/2)\varphi\big{)}\,dx\,+$
$\displaystyle\qquad+\
{\dddot{u}}^{+}(-1)\varphi^{+}(-1)-\ddot{u}^{+}(-1)\dot{\varphi}^{+}(-1)-{\dddot{u}}^{-}(1)\varphi^{-}(1)+\ddot{u}^{-}(1)\dot{\varphi}^{-}(1)+$
$\displaystyle\qquad+\ \,\eta\\!\sum_{\left(S_{u}\cap A\right)\setminus\\{\pm
1\\}}\\!\Big{(}\mathbf{\big{[}\kern-5.0pt\big{[}}\dddot{u}\varphi\mathbf{\big{]}\kern-5.0pt\big{]}}-\mathbf{\big{[}\kern-5.0pt\big{[}}\ddot{u}\dot{\varphi}\mathbf{\big{]}\kern-5.0pt\big{]}}\Big{)}\bigg{)}\
+\ o(\varepsilon)\,.$

As in Step 4, let $\varphi\in\mathbf{H}^{2}\cap C^{2}([t_{l},t_{l+1}])$,
$l=0,...,\mathsf{T}$, $\mbox{{spt}}(\varphi)\subset A$, with $A$ Borel subset
of $(-1,1)$ and $S_{\varphi}=\emptyset=(S_{u}\setminus S_{\dot{u}})\cap A$.
Then, up to a finite set of possible values of $\varepsilon$ entailing
cancelation of $\dot{u}$ discontinuity, we can choose $\varepsilon$
arbitrarily small such that $(S_{u+\varepsilon\varphi}\cup
S_{\dot{u}+\varepsilon\dot{\varphi}})\cap
A=S_{\dot{u}+\varepsilon\dot{\varphi}}\cap A=S_{\dot{u}}\,$; thus, by Green
formula (3.44)

(3.60) $\displaystyle 0\\!\\!$ $\displaystyle\leq$
$\displaystyle\\!\\!E_{1}(u+\varepsilon\varphi,A)-E_{1}(u,A)\,\leq$
$\displaystyle\leq$
$\displaystyle\\!\\!\beta\big{(}\sharp\,(S_{\dot{u}+\varepsilon\dot{\varphi}}\cap
A)-\sharp\ (S_{\dot{u}}\cap A)\big{)}\,+$ $\displaystyle\qquad+\
2\varepsilon\,\bigg{(}\sum_{l=0}^{\mathsf{T}}\int_{t_{l}}^{t_{l+1}}\big{(}\eta\,u^{\prime\prime}\varphi^{\prime\prime}+\mu(u-w)\varphi-(f/2)\varphi\big{)}\,dx\bigg{)}\,+\,o(\varepsilon)\,=$
$\displaystyle=$
$\displaystyle\\!\\!2\varepsilon\,\bigg{(}\sum_{l=0}^{\mathsf{T}}\int_{t_{l}}^{t_{l+1}}\Big{(}\eta\,u^{\prime\prime\prime\prime}+\mu(u-w)-(f/2)\Big{)}\,\varphi\
dx\ +$ $\displaystyle\qquad+\
{\dddot{u}}^{+}(-1)\varphi^{+}(-1)-\ddot{u}^{+}(-1)\dot{\varphi}^{+}(-1)-{\dddot{u}}^{-}(1)\varphi^{-}(1)+\ddot{u}^{-}(1)\dot{\varphi}^{-}(1)\,+$
$\displaystyle\qquad+\ \eta\sum_{\left(S_{\dot{u}}\cap A\right)\setminus\\{\pm
1\\}}\Big{(}\mathbf{\big{[}\kern-5.0pt\big{[}}+\dddot{u}\varphi\mathbf{\big{]}\kern-5.0pt\big{]}}-\mathbf{\big{[}\kern-5.0pt\big{[}}\ddot{u}\dot{\varphi}\mathbf{\big{]}\kern-5.0pt\big{]}}\Big{)}\bigg{)}\
+\ o(\varepsilon)\ .$

In all cases now $\varphi=v-u$ with $v\in K$.
By all choices of open sets $A$ and $v\in K$ fulfilling $\mathop{{\rm
spt}}\nolimits(v-u)\subset A\subset\subset(-1,1)\\!\setminus\\!(S_{u}\cup
S_{\dot{u}})$ we get,

$\displaystyle\displaystyle\int_{(-1,1)\setminus\left(S_{u}\cup
S_{\dot{u}}\right)}\\!\\!\big{(}\eta
u^{\prime\prime\prime\prime}+\mu(u-w)-f/2\big{)}(v-u)\,dx=$
$\displaystyle\displaystyle=\ \int_{(-1,1)\setminus\left(S_{u}\cup
S_{\dot{u}}\right)}\\!\\!\big{(}\eta
u^{\prime\prime}(v-u)^{\prime\prime}+\mu(u-w)-f/2\big{)}(v-u)\,dx\ \geq\ 0$

say (3.49). Then, by inserting (3.49) in (3.59),(3.60), we single out the
conditions at every point of singular set. Outside the contact set
$\\{u^{+}=w\\}\\!\cup\\!\\{u^{-}=w\\}$ we can repeat the discussion made in
the proof of Proposition 3.9, since $\varphi^{\pm}=(v-u)^{\pm}$ and
$\dot{\varphi}^{\pm}=(\dot{v}-\dot{u})^{\pm}$ are allowed to achieve both
positive and negative values outside the contact set.
Up to a finite set of possible values of $\varepsilon$ entailing cancellation
of discontinuity, we have $S_{u+\varepsilon\varphi}\cap A=S_{u}\cap A$. Then
by discarding such values we can choose arbitrarily small $\varepsilon$
satisfying

$\sharp\,((S_{\dot{\varphi}}\setminus S_{u+\varepsilon\varphi})\cap
A)=\sharp\,((S_{\dot{\varphi}}\setminus S_{u})\cap A)=0$

By taking into account (3.23) and the arbitrariness of the two traces of
$\varphi$ and $\dot{\varphi}$ on the two sides of points in $S_{u}$, for small
$\varepsilon$, we can choose $\varphi$ with $\varphi^{\pm}=0$, and
$\dot{\varphi}^{+}=0$ together with $\dot{\varphi}^{-}$ arbitrary, or
viceversa to get $\,\ddot{u}^{\pm}=0$ on $S_{u}\setminus\\{\pm 1\\}$.
Similarly, we obtain $\,\dddot{u}^{\pm}=0$ on $S_{u}\setminus\\{\pm 1\\}$ by
choosing $\dot{\varphi}^{\pm}=0$, and $\varphi^{+}=0$ together with
$\varphi^{-}$ arbitrary or vice-versa. So (3.50) is proved.
If some clamped endpoint ($-1$ and/or $+1$) belong to $S_{u}$, then (3.54) is
obtained as above, but taking into account that $\varphi\equiv 0$ outside
$[-1,1]$.
Summarizing, we obtain (3.51),(3.53),(3.54),(3.55), hence (3.52) and (3.56),
by the same argument of Steps 3 - 7. On the contact set $\\{u=w\\}$ we can
repeat again the discussion made in the proof of Proposition 3.9, but here the
coefficient of $2\varepsilon$ in (3.60) must be only nonnegative, thus we get
inequalities in place of equalities. Moreover, since
$\dot{\varphi}^{+}=(\dot{v}-\dot{u})^{+}$ is allowed to achieve only positive
values and $\dot{\varphi}^{-}=(\dot{v}-\dot{u})^{-}$ is allowed to achieve
only negative values, whereas left and right values have always opposite sign,
we deduce (3.57), (3.58). On the other hand, on the contact set
$\varphi^{\pm}=(v-u)^{\pm}$ is always null; therefore, we get no condition on
every term whose multiplier is $\varphi^{+}$ or $\varphi^{-}$. $\square$

######  Proposition 3.12.

. (Variational conditions for the minimizers of $F_{1}$ under unilateral
constraint)
Every minimizing pair $(u_{r},u_{p})$ of $F_{1}$ over the convex set

$\mathbf{K}\\!:=\\!\left\\{(v_{r},v_{p})\\!\in\\!\mathbf{H}^{2}(-2,2)\\!\times\\!H^{2}(-2,2)\\!:\,v_{r}\\!\geq\\!v_{p}\,\hbox{on}\,[-1,1],\,v_{r}\\!=\\!v_{p}\\!=\\!w\,\,\hbox{on}\,(-2,-1)\\!\cup\\!(1,2)\right\\}$

fulfils the quasi-variational inequalities

(3.61) $\left\\{\begin{array}[]{l}u_{r}\in\mathbf{H}^{2}(-2,2):\
v_{r}\\!\geq\\!v_{p}\,\hbox{on}\,[-1,1],\,v_{r}\\!=\\!v_{p}\\!=\\!w\,\,\hbox{on}\,(-2,-1)\\!\cup\\!(1,2)\hbox{
and}\vspace{0.1cm}\\\
\displaystyle\\!\\!\int_{(-1,1)\setminus\left(S_{u_{r}}\cup
S_{\dot{u}_{r}}\right)}\\!\\!\\!\\!\\!\\!\\!\big{(}\eta\,u^{\prime\prime\prime\prime}_{r}\\!+\mu(u_{r}-u_{p})-f_{r}/2\big{)}\,\big{(}u_{r}-v\big{)}\,\leq\,0\quad\forall
v:(u_{r},v)\\!\in\\!\mathbf{K},\end{array}\right.$ (3.62)
$\left\\{\begin{array}[]{l}u_{p}\in H^{2}(-2,2):\
v_{p}\\!\leq\\!v_{r}\,\hbox{on}\,[-1,1],\,v_{r}\\!=\\!v_{p}\\!=\\!w\,\,\hbox{on}\,(-2,-1)\\!\cup\\!(1,2)\hbox{
and}\vspace{0.1cm}\\\
\displaystyle\\!\\!\int_{(-1,1)\setminus\left(S_{u_{r}}\cup
S_{\dot{u}_{r}}\right)}\\!\\!\\!\\!\\!\\!\\!\\!\big{(}\gamma\,u^{\prime\prime\prime\prime}_{p}\\!+\mu(u_{p}-u_{r})-f_{p}/2\big{)}\,\big{(}u_{p}-z\big{)}\,\leq\,0\quad\forall
z:(z,u_{p})\\!\in\\!\mathbf{K},\end{array}\right.$

together with the standard bilateral conditions (say (3.50)-(3.55) with
$u_{r},u_{p}$ replacing respectively $u,w$) at the free discontinuity and
free-gradient discontinuity set where the contact does not play a role, and
the unilateral conditions at the free discontinuity and free-gradient
discontinuity set where the contact with the obstacle plays a role:

(3.63) $\ddot{u}_{r}^{+}\\!\geq 0\qquad\hbox{on }\big{(}(S_{u_{r}}\cup
S_{\dot{u}_{r}})\\!\setminus\\!\\{+1\\}\big{)}\cap\\{u_{r}^{+}\\!=\\!u_{p}\\}\,,$
(3.64) $\ddot{u}_{r}^{-}\\!\geq 0\qquad\hbox{on }\big{(}(S_{u_{r}}\cup
S_{\dot{u}_{r}})\\!\setminus\\!\\{-1\\}\big{)}\cap\\{u_{r}^{-}\\!=\\!u_{p}\\}\,.$

No condition on $\dddot{u}_{r}^{\pm}$ is present on $S_{u_{r}}\cup
S_{\dot{u}_{r}}$.

Proof -  Repetition of the steps of the last proof provides the proof the
claims about minimizers $(u_{r},u_{p})$ of $F_{1}$ with unilateral implicit
constraint, by performing all the admissible variations of $u_{r}$ which are
of the kind $u_{r}+\varepsilon(v-u_{r})$, with $\varepsilon\in[0,1]$ and
$(v,u_{p})\in\mathbf{K}$ and $v_{r}+\varepsilon(z-v_{r})$, with
$\varepsilon\in[0,1]$ and and $(u_{r},z)\in\mathbf{K}$. $\square$

##  4\. Hard-device reinforcement of flexural plate

In this section we deduce the existence statement in the case of hard-device
reinforcement: minimization of functional $E$ defined by (2.4). Proof of
Theorem 2.1 \- After noticing that by

$\,E(\emptyset,w)\,=\,\eta\|D^{2}w\|_{L^{2}(\Omega)}^{2}-\int_{\Omega}fw\,d{\bf
x}\,<\,+\infty\ .$

the domain of $E$ is not empty, and by

(4.1)
$\int_{\Omega}\\!\\!\big{(}\,\mu\,|v-w|^{2}\,-fv\,\big{)}\,dx=\int_{\Omega}\\!\\!\mu\Big{(}v-\big{(}w+f/(2\mu)\big{)}\Big{)}^{2}dx\,-\int_{\Omega}\\!\\!\Big{(}fw+f^{2}/(4\mu)\Big{)}dx$

where the last summand on the right-hand side is a constant, we have that the
functional $E$ is bounded from below since beside such constant all other
terms are nonnegative.

The notion of essential admissible pairs, set by (2.5), selects ([19]) those
pairs $(K,v)$ which are cleansed of every spurious artifact that does not
affect the functional value and are good representatives in equivalence
classes of admissible pairs. This definition of admissible pair prevents
diffused damage but allows to prove partial regularity of displacements $v$:
free discontinuity (crack) and free gradient discontinuity (folds) are allowed
in competing configurations of the structure.
Thus, the claims of present Theorem 2.1 follow from Theorem 2.3 in [12] and
[16] about functional (2.2) defined therein, by setting $g=w+f/(2\mu)$, a
datum which belongs to $L^{4}(\Omega)$ due to present assumptions. Precisely
we can choose $\widetilde{\Omega}=\Omega_{p}$, $\alpha=\beta=1$; hence
(2.3),(2.4),(2.5) and (2.20) of [16] are fulfilled thanks to the conditions
(2.1),(2.3) assumed here. Moreover $D^{2}w\in L^{\infty}(A)$ for any open set
s.t. $\Omega\subset\\!\subset A\subset\\!\subset\Omega_{p}$ and we have that
the sets $M,T_{0},T_{1}$ (as denoted in [16]) are empty, hence (2.6)-(2.11) of
[16] hold true thanks to the assumption (2.2) made here.

$\square$

##  5\. Strengthening reinforcement of flexural plate

In this section we deduce the existence statement in the case of strengthening
reinforcement: minimization of functional $F$ defined by (2.7).
To deal with the case of strengthening reinforcement we need a relaxed
formulation of functional (2.7), as it is usual in the analysis of free
discontinuity problems. We list standard notations (see [3],[12],[14],[18]):
$B_{\varrho}(\mathbf{x})$ denotes the open ball
$\\{\,\mathbf{y}\\!\in\\!\mathbb{R}^{2}\\!:\,|\mathbf{y}-\mathbf{x}|\\!<\\!\varrho\,\\}$;

$\mathcal{H}^{1}(A)$ and $|A|$ denote respectively, the 1-dimensional
Hausdorff measure and the outer Lebesgue measure of a subset
$A\subset\mathbb{R}^{2}$; for every Borel function $v:\Omega\to\mathbb{R}$ and
$\mathbf{x}\in\Omega$,
$z\in\overline{\mathbb{R}}:=\mathbb{R}\cup\\{-\infty,+\infty\\}$, we set
$\displaystyle{z}={\rm ap}\lim_{\mathbf{y}\to\mathbf{x}}v(\mathbf{y})\,$
(notation for the approximate limit of $v$ at $\mathbf{x}$) if, for every
$g\in C^{0}(\overline{\mathbb{R}}),$

$g({z})\ =\ \lim_{\varrho\to
0}\,|B_{\varrho}(\mathbf{0})|^{-1}\\!\\!\int_{B_{\varrho}(\mathbf{0})}\\!\\!g(v(\mathbf{x}+\mbox{\boldmath$\xi$}))\,d\mbox{\boldmath$\xi$}\,;$

the function $\displaystyle\widetilde{v}(\mathbf{x})=\mathop{\rm
ap\,lim}\limits_{\mathbf{y}\to\mathbf{x}}v(\mathbf{y})$ is called
representative of $v\,;$

$\displaystyle S_{v}=\\{\mathbf{x}\in\Omega:\,\not\\!\exists z\ \mbox{ such
that }\ \mathop{\rm
ap\,lim}\limits_{\mathbf{y}\to\mathbf{x}}v(\mathbf{\mathbf{y}})=z\\}\quad\hbox{is
the {\sl singular set} of }v\,.$

A Borel function $v:\\!\Omega\\!\to\\!\mathbb{R}$ is approximately continuous
at $\mathbf{x}\\!\in\\!\Omega$ iff $\displaystyle
v({\mathbf{x}})\\!=\\!\mathop{\rm
ap\,lim}\limits_{\mathbf{y}\to\mathbf{x}}v(\mathbf{y})$.
As usual, $Dv$ denotes the distributional gradient of $v$ and $\nabla
v(\mathbf{x})$ denotes the approximate gradient of $v\,,$ say $v$ is
approximately differentiable at $x$ if there exists a vector $\nabla v({\bf
x})\in\mathbb{R}^{2}$ (the approximate gradient of $v$ at ${\bf x}$) such that

$\mathop{\rm
ap\,lim}\limits_{\mathbf{y}\to\mathbf{x}}\frac{|v(\mathbf{y})-\widetilde{v}(\mathbf{x})-\nabla
v(\mathbf{x})\cdot(\mathbf{y}-\mathbf{x})|}{|\mathbf{y}-\mathbf{x}|}=0.$

A function $u\\!\in\\!BV(\Omega)$ is approximately differentiable a.e.,
moreover for ${\mathcal{H}}^{1}$ almost every $\mathbf{x}\\!\in\\!S_{u}$ there
exist $\nu(\mathbf{x})\\!\in\partial\\!B_{1}$,
$v_{+}(\mathbf{x})\\!\in\\!\mathbb{R}$, $v_{-}(\mathbf{x})\\!\in\\!\mathbb{R}$
with $v_{+}(\mathbf{x})\\!>\\!v_{-}(\mathbf{x})$ such that

$\lim_{\varrho\to 0}\ \varrho^{-n}\\!\int_{\\{\mathbf{y}\in
B_{\varrho};\,\mathbf{y}\cdot\nu(\mathbf{x})>0\\}}|v(\mathbf{x}+\mathbf{y})-v_{+}(\mathbf{x})|\,d\mathbf{y}=0,$
$\lim_{\varrho\to 0}\ \varrho^{-n}\\!\int_{\\{\mathbf{y}\in
B_{\varrho};\,\mathbf{y}\cdot\nu(\mathbf{x})<0\\}}|v(\mathbf{x}+\mathbf{y})-v_{-}(\mathbf{x})|\,d\mathbf{y}=0.$

$S\kern-2.0ptB\kern-1.0ptV(\Omega)$ denotes the De Giorgi class of functions
$v\in BV(\Omega)$ such that

$\int_{\Omega}|Dv|=\int_{\Omega}|\nabla
v|\,d\mathbf{x}+\int_{S_{v}}|v^{+}-v^{-}|\,d{\mathcal{H}}^{1}.$

We introduce:

$S\kern-2.0ptB\kern-1.0ptV_{loc}(\Omega)\ :=\ \left\\{v\in
S\kern-2.0ptB\kern-1.0ptV(\Omega^{\prime});\
\forall\,\Omega^{\prime}\subset\kern-1.0pt\subset\Omega\right\\},$ (5.1)
$G\kern-1.0ptS\kern-1.5ptBV(\Omega)\ :=\ \big{\\{}v:{}\Omega\to\mathbb{R}\
\hbox{Borel function};-k\lor v\land k\in
S\kern-2.0ptB\kern-1.0ptV_{loc}(\Omega)\ \forall k\in\mathbb{N}\big{\\}}.$
(5.2) $G\kern-1.0ptS\kern-1.5ptBV^{2}(\Omega)\ :=\ \big{\\{}\,v\in
G\kern-1.0ptS\kern-1.5ptBV(\Omega),~{}\nabla
v\in\big{(}G\kern-1.0ptS\kern-1.5ptBV(\Omega)\big{)}^{2}\big{\\}}.$

If $v\in G\kern-1.0ptS\kern-1.5ptBV(\Omega)$ then $\nabla v$ exists a.e., and
for $v\in G\kern-1.0ptS\kern-1.5ptBV^{2}(\Omega)$ we set
$\nabla^{2}v=\nabla(\nabla v)$.

Eventually, we introduce the weak formulation $\mathcal{F}$ of functional $F$
defined by (2.7):

(5.3) $\left\\{\begin{array}[]{l}\mathcal{F}(v_{r},v_{p})\,=\,\mathcal{F}(V)\
:=\\\ \displaystyle\ \
\mathcal{H}^{1}\\!\left(S_{v_{r}}\right)+\,\eta\\!\int_{\Omega}\\!|\nabla^{2}v_{r}|^{2}d{\bf
x}+\mu\\!\int_{\Omega}\\!|v_{r}-v_{p}|^{q}\,d{\bf
x}+\\!\int_{\Omega}\\!\\!\\!\left(\gamma\,|D^{2}v_{p}|^{2}-f\,v_{p}\right)d{\bf
x}\ ,\vspace{0.1cm}\\\ \ \
\forall\,V=(v_{r},v_{p})\in\mathbb{X}:=\Big{(}GSBV^{2}(\Omega_{p})\cap
L^{2}(\Omega_{p})\Big{)}\\!\times\\!H^{2}(\Omega_{p})\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\hbox{s.t.}\ \ v_{r}=v_{p}=w\ \hbox{
a.e. }\Omega_{p}\setminus\overline{\Omega}\,.\end{array}\right.$

We emphasize that, since $v_{p}=w$ in $\Omega_{p}\setminus\overline{\Omega}$
and $w\in C^{2}(\Omega_{p}\setminus\overline{\Omega}),$ we get $S_{v_{p}}\cup
S_{\nabla v_{p}}=\emptyset$ and

(5.4) $\mathcal{F}(v_{r},v_{p})\ =\
\mathcal{F}_{r}(v_{r})+M(v_{r}-v_{p})+F_{p}(v_{p})$

where $M$, $F_{p}$ and $\mathcal{F}_{r}$ are defined by (2.12), (2.13) and

(5.5) $\mathcal{F}_{r}(v_{r})\ :=\ \mathcal{H}^{1}\left(S_{v_{r}}\cup
S_{\nabla
v_{r}}\right)\,+\,\int_{\Omega}\Big{(}\eta|\nabla^{2}v_{r}|^{2}-f_{r}v_{r}\Big{)}\,d{\bf
x}\ .$

######  Theorem 5.1.

Assume (2.1),(2.2),(2.3) and (2.6).
Then the functional $\mathcal{F}$ achieves a finite minimum over
$\mathbb{X}\,.$

Proof - First we notice that $\mathcal{F}$ has non empty domain: in fact (2.2)
entails $S_{w}=S_{\nabla w}=\emptyset$ and

(5.6)
$\mathcal{F}(w,w)\,<\,(\eta+\gamma)\|D^{2}w\|_{L^{2}(\Omega)}^{2}-\int_{\Omega}(f_{r}+f_{p})\,w\
d{\bf x}\,<\,+\infty\ .$

We have the identity

(5.7)
$\int_{\Omega}\\!\\!\big{(}\,\mu\,|v_{r}-v_{p}|^{2}-f_{r}v\,\big{)}dx=\\!\int_{\Omega}\\!\\!\mu\Big{(}v_{r}-\big{(}v_{p}+f_{r}/(2\mu)\big{)}\Big{)}^{2}dx-\\!\int_{\Omega}\\!\\!\Big{(}f_{r}v_{p}+f_{r}^{2}/(4\mu)\Big{)}dx.$

If $K_{\Omega}$ denotes the best Poincaré inequality constant in
$H^{2}_{0}(\Omega)$, namely

(5.8)
$\|v\|_{L^{2}(\Omega)}^{2}\,\leq\,K_{\Omega}\,\|v^{\prime\prime}\|_{L^{2}(\Omega)}^{2}\qquad\forall\,v\\!\in\\!H^{2}_{0}(\Omega)\,,$

and, arguing as like as in (3.16) we get, for everyy
$V=(v_{r},v_{p})\in\mathbb{X}$

(5.9) $F(v_{r},v_{p})\geq
J(v_{r})+\int_{-1}^{1}\\!\\!\Big{(}\eta|\ddot{v}_{r}|^{2}+(\gamma/2)\,|v_{p}^{\prime\prime}|^{2}\Big{)}\,dx\,-C(\mu,\gamma,f_{r},f_{p},K_{\Omega})\
.$

Then the functional $F$ is bounded from below on its domain. Hence we can
select a minimizing sequence $V_{h}=\big{(}(v_{r})_{h},(v_{p})_{h}\big{)}$ for
$\mathcal{F}$ : $\lim_{h}\mathcal{F}(V_{h})=\inf\mathcal{F}\in\mathbb{R}$ .
Thanks to (5.6), we may suppose that

(5.10) $c\,\leq\,\mathcal{F}(V_{h})\leq C:=\mathcal{F}(w,w)<+\infty\,.$

Summarizing $F_{p}\big{(}(v_{p})_{h}\big{)}\leq C$,
$F_{r}\big{(}(v_{r})_{h}\big{)}\leq C$ and $(v_{p})_{h}$ is bounded in
$H^{2}(\Omega)$. Moreover there is $u_{p}\in H^{2}(\Omega_{p})$ such that, up
to subsequences and without relabelling, $(v_{p})_{h}$ is converging to
$u_{p}$ weakly in $H^{2}(\Omega)$ and strongly in $L^{2}(\Omega)$, and
$\|(v_{p})^{\prime\prime}_{h}\|_{L^{2}}\to\|u_{p}^{\prime\prime}\|_{L^{2}}$ .
By using any fixed $(v_{p})_{h}$ chosen from the sequence (which is bounded in
$H^{2}$) as datum we find a minimizer, denoted by $z_{h}$, in
$GSBV^{2}(\Omega_{p})\cap L^{q}(\Omega_{p})$ of

$v\ \mapsto\ \mathcal{F}_{r}(v)+M(v-(v_{p})_{h})$

since this problem is equivalent to the minimization of Blake & Zisserman
functional for image segmentation with gray-level datum
$g=(v_{p})_{h}+f_{r}/(2\mu)$ and Dirichlet boundary condition, referring to
notation of Theorem 3.1 in [16]. Then

(5.11) $\mathcal{F}_{r}\big{(}z_{h}\big{)}+M\big{(}z_{h}-(v_{p})_{h}\big{)}\
\leq\mathcal{F}_{r}\big{(}(v_{r})_{h}\big{)}+M((v_{r})_{h}-(v_{p})_{h})\qquad\forall
h\,.$

Hence, by (5.4) and standard lower semicontinuity of $F_{p}$, the sequence of
pairs $(z_{h},(v_{p})_{h})$ is a minimizing sequence for $\mathcal{F}$ too.
Moreover, by (5.10),(5.11) we get

$\mathcal{F}_{r}\big{(}z_{h}\big{)}+M\big{(}z_{h}-(u_{p})_{h}\big{)}\leq C$

By compactness property of Theorem 8 in [11], there are
$u_{r}\\!\in\\!GSBV^{2}(\Omega_{p})\cap L^{2}(\Omega_{p})$ and a subsequence
s.t., again by extracting without relabeling, $z_{h}\to u_{r}$. Moreover by
lower semi-continuity property of Theorem 10 in [11], we get: $\,z_{h}\to
u_{r}$ a.e., $z_{h}\rightharpoonup u_{r}$ $L^{2}$ and strongly in $L^{s},$
$1\\!\leq\\!s<2,$ $\nabla z_{h}\to\nabla u_{r}$ a.e.,
$\nabla^{2}z_{h}\to\nabla^{2}u_{r}$ a.e., $\nabla^{2}z_{h}\to\nabla^{2}z$
weakly in $L^{2}$ and

$\mathcal{F}_{r}(u_{r})+M(u_{r}-{u_{p}})\leq\liminf_{h}\mathcal{F}_{r}(z_{h})+M\big{(}z_{h}-(v_{p})_{h}\big{)}\leq\,C\
.$

Thus the pair $(u_{r},u_{p})$ is a minimizer of relaxed functional
$\mathcal{F}$. $\square$ Proof of Theorem 2.2 \- Let
$V=(u_{r},u_{p})\in\mathop{{\rm argmin}}\nolimits\mathcal{F}$ (the existence
of at least one such $V$ is warranted by Theorem 5.1). Then $u_{p}$ minimizes
$z\mapsto F_{p}(z)+M(v_{r}-z)$ among $z\in H^{2}(\Omega_{p})$ s.t. $z=w$ a.e.
$\Omega_{p}\setminus\overline{\Omega}$. So, due to (2.2) and (2.1), $u_{p}\in
C^{2}\cap L^{\infty}(\Omega)$.
Moreover, if $V=(u_{r},u_{p})\in\mathop{{\rm argmin}}\nolimits\mathcal{F}$,
then $u_{p}\in H^{4}(\Omega)$ and, referring to (2.12) and (2.13), $u_{r}$
minimizes $v\mapsto\mathcal{F}_{r}(v)+M(v-u_{p})$ among $v\in
G\kern-1.0ptS\kern-1.5ptBV^{2}(\Omega_{p})\cap L^{2}(\Omega_{p}))$ s.t.
$v=u_{p}=w$ a.e. $\Omega_{p}\setminus\overline{\Omega}$. Thus, exploiting the
identity (5.7), by Theorem 2.2 of [16] with the choices $\alpha=\beta=1$,
$g=u_{p}+f_{r}/(2\mu)\in L^{4}(\Omega_{p})$ and $M=T_{0}=T_{1}=\emptyset$, and
setting $Z=\overline{S_{u_{r}}\cup S_{\nabla u_{r}}}$, we obtain that the
triplet $(Z,\widetilde{u_{r}},u_{p})$ is an essential admissible triplet that
minimizes $F$.

By applying the regularization argument detailed in [12],[16],[18] we obtain
that $\widetilde{u_{r}}\in C^{2}(\Omega_{p}\setminus Z)$, where $Z$ is the
smallest closed subset of $\Omega_{p}$ containing the region where $C^{2}$
regularity of $\widetilde{u_{r}}$ is missing, and
$\mathcal{H}^{1}(Z\setminus(S_{u_{r}}\cup S_{\nabla u_{r}})=0$. Eventually

$\begin{array}[]{l}\hskip-5.69046pt\mathcal{F}_{r}(\widetilde{u_{r}})+G(\widetilde{u_{r}}-u_{p})+F_{p}(u_{p})\leq\mathcal{F}_{r}({u_{r}})+G({u_{r}}-u_{p})+F_{p}(u_{p})\leq\\\
\qquad\qquad\qquad\qquad\qquad\qquad\quad\,\leq\liminf_{h}\left(\
\mathcal{F}_{r}(z_{h})+G(z_{h}-u_{h})+F_{p}(u_{h})\
\right)=\inf_{\mathbb{X}}\mathcal{F}\end{array}$

hence
$\mathcal{F}(\,\widetilde{u_{r}}\,,\,u_{p}\,)\,=\,\min_{\mathbb{X}}\,\mathcal{F}\,.$

Summarizing
$F\big{(}\,Z,\,\widetilde{u_{r}},\,u_{p}\,\big{)}=\min\\{\,F(K,v_{r},v_{p}):(K,v_{r},v_{p})\hbox{
admissible triplet}\,\\}\,.$ $\square$

######  Remark 5.2.

We emphasize that also the non-interpenetration between plate and
reinforcement could be taken into account: e.g., adding the constraint $v\geq
w\ a.e\ \Omega_{p}$ to the essential admissible pairs for hard-device
reinforcement and adding the constraint $v_{r}\geq v_{p}\ a.e\ \Omega_{p}$ to
the essential admissible triplets for strengthening reinforcement. Notice that
here Remark 3.6 does not apply: competing functions are functions defined only
almost everywhere, therefore the unilateral constraints act in the almost
everywhere sense only. These unilateral constraints do not introduce any
additional difficulty in the study of the weak formulations of both $F$ and
$E$, since inequalities are preserved by compactness properties of minimizing
sequences. Therefore Theorem 5.1 holds true also under the additional
constraint $v_{r}\geq v_{p}$.
But the subsequent step required to show Theorem 2.2, say the proof of partial
regularity for weak minimizers, would be not straightforward.
For this reason in this short note we skip this substantial difficulty,
postponing the analysis of the 2 dimensional problems with unilateral
constraints to a forthcoming paper.
However, in 1 dimension the strong and weak formulation do coincide, so the
analogous of Theorems 2.1 and 2.2 hold true with or without the non-
interpenetration constraint for beams: we have taken into account these
constraints in the one-dimensional case by Theorems 3.7, 3.8 and Propositions
3.11, 3.12.

##  6\. Elastic-plastic reinforcement of flexural plate

In this section we deduce the existence statement in the case of strengthening
reinforcement: minimization of functional $G$ defined by (2.3). Proof of
Theorem 2.3 \- By
$G(\emptyset,w,w)\\!=\\!(\eta+\gamma)\|D^{2}w\|^{2}_{L^{2}(\Omega)}\\!-\int_{\Omega}(f_{r}+f_{p})w\,d{\bf
x}<+\infty$, we know that the functional $G$ has nonempty domain. Moreover
(5.9) warrants that the functional $G$ is bounded from below.
The existence of a minimizer of $G$ over over essential admissible triplets
$(K,v_{r},v_{p})$, namely triplets fulfilling (2.10), can be achieved by
repetition of the direct method approach with the techniques of [10].
Actually here, about minimization with respect to $v_{r}$, we have these
differences with respect to [10]: presence of the additional coupling term
$\mu\int_{\Omega}|v_{r}-v_{p}|^{2}dx$; there are neither vanishing moments nor
a safe load condition for the load $f_{r}$; last, there is a Dirichlet datum
$w$ at the boundary.
However vanishing moments and load $f$ were exploited in [10] only to achieve
the boundedness from below of the functional, whereas here such boundedness is
already warranted by (5.9). Moreover the additional term is a lower order
perturbation, not affecting the existence of weak minimizers (thanks to the
identity (5.7), still valid in present case), but requiring a technical
correction in the proof of strong solutions by regularization of weak
solutions.
Precisely, first step (existence of weak solutions) requires no change: we
introduce the space $SBH$ of Special Bounded hessian functions

$SBH(\widetilde{\Omega}):=\\{v\in H^{1,1}(\widetilde{\Omega}):Dv\in
SBV(\Omega),\,v=w\hbox{ on }\widetilde{\Omega}\\!\setminus\\!\Omega)\\}\,,$

here $SBV$ is the space of bounded variation functions whose derivative has no
Cantor part ([3]); then we set the weak formulation of functional $G$ defined
in (2.3), defined on $v\in SBH(\Omega)$:

$\displaystyle\mathcal{G}(v_{r},v_{p})\\!\\!$
$\displaystyle:=\,\mathcal{H}^{1}\left(S_{Dv_{r}}\right)\,+\,\sigma\int_{S_{Dv_{r}}}\\!\\!|[Dv_{r}]\,|\,d\mathcal{H}^{1}\,+\int_{\Omega}\\!\big{(}\eta|\nabla^{2}v_{r}|^{2}-f_{r}v_{r}\big{)}\,d{\bf
x}+$ $\displaystyle\qquad\,+\,\mu\int_{\Omega}|v_{r}-v_{p}|^{2}\,d{\bf
x}+\,\int_{\Omega}\\!\big{(}\gamma|D^{2}v_{p}|^{2}-f_{p}v_{p}\big{)}\,d{\bf
x}$

where $[z]$ denotes the jump of $z$ and $\nabla z$ denotes the approximate
gradient of $z$, say the absolutely continuous part of $Dz$.
The existence of a minimizing pair $(u_{r},u_{p})$ for $\mathcal{G}$ follows
by the same argument of present Theorem 5.1, taking into account of Theorem
2.9 in [10]. The proof of partial regularity in $\Omega$ for weak minimizers
is achieved by exploiting blow-up and quasi-minimizers as in Theorem 4.15 in
[10]: only Lemma 4.3 of [10] must be adapted as detailed below, to take into
account of the additional glueing term.
Still by identity (5.7), the load and glue terms together are represented (up
to the addition of a constant irrelevant in minimization) by
$\displaystyle\mu\int_{\Omega}\\!(v-h)^{2}dx$ where $h:=u_{p}+f_{r}/(2\mu)$
and $h\in L^{s},s>2$: this contribution replaces here the term
$-\int_{\Omega}gv$ of [10], however this does not affect regularization of
weak solutions, since, setting
$\mathcal{E}(v)=\mathcal{G}(v)-\mu\int_{\Omega}(v-h)^{2},$ every local
minimizer of $\mathcal{G}$ is a local quasi minimizer of $\mathcal{E}$, due to
the excess estimate (consequence of $u,v\in SBH(\Omega)\subset
L^{\infty}(\Omega)$, $h\in L^{s}(\Omega)$, $s>2$ and Hölder inequality):

$\int_{B_{\varrho}({\bf
x})}\Big{(}(v-h)^{2}-(u-h)^{2}\Big{)}\,dx\,=\,\int_{B_{\varrho}({\bf
x})}\Big{(}u^{2}-v^{2}-h(v-u)\Big{)}\,\leq\,C\,\varrho^{2-2/s}$

valid for $\overline{B_{\varrho}({\bf x})}\subset\Omega$, $0<\varrho<1$ and
$u,v\in SBH(\Omega)$ s.t. $v=u$ on $\Omega\\!\setminus\\!B_{\varrho}({\bf
x})$.
Partial regularity at the boundary under Dirichlet condition, can be achieved
by the same argument of [16], taking into account of the simplifications due
to the fact that here the competing functions are not only in
$GSBV^{2}(\Omega)$, but they belong to $SBH(\Omega)$, hence they are globally
continuous.
Summarizing a minimizing pair $(u_{r},u_{p})$ of $\mathcal{G}$ leads to an
essential minimizing triplet
$\big{(}\overline{S_{\widetilde{Du_{r}}}},u_{r},u_{p}\big{)}$ of $G$.
$\square$

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