# Asymptotic analysis of linearly elastic flexural shells subjected to an
obstacle in absence of friction

Paolo Piersanti  Department of Mathematics and Institute for Scientific
Computing and Applied Mathematics, Indiana University Bloomington, 729 East
Third Street, Bloomington, Indiana, USA<EMAIL_ADDRESS>
###### Abstract.

In this paper we identify a set of two-dimensional variational inequalities
that model the displacement of a linearly elastic flexural shell subjected to
a confinement condition, expressing that all the points of the admissible
deformed configurations remain in a given half-space. The action of friction
is neglected.

##  1\. Introduction

Unilateral contact problems arise in many fields such as medicine,
engineering, biology and material science. For instance, the description of
the motion inside the human heart of the three Aorta valves, which can be
regarded as linearly elastic shells, is governed by a mathematical model built
up in a way such that each valve remains confined in a certain portion of
space without penetrating, or being penetrated, by the other two valves. In
this direction we cite the recent references [34], [35] and [40].

The displacement of a linearly elastic shell is modelled, in general, via the
classical equations of three-dimensional linearized elasticity (cf., e.g.,
[6]). The intrinsic complexity of this model, however, prevents certain
situations from being amenably studied like, for instance, when the shell is
non-homogeneous and anisotropic (cf., e.g., [4] and [5]), or its thickness
varies periodically (cf., e.g., [38] and [39]). It might thus be useful to
perform a _dimension reduction_ in order to obtain approximate models which
are more amenable to analyze.

The identification of two-dimensional limit models for time-independent
linearly elastic shells was extensively treated by Ciarlet and his associates
in the seminal papers [11, 12, 13, 14, 15, 18] for the purpose of justifying
Koiter’s model. For the justification of Koiter’s model in the time-dependent
case where the action of temperature is considered, we refer the reader to
[28]. In the aforementioned papers no confinement conditions were imposed. The
recent papers [32] and [33] provide examples of variational-inequalities-based
models arising in biology.

In the recent papers [16, 17, 19, 20], Ciarlet and his associates fully
justified Koiter’s model in the case where the linearly elastic shell under
consideration is an elliptic membrane shell subject to the aforementioned
confinement condition.

The confinement condition we are here considering considerably departs from
the Signorini condition usually considered in the existing literature, where
only the “lower face” of the shell is required to remain above the
“horizontal” plane. Such a confinement condition renders the asymptotic
analysis considerably more difficult as the constraint now bears on a vector
field, the displacement vector field of the reference configuration, instead
of on only a single component of this field.

The recovery of a set of two-dimensional variational inequalities as a result
of a rigorous asymptotic analysis conducted on a _ad hoc_ three-dimensional
model based on the classical equations of three-dimensional linearized
elasticity in the case where the linearly elastic shell under consideration is
a flexural shell has only been addressed, to our best knowledge, by Léger and
Miara in the paper [25] in a very amenable geometrical framework. In the paper
[25] the authors do not consider the action of friction. The method proposed
in the paper [25], which makes use of a very lengthy argument exploiting the
properties of cones, is suitable for treating the case where only one linearly
elastic flexural shell is considered. In the context of multi-physics multi-
scale problems, like for instance those considered by Quarteroni and his
associates in [34], [35] and [40], it is not however clear whether the
approach proposed by Léger and Miara is suitable. The purpose of this paper is
to remedy this situation by presenting a new method, based on the rigorous
asymptotic analysis technique developed by Ciarlet, Lods and Miara [18] and
the properties of the penalty method for constrained optimization problem
(cf., e.g., [7]), for recovering the same set of two-dimensional variational
inequalities for linearly elastic flexural shells that were recovered in the
recent papers [16, 17] in a more general geometrical framework. It is also
worth mentioning that the method we present in this paper makes use of
considerably less lengthy computations than those in [25].

More precisely, in this paper we recover, via a rigorous asymptotic analysis
as the thickness approaches zero over a _ad hoc_ three-dimensional model
(three-dimensional in the sense that it is defined over a three-dimensional
subset of $\mathbb{R}^{3}$), a set of two-dimensional (two-dimensional in the
sense that it is defined over a two-dimensional subset of $\mathbb{R}^{2}$)
variational inequalities governing the displacement of a linearly elastic
flexural shell subject to remain confined in a half-space in absence of
friction. The problem under consideration is an _obstacle problem_.

The paper is divided into five sections (including this one). In section 2 we
recall some background and notation. In section 3 we recall the formulation
and the properties of a three-dimensional obstacle problem for “general”
linearly elastic shell. In section 4 we specialise the formulation presented
in the previous section to the case where the linearly elastic shell under
consideration is a flexural shell, and then we scale the three-dimensional
obstacle problem in a way such that the integration domain becomes independent
of the thickness parameter. The penalized version of the three-dimensional
obstacle problem is introduced at the end of this section. Finally, in section
5, a rigorous asymptotic analysis is carried out and the desired set of two-
dimensional variational inequalities is recovered.

##  2\. Geometrical preliminaries

For details about the classical notions of differential geometry used in this
section and the next one, see, e.g. [8] or [9].

Greek indices, except $\varepsilon$, take their values in the set $\\{1,2\\}$,
while Latin indices, except when they are used for indexing sequences, take
their values in the set $\\{1,2,3\\}$, and the summation convention with
respect to repeated indices is systematically used in conjunction with these
two rules. The notation $\mathbb{E}^{3}$ designates the three-dimensional
Euclidean space whose origin is denoted by $\bm{O}$; the Euclidean inner
product and the vector product of $\bm{u},\bm{v}\in\mathbb{E}^{3}$ are denoted
$\bm{u}\cdot\bm{v}$ and $\bm{u}\times\bm{v}$; the Euclidean norm of
$\bm{u}\in\mathbb{E}^{3}$ is denoted $\left|\bm{u}\right|$. The notation
$\delta^{j}_{i}$ designates the Kronecker symbol.

Given an open subset $\Omega$ of $\mathbb{R}^{n}$, notations such as
$L^{2}(\Omega)$, $H^{1}(\Omega)$, or $H^{2}(\Omega)$, designate the usual
Lebesgue and Sobolev spaces, and the notation $\mathcal{D}(\Omega)$ designates
the space of all functions that are infinitely differentiable over $\Omega$
and have compact supports in $\Omega$. The notation $\left\|\cdot\right\|_{X}$
designates the norm in a normed vector space $X$. Spaces of vector-valued
functions are denoted with boldface letters.

The positive and negative parts of a function $f:\Omega\to\mathbb{R}$ are
respectively denoted by:

$f^{+}(x):=\max\\{f(x),0\\}\quad\textup{ and }\quad
f^{-}(x):=-\min\\{f(x),0\\}\quad x\in\Omega.$

The boundary $\Gamma$ of an open subset $\Omega$ in $\mathbb{R}^{n}$ is said
to be Lipschitz-continuous if the following conditions are satisfied (cf.,
e.g., Section 1.18 of [10]): Given an integer $s\geq 1$, there exist constants
$\alpha_{1}>0$ and $L>0$, and a finite number of local coordinate systems,
with coordinates
$\bm{\phi}^{\prime}_{r}=(\phi_{1}^{r},\dots,\phi_{n-1}^{r})\in\mathbb{R}^{n-1}$
and $\phi_{r}=\phi_{n}^{r}$, sets
$\tilde{\omega}_{r}:=\\{\bm{\phi}_{r}\in\mathbb{R}^{n-1};|\bm{\phi}_{r}|<\alpha_{1}\\}$,
$1\leq r\leq s$, and corresponding functions

$\tilde{\theta}_{r}:\tilde{\omega}_{r}\to\mathbb{R},$

such that

$\Gamma=\bigcup_{r=1}^{s}\\{(\bm{\phi}^{\prime}_{r},\phi_{r});\bm{\phi}^{\prime}_{r}\in\tilde{\omega}_{r}\textup{
and }\phi_{r}=\tilde{\theta}_{r}(\bm{\phi}^{\prime}_{r})\\},$

and

$|\tilde{\theta}_{r}(\bm{\phi}^{\prime}_{r})-\tilde{\theta}_{r}(\bm{\upsilon}^{\prime}_{r})|\leq
L|\bm{\phi}^{\prime}_{r}-\bm{\upsilon}^{\prime}_{r}|,\textup{ for all
}\bm{\phi}^{\prime}_{r},\bm{\upsilon}^{\prime}_{r}\in\tilde{\omega}_{r},\textup{
and all }1\leq r\leq s.$

We observe that the second last formula takes into account overlapping local
charts, while the last set of inequalities express the Lipschitz continuity of
the mappings $\tilde{\theta}_{r}$.

An open set $\Omega$ is said to be locally on the same side of its boundary
$\Gamma$ if, in addition, there exists a constant $\alpha_{2}>0$ such that

$\displaystyle\\{(\bm{\phi}^{\prime}_{r},\phi_{r});\bm{\phi}^{\prime}_{r}\in\tilde{\omega}_{r}\textup{
and
}\tilde{\theta}_{r}(\bm{\phi}^{\prime}_{r})<\phi_{r}<\tilde{\theta}_{r}(\bm{\phi}^{\prime}_{r})+\alpha_{2}\\}$
$\displaystyle\subset\Omega,\quad\textup{ for all }1\leq r\leq s,$
$\displaystyle\\{(\bm{\phi}^{\prime}_{r},\phi_{r});\bm{\phi}^{\prime}_{r}\in\tilde{\omega}_{r}\textup{
and
}\tilde{\theta}_{r}(\bm{\phi}^{\prime}_{r})-\alpha_{2}<\phi_{r}<\tilde{\theta}_{r}(\bm{\phi}^{\prime}_{r})\\}$
$\displaystyle\subset\mathbb{R}^{n}\setminus\overline{\Omega},\quad\textup{
for all }1\leq r\leq s.$

A _domain in_ $\mathbb{R}^{n}$ is a bounded and connected open subset $\Omega$
of $\mathbb{R}^{n}$, whose boundary $\partial\Omega$ is Lipschitz-continuous,
the set $\Omega$ being locally on a single side of $\partial\Omega$.

Let $\omega$ be a domain in $\mathbb{R}^{2}$, let $y=(y_{\alpha})$ denote a
generic point in $\overline{\omega}$, and let
$\partial_{\alpha}:=\partial/\partial y_{\alpha}$ and
$\partial_{\alpha\beta}:=\partial^{2}/\partial y_{\alpha}\partial y_{\beta}$.
A mapping $\bm{\theta}\in\mathcal{C}^{1}(\overline{\omega};\mathbb{E}^{3})$ is
an _immersion_ if the two vectors

$\bm{a}_{\alpha}(y):=\partial_{\alpha}\bm{\theta}(y)$

are linearly independent at each point $y\in\overline{\omega}$. Then the image
$\bm{\theta}(\overline{\omega})$ of the set $\overline{\omega}$ under the
mapping $\bm{\theta}$ is a _surface in_ $\mathbb{E}^{3}$, equipped with
$y_{1},y_{2}$ as its _curvilinear coordinates_. Given any point
$y\in\overline{\omega}$, the vectors $\bm{a}_{\alpha}(y)$ span the _tangent
plane_ to the surface $\bm{\theta}(\overline{\omega})$ at the point
$\bm{\theta}(y)$, the unit vector

$\bm{a}_{3}(y):=\frac{\bm{a}_{1}(y)\wedge\bm{a}_{2}(y)}{|\bm{a}_{1}(y)\wedge\bm{a}_{2}(y)|}$

is normal to $\bm{\theta}(\overline{\omega})$ at $\bm{\theta}(y)$, the three
vectors $\bm{a}_{i}(y)$ form the _covariant_ basis at $\bm{\theta}(y)$, and
the three vectors $\bm{a}^{j}(y)$ defined by the relations

$\bm{a}^{j}(y)\cdot\bm{a}_{i}(y)=\delta^{j}_{i}$

form the _contravariant_ basis at $\bm{\theta}(y)$; note that the vectors
$\bm{a}^{\beta}(y)$ also span the tangent plane to
$\bm{\theta}(\overline{\omega})$ at $\bm{\theta}(y)$ and that
$\bm{a}^{3}(y)=\bm{a}_{3}(y)$.

The _first fundamental form_ of the surface $\bm{\theta}(\overline{\omega})$
is then defined by means of its _covariant components_

$a_{\alpha\beta}:=\bm{a}_{\alpha}\cdot\bm{a}_{\beta}=a_{\beta\alpha}\in\mathcal{C}^{0}(\overline{\omega}),$

or by means of its _contravariant components_

$a^{\alpha\beta}:=\bm{a}^{\alpha}\cdot\bm{a}^{\beta}=a^{\beta\alpha}\in\mathcal{C}^{0}(\overline{\omega}).$

Note that the symmetric matrix field $(a^{\alpha\beta})$ is then the inverse
of the matrix field $(a_{\alpha\beta})$, that
$\bm{a}^{\beta}=a^{\alpha\beta}\bm{a}_{\alpha}$ and
$\bm{a}_{\alpha}=a_{\alpha\beta}\bm{a}^{\beta}$, and that the _area element_
along $\bm{\theta}(\overline{\omega})$ is given at each point
$\bm{\theta}(y),\,y\in\overline{\omega}$, by $\sqrt{a(y)}\,\mathrm{d}y$, where

$a:=\det(a_{\alpha\beta})\in\mathcal{C}^{0}(\overline{\omega}).$

Given an immersion
$\bm{\theta}\in\mathcal{C}^{2}(\overline{\omega};\mathbb{E}^{3})$, the _second
fundamental form_ of the surface $\bm{\theta}(\overline{\omega})$ is defined
by means of its _covariant components_

$b_{\alpha\beta}:=\partial_{\alpha}\bm{a}_{\beta}\cdot\bm{a}_{3}=-\bm{a}_{\beta}\cdot\partial_{\alpha}\bm{a}_{3}=b_{\beta\alpha}\in\mathcal{C}^{0}(\overline{\omega}),$

or by means of its _mixed components_

$b^{\beta}_{\alpha}:=a^{\beta\sigma}b_{\alpha\sigma}\in\mathcal{C}^{0}(\overline{\omega}),$

and the _Christoffel symbols_ associated with the immersion $\bm{\theta}$ are
defined by

$\Gamma^{\sigma}_{\alpha\beta}:=\partial_{\alpha}\bm{a}_{\beta}\cdot\bm{a}^{\sigma}=\Gamma^{\sigma}_{\beta\alpha}\in\mathcal{C}^{0}(\overline{\omega}).$

The _Gaussian curvature_ at each point
$\bm{\theta}(y),\,y\in\overline{\omega}$, of the surface
$\bm{\theta}(\overline{\omega})$ is defined by

$K(y):=\frac{\det(b_{\alpha\beta}(y))}{\det(a_{\alpha\beta}(y))}=\det\left(b^{\beta}_{\alpha}(y)\right)$

(the denominator in the above relation does not vanish since $\bm{\theta}$ is
assumed to be an immersion). Note that the Gaussian curvature $K(y)$ at the
point $\bm{\theta}(y)$ is also equal to the inverse of the product of the two
principal radii of curvature at this point.

Given an immersion
$\bm{\theta}\in\mathcal{C}^{2}(\overline{\omega};\mathbb{E}^{3})$ and a vector
field
$\bm{\eta}=(\eta_{i})\in\mathcal{C}^{1}(\overline{\omega};\mathbb{R}^{3})$,
the vector field

$\tilde{\bm{\eta}}:=\eta_{i}\bm{a}^{i}$

can be viewed as a _displacement field of the surface_
$\bm{\theta}(\overline{\omega})$, thus defined by means of its _covariant
components_ $\eta_{i}$ over the vectors $\bm{a}^{i}$ of the contravariant
bases along the surface. If the norms
$\left\|\eta_{i}\right\|_{\mathcal{C}^{1}(\overline{\omega})}$ are small
enough, the mapping
$(\bm{\theta}+\eta_{i}\bm{a}^{i})\in\mathcal{C}^{1}(\overline{\omega};\mathbb{E}^{3})$
is also an immersion, so that the set
$(\bm{\theta}+\eta_{i}\bm{a}^{i})(\overline{\omega})$ is also a surface in
$\mathbb{E}^{3}$, equipped with the same curvilinear coordinates as those of
the surface $\bm{\theta}(\overline{\omega})$, called the _deformed surface_
corresponding to the displacement field
$\tilde{\bm{\eta}}=\eta_{i}\bm{a}^{i}$. One can then define the first
fundamental form of the deformed surface by means of its covariant components

$\displaystyle a_{\alpha\beta}(\bm{\eta}):=$
$\displaystyle(\bm{a}_{\alpha}+\partial_{\alpha}\tilde{\bm{\eta}})\cdot(\bm{a}_{\beta}+\partial_{\beta}\tilde{\bm{\eta}})$
$\displaystyle=$ $\displaystyle
a_{\alpha\beta}+\bm{a}_{\alpha}\cdot\partial_{\beta}\tilde{\bm{\eta}}+\partial_{\alpha}\tilde{\bm{\eta}}\cdot\bm{a}_{\beta}+\partial_{\alpha}\tilde{\bm{\eta}}\cdot\partial_{\beta}\tilde{\bm{\eta}}.$

The _linear part with respect to_ $\tilde{\bm{\eta}}$ in the difference
$\dfrac{1}{2}(a_{\alpha\beta}(\bm{\eta})-a_{\alpha\beta})$ is called the
_linearized change of metric tensor_ associated with the displacement field
$\eta_{i}\bm{a}^{i}$, the covariant components of which are thus defined by

$\gamma_{\alpha\beta}(\bm{\eta}):=\dfrac{1}{2}\left(\bm{a}_{\alpha}\cdot\partial_{\beta}\tilde{\bm{\eta}}+\partial_{\alpha}\tilde{\bm{\eta}}\cdot\bm{a}_{\beta}\right)=\frac{1}{2}(\partial_{\beta}\eta_{\alpha}+\partial_{\alpha}\eta_{\beta})-\Gamma^{\sigma}_{\alpha\beta}\eta_{\sigma}-b_{\alpha\beta}\eta_{3}=\gamma_{\beta\alpha}(\bm{\eta}).$

The _linear part with respect to_ $\tilde{\bm{\eta}}$ in the difference
$\dfrac{1}{2}(b_{\alpha\beta}(\bm{\eta})-b_{\alpha\beta})$ is called the
_linearized change of curvature tensor_ associated with the displacement field
$\eta_{i}\bm{a}^{i}$, the covariant components of which are thus defined by

$\displaystyle\rho_{\alpha\beta}(\bm{\eta})$
$\displaystyle:=(\partial_{\alpha\beta}\tilde{\bm{\eta}}-\Gamma_{\alpha\beta}^{\sigma}\partial_{\sigma}\tilde{\bm{\eta}})\cdot\bm{a}_{3}=\rho_{\beta\alpha}(\bm{\eta})$
$\displaystyle=\partial_{\alpha\beta}\eta_{3}-\Gamma_{\alpha\beta}^{\sigma}\partial_{\sigma}\eta_{3}-b_{\alpha}^{\sigma}b_{\sigma\beta}\eta_{3}+b_{\alpha}^{\sigma}(\partial_{\beta}\eta_{\sigma}-\Gamma_{\beta\sigma}^{\tau}\eta_{\tau})+b_{\beta}^{\tau}(\partial_{\alpha}\eta_{\tau}-\Gamma_{\alpha\tau}^{\sigma}\eta_{\sigma})+(\partial_{\alpha}b_{\beta}^{\tau}+\Gamma_{\alpha\sigma}^{\tau}b_{\beta}^{\sigma}-\Gamma_{\alpha\beta}^{\sigma}b_{\sigma}^{\tau})\eta_{\tau}.$

It turns out that, when a _generic surface_ is subjected to a displacement
field $\eta_{i}\bm{a}^{i}$ whose _tangential covariant components
$\eta_{\alpha}$ vanish on a non-zero length portion of boundary of the domain
$\omega$_, denoted $\gamma_{0}$ in the statement of the next result, the
following inequality holds (this inequality plays an essential role in our
convergence analysis; cf. the proof of Theorem 5.1). Note that the components
of the displacement fields, linearized change of metric tensor and linearized
change of curvature tensor appearing in the next theorem are no longer assumed
to be continuously differentiable functions; they are instead to be understood
in a generalized sense, since they now belong to ad hoc Lebesgue or Sobolev
spaces. Throughout the paper the symbol $\partial_{\nu}$ denotes the _outer
unit normal derivative operator along the boundary $\gamma$._

######  Theorem 2.1.

Let $\omega$ be a domain in $\mathbb{R}^{2}$ and let an immersion
$\bm{\theta}\in\mathcal{C}^{3}(\overline{\omega};\mathbb{E}^{3})$ be given.
Define the space

$\displaystyle\bm{V}_{K}(\omega)$ $\displaystyle:=\\{\bm{\eta}=(\eta_{i})\in
H^{1}(\omega)\times H^{1}(\omega)\times
H^{2}(\omega);\eta_{i}=\partial_{\nu}\eta_{3}=0\textup{ on }\gamma_{0}\\}.$

Then there exists a constant $c_{0}=c_{0}(\omega,\gamma_{0},\bm{\theta})>0$
such that

$\left\\{\sum_{\alpha}\left\|\eta_{\alpha}\right\|^{2}_{H^{1}(\omega)}+\left\|\eta_{3}\right\|^{2}_{H^{2}(\omega)}\right\\}^{1/2}\leq
c_{0}\left\\{\sum_{\alpha,\beta}\left\|\gamma_{\alpha\beta}(\bm{\eta})\right\|_{L^{2}(\omega)}^{2}+\sum_{\alpha,\beta}\left\|\rho_{\alpha\beta}(\bm{\eta})\right\|_{L^{2}(\omega)}^{2}\right\\}^{1/2}$

for all $\bm{\eta}=(\eta_{i})\in\bm{V}_{K}(\omega)$. ∎

The above inequality, which is due to [1] and was later on improved by [2]
(see also Theorem 2.6-4 of [8]), constitutes an example of a _Korn inequality
on a general surface_ , in the sense that it provides an estimate of an
appropriate norm of a displacement field defined on a surface in terms of an
appropriate norm of a specific “measure of strain” (here, the linearized
change of metric tensor and the linearized change of curvature tensor)
corresponding to the displacement field considered.

##  3\. The three-dimensional obstacle problem for a “general” linearly
elastic shell

Let $\omega$ be a domain in $\mathbb{R}^{2}$, let $\gamma:=\partial\omega$,
and let $\gamma_{0}$ be a non-empty relatively open subset of $\gamma$. For
each $\varepsilon>0$, we define the sets

$\Omega^{\varepsilon}=\omega\times\left]-\varepsilon,\varepsilon\right[\quad\textup{
and
}\quad\Gamma^{\varepsilon}_{0}:=\gamma_{0}\times\left[-\varepsilon,\varepsilon\right],$

we let $x^{\varepsilon}=(x^{\varepsilon}_{i})$ designate a generic point in
the set $\overline{\Omega^{\varepsilon}}$, and we let
$\partial^{\varepsilon}_{i}:=\partial/\partial x^{\varepsilon}_{i}$. Hence we
also have $x^{\varepsilon}_{\alpha}=y_{\alpha}$ and
$\partial^{\varepsilon}_{\alpha}=\partial_{\alpha}$.

Given an immersion
$\bm{\theta}\in\mathcal{C}^{3}(\overline{\omega};\mathbb{E}^{3})$ and
$\varepsilon>0$, consider a _shell_ with _middle surface_
$\bm{\theta}(\overline{\omega})$ and with _constant thickness_ $2\varepsilon$.
This means that the _reference configuration_ of the shell is the set
$\bm{\Theta}(\overline{\Omega^{\varepsilon}})$, where the mapping
$\bm{\Theta}:\overline{\Omega^{\varepsilon}}\to\mathbb{E}^{3}$ is defined by

$\bm{\Theta}(x^{\varepsilon}):=\bm{\theta}(y)+x^{\varepsilon}_{3}\bm{a}^{3}(y)\text{
at each point
}x^{\varepsilon}=(y,x^{\varepsilon}_{3})\in\overline{\Omega^{\varepsilon}}.$

One can then show (cf., e.g., Theorem 3.1-1 of [8]) that, if $\varepsilon>0$
is small enough, such a mapping
$\bm{\Theta}\in\mathcal{C}^{2}(\overline{\Omega^{\varepsilon}};\mathbb{E}^{3})$
is an _immersion_ , in the sense that the three vectors

$\bm{g}^{\varepsilon}_{i}(x^{\varepsilon}):=\partial^{\varepsilon}_{i}\bm{\Theta}(x^{\varepsilon}),$

are linearly independent at each point
$x^{\varepsilon}\in\overline{\Omega^{\varepsilon}}$; these vectors then
constitute the _covariant basis_ at the point $\bm{\Theta}(x^{\varepsilon})$,
while the three vectors $\bm{g}^{j,\varepsilon}(x^{\varepsilon})$ defined by
the relations

$\bm{g}^{j,\varepsilon}(x^{\varepsilon})\cdot\bm{g}^{\varepsilon}_{i}(x^{\varepsilon})=\delta^{j}_{i},$

constitute the _contravariant basis_ at the same point. It will be implicitly
assumed in the sequel that $\varepsilon>0$ _is small enough so that
$\bm{\Theta}:\overline{\Omega^{\varepsilon}}\to\mathbb{E}^{3}$_ is an
_immersion_.

One then defines the _metric tensor associated with the immersion_
$\bm{\Theta}$ by means of its _covariant components_

$g^{\varepsilon}_{ij}:=\bm{g}^{\varepsilon}_{i}\cdot\bm{g}^{\varepsilon}_{j}\in\mathcal{C}^{1}(\overline{\Omega^{\varepsilon}}),$

or by means of its _contravariant components_

$g^{ij,\varepsilon}:=\bm{g}^{i,\varepsilon}\cdot\bm{g}^{i,\varepsilon}\in\mathcal{C}^{1}(\overline{\Omega^{\varepsilon}}).$

Note that the symmetric matrix field $(g^{ij,\varepsilon})$ is then the
inverse of the matrix field $(g^{\varepsilon}_{ij})$, that
$\bm{g}^{j,\varepsilon}=g^{ij,\varepsilon}\bm{g}^{\varepsilon}_{i}$ and
$g^{\varepsilon}_{i}=g^{\varepsilon}_{ij}\bm{g}^{j,\varepsilon}$, and that the
_volume element_ in $\bm{\Theta}(\overline{\Omega^{\varepsilon}})$ is given at
each point $\bm{\Theta}(x^{\varepsilon})$,
$x^{\varepsilon}\in\overline{\Omega^{\varepsilon}}$, by
$\sqrt{g^{\varepsilon}(x^{\varepsilon})}\,\mathrm{d}x^{\varepsilon}$, where

$g^{\varepsilon}:=\det(g^{\varepsilon}_{ij})\in\mathcal{C}^{1}(\overline{\Omega^{\varepsilon}}).$

One also defines the _Christoffel symbols_ associated with the immersion
$\bm{\Theta}$ by

$\Gamma^{p,\varepsilon}_{ij}:=\partial_{i}\bm{g}^{\varepsilon}_{j}\cdot\bm{g}^{p,\varepsilon}=\Gamma^{p,\varepsilon}_{ji}\in\mathcal{C}^{0}(\overline{\Omega^{\varepsilon}}).$

Note that $\Gamma^{3,\varepsilon}_{\alpha 3}=\Gamma^{p,\varepsilon}_{33}=0$.

Given a vector field
$\bm{v}^{\varepsilon}=(v^{\varepsilon}_{i})\in\mathcal{C}^{1}(\overline{\Omega^{\varepsilon}};\mathbb{R}^{3})$,
the associated vector field

$\tilde{\bm{v}}^{\varepsilon}:=v^{\varepsilon}_{i}\bm{g}^{i,\varepsilon},$

can be viewed as a _displacement field_ of the reference configuration
$\bm{\Theta}(\overline{\Omega^{\varepsilon}})$ of the shell, thus defined by
means of its _covariant components_ $v^{\varepsilon}_{i}$ over the vectors
$\bm{g}^{i,\varepsilon}$ of the contravariant bases in the reference
configuration.

If the norms
$\left\|v^{\varepsilon}_{i}\right\|_{\mathcal{C}^{1}(\overline{\Omega^{\varepsilon}})}$
are small enough, the mapping
$(\bm{\Theta}+v^{\varepsilon}_{i}\bm{g}^{i,\varepsilon})$ is also an
immersion, so that one can also define the metric tensor of the _deformed
configuration_
$(\bm{\Theta}+v^{\varepsilon}_{i}\bm{g}^{i,\varepsilon})(\overline{\Omega^{\varepsilon}})$
by means of its covariant components

$\displaystyle
g^{\varepsilon}_{ij}(v^{\varepsilon}):=(\bm{g}^{\varepsilon}_{i}+\partial^{\varepsilon}_{i}\tilde{\bm{v}}^{\varepsilon})\cdot(\bm{g}^{\varepsilon}_{j}+\partial^{\varepsilon}_{j}\tilde{\bm{v}}^{\varepsilon})=g^{\varepsilon}_{ij}+\bm{g}^{\varepsilon}_{i}\cdot\partial_{j}\tilde{\bm{v}}^{\varepsilon}+\partial^{\varepsilon}_{i}\tilde{\bm{v}}^{\varepsilon}\cdot\bm{g}^{\varepsilon}_{j}+\partial_{i}\tilde{\bm{v}}^{\varepsilon}\cdot\partial_{j}\tilde{\bm{v}}^{\varepsilon}.$

The linear part with respect to $\tilde{\bm{v}}^{\varepsilon}$ in the
difference
$\dfrac{1}{2}(g^{\varepsilon}_{ij}(\bm{v}^{\varepsilon})-g^{\varepsilon}_{ij})$
is then called the _linearized strain tensor_ associated with the displacement
field $v^{\varepsilon}_{i}\bm{g}^{i,\varepsilon}$, the covariant components of
which are thus defined by

$e^{\varepsilon}_{i\|j}(\bm{v}^{\varepsilon}):=\frac{1}{2}\left(\bm{g}^{\varepsilon}_{i}\cdot\partial_{j}^{\varepsilon}\tilde{\bm{v}}^{\varepsilon}+\partial^{\varepsilon}_{i}\tilde{\bm{v}}^{\varepsilon}\cdot\bm{g}^{\varepsilon}_{j}\right)=\frac{1}{2}(\partial^{\varepsilon}_{j}v^{\varepsilon}_{i}+\partial^{\varepsilon}_{i}v^{\varepsilon}_{j})-\Gamma^{p,\varepsilon}_{ij}v^{\varepsilon}_{p}=e_{j\|i}^{\varepsilon}(\bm{v}^{\varepsilon}).$

The functions $e^{\varepsilon}_{i\|j}(\bm{v}^{\varepsilon})$ are called the
_linearized strains in curvilinear coordinates_ associated with the
displacement field $v^{\varepsilon}_{i}\bm{g}^{i,\varepsilon}$.

We assume throughout this paper that, for each $\varepsilon>0$, the reference
configuration $\bm{\Theta}(\overline{\Omega^{\varepsilon}})$ of the shell is a
_natural state_ (i.e., stress-free) and that the material constituting the
shell is _homogeneous_ , _isotropic_ , and _linearly elastic_. The behavior of
such an elastic material is thus entirely governed by its two _Lamé constants_
$\lambda\geq 0$ and $\mu>0$ (for details, see, e.g., Section 3.8 of [6]).

We will also assume that the shell is subjected to _applied body forces_ whose
density per unit volume is defined by means of its covariant components
$f^{i,\varepsilon}\in L^{2}(\Omega^{\varepsilon})$, and to a _homogeneous
boundary condition of place_ along the portion $\Gamma^{\varepsilon}_{0}$ of
its lateral face (i.e., the displacement vanishes on
$\Gamma^{\varepsilon}_{0}$).

For what concerns surface traction forces, the mathematical models
characterised by the confinement condition considered in this paper
(confinement condition which is also considered in [24] in a more amenable
geometrical framework) do not take any surface traction forces into account.
Indeed, there could be no surface traction forces applied to the portion of
the three-dimensional shell boundary that engages contact with the obstacle.

The confinement condition considered in this paper is more suitable in the
context of multi-scales multi-bodies problems like, for instance, the study of
the motion of the human heart valves, conducted by Quarteroni and his
associates in [34, 35, 40] and the references therein.

In this paper we consider a specific _obstacle problem_ for such a shell, in
the sense that the shell is also subjected to a _confinement condition_ ,
expressing that any _admissible displacement vector field_
$v^{\varepsilon}_{i}\bm{g}^{i,\varepsilon}$ must be such that all the points
of the corresponding deformed configuration remain in a _half-space_ of the
form

$\mathbb{H}:=\\{x\in\mathbb{E}^{3};\,\bm{Ox}\cdot\bm{q}\geq 0\\},$

where $\bm{q}$ is a _nonzero vector_ given once and for all. In other words,
any admissible displacement field must satisfy

$\left(\bm{\Theta}(x^{\varepsilon})+v^{\varepsilon}_{i}(x^{\varepsilon})\bm{g}^{i,\varepsilon}(x^{\varepsilon})\right)\cdot\bm{q}\geq
0,$

for all $x^{\varepsilon}\in\overline{\Omega^{\varepsilon}}$, or possibly only
for almost all (a.a. in what follows) $x^{\varepsilon}\in\Omega^{\varepsilon}$
when the covariant components $v^{\varepsilon}_{i}$ are required to belong to
the Sobolev space $H^{1}(\Omega^{\varepsilon})$ as in Theorem 3.1 below.

We will of course assume that the reference configuration satisfies the
confinement condition, i.e., that

$\bm{\Theta}(\overline{\Omega^{\varepsilon}})\subset\mathbb{H}.$

It is to be emphasized that the above confinement condition _considerably
departs_ from the usual _Signorini condition_ favored by most authors, who
usually require that only the points of the undeformed and deformed “lower
face” $\omega\times\\{-\varepsilon\\}$ of the reference configuration satisfy
the confinement condition (see, e.g., [24], [25], [36]). Clearly, the
confinement condition considered in the present paper is more physically
realistic, since a Signorini condition imposed only on the lower face of the
reference configuration does not prevent – at least “mathematically” – other
points of the deformed reference configuration to “cross” the plane
$\\{x\in\mathbb{E}^{3};\;\bm{Ox}\cdot\bm{q}=0\\}$ and then to end up on the
“other side” of this plane. It is evident that the vector $\bm{q}$ is thus
orthogonal to the plane associated with the half-space where the linearly
elastic shell is required to remain confined.

Such a confinement condition renders the asymptotic analysis considerably more
difficult, however, as the constraint now bears on a vector field, the
displacement vector field of the reference configuration, instead of on only a
single component of this field.

The mathematical modeling of such an _obstacle problem for a linearly elastic
shell_ is then clear; since, _apart from_ the confinement condition, the rest,
i.e., the _function space_ and the expression of the quadratic _energy_
$J^{\varepsilon}$, is classical (see, e.g. [8]). More specifically, let

$A^{ijk\ell,\varepsilon}:=\lambda
g^{ij,\varepsilon}g^{k\ell,\varepsilon}+\mu\left(g^{ik,\varepsilon}g^{j\ell,\varepsilon}+g^{i\ell,\varepsilon}g^{jk,\varepsilon}\right)=A^{jik\ell,\varepsilon}=A^{k\ell
ij,\varepsilon},$

denote the contravariant components of the _elasticity tensor_ of the elastic
material constituting the shell. Then the unknown of the problem, which is the
vector field $\bm{u}^{\varepsilon}=(u^{\varepsilon}_{i})$ where the functions
$u^{\varepsilon}_{i}:\overline{\Omega^{\varepsilon}}\to\mathbb{R}$ are the
three covariant components of the unknown “three-dimensional” displacement
vector field $u^{\varepsilon}_{i}\bm{g}^{i,\varepsilon}$ of the reference
configuration of the shell, should minimize the _energy_
$J^{\varepsilon}:\bm{H}^{1}(\Omega^{\varepsilon})\to\mathbb{R}$ defined by

$J^{\varepsilon}(\bm{v}^{\varepsilon}):=\frac{1}{2}\int_{\Omega^{\varepsilon}}A^{ijk\ell,\varepsilon}e^{\varepsilon}_{k\|\ell}(\bm{v}^{\varepsilon})e^{\varepsilon}_{i\|j}(\bm{v}^{\varepsilon})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}-\int_{\Omega^{\varepsilon}}f^{i,\varepsilon}v^{\varepsilon}_{i}\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon},$

for each
$\bm{v}^{\varepsilon}=(v^{\varepsilon}_{i})\in\bm{H}^{1}(\Omega^{\varepsilon})$
over the _set of admissible displacements_ defined by:

$\bm{U}(\Omega^{\varepsilon}):=\\{\bm{v}^{\varepsilon}=(v^{\varepsilon}_{i})\in\bm{H}^{1}(\Omega^{\varepsilon});\;\bm{v}^{\varepsilon}=\textbf{0}\text{
on
}\Gamma^{\varepsilon}_{0},(\bm{\Theta}(x^{\varepsilon})+v^{\varepsilon}_{i}(x^{\varepsilon})\bm{g}^{i,\varepsilon}(x^{\varepsilon}))\cdot\bm{q}\geq
0\textup{ for a.a. }x^{\varepsilon}\in\Omega^{\varepsilon}\\}.$

The solution to this _minimization problem_ exists and is unique, and it can
be also characterized as the unique solution of the following problem:

######  Problem $\mathcal{P}(\Omega^{\varepsilon})$.

Find
$\bm{u}^{\varepsilon}=(u_{i}^{\varepsilon})\in\bm{U}(\Omega^{\varepsilon})$
that satisfies the following variational inequalities:

$\int_{\Omega^{\varepsilon}}A^{ijk\ell,\varepsilon}e^{\varepsilon}_{k\|\ell}(\bm{u}^{\varepsilon})\left(e^{\varepsilon}_{i\|j}(\bm{v}^{\varepsilon})-e^{\varepsilon}_{i\|j}(\bm{u}^{\varepsilon})\right)\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}\geq\int_{\Omega^{\varepsilon}}f^{i,\varepsilon}(v^{\varepsilon}_{i}-u^{\varepsilon}_{i})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon},$

for all
$\bm{v}^{\varepsilon}=(v^{\varepsilon}_{i})\in\bm{U}(\Omega^{\varepsilon})$.
$\blacksquare$

The following result can be thus straightforwardly proved.

######  Theorem 3.1.

The quadratic minimization problem: Find a vector field
$\bm{u}^{\varepsilon}\in\bm{U}(\Omega^{\varepsilon})$ such that

$J^{\varepsilon}(\bm{u}^{\varepsilon})=\inf_{\bm{v}^{\varepsilon}\in\bm{U}(\Omega^{\varepsilon})}J^{\varepsilon}(\bm{v}^{\varepsilon}),$

has one and only one solution. Besides, $\bm{u}^{\varepsilon}$ is also the
unique solution of Problem $\mathcal{P}(\Omega^{\varepsilon})$.

###### Proof.

Define the space

$\bm{V}(\Omega^{\varepsilon}):=\\{\bm{v}^{\varepsilon}=(v^{\varepsilon}_{i})\in\bm{H}^{1}(\Omega^{\varepsilon});\;\bm{v}^{\varepsilon}=\textbf{0}\text{
on }\Gamma^{\varepsilon}_{0}\\}.$

Then, thanks to the uniform positive-definiteness of the elasticity tensor
$(A^{ijk\ell,\varepsilon})$ [6], and to the boundary condition of place
satisfied on
$\Gamma^{\varepsilon}_{0}=\gamma_{0}\times\left[-\varepsilon,\varepsilon\right]$
(recall that $\lambda\geq 0,\,\mu>0$, and that $\gamma_{0}$ is a non-empty
relatively open subset of $\gamma=\partial\omega$), it can be shown (see
Theorems 3.8-3 and 3.9-1 of [9]) that the continuous and symmetric bilinear
form

$(\bm{v}^{\varepsilon},\bm{w}^{\varepsilon})\in\bm{H}^{1}(\Omega^{\varepsilon})\times\bm{H}^{1}(\Omega^{\varepsilon})\mapsto\int_{\Omega^{\varepsilon}}A^{ijk\ell,\varepsilon}e^{\varepsilon}_{k\|\ell}(\bm{v}^{\varepsilon})e^{\varepsilon}_{i\|j}(\bm{w}^{\varepsilon})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon},$

is $\bm{V}(\Omega^{\varepsilon})$-elliptic; besides, the linear form

$\bm{v}^{\varepsilon}\in\bm{H}^{1}(\Omega^{\varepsilon})\mapsto\int_{\Omega^{\varepsilon}}f^{i,\varepsilon}v^{\varepsilon}_{i}\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon},$

is clearly continuous. Finally, the set $\bm{U}(\Omega^{\varepsilon})$ is
nonempty (by assumption), closed in $\bm{H}^{1}(\Omega^{\varepsilon})$ (any
convergent sequence in $\bm{V}(\Omega^{\varepsilon})$ contains a subsequence
that pointwise converges almost everywhere to its limit), and convex (as is
immediately verified).

The existence and uniqueness of the solution to the minimization problem and
its characterization by means of variational inequalities is then classical
(see, e.g., [10], [21] or [22]). ∎

Since
$\bm{\theta}(\overline{\omega})\subset\bm{\Theta}(\overline{\Omega^{\varepsilon}})$,
it evidently follows that $\bm{\theta}(y)\cdot\bm{q}\geq 0$ for all
$y\in\overline{\omega}$. But in fact, a stronger property holds (cf. Lemma 2.1
of [20]).

######  Lemma 3.1.

Let $\omega$ be a domain in $\mathbb{R}^{2}$, let
$\bm{\theta}\in\mathcal{C}^{1}(\overline{\omega};\mathbb{E}^{3})$ be an
immersion, let $\bm{q}\in\mathbb{E}^{3}$ be a non-zero vector, and let
$\varepsilon>0$. Then the inclusion

$\bm{\Theta}(\overline{\Omega^{\varepsilon}})\subset\mathbb{H}=\\{x\in\mathbb{E}^{3};\;\bm{Ox}\cdot\bm{q}\geq
0\\}$

implies that

$\inf_{y\in\overline{\omega}}(\bm{\theta}(y)\cdot\bm{q})>0.$

∎

Clearly, the assumed inclusion
$\bm{\Theta}(\overline{\Omega^{\varepsilon}})\subset\mathbb{H}$ implies that
$\inf_{y\in\overline{\omega}}(\bm{\theta}(y)\cdot\bm{q})$ depends on
$\varepsilon>0$ and approaches zero as $\varepsilon$ approaches zero.

We now consider the “penalized” version of Problem
$\mathcal{P}(\Omega^{\varepsilon})$. One such penalization transforms the set
of variational inequalities in Problem $\mathcal{P}(\Omega^{\varepsilon})$
into a set of nonlinear variational equations, where the nonlinearity is
defined in terms of the measure of the “violation” of the constraint. Let
$\kappa>0$ denote a _penalty parameter_. The “penalized” variational
formulation corresponding to Problem $\mathcal{P}(\Omega^{\varepsilon})$ takes
the following form:

######  Problem $\mathcal{P}_{\kappa}(\Omega^{\varepsilon})$.

Find
$\bm{u}_{\kappa}^{\varepsilon}=(u_{i,\kappa}^{\varepsilon})\in\bm{V}(\Omega^{\varepsilon})$
that satisfies the following variational equations:

$\displaystyle\int_{\Omega^{\varepsilon}}A^{ijk\ell,\varepsilon}e^{\varepsilon}_{k\|\ell}(\bm{u}_{\kappa}^{\varepsilon})e^{\varepsilon}_{i\|j}(\bm{v}^{\varepsilon})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}-\dfrac{\varepsilon}{\kappa}\int_{\Omega^{\varepsilon}}\left\\{[\bm{\Theta}+u_{i,\kappa}^{\varepsilon}\bm{g}^{i,\varepsilon}]\cdot\bm{q}\right\\}^{-}(v_{i}^{\varepsilon}\bm{g}^{i,\varepsilon}\cdot\bm{q})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}=\int_{\Omega^{\varepsilon}}f^{i,\varepsilon}v^{\varepsilon}_{i}\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon},$

for all
$\bm{v}^{\varepsilon}=(v^{\varepsilon}_{i})\in\bm{V}(\Omega^{\varepsilon})$.
$\blacksquare$

In order to show the existence and uniqueness of solution for Problem
$\mathcal{P}_{\kappa}(\Omega^{\varepsilon})$, we have to show that the
negative part operator is monotone.

######  Lemma 3.2.

Let $\Omega\subset\mathbb{R}^{n}$, with $n\geq 1$ an integer, be an open set.
The operator $-\\{\cdot\\}:L^{2}(\Omega)\to L^{2}(\Omega)$ defined by

$f\in L^{2}(\Omega)\mapsto-\\{f\\}^{-}:=\min\\{f,0\\}\in L^{2}(\Omega),$

is monotone, bounded and Lipschitz continuous with Lipschitz constant equal to
$1$.

###### Proof.

Let $f$ and $g$ be arbitrarily given in $L^{2}(\Omega)$. In what follows, sets
of the form $\\{f\geq 0\\}$ read

$\\{x\in\Omega;f(x)\geq 0\\}.$

Recall that the negative part of a function is also given by

$f^{-}=\dfrac{|f|-f}{2}.$

We have that

$\displaystyle\int_{\Omega}\left\\{(-\\{f\\}^{-})-(-\\{g\\}^{-})\right\\}(f-g)\,\mathrm{d}x$
$\displaystyle\geq\int_{\Omega}|-\\{f\\}^{-}|^{2}\,\mathrm{d}x+\int_{\Omega}|-\\{g\\}^{-}|^{2}\,\mathrm{d}x-2\int_{\\{f\leq
0\\}\cap\\{g\leq 0\\}}(-\\{f\\}^{-})(-\\{g\\}^{-})\,\mathrm{d}x$
$\displaystyle\geq\int_{\\{f\leq 0\\}\cap\\{g\leq
0\\}}\left|(-\\{f\\}^{-})-(-\\{g\\}^{-})\right|^{2}\,\mathrm{d}x\geq 0,$

and the monotonicity is thus established.

Let us show that the operator under consideration is bounded, in the sense
that maps bounded sets onto bounded sets. Let $\mathscr{F}\subset
L^{2}(\Omega)$ be bounded. By Hölder’s inequality, we have that for each
$f\in\mathscr{F}$,

$\sup_{\begin{subarray}{c}v\in L^{2}(\Omega)\\\ v\neq
0\end{subarray}}\dfrac{\left|\int_{\Omega}(-\\{f\\}^{-})v\,\mathrm{d}x\right|}{\|v\|_{L^{2}(\Omega)}}\leq\|f\|_{L^{2}(\Omega)},$

and the boundedness of $\mathscr{F}$ implies the boundedness of the supremum.
The boundedness property is thus established.

Finally, to establish the Lipschitz continuity property, evaluate

$\displaystyle\left(\int_{\Omega}|(-\\{f\\}^{-})-(-\\{g\\}^{-})|^{2}\,\mathrm{d}x\right)^{1/2}=\left(\int_{\Omega}\left|\dfrac{f-|f|}{2}-\dfrac{g-|g|}{2}\right|^{2}\,\mathrm{d}x\right)^{1/2}$
$\displaystyle=\dfrac{1}{2}\left(\int_{\Omega}|(f-g)-(|f|-|g|)|^{2}\,\mathrm{d}x\right)^{1/2}\leq\dfrac{1}{2}\\{\|f-g\|_{L^{2}(\Omega)}+\||f|-|g|\|_{L^{2}(\Omega)}\\}$
$\displaystyle=\dfrac{1}{2}\|f-g\|_{L^{2}(\Omega)}+\dfrac{1}{2}\left(\int_{\Omega}\left||f|-|g|\right|^{2}\,\mathrm{d}x\right)^{1/2}\leq\dfrac{1}{2}\|f-g\|_{L^{2}(\Omega)}+\dfrac{1}{2}\left(\int_{\Omega}|f-g|^{2}\,\mathrm{d}x\right)^{1/2}$
$\displaystyle=\|f-g\|_{L^{2}(\Omega)},$

where the inequality holds thanks to the Minkowski inequality. In conclusion,
we have shown that

$\|(-\\{f\\}^{-})-(-\\{g\\}^{-})\|_{L^{2}(\Omega)}\leq\|f-g\|_{L^{2}(\Omega)},$

and the arbitrariness of $f$ and $g$ gives the desired Lipschitz continuity
property. This completes the proof. ∎

The existence and uniqueness of the solution for Problem
$\mathcal{P}_{\kappa}(\Omega^{\varepsilon})$ is classical too (cf., e.g.,
Theorem 3.15 in [23], or [26]).

######  Theorem 3.2.

Problem $\mathcal{P}_{\kappa}(\Omega^{\varepsilon})$ has a unique solution.

###### Proof.

Observe that the linear form

$\bm{v}^{\varepsilon}\in\bm{H}^{1}(\Omega^{\varepsilon})\mapsto\int_{\Omega^{\varepsilon}}f^{i,\varepsilon}v^{\varepsilon}_{i}\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon},$

is continuous (cf. Theorem 3.1). The mapping
$A:\bm{V}(\Omega^{\varepsilon})\to\bm{V}^{\prime}(\Omega^{\varepsilon})$
defined by

$\langle
A\bm{u},\bm{v}\rangle_{\bm{V}^{\prime}(\Omega^{\varepsilon}),\bm{V}(\Omega^{\varepsilon})}:=\int_{\Omega^{\varepsilon}}A^{ijk\ell,\varepsilon}e^{\varepsilon}_{k\|\ell}(\bm{u})e^{\varepsilon}_{i\|j}(\bm{v})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}-\dfrac{\varepsilon}{\kappa}\int_{\Omega^{\varepsilon}}\left\\{[\bm{\Theta}+u_{i}\bm{g}^{i,\varepsilon}]\cdot\bm{q}\right\\}^{-}(v_{i}\bm{g}^{i,\varepsilon}\cdot\bm{q})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon},$

is continuous and, thanks to an _ad hoc_ Korn’s inequality in curvilinear
coordinates (cf., e.g., Theorem 1.7-4 of [8]) and Lemma 3.2, strictly monotone
and coercive. Therefore, the conclusion follows by the Minty-Browder theorem
(cf., e.g., Theorem 9.14-1 of [10]). ∎

By means of a different classical approach based on energy estimates [26, 37]
(see also part (iv) of Theorem 9.14-1 of [10]), one can prove Theorem 3.2 by
establishing that for each $\varepsilon>0$

(1)
$\bm{u}^{\varepsilon}_{\kappa}\rightharpoonup\bm{u}^{\varepsilon},\quad\textup{
in }\bm{V}(\Omega^{\varepsilon}),$

and that, thanks to the monotonicity of the penalty term established in Lemma
3.2, the weak limit satisfies Problem $\mathcal{P}(\Omega^{\varepsilon})$. As
a result of the convergence (1) and the continuity of the linear operator
$e^{\varepsilon}_{i\|j}:\bm{H}^{1}(\Omega^{\varepsilon})\to
L^{2}(\Omega^{\varepsilon})$, we have that

$e^{\varepsilon}_{i\|j}(\bm{u}^{\varepsilon}_{\kappa})\rightharpoonup
e^{\varepsilon}_{i\|j}(\bm{u}^{\varepsilon}),\quad\textup{ in
}L^{2}(\Omega^{\varepsilon}).$

Actually, a stronger conclusion holds: Given $\varepsilon>0$, the following
_strong_ convergence holds

$\bm{u}^{\varepsilon}_{\kappa}\to\bm{u}^{\varepsilon},\quad\textup{ in
}\bm{V}(\Omega^{\varepsilon}).$

To see this, observe that the uniform positive-definiteness of the elasticity
tensor $(A^{ijk\ell,\varepsilon})$ (cf., e.g., [6]), Korn’s inequality (viz.
Theorem 1.7-4 of [8]), the monotonicity of the penalty term (Lemma 3.2), the
continuity of the operators $e_{i\|j}^{\varepsilon}$, and the weak convergence
(1) give

$\displaystyle\|\bm{u}^{\varepsilon}_{\kappa}-\bm{u}^{\varepsilon}\|_{\bm{H}^{1}(\Omega^{\varepsilon})}$
$\displaystyle\leq\dfrac{1}{C_{0}}\int_{\Omega^{\varepsilon}}A^{ijk\ell,\varepsilon}e^{\varepsilon}_{k\|\ell}(\bm{u}^{\varepsilon}_{\kappa}-\bm{u}^{\varepsilon})e^{\varepsilon}_{i\|j}(\bm{u}^{\varepsilon}_{\kappa}-\bm{u}^{\varepsilon})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}$
$\displaystyle=\dfrac{\varepsilon}{C_{0}\kappa}\int_{\Omega^{\varepsilon}}\left\\{[\bm{\Theta}+u^{\varepsilon}_{i,\kappa}\bm{g}^{i,\varepsilon}]\cdot\bm{q}\right\\}^{-}((u^{\varepsilon}_{i,\kappa}-u^{\varepsilon}_{i})\bm{g}^{i,\varepsilon}\cdot\bm{q})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}$
$\displaystyle\quad+\dfrac{1}{C_{0}}\int_{\Omega^{\varepsilon}}f^{i,\varepsilon}(u^{\varepsilon}_{i,\kappa}-u^{\varepsilon}_{i})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}$
$\displaystyle\quad-\dfrac{1}{C_{0}}\int_{\Omega^{\varepsilon}}A^{ijk\ell,\varepsilon}e^{\varepsilon}_{k\|\ell}(\bm{u}^{\varepsilon})e^{\varepsilon}_{i\|j}(\bm{u}^{\varepsilon}_{\kappa}-\bm{u}^{\varepsilon})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}$
$\displaystyle\leq\dfrac{1}{C_{0}}\int_{\Omega^{\varepsilon}}f^{i,\varepsilon}(u^{\varepsilon}_{i,\kappa}-u^{\varepsilon}_{i})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}$
$\displaystyle\quad-\dfrac{1}{C_{0}}\int_{\Omega^{\varepsilon}}A^{ijk\ell,\varepsilon}e^{\varepsilon}_{k\|\ell}(\bm{u}^{\varepsilon})e^{\varepsilon}_{i\|j}(\bm{u}^{\varepsilon}_{\kappa}-\bm{u}^{\varepsilon})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}\to
0,$

as $\kappa\to 0$. In particular, we have that for each $\varepsilon>0$ and for
each $\delta>0$, we can find a number
$\kappa_{0}=\kappa_{0}(\delta,\varepsilon)>0$ such that, for each
$0<\kappa<\kappa_{0}$, it results

(2)
$\|\bm{u}^{\varepsilon}-\bm{u}^{\varepsilon}_{\kappa}\|_{\bm{H}^{1}(\Omega^{\varepsilon})}<\dfrac{\delta}{2},$

for all $\varepsilon>0$, where $\bm{u}^{\varepsilon}$ and
$\bm{u}^{\varepsilon}_{\kappa}$ respectively denote the solutions of Problem
$\mathcal{P}(\Omega^{\varepsilon})$ and Problem
$\mathcal{P}_{\kappa}(\Omega^{\varepsilon})$.

In all what follows we make the assumption that $\kappa_{0}$ is independent of
$\varepsilon$. This kind of property was rephrased in a different context in
the paper [20] where, in the remark following Lemma 2.1, the authors assumed
the quantity $\inf_{y\in\overline{\omega}}(\bm{\theta}(y)\cdot\bm{q})$ was
independent of the thickness $\varepsilon$. Note that the assumption that
$\inf_{y\in\overline{\omega}}(\bm{\theta}(y)\cdot\bm{q})$ was independent of
the thickness $\varepsilon$ was not made in this paper.

##  4\. The scaled three-dimensional problem for a family of flexural shells

In section 3, we considered an obstacle problem for “general” linearly elastic
shells. From now on, we will restrict ourselves to a specific class of shells,
according to the following definition (proposed in [18]; see also [8]).

Consider a linearly elastic shell, subjected to the various assumptions set
forth in section 3. Such a shell is said to be a _linearly elastic flexural
shell_ if the following two additional assumptions are satisfied: _first_ ,
$\emptyset\neq\gamma_{0}\subset\gamma$, i.e., the homogeneous boundary
condition of place is imposed over a _nonzero area portion of the entire
lateral face_ $\gamma_{0}\times\left[-\varepsilon,\varepsilon\right]$ of the
shell, and _second_ , the space

$\displaystyle\bm{V}_{F}(\omega):=\\{\bm{\eta}=(\eta_{i})\in
H^{1}(\omega)\times H^{1}(\omega)\times
H^{2}(\omega);\gamma_{\alpha\beta}(\bm{\eta})=0\textup{ in }\omega\textup{ and
}\eta_{i}=\partial_{\nu}\eta_{3}=0\textup{ on }\gamma_{0}\\}$

contains nonzero functions, i.e., $\bm{V}_{F}(\omega)\neq\\{\bf 0\\}$.

In this paper, we consider the _obstacle problem_ as defined in section 3 _for
a family of linearly elastic flexural shells_ , all sharing the _same middle
surface_ and whose thickness $2\varepsilon>0$ is considered as a “small”
parameter approaching zero. Our objective then consists in performing an
_asymptotic analysis as_ $\varepsilon\to 0$, so as to seek whether we can
identify a _limit two-dimensional problem_. To this end, we shall resort to a
(by now standard) methodology first proposed by Ciarlet, Lods and Miara (cf.
Theorem 5.1 of [18] and Theorem 6.2-1 of [8]): To begin with, we “scale” each
problem $\mathcal{P}(\Omega^{\varepsilon}),\,\varepsilon>0$, over a _fixed
domain_ $\Omega$, using appropriate _scalings on the unknowns_ and
_assumptions on the data_. Note that these scalings and assumptions definitely
_depend_ on the type of shells that are considered; for instance, those used
for the _linearly elastic elliptic membrane shells_ considered elsewhere (cf.
[19] and also [20]) are different.

More specifically, let

$\Omega:=\omega\times\left]-1,1\right[,$

let $x=(x_{i})$ denote a generic point in the set $\overline{\Omega}$, and let
$\partial_{i}:=\partial/\partial x_{i}$. With each point
$x=(x_{i})\in\overline{\Omega}$, we associate the point
$x^{\varepsilon}=(x^{\varepsilon}_{i})$ defined by

$x^{\varepsilon}_{\alpha}:=x_{\alpha}=y_{\alpha}\quad\text{ and }\quad
x^{\varepsilon}_{3}:=\varepsilon x_{3},$

so that $\partial^{\varepsilon}_{\alpha}=\partial_{\alpha}$ and
$\partial^{\varepsilon}_{3}=\displaystyle\frac{1}{\varepsilon}\partial_{3}$.
To the unknown $\bm{u}^{\varepsilon}=(u^{\varepsilon}_{i})$ and to the vector
fields $\bm{v}^{\varepsilon}=(v^{\varepsilon}_{i})$ appearing in the
formulation of Problem $\mathcal{P}(\Omega^{\varepsilon})$ corresponding to a
linearly elastic flexural shell, we then associate the _scaled unknown_
$\bm{u}(\varepsilon)=(u_{i}(\varepsilon))$ and the _scaled vector fields_
$\bm{v}=(v_{i})$ by letting

$u_{i}(\varepsilon)(x):=u^{\varepsilon}_{i}(x^{\varepsilon})\text{ and
}v_{i}(x):=v^{\varepsilon}_{i}(x^{\varepsilon}),$

at each $x\in\overline{\Omega}$. Finally, we _assume_ that there exist
functions $f^{i}\in L^{2}(\Omega)$ _independent on_ $\varepsilon$ such that
the following _assumptions on the data_ hold:

$f^{i,\varepsilon}(x^{\varepsilon})=\varepsilon^{2}f^{i}(x)\text{ at each
}x\in\Omega.$

Note that the independence on $\varepsilon$ of the Lamé constants assumed in
section 3 in the formulation of Problem $\mathcal{P}(\Omega^{\varepsilon})$
implicitly constituted another _assumption on the data_.

In view of the proposed scaling, we define the “scaled” version of the
geometrical entities introduced in section 2:

$\displaystyle\bm{g}^{i}(\varepsilon)(x)$
$\displaystyle:=\bm{g}^{i,\varepsilon}(x^{\varepsilon})\textup{ at each
}x\in\overline{\Omega},$ $\displaystyle g(\varepsilon)(x)$
$\displaystyle:=g^{\varepsilon}(x^{\varepsilon})\text{ and
}A^{ijk\ell}(\varepsilon)(x):=A^{ijk\ell,\varepsilon}(x^{\varepsilon})\text{
at each }x\in\overline{\Omega},$ $\displaystyle
e_{\alpha\|\beta}(\varepsilon;\bm{v})$
$\displaystyle:=\frac{1}{2}(\partial_{\beta}v_{\alpha}+\partial_{\alpha}v_{\beta})-\Gamma^{k}_{\alpha\beta}(\varepsilon)v_{k}=e_{\beta\|\alpha}(\varepsilon;\bm{v}),$
$\displaystyle
e_{\alpha\|3}(\varepsilon;\bm{v})=e_{3\|\alpha}(\varepsilon;\bm{v})$
$\displaystyle:=\frac{1}{2}\left(\frac{1}{\varepsilon}\partial_{3}v_{\alpha}+\partial_{\alpha}v_{3}\right)-\Gamma^{\sigma}_{\alpha
3}(\varepsilon)v_{\sigma},$ $\displaystyle e_{3\|3}(\varepsilon;\bm{v})$
$\displaystyle:=\frac{1}{\varepsilon}\partial_{3}v_{3},$

where

$\Gamma^{p}_{ij}(\varepsilon)(x):=\Gamma^{p,\varepsilon}_{ij}(x^{\varepsilon})\textup{
at each }x\in\overline{\Omega}.$

Define the space

$\bm{V}(\Omega):=\\{\bm{v}=(v_{i})\in\bm{H}^{1}(\Omega);\;\bm{v}=\textbf{0}\textup{
on }\gamma_{0}\times\left[-1,1\right]\\},$

and define, for each $\varepsilon>0$, the set

$\displaystyle\bm{U}(\varepsilon;\Omega)$
$\displaystyle:=\\{\bm{v}=(v_{i})\in\bm{V}(\Omega);\big{(}\bm{\theta}(y)+\varepsilon
x_{3}\bm{a}_{3}(y)+v_{i}(x)\bm{g}^{i}(\varepsilon)(x)\big{)}\cdot\bm{q}\geq
0\textup{ for a.a. }x=(y,x_{3})\in\Omega\\}.$

We are thus in a position to introduce the “scaled” version of Problem
$\mathcal{P}(\Omega^{\varepsilon})$, that will be denoted in what follows by
$\mathcal{P}(\varepsilon;\Omega)$.

######  Problem $\mathcal{P}(\varepsilon;\Omega)$.

Find $\bm{u}(\varepsilon)=(u_{i}(\varepsilon))\in\bm{U}(\varepsilon;\Omega)$
that satisfies the following variational inequalities:

$\displaystyle\int_{\Omega}A^{ijk\ell}(\varepsilon)e_{k\|\ell}(\varepsilon;\bm{u}(\varepsilon))\left(e_{i\|j}(\varepsilon;\bm{v})-e_{i\|j}(\varepsilon;\bm{u}(\varepsilon))\right)\sqrt{g(\varepsilon)}\,\mathrm{d}x\geq\varepsilon^{2}\int_{\Omega}f^{i}(v_{i}-u_{i}(\varepsilon))\sqrt{g(\varepsilon)}\,\mathrm{d}x,$

for all $\bm{v}=(v_{i})\in\bm{U}(\varepsilon;\Omega)$. $\blacksquare$

######  Theorem 4.1.

The scaled unknown $\bm{u}(\varepsilon)$ is the unique solution of the
variational Problem $\mathcal{P}(\varepsilon;\Omega)$.

###### Proof.

The variational Problem $\mathcal{P}(\varepsilon;\Omega)$ simply constitutes a
re-writing of the variational Problem $\mathcal{P}(\Omega^{\varepsilon})$,
this time in terms of the scaled unknown $\bm{u}(\varepsilon)$, of the vector
fields $\bm{v}$, and of the functions $f^{i}$, which are now all defined over
the domain ${\Omega}$. Then the assertion follows from this observation. ∎

The functions $e_{i\|j}(\varepsilon;\bm{v})$ appearing in Problem
$\mathcal{P}(\varepsilon;\Omega)$ are called the _scaled linearized strains in
curvilinear coordinates_ associated with the scaled displacement vector field
$v_{i}\bm{g}^{i}(\varepsilon)$.

For later purposes (like in Lemma 4.1 below), we also let

$\bm{g}_{i}(\varepsilon)(x):=g^{\varepsilon}_{i}(x^{\varepsilon})\textup{ at
each }x\in\overline{\Omega}.$

Likewise, one can introduce the “scaled” version of Problem
$\mathcal{P}_{\kappa}(\Omega^{\varepsilon})$, that will be denoted in what
follows by $\mathcal{P}_{\kappa}(\varepsilon;\Omega)$.

######  Problem $\mathcal{P}_{\kappa}(\varepsilon;\Omega)$.

Find
$\bm{u}_{\kappa}(\varepsilon)=(u_{i,\kappa}(\varepsilon))\in\bm{V}(\Omega)$
that satisfies the following variational equations:

$\displaystyle\int_{\Omega}A^{ijk\ell}(\varepsilon)e_{k\|\ell}(\varepsilon;\bm{u}_{\kappa}(\varepsilon))e_{i\|j}(\varepsilon;\bm{v})\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad-\dfrac{\varepsilon}{\kappa}\int_{\Omega}\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}(v_{i}\bm{g}^{i}(\varepsilon)\cdot\bm{q})\sqrt{g(\varepsilon)}\,\mathrm{d}x=\varepsilon^{2}\int_{\Omega}f^{i}v_{i}\sqrt{g(\varepsilon)}\,\mathrm{d}x,$

for all $\bm{v}=(v_{i})\in\bm{V}(\Omega)$. $\blacksquare$

The following existence and uniqueness result can be thus easily proved in the
same fashion as Theorem 3.2.

######  Theorem 4.2.

The scaled unknown $\bm{u}_{\kappa}(\varepsilon)$ is the unique solution of
the variational Problem $\mathcal{P}_{\kappa}(\varepsilon;\Omega)$.

###### Proof.

The variational Problem $\mathcal{P}_{\kappa}(\varepsilon;\Omega)$ simply
constitutes a re-writing of the variational Problem
$\mathcal{P}_{\kappa}(\Omega^{\varepsilon})$, this time in terms of the scaled
unknown $\bm{u}_{\kappa}(\varepsilon)$, of the vector fields $\bm{v}$, and of
the functions $f^{i}$, which are now all defined over the domain $\Omega$.
Then the assertion follows from this observation. ∎

By means of an analogous reasoning, a condition similar to (2) can be derived,
i.e., for each $\delta>0$ we can find a number
$\kappa_{0}=\kappa_{0}(\delta)>0$ (recall that we assumed $\kappa_{0}$ to be
independent of $\varepsilon$) such that, for each $0<\kappa<\kappa_{0}$, it
results

(3)
$\|\bm{u}(\varepsilon)-\bm{u}_{\kappa}(\varepsilon)\|_{\bm{H}^{1}(\Omega)}<\dfrac{\delta}{2},$

for each $\varepsilon>0$, where $\bm{u}(\varepsilon)$ and
$\bm{u}_{\kappa}(\varepsilon)$ respectively denote the solutions of Problem
$\mathcal{P}(\varepsilon;\Omega)$ and Problem
$\mathcal{P}_{\kappa}(\varepsilon;\Omega)$.

Without loss of generality (cf., e.g., [37]), given any $\varepsilon>0$, we
restrict ourselves to considering penalty parameters with the following
property:

(4) $0<\kappa=\sqrt{\varepsilon}.$

It is straightforward to observe that $\kappa\to 0$ as $\varepsilon\to 0$. We
observe that the variational Problem $\mathcal{P}(\varepsilon;\Omega)$ could
have been equivalently written as a _minimization problem_ , thus mimicking
that found in Theorem 3.1. It turns out, however, that its formulation in
Theorem 4.2 as a set of _penalized variational equations_ is more convenient
for the asymptotic analysis undertaken in section 5.

It is immediately verified (cf., e.g., [8]) that _other_ assumptions on the
data are possible that would give rise to the _same_ problem over the fixed
domain $\Omega$. For instance, should the Lamé constants (now denoted)
$\lambda^{\varepsilon}$ and $\mu^{\varepsilon}$ appearing in Problem
$\mathcal{P}(\Omega^{\varepsilon})$ be of the form
$\lambda^{\varepsilon}=\varepsilon^{t}\lambda$ and
$\mu^{\varepsilon}=\varepsilon^{t}\mu$, where $\lambda\geq 0$ and $\mu$ are
constants independent of $\varepsilon$ and $t$ is an arbitrary real number,
the _same_ Problem $\mathcal{P}(\varepsilon;\Omega)$ arises if we assume that
the components of the applied body force density are now of the form

$f^{i,\varepsilon}(x^{\varepsilon})=\varepsilon^{2+t}f^{i}(x)\text{ at each
}x\in\Omega,$

where the functions $f^{i}\in L^{2}(\Omega)$ are independent of $\varepsilon$.

The next lemma assembles various asymptotic properties as $\varepsilon\to 0$
of functions and vector fields appearing in the formulation of Problem
$\mathcal{P}(\varepsilon;\Omega)$; these properties will be repeatedly used in
the proof of the convergence theorem (Theorem 5.1).

In the statement of the next preparatory lemma (cf., e.g., Theorems 3.3-1 and
3.3-2 of [8]), the notation “$O(\varepsilon)$”, or “$O(\varepsilon^{2})$”,
stands for a remainder that is of order $\varepsilon$, or $\varepsilon^{2}$,
with respect to the sup-norm over the set $\overline{\Omega}$, and any
function, or vector-valued function, of the variable $y\in\overline{\omega}$,
such as $a^{\alpha\beta},b_{\alpha\beta},\bm{a}^{i}$, etc. (all these are
defined in section 2) is identified with the function, or vector-valued
function, of
$x=(y,x_{3})\in\overline{\Omega}=\overline{\omega}\times\left[-1,1\right]$
that takes the same value at $x_{3}=0$ and is independent of
$x_{3}\in\left[-1,1\right]$; for brevity, this extension from
$\overline{\omega}$ to $\overline{\Omega}$ is designated with the same
notation.

Recall that $\varepsilon>0$ is implicitly assumed to be small enough so that
$\bm{\Theta}:\overline{\Omega^{\varepsilon}}\to\mathbb{E}^{3}$ is an
immersion.

######  Lemma 4.1.

Let $\bm{\theta}\in\mathcal{C}^{3}(\overline{\omega};\mathbb{R}^{3})$ be an
immersion. Let $\varepsilon_{0}$ be defined as in Theorem 3.1-1 of [8]. The
functions $A^{ijk\ell}(\varepsilon)=A^{jik\ell}(\varepsilon)=A^{k\ell
ij}(\varepsilon)$ have the following properties:

$A^{ijk\ell}(\varepsilon)=A^{ijk\ell}(0)+O(\varepsilon),\quad
A^{\alpha\beta\sigma 3}(\varepsilon)=A^{\alpha 333}(\varepsilon)=0,$

for all $0<\varepsilon\leq\varepsilon_{0}$, where

$\displaystyle A^{\alpha\beta\sigma\tau}(0)$ $\displaystyle=\lambda
a^{\alpha\beta}a^{\sigma\tau}+\mu(a^{\alpha\sigma}a^{\beta\tau}+a^{\alpha\tau}a^{\beta\sigma}),$
$\displaystyle A^{\alpha\beta 33}(0)$ $\displaystyle=\lambda
a^{\alpha\beta},\quad A^{\alpha 3\sigma 3}(0)=\mu a^{\alpha\sigma},\quad
A^{3333}(0)=\lambda+2\mu,$

and there exists a constant $C_{0}>0$ such that

$\sum_{i,j}\left|t_{ij}\right|^{2}\leq
C_{0}A^{ijk\ell}(\varepsilon)(x)t_{k\ell}t_{ij}$

for all $0<\varepsilon\leq\varepsilon_{0}$, all $x\in\overline{\Omega}$, and
all symmetric matrices $(t_{ij})$.

The functions $\Gamma^{p}_{ij}(\varepsilon)$ and $g(\varepsilon)$ have the
following properties:

$\displaystyle\Gamma^{\sigma}_{\alpha\beta}(\varepsilon)$
$\displaystyle=\Gamma^{\sigma}_{\alpha\beta}-\varepsilon
x_{3}(\partial_{\alpha}b^{\sigma}_{\beta}+\Gamma^{\sigma}_{\alpha\tau}b^{\tau}_{\beta}-\Gamma^{\tau}_{\alpha\beta}b^{\sigma}_{\tau})+O(\varepsilon^{2}),$
$\displaystyle\Gamma^{3}_{\alpha\beta}(\varepsilon)$
$\displaystyle=b_{\alpha\beta}-\varepsilon
x_{3}b^{\sigma}_{\alpha}b_{\sigma\beta},\quad\partial_{3}\Gamma^{p}_{\alpha\beta}(\varepsilon)=O(\varepsilon),$
$\displaystyle\Gamma^{\sigma}_{\alpha 3}(\varepsilon)$
$\displaystyle=-b^{\sigma}_{\alpha}-\varepsilon
x_{3}b^{\tau}_{\alpha}b^{\sigma}_{\tau}+O(\varepsilon^{2}),\quad\Gamma^{3}_{\alpha
3}(\varepsilon)=\Gamma^{p}_{33}(\varepsilon)=0,$ $\displaystyle
g(\varepsilon)$ $\displaystyle=a+O(\varepsilon),$

for all $0<\varepsilon\leq\varepsilon_{0}$ and all $x\in\overline{\Omega}$. In
particular then, there exist constants $g_{0}$ and $g_{1}$ such that

$0<g_{0}\leq g(\varepsilon)(x)\leq g_{1}\textup{ for all
}0<\varepsilon\leq\varepsilon_{0}\textup{ and all }x\in\overline{\Omega}.$

The vector fields $\bm{g}_{i}(\varepsilon)$ and $\bm{g}^{j}(\varepsilon)$ have
the following properties:

$\displaystyle\bm{g}_{\alpha}(\varepsilon)$
$\displaystyle=\bm{a}_{\alpha}-\varepsilon
x_{3}b^{\sigma}_{\alpha}\bm{a}_{\sigma},\quad\bm{g}_{3}(\varepsilon)=\bm{a}_{3},$
$\displaystyle\bm{g}^{\alpha}(\varepsilon)$
$\displaystyle=\bm{a}^{\alpha}+\varepsilon
x_{3}b^{\alpha}_{\sigma}\bm{a}^{\sigma}+O(\varepsilon^{2}),\quad\bm{g}^{3}(\varepsilon)=\bm{a}^{3}.$

∎

We recall (cf., e.g., [8]), that the various relations and estimates in Lemma
4.1 hold in fact for _any_ family of linearly elastic shells, i.e.,
irrespective of whether these shells are flexural ones or not.

When one considers a family of linearly elastic _flexural shells_ whose
thickness $2\varepsilon$ approaches zero, a specific _Korn’s inequality in
curvilinear coordinates_ (cf., e.g., Theorem 4.1 of [18] or Theorem 5.3-1 of
[8]) holds over the _fixed_ domain $\Omega=\omega\times\left]-1,1\right[$,
according to the following theorem. That the constant $C_{1}$ that appears in
this inequality is _independent of_ $\varepsilon>0$ plays a key role in the
asymptotic analysis of such a family (see part (i) of the proof of Theorem
5.1).

######  Theorem 4.3.

Let $\bm{\theta}\in\mathcal{C}^{3}(\overline{\omega};\mathbb{R}^{3})$ be an
immersion. Let there be given a family of linearly elastic flexural shells
with the same middle surface $\bm{\theta}(\overline{\omega})$ and thickness
$2\varepsilon>0$. Define the space

$\bm{V}(\Omega):=\\{\bm{v}=(v_{i})\in\bm{H}^{1}(\Omega);\;\bm{v}=\bm{0}\textup{
on }\gamma_{0}\times\left[-1,1\right]\\}.$

Then there exist constants $\varepsilon_{1}>0$ and $C_{1}>0$ such that

$\left\\{\sum_{i}\left\|v_{i}\right\|^{2}_{H^{1}(\Omega)}\right\\}^{1/2}\leq\dfrac{C}{\varepsilon}\left\\{\sum_{i,j}\left\|e_{i\|j}(\varepsilon;\bm{v})\right\|^{2}_{L^{2}(\Omega)}\right\\}^{1/2}$

for all $0<\varepsilon\leq\varepsilon_{1}$ and all
$\bm{v}=(v_{i})\in\bm{V}(\Omega)$. ∎

##  5\. Rigorous asymptotic analysis

The ultimate goal of this paper is to show, in the same spirit as [18] (see
also Theorem 6.2-1 of [8]), that _the solutions $\bm{u}(\varepsilon)$ of the_
(scaled) _three-dimensional problems $\mathcal{P}(\Omega^{\varepsilon})$
converge - as $\varepsilon$ approaches zero - to the solution of a two-
dimensional problem_, denoted $\mathcal{P}_{F}(\omega)$ in what follows.

The proof proposed in [18] (see also Theorem 6.2-1 of [8]) resorts, however,
to the usage of a specific vector field that was first introduced by Miara and
Sanchez-Palencia [27]. This argument is, in general, not easy to generalize to
the context of variational inequalities, for which the test functions are
chosen in a nonempty, closed and convex subset of a certain space.

In order to overcome this difficulty, we first prove that, under the
assumption (4), _the solutions $\bm{u}_{\kappa}(\varepsilon)$ of_ Problem
$\mathcal{P}_{\kappa}(\varepsilon;\Omega)$ converge - as $\varepsilon$
approaches zero - to the solution of the variational problem
$\mathcal{P}_{F}(\omega)$.

Define the set

$\displaystyle\bm{U}_{F}(\omega)$
$\displaystyle:=\\{\bm{\eta}=(\eta_{i})\in\bm{V}_{F}(\omega);\big{(}\bm{\theta}(y)+\eta_{i}(y)\bm{a}^{i}(y)\big{)}\cdot\bm{q}\geq
0\textup{ for a.a. }y\in\omega\\},$

where the space $\bm{V}_{F}(\omega)$ has been defined in section 4. We observe
that the set $\bm{U}_{F}(\omega)$ is nonempty, closed and convex in the space
$\bm{V}_{F}(\omega)$. The vector fields $\bm{a}^{i}$ and the functions
$\Gamma^{\sigma}_{\alpha\beta},b_{\alpha\beta},a^{\alpha\beta},a$, and
$\gamma_{\alpha\beta}(\bm{\eta})$, have been defined in section 2. We are thus
in a position to define the two-dimensional problem $\mathcal{P}_{F}(\omega)$
as follows:

######  Problem $\mathcal{P}_{F}(\omega)$.

Find $\bm{\zeta}\in\bm{U}_{F}(\omega)$ that satisfies the following
variational inequalities:

$\dfrac{1}{3}\int_{\omega}a^{\alpha\beta\sigma\tau}\rho_{\sigma\tau}(\bm{\zeta})\rho_{\alpha\beta}(\bm{\eta}-\bm{\zeta})\sqrt{a}\,\mathrm{d}y\geq\int_{\omega}p^{i}(\eta_{i}-\zeta_{i})\sqrt{a}\,\mathrm{d}y,$

for all $\bm{\eta}=(\eta_{i})\in\bm{U}_{F}(\omega)$, where

$a^{\alpha\beta\sigma\tau}:=\frac{4\lambda\mu}{\lambda+2\mu}a^{\alpha\beta}a^{\sigma\tau}+2\mu\left(a^{\alpha\sigma}a^{\beta\tau}+a^{\alpha\tau}a^{\beta\sigma}\right)\textup{
and }p^{i}:=\int^{1}_{-1}f^{i}\,\mathrm{d}x_{3}.$

$\blacksquare$

In the same spirit as Theorem 3.1, it can be show that Problem
$\mathcal{P}_{F}(\omega)$ admits one and only one solution.

We are now ready to show that, under the assumption (4) according to which
$\kappa\leq\varepsilon$, _the solutions $\bm{u}_{\kappa}(\varepsilon)$ of_
Problem $\mathcal{P}_{\kappa}(\varepsilon;\Omega)$ converge - as $\varepsilon$
approaches zero - to the solution of Problem $\mathcal{P}_{F}(\omega)$.

Differently from the linearly elastic elliptic membrane case, we will see that
problems arise when one has to define an appropriate test vector field for
recovering the limit model, which is the main object of Theorem 5.1 (cf. part
(iv)). Since this test vector field will have to depend on the partial
derivatives of the displacement, it is hard to find an expression for it that
takes into account the geometrical constraint as well. For this reason, the
penalty method seems to be the most convenient technique to attack this
problem.

######  Theorem 5.1.

Let $\omega$ be a domain in $\mathbb{R}^{2}$, let
$\bm{\theta}\in\mathcal{C}^{3}(\overline{\omega};\mathbb{E}^{3})$ be the
middle surface of a _flexural shell_ , let $\gamma_{0}$ be a non-zero length
portion of the boundary $\gamma$ (cf. section 4) and let
$\bm{q}\in\mathbb{E}^{3}$ be a non-zero vector given once and for all. Let us
consider the _non-trivial_ space (cf. section 4)

$\bm{V}_{F}(\omega):=\\{\bm{\eta}=(\eta_{i})\in H^{1}(\omega)\times
H^{1}(\omega)\times H^{2}(\omega);\gamma_{\alpha\beta}(\bm{\eta})=0\textup{ in
}\omega\textup{ and }\eta_{i}=\partial_{\nu}\eta_{3}=0\textup{ on
}\gamma_{0}\\},$

and let us define the set

$\displaystyle\bm{U}_{F}(\omega)$
$\displaystyle:=\\{\bm{\eta}=(\eta_{i})\in\bm{V}_{F}(\omega);\big{(}\bm{\theta}(y)+\eta_{i}(y)\bm{a}^{i}(y)\big{)}\cdot\bm{q}\geq
0\textup{ for a.a. }y\in\omega\\}.$

Let there be given a family of linearly elastic flexural shells with the same
middle surface $\bm{\theta}(\overline{\omega})$ and thickness
$2\varepsilon>0$, and let $\bm{u}_{\kappa}(\varepsilon)\in\bm{V}(\Omega)$
denote for each $\varepsilon>0$ the unique solution of Problem
$\mathcal{P}_{\kappa}(\varepsilon;\Omega)$, where the penalty parameter
$\kappa$ is assumed to be as in (4).

Then there exists $\bm{u}\in\bm{H}^{1}(\Omega)$ independent of the variable
$x_{3}$ and satisfying

$\displaystyle{\bm{u}=\bf 0}\textup{ on
}\Gamma_{0}=\gamma_{0}\times\left[-1,1\right],$
$\displaystyle\bm{u}_{\kappa}(\varepsilon)\to\bm{u}\textup{ in
}\bm{H}^{1}(\Omega)\textup{ as }\varepsilon\to 0.$

Define the average

$\overline{\bm{u}}=(\overline{u}_{i}):=\frac{1}{2}\int^{1}_{-1}\bm{u}\,\mathrm{d}x_{3}.$

Then

$\overline{\bm{u}}=\bm{\zeta},$

where $\bm{\zeta}$ is the unique solution to the two-dimensional variational
Problem $\mathcal{P}_{F}(\omega)$.

###### Proof.

Strong and weak convergences as $\varepsilon\to 0$ are respectively denoted by
$\to$ and $\rightharpoonup$. For brevity, we let

$e_{i\|j}(\varepsilon):=e_{i\|j}(\varepsilon;\bm{u}_{\kappa}(\varepsilon)).$

The _outline_ of the proof is to a large extent inspired by the proof of
Theorem 6.2-1 of [8] (itself adapted from Theorem 5.1 in Ciarlet, Lods and
Miara [18]), where no confinement condition was imposed. This is why some
parts of the proof are reminiscent of those in [8]; otherwise, considering the
confinement condition requires extra care.

(i) _There exists a subsequence, still denoted
$(\bm{u}_{\kappa}(\varepsilon))_{\varepsilon>0}$, and there exists
$\bm{u}\in\bm{H}^{1}(\Omega)$ and there exist $e_{i\|j}^{1}\in L^{2}(\Omega)$
satisfying_

${\bm{u}=\bf 0}\textup{ on }\Gamma_{0}=\gamma_{0}\times\left[-1,1\right]$

_and such that_

$\displaystyle\bm{u}_{\kappa}(\varepsilon)\rightharpoonup\bm{u}\textup{ in
}\bm{H}^{1}(\Omega)\textup{ and thus
}\bm{u}_{\kappa}(\varepsilon)\to\bm{u}\textup{ in }\bm{L}^{2}(\Omega),$
$\displaystyle(\bm{\theta}(y)+u_{i}(y,x_{3})\bm{a}^{i}(y))\cdot\bm{q}\geq
0\textup{ for a.a. }x=(y,x_{3})\in\Omega,$
$\displaystyle\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon)\rightharpoonup
e_{i\|j}^{1}\textup{ in }L^{2}(\Omega).$

Letting $\bm{v}=\bm{u}_{\kappa}(\varepsilon)$ in the variational equations of
Problem $\mathcal{P}_{\kappa}(\varepsilon;\Omega)$. Combining the uniform
positive-definiteness of the tensor $(A^{ijk\ell}(\varepsilon))$, the Korn
inequality of Theorem 4.3, the asymptotic behavior of the function
$g(\varepsilon)$ (Lemma 4.1), and the fact that (see Lemmas 3.1 and 3.2)

$\displaystyle-\dfrac{\varepsilon}{\kappa}\int_{\Omega}\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}(u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)\cdot\bm{q})\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle=-\dfrac{\varepsilon}{\kappa}\int_{\Omega}\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}[(\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon))\cdot\bm{q}]\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad+\dfrac{\varepsilon}{\kappa}\int_{\Omega}\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}[(\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3})\cdot\bm{q}]\sqrt{g(\varepsilon)}\,\mathrm{d}x\geq
0,\quad\textup{ for all }\kappa>0\textup{ and all }\varepsilon>0,$

we obtain for $\varepsilon>0$ sufficiently small:

$\displaystyle
C^{-2}_{1}\varepsilon^{2}\sum_{i}\left\|u_{i,\kappa}(\varepsilon)\right\|^{2}_{H^{1}(\Omega)}$
$\displaystyle\leq\sum_{i,j}\left\|e_{i\|j}(\varepsilon)\right\|^{2}_{L^{2}(\Omega)}\leq\dfrac{C_{0}}{\sqrt{g_{0}}}\int_{\Omega}A^{ijk\ell}(\varepsilon)e_{k\|\ell}(\varepsilon)e_{i\|j}(\varepsilon)\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad-\dfrac{C_{0}\varepsilon}{\kappa\sqrt{g_{0}}}\int_{\Omega}\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}(u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)\cdot\bm{q})\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle=\varepsilon^{2}\frac{C_{0}}{\sqrt{g_{0}}}\int_{\Omega}f^{i}u_{i,\kappa}(\varepsilon)\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\leq\varepsilon^{2}C_{0}\sqrt{\frac{g_{1}}{g_{0}}}\Big{\\{}\sum_{i}\|f^{i}\|^{2}_{L^{2}(\Omega)}\Big{\\}}^{1/2}\Big{\\{}\sum_{i}\left\|u_{i,\kappa}(\varepsilon)\right\|^{2}_{L^{2}(\Omega)}\Big{\\}}^{1/2}.$

This chain of inequalities first shows that the norms
$\left(\left\|u_{i,\kappa}(\varepsilon)\right\|_{H^{1}(\Omega)}\right)_{\varepsilon>0}$
are bounded independently of $\varepsilon$, secondly, that the terms
$\left(\varepsilon^{-1}\left\|e_{i\|j}(\varepsilon)\right\|_{L^{2}(\Omega)}\right)_{\varepsilon>0}$
are bounded uniformly with respect to $\varepsilon$ and, finally, that the
terms

(5)
$\left(\dfrac{1}{\sqrt{\varepsilon}}\left\|\dfrac{\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}}{\kappa}\right\|_{L^{2}(\Omega)}^{2}\right)_{\varepsilon>0}$

are uniformly bounded with respect to $\varepsilon$ as well.

Hence there exist a subsequence, a vector field $\bm{u}\in\bm{H}^{1}(\Omega)$,
and functions $e_{i\|j}^{1}\in L^{2}(\Omega)$, such that:

(6) $\displaystyle\bm{u}_{\kappa}(\varepsilon)$
$\displaystyle\rightharpoonup\bm{u}\textup{ in }\bm{H}^{1}(\Omega),$
$\displaystyle\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon)$
$\displaystyle\rightharpoonup e_{i\|j}^{1}\textup{ in }L^{2}(\Omega),$
$\displaystyle\dfrac{1}{\kappa}\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}$
$\displaystyle\to 0\textup{ in }L^{2}(\Omega),$
$\displaystyle\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}$
$\displaystyle\to 0\textup{ in }L^{2}(\Omega).$

The fact that $\bm{u}_{\kappa}(\varepsilon)\to\bm{u}\textup{ in
}\bm{L}^{2}(\Omega)$ is a consequence of the Rellich-Kondrašov Theorem (viz.,
e.g., Theorem 6.6-3 of [10]). Note that, by the definition of limit, the
latter means that for any given $\delta>0$ there exists a number
$\varepsilon_{2}=\varepsilon_{2}(\delta)>0$ such that for all
$0<\varepsilon<\varepsilon_{2}$ it results:

(7) $\|\bm{u}_{\kappa}(\varepsilon)-\bm{u}\|_{\bm{L}^{2}(\Omega)}<\delta.$

Note that the third convergence in (6) means that for any given $\delta>0$
there exists a number
$\check{\varepsilon}_{2}=\check{\varepsilon}_{2}(\delta)>0$ such that for all
$0<\varepsilon<\check{\varepsilon}_{2}$ it results:

(8) $\left\|\dfrac{\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}}{\kappa}\right\|_{L^{2}(\Omega)}<\delta.$

Moreover, recall that the term (5) is uniformly bounded with respect to
$\varepsilon$, in the sense that there exists a constant
$C=C(C_{0},C_{1},g_{0},g_{1},\omega,\gamma_{0},\bm{\theta},\bm{f})>0$ such
that:

(9) $\left\|\dfrac{\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}}{\kappa}\right\|_{L^{2}(\Omega)}\leq
C\varepsilon^{1/4},$

so that $\check{\varepsilon}_{2}=\delta^{4}/C$.

Note that the fourth convergence in (6) means that for any given $\delta>0$
there exists a number $\hat{\varepsilon}_{2}=\hat{\varepsilon}_{2}(\delta)>0$
such that for all $0<\varepsilon<\hat{\varepsilon}_{2}$ it results:

(10) $\left\|\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}\right\|_{L^{2}(\Omega)}<\delta.$

Moreover, recall that the term (5) is uniformly bounded with respect to
$\varepsilon$, in the sense that there exists a constant
$C=C(C_{0},C_{1},g_{0},g_{1},\omega,\gamma_{0},\bm{\theta},\bm{f})>0$ such
that:

(11) $\left\|\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}\right\|_{L^{2}(\Omega)}\leq
C\kappa\varepsilon^{1/4}=C\varepsilon^{3/4},$

so that $\hat{\varepsilon}_{2}=\delta^{4/3}/C$ is a suitable threshold number
for verifying the definition of limit.

Combining the convergence (7) with the last convergence in (6), the continuity
of the negative part operator (Lemma 3.2), and the fact that, for a generic
function,

$f^{-}=\dfrac{|f|-f}{2},$

gives:

$(\bm{\theta}(y)+u_{i}(y,x_{3})\bm{a}^{i}(y))\cdot\bm{q}\geq 0,\quad\textup{
for a.a. }x=(y,x_{3})\in\Omega.$

That $\bm{u}=\bf 0$ on $\gamma_{0}\times\left[-1,1\right]$ follows from the
continuity of the trace operator $\mathop{\mathrm{tr}}:H^{1}(\Omega)\to
L^{2}(\gamma\times\left[-1,1\right])$. Indeed, for all $i$, we have that for
all $v\in H^{1}(\Omega)$ such that $v=0$ on $\Gamma\setminus\Gamma_{0}$,

$0=\int_{\Gamma_{0}}u_{i,\kappa}(\varepsilon)v\,\mathrm{d}\Gamma\to\int_{\Gamma_{0}}u_{i}v\,\mathrm{d}\Gamma,$

so that a density result proved by Bernard [3] (see also Theorem 6.7-3 of
[10]) gives $u_{i}=0$ on $\Gamma_{0}$.

(ii) _The weak limits $u_{i}\in H^{1}(\Omega)$ found in_ (i) _are independent
of the variable $x_{3}\in\left[-1,1\right]$, in the sense that they satisfy_

$\partial_{3}u_{i}=0\textup{ in }L^{2}(\Omega).$

_Besides, the average $\overline{\bm{u}}$ satisfies
$\overline{\bm{u}}\in\bm{U}_{F}(\omega)$, namely,_

$\displaystyle\overline{\bm{u}}=(\overline{u}_{i})\in H^{1}(\omega)\times
H^{1}(\omega)\times$ $\displaystyle H^{2}(\omega)\textup{ and
}\overline{u}_{i}=\partial_{\nu}\overline{u}_{3}=0\textup{ on }\gamma_{0},$
$\displaystyle\gamma_{\alpha\beta}(\overline{\bm{u}})$
$\displaystyle=0\textup{ in }\omega,$
$\displaystyle\big{(}\bm{\theta}(y)+\overline{u}_{i}(y)\bm{a}^{i}(y)\big{)}$
$\displaystyle\cdot\bm{q}\geq 0\textup{ for a.a. }y\in\omega.$

Apart from the latter property, the proof is identical to that of part (ii) of
the proof of Theorem 6.2-1 in [8]. Let us thus prove that

$\big{(}\bm{\theta}(y)+\overline{u}_{i}(y)\bm{a}^{i}(y)\big{)}\cdot\bm{q}\geq
0\textup{ for a.a. }y\in\omega.$

By part (i), we have that

$(\bm{\theta}(y)+u_{i}(y,x_{3})\bm{a}^{i}(y))\cdot\bm{q}\geq 0\quad\textup{
for a.a. }x=(y,x_{3})\in\Omega.$

Since $\bm{u}=(u_{i})$ is independent of $x_{3}$, we have that an application
of Theorem 4.2-1 (a) of [8] and part (i) gives

$0\leq(\bm{\theta}(y)+u_{i}(y,x_{3})\bm{a}^{i}(y))\cdot\bm{q}=\left(\bm{\theta}(y)+\dfrac{1}{2}\int_{-1}^{1}u_{i}(y,x_{3})\,\mathrm{d}x_{3}\bm{a}^{i}(y)\right)\cdot\bm{q},\quad\textup{
for a.a. }y\in\omega,$

so that $\overline{\bm{u}}=(\overline{u}_{i})\in\bm{U}_{F}(\omega)$.

(iii) _The weak limits $e_{i\|j}^{1}\in L^{2}(\Omega)$ and
$\bm{u}\in\bm{H}^{1}(\Omega)$ found in_ (i) _satisfy_

$\displaystyle-\partial_{3}e_{\alpha\|\beta}^{1}=\rho_{\alpha\beta}(\bm{u})\textup{
in }L^{2}(\Omega),$ $\displaystyle e_{\alpha\|3}^{1}=0\quad\textup{ and }\quad
e_{3\|3}^{1}=-\frac{\lambda}{\lambda+2\mu}a^{\alpha\beta}e_{\alpha\|\beta}^{1}\textup{
in }\Omega.$

The equality $-\partial_{3}e_{\alpha\|\beta}^{1}=\rho_{\alpha\beta}(\bm{u})$
in $L^{2}(\Omega)$ follows from Theorem 5.2-2 of [8]. Let
$\bm{v}=(v_{i})\in\bm{V}(\Omega)$ be arbitrarily chosen. It is known (cf.,
e.g., part (iii) of Theorem 6.2-1 of [8]) that

$\displaystyle\varepsilon e_{\alpha\|\beta}(\varepsilon;\bm{v})$
$\displaystyle\to 0\textup{ in }L^{2}(\Omega),$ $\displaystyle\varepsilon
e_{\alpha\|3}(\varepsilon;\bm{v})$
$\displaystyle\to\dfrac{1}{2}\partial_{3}v_{\alpha}\textup{ in
}L^{2}(\Omega),$ $\displaystyle\varepsilon e_{3\|3}(\varepsilon;\bm{v})$
$\displaystyle=\partial_{3}v_{3}\textup{ for all }\varepsilon>0.$

These relations, combined with the boundedness of the terms
$\left(\varepsilon^{-1}\left\|e_{i\|j}(\varepsilon)\right\|_{L^{2}(\Omega)}\right)_{\varepsilon>0}$
independently of $\varepsilon>0$ (part (i)) and the asymptotic behavior of the
functions $A^{ijk\ell}(\varepsilon)$ and $\sqrt{g(\varepsilon)}$ as
$\varepsilon\to 0$ (Lemma 4.1), give

$\displaystyle\int_{\Omega}\left(A^{\alpha\beta\sigma\tau}(\varepsilon)\dfrac{e_{\sigma\|\tau}(\varepsilon)}{\varepsilon}+A^{\alpha\beta
33}(\varepsilon)\dfrac{e_{3\|3}(\varepsilon)}{\varepsilon}\right)\varepsilon
e_{\alpha\|\beta}(\varepsilon;\bm{v})\sqrt{g(\varepsilon)}\,\mathrm{d}x\to
0\textup{ as }\varepsilon\to 0,$ $\displaystyle\int_{\Omega}\left(4A^{\alpha
3\sigma
3}(\varepsilon)\dfrac{e_{\sigma\|3}(\varepsilon)}{\varepsilon}\right)\varepsilon
e_{\alpha\|3}(\varepsilon;\bm{v})\sqrt{g(\varepsilon)}\,\mathrm{d}x\to\int_{\Omega}2\mu
a^{\alpha\sigma}e_{\sigma\|3}^{1}\partial_{3}v_{\alpha}\sqrt{a}\,\mathrm{d}x\textup{
as }\varepsilon\to 0,$
$\displaystyle\int_{\Omega}\left(A^{33\sigma\tau}(\varepsilon)\dfrac{e_{\sigma\|\tau}(\varepsilon)}{\varepsilon}+A^{3333}(\varepsilon)\dfrac{e_{3\|3}(\varepsilon)}{\varepsilon}\right)\varepsilon
e_{3\|3}(\varepsilon;\bm{v})\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad\to\int_{\Omega}\left(\lambda
a^{\sigma\tau}e_{\sigma\|\tau}^{1}+(\lambda+2\mu)e_{3\|3}^{1}\right)\partial_{3}v_{3}\sqrt{a}\,\mathrm{d}x\textup{
as }\varepsilon\to 0,$
$\displaystyle-\dfrac{\varepsilon}{\kappa}\int_{\Omega}\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}(v_{i}\bm{g}^{i}(\varepsilon)\cdot\bm{q})\sqrt{g(\varepsilon)}\,\mathrm{d}x\to
0\textup{ as }\varepsilon\to 0\,(\textup{cf.
}\eqref{kappa}\textup{--}\eqref{process1}),$
$\displaystyle\varepsilon^{2}\int_{\Omega}f^{i}v_{i}\sqrt{g(\varepsilon)}\,\mathrm{d}x\to
0\textup{ as }\varepsilon\to 0.$

Consequently,

$\int_{\Omega}\left((2\mu
a^{\alpha\sigma}e_{\sigma\|3}^{1})\partial_{3}v_{\alpha}+\big{(}\lambda
a^{\sigma\tau}e_{\sigma\|\tau}^{1}+(\lambda+2\mu)e_{3\|3}^{1}\big{)}\partial_{3}v_{3}\right)\sqrt{a}\,\mathrm{d}x=0.$

Since this inequality holds for any vector field
$\bm{v}=(v_{i})\in\bm{V}(\Omega)$, it follows by Theorem 3.4-1 of [8] that

$e_{\sigma\|3}^{1}=0\quad\textup{ and }\quad\lambda
a^{\sigma\tau}e_{\sigma\|\tau}^{1}+(\lambda+2\mu)e_{3\|3}^{1}=0\textup{ in
}L^{2}(\Omega).$

In particular, the latter gives:

$e_{3\|3}^{1}=-\frac{\lambda}{\lambda+2\mu}a^{\alpha\beta}e_{\alpha\|\beta}^{1}\textup{
in }L^{2}(\Omega).$

(iv) _The weak limit $\overline{\bm{u}}=(\overline{u}_{i})$ is the solution of
Problem $\mathcal{P}_{F}(\omega)$._

Given any $\bm{\eta}\in\bm{U}_{F}(\omega)$, define the vector field
$\bm{w}(\varepsilon;\bm{\eta})=(w_{i}(\varepsilon;\bm{\eta}))$ by:

(12) $\displaystyle w_{\alpha}(\varepsilon;\bm{\eta})$
$\displaystyle:=\eta_{\alpha}-\varepsilon
x_{3}(\partial_{\alpha}\eta_{3}+b_{\alpha}^{\sigma}\eta_{\sigma}),$
$\displaystyle w_{3}(\varepsilon;\bm{\eta})$ $\displaystyle:=\eta_{3}.$

We have that $\bm{w}(\varepsilon;\bm{\eta})\in\bm{V}(\Omega)$ and
$e_{3\|3}(\varepsilon;\bm{w}(\varepsilon;\bm{\eta}))=0$ for all
$\varepsilon>0$. Following the same steps as in part (iv) of [8], observe
that:

(13) $\displaystyle\bm{w}(\varepsilon;\bm{\eta})\to$
$\displaystyle\bm{\eta}\textup{ in }\bm{H}^{1}(\Omega),$
$\displaystyle\dfrac{1}{\varepsilon}e_{\alpha\|\beta}(\varepsilon;\bm{w}(\varepsilon;\bm{\eta}))\to$
$\displaystyle\\{-x_{3}\rho_{\alpha\beta}(\bm{\eta})\\}\textup{ in
}L^{2}(\Omega),$ $\displaystyle\textup{The sequence
}\left(\dfrac{1}{\varepsilon}e_{\alpha\|3}(\varepsilon;\bm{w}(\varepsilon;\bm{\eta}))\right)_{\varepsilon>0}$
$\displaystyle\textup{ converges in }L^{2}(\Omega).$

Fix $\varepsilon>0$ and define the number

$\Lambda(\varepsilon):=\int_{\Omega}A^{ijk\ell}(\varepsilon)\left\\{\dfrac{1}{\varepsilon}e_{k\|\ell}(\varepsilon)-e_{k\|\ell}^{1}\right\\}\left\\{\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon)-e_{i\|j}^{1}\right\\}\sqrt{g(\varepsilon)}\,\mathrm{d}x,$

so that, combining the uniform positive-definiteness of the three-dimensional
elasticity tensor $A^{ijk\ell}(\varepsilon)$ and the asymptotic behavior of
the function $g(\varepsilon)$ (Lemma 4.1), we obtain:

$0\leq\sum_{i,j}\left\|\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon)-e_{i\|j}^{1}\right\|_{L^{2}(\Omega)}^{2}\leq\dfrac{C_{0}}{\sqrt{g_{0}}}\Lambda(\varepsilon).$

An application of the latter to the variational equations of Problem
$\mathcal{P}_{\kappa}(\varepsilon;\Omega)$ under the specialization
$\bm{v}=(\bm{w}(\varepsilon;\bm{\eta})-\bm{u}_{\kappa}(\varepsilon))$, with
$\bm{\eta}\in\bm{U}_{F}(\omega)$ gives:

(14) $\displaystyle 0\leq\Lambda(\varepsilon)$
$\displaystyle=\int_{\Omega}A^{ijk\ell}(\varepsilon)\left\\{\dfrac{1}{\varepsilon}e_{k\|\ell}(\varepsilon)\right\\}\left\\{\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon)\right\\}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad-2\int_{\Omega}A^{ijk\ell}(\varepsilon)\left\\{\dfrac{1}{\varepsilon}e_{k\|\ell}(\varepsilon)\right\\}e_{i\|j}^{1}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad+\int_{\Omega}A^{ijk\ell}(\varepsilon)e_{k\|\ell}^{1}e_{i\|j}^{1}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}A^{ijk\ell}(\varepsilon)\left\\{\dfrac{1}{\varepsilon}e_{k\|\ell}(\varepsilon)\right\\}\left\\{\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon;\bm{w}(\varepsilon;\bm{\eta}))\right\\}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad-\dfrac{1}{\varepsilon\kappa}\int_{\Omega}\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}\left((w_{i}(\varepsilon;\bm{\eta})-u_{i,\kappa}(\varepsilon))\bm{g}^{i}(\varepsilon)\cdot\bm{q}\right)\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad-2\int_{\Omega}A^{ijk\ell}(\varepsilon)\left\\{\dfrac{1}{\varepsilon}e_{k\|\ell}(\varepsilon)\right\\}e_{i\|j}^{1}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad+\int_{\Omega}A^{ijk\ell}(\varepsilon)e_{k\|\ell}^{1}e_{i\|j}^{1}\sqrt{g(\varepsilon)}\,\mathrm{d}x-\int_{\Omega}f^{i}(w_{i}(\varepsilon;\bm{\eta})-u_{i,\kappa}(\varepsilon))\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}A^{\alpha\beta\sigma\tau}(\varepsilon)\left\\{\dfrac{1}{\varepsilon}e_{\sigma\|\tau}(\varepsilon)\right\\}\left\\{\dfrac{1}{\varepsilon}e_{\alpha\|\beta}(\varepsilon;\bm{w}(\varepsilon;\bm{\eta}))\right\\}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad+\int_{\Omega}A^{\alpha\beta
33}(\varepsilon)\left\\{\dfrac{1}{\varepsilon}e_{3\|3}(\varepsilon)\right\\}\left\\{\dfrac{1}{\varepsilon}e_{\alpha\|\beta}(\varepsilon;\bm{w}(\varepsilon;\bm{\eta}))\right\\}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad+4\int_{\Omega}A^{\alpha 3\sigma
3}(\varepsilon)\left\\{\dfrac{1}{\varepsilon}e_{\sigma\|3}(\varepsilon)\right\\}\left\\{\dfrac{1}{\varepsilon}e_{\alpha\|3}(\varepsilon;\bm{w}(\varepsilon;\bm{\eta}))\right\\}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad+\int_{\Omega}A^{33\sigma\tau}(\varepsilon)\left\\{\dfrac{1}{\varepsilon}e_{\sigma\|\tau}(\varepsilon)\right\\}\left\\{\dfrac{1}{\varepsilon}e_{3\|3}(\varepsilon;\bm{w}(\varepsilon;\bm{\eta}))\right\\}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad+\int_{\Omega}A^{3333}(\varepsilon)\left\\{\dfrac{1}{\varepsilon}e_{3\|3}(\varepsilon)\right\\}\left\\{\dfrac{1}{\varepsilon}e_{3\|3}(\varepsilon;\bm{w}(\varepsilon;\bm{\eta}))\right\\}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad-\dfrac{1}{\varepsilon\kappa}\int_{\Omega}\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}\left((w_{i}(\varepsilon;\bm{\eta})-u_{i,\kappa}(\varepsilon))\bm{g}^{i}(\varepsilon)\cdot\bm{q}\right)\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad-2\int_{\Omega}A^{ijk\ell}(\varepsilon)\left\\{\dfrac{1}{\varepsilon}e_{k\|\ell}(\varepsilon)\right\\}e_{i\|j}^{1}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad+\int_{\Omega}A^{ijk\ell}(\varepsilon)e_{k\|\ell}^{1}e_{i\|j}^{1}\sqrt{g(\varepsilon)}\,\mathrm{d}x-\int_{\Omega}f^{i}(w_{i}(\varepsilon;\bm{\eta})-u_{i,\kappa}(\varepsilon))\sqrt{g(\varepsilon)}\,\mathrm{d}x.$

By virtue of the definition of the vector field
$\bm{w}(\varepsilon;\bm{\eta})=(w_{i}(\varepsilon;\bm{\eta}))$ introduced in
(12) and the asymptotic behaviour of the contravariant basis vectors
$\bm{g}^{i}(\varepsilon)$ established in Lemma 4.1, we have that

(15)
$\displaystyle-\dfrac{1}{\varepsilon}\int_{\Omega}\dfrac{\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}}{\kappa}[(w_{i}(\varepsilon;\bm{\eta})-u_{i,\kappa}(\varepsilon))\bm{g}^{i}(\varepsilon)\cdot\bm{q}]\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle=-\dfrac{1}{\varepsilon}\int_{\Omega}\dfrac{\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}}{\kappa}[(\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+w_{i}(\varepsilon;\bm{\eta})\bm{g}^{i}(\varepsilon))\cdot\bm{q}]\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad+\dfrac{1}{\varepsilon}\int_{\Omega}\dfrac{\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}}{\kappa}\left([\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right)\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle=-\dfrac{1}{\varepsilon}\int_{\Omega}\dfrac{\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}}{\kappa}[(\bm{\theta}+\eta_{i}\bm{a}^{i})\cdot\bm{q}]\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad-\dfrac{1}{\varepsilon}\int_{\Omega}\dfrac{\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}}{\kappa}[\varepsilon
x_{3}(\eta_{\alpha}b_{\sigma}^{\alpha}\bm{a}^{\sigma})\cdot\bm{q}]\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad-\dfrac{1}{\varepsilon}\int_{\Omega}\dfrac{\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}}{\kappa}[\varepsilon
x_{3}(\bm{a}_{3}-(\partial_{\alpha}\eta_{3}+b_{\alpha}^{\sigma}\eta_{\sigma})\bm{a}^{\alpha})\cdot\bm{q}]\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad+\dfrac{1}{\varepsilon}\int_{\Omega}\dfrac{\left\\{[\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right\\}^{-}}{\kappa}\left([\bm{\theta}+\varepsilon
x_{3}\bm{a}_{3}+u_{i,\kappa}(\varepsilon)\bm{g}^{i}(\varepsilon)]\cdot\bm{q}\right)\sqrt{g(\varepsilon)}\,\mathrm{d}x+O(\varepsilon)$

has $\limsup$ less or equal than zero when $\varepsilon\to 0$ since the first
addend in the last equality is less or equal than zero for all $\varepsilon>0$
being $\bm{\eta}\in\bm{U}_{F}(\omega)$, the second and the third addends tend
to zero as $\varepsilon\to 0$ thanks to the third convergence of (6), and the
fourth addend is less or equal than zero thansk to the monotonicity of
$-\\{\cdot\\}^{-}$ established in Lemma 3.2.

The asymptotic behavior of the functions $\bm{w}(\varepsilon;\bm{\eta})$ and
$\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon;\bm{w}(\varepsilon;\bm{\eta}))$
exhibited in (13), the asymptotic behavior of the three-dimensional elasticity
tensor $A^{ijk\ell}(\varepsilon)$ and $g(\varepsilon)$ (Lemma 4.1), the weak
convergences $\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon)\rightharpoonup
e_{i\|j}^{1}$ in $L^{2}(\Omega)$ established in part (i), and the relations
satisfied by $e_{i\|3}^{1}$ (part (iii)) together give:

(16) $\displaystyle\int_{\Omega}A^{ijk\ell}(\varepsilon)$
$\displaystyle\left\\{\dfrac{1}{\varepsilon}e_{k\|\ell}(\varepsilon)\right\\}\left\\{\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon;\bm{w}(\varepsilon;\bm{\eta}))\right\\}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\to\int_{\Omega}(A^{\alpha\beta\sigma\tau}(0)e_{\sigma\|\tau}^{1}+A^{\alpha\beta
33}(0)e_{3\|3}^{1})\\{-x_{3}\rho_{\alpha\beta}(\bm{\eta})\\}\sqrt{a}\,\mathrm{d}x,$
$\displaystyle 2\int_{\Omega}A^{ijk\ell}(\varepsilon)$
$\displaystyle\left\\{\dfrac{1}{\varepsilon}e_{k\|\ell}(\varepsilon)\right\\}e_{i\|j}^{1}\sqrt{g(\varepsilon)}\,\mathrm{d}x-\int_{\Omega}A^{ijk\ell}(\varepsilon)e_{k\|\ell}^{1}e_{i\|j}^{1}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\to\int_{\Omega}A^{ijk\ell}(0)e_{k\|\ell}^{1}e_{i\|j}^{1}\sqrt{a}\,\mathrm{d}x=\int_{\Omega}a^{\alpha\beta\sigma\tau}e_{\sigma\|\tau}^{1}e_{\alpha\|\beta}^{1}\sqrt{a}\,\mathrm{d}x,$
$\displaystyle\int_{\Omega}f^{i}(w_{i}(\varepsilon;\bm{\eta}$
$\displaystyle)-u_{i,\kappa}(\varepsilon))\sqrt{g(\varepsilon)}\,\mathrm{d}x\to\int_{\Omega}f^{i}(\eta_{i}-u_{i})\sqrt{a}\,\mathrm{d}x=\int_{\omega}p^{i}(\eta_{i}-\overline{u}_{i})\sqrt{a}\,\mathrm{d}y.$

We have yet to take into account the relations
$\rho_{\alpha\beta}(\bm{u})=-\partial_{3}e_{\alpha\|\beta}^{1}$ in
$L^{2}(\Omega)$ established in part (iii). Since $\bm{u}$ is independent of
$x_{3}$ (part (ii)), these relations show that the functions
$e_{\alpha\|\beta}^{1}$ are of the form

$e_{\alpha\|\beta}^{1}=\Upsilon_{\alpha\beta}-x_{3}\rho_{\alpha\beta}(\overline{\bm{u}})\quad\textup{
with }\Upsilon_{\alpha\beta}\in L^{2}(\omega).$

The asymptotic behaviors observed in (14)–(16), Lemma 3.1, and the uniform
positive-definiteness of the fourth order two-dimensional elasticity tensor
$(a^{\alpha\beta\sigma\tau})$ in turn imply:

$\displaystyle 0$ $\displaystyle\leq\limsup_{\varepsilon\to
0}\Lambda(\varepsilon)\leq\lim_{\varepsilon\to
0}\int_{\Omega}A^{ijk\ell}(\varepsilon)\left\\{\dfrac{1}{\varepsilon}e_{k\|\ell}(\varepsilon)\right\\}\left\\{\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon;\bm{w}(\varepsilon;\bm{\eta}))\right\\}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad-2\lim_{\varepsilon\to
0}\int_{\Omega}A^{ijk\ell}(\varepsilon)\left\\{\dfrac{1}{\varepsilon}e_{k\|\ell}(\varepsilon)\right\\}e_{i\|j}^{1}\sqrt{g(\varepsilon)}\,\mathrm{d}x+\lim_{\varepsilon\to
0}\int_{\Omega}A^{ijk\ell}(\varepsilon)e_{k\|\ell}^{1}e_{i\|j}^{1}\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle\quad-\lim_{\varepsilon\to
0}\int_{\Omega}f^{i}(w_{i}(\varepsilon;\bm{\eta})-u_{i,\kappa}(\varepsilon))\sqrt{g(\varepsilon)}\,\mathrm{d}x$
$\displaystyle=-\int_{\omega}p^{i}(\eta_{i}-\overline{u}_{i})\sqrt{a}\,\mathrm{d}y-\int_{\Omega}A^{ijk\ell}(0)e_{k\|\ell}^{1}e_{i\|j}^{1}\sqrt{a}\,\mathrm{d}x+\int_{\Omega}A^{ijk\ell}(0)e_{k\|\ell}^{1}\\{-x_{3}\rho_{\alpha\beta}(\bm{\eta})\\}\sqrt{a}\,\mathrm{d}x$
$\displaystyle=-\int_{\omega}p^{i}(\eta_{i}-\overline{u}_{i})\sqrt{a}\,\mathrm{d}y-\dfrac{1}{2}\int_{\Omega}a^{\alpha\beta\sigma\tau}\Upsilon_{\sigma\tau}\Upsilon_{\alpha\beta}\sqrt{a}\,\mathrm{d}x+\dfrac{1}{2}\int_{\Omega}a^{\alpha\beta\sigma\tau}x_{3}^{2}\rho_{\sigma\tau}(\overline{\bm{u}})\rho_{\alpha\beta}(\bm{\eta}-\overline{\bm{u}})\sqrt{a}\,\mathrm{d}x$
$\displaystyle\leq\dfrac{1}{3}\int_{\omega}a^{\alpha\beta\sigma\tau}\rho_{\sigma\tau}(\overline{\bm{u}})\rho_{\alpha\beta}(\bm{\eta}-\overline{\bm{u}})\,\mathrm{d}y-\int_{\omega}p^{i}(\eta_{i}-\overline{u}_{i})\sqrt{a}\,\mathrm{d}y.$

In conclusion, the latter yields

$0\leq\dfrac{1}{3}\int_{\omega}a^{\alpha\beta\sigma\tau}\rho_{\sigma\tau}(\overline{\bm{u}})\rho_{\alpha\beta}(\bm{\eta}-\overline{\bm{u}})\,\mathrm{d}y-\int_{\omega}p^{i}(\eta_{i}-\overline{u}_{i})\sqrt{a}\,\mathrm{d}y,$

which establishes that $\overline{\bm{u}}$ is the unique solution for Problem
$\mathcal{P}_{F}(\omega)$ since $\bm{\eta}=(\eta_{i})\in\bm{U}_{F}(\omega)$ is
arbitrarily chosen.

(v) _The weak convergence $\bm{u}_{\kappa}(\varepsilon)\rightharpoonup\bm{u}$
in $\bm{H}^{1}(\Omega)$ established in part (i) is in fact strong, i.e.,_

$\bm{u}_{\kappa}(\varepsilon)\to\bm{u}\quad\textup{ in }\bm{H}^{1}(\Omega),$

_and holds for the whole sequence
$(\bm{u}_{\kappa}(\varepsilon))_{\varepsilon>0}$. _

The proof is identical to that of part (vi) of the proof of Theorem 6.2-1 of
[8] and for this reason is omitted. The assertion means that for all
$\delta>0$ there exists a number $\varepsilon_{2}=\varepsilon_{2}(\delta)>0$
such that for all $0<\varepsilon<\varepsilon_{2}$ it results

$\|\bm{u}_{\kappa}(\varepsilon)-\bm{u}\|_{\bm{H}^{1}(\Omega)}<\delta.$

(vi) _The weak convergences
$\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon)\rightharpoonup e_{i\|j}^{1}$ in
$L^{2}(\Omega)$ established in part _(i)_ are in fact strong, i.e.,_

$\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon)\to e_{i\|j}^{1}\quad\textup{ in
}L^{2}(\Omega).$

_Besides, the limits $e_{i\|j}^{1}$ are unique; hence these convergences hold
for the whole family
$\left(\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon)\right)_{\varepsilon>0}$._

Let $\Lambda(\varepsilon)$ be as in part (iv). Since
$\overline{\bm{u}}\in\bm{U}_{F}(\omega)$, we define a vector field
$\tilde{\bm{w}}(\varepsilon)\in\bm{V}(\Omega)$ as follows:

$\displaystyle\tilde{w}_{\alpha}(\varepsilon)$
$\displaystyle:=\overline{u}_{\alpha}-\varepsilon
x_{3}(\partial_{\alpha}\overline{u}_{3}+b_{\alpha}^{\sigma}\overline{u}_{\sigma}),$
$\displaystyle\tilde{w}_{3}(\varepsilon)$ $\displaystyle:=\overline{u}_{3}.$

We have that, in the same spirit as (13):

(17) $\displaystyle\tilde{\bm{w}}(\varepsilon)\to$
$\displaystyle\overline{\bm{u}}\textup{ in }\bm{H}^{1}(\Omega),$
$\displaystyle\dfrac{1}{\varepsilon}e_{\alpha\|\beta}(\varepsilon;\tilde{\bm{w}}(\varepsilon))\to$
$\displaystyle\\{-x_{3}\rho_{\alpha\beta}(\overline{\bm{u}})\\}\textup{ in
}L^{2}(\Omega),$ $\displaystyle\textup{The sequence
}\left(\dfrac{1}{\varepsilon}e_{\alpha\|3}(\varepsilon;\tilde{\bm{w}}(\varepsilon))\right)_{\varepsilon>0}$
$\displaystyle\textup{ converges in }L^{2}(\Omega).$

Specializing
$\bm{v}=(\tilde{\bm{w}}(\varepsilon)-\bm{u}_{\kappa}(\varepsilon))$ in the
variational equations of Problem $\mathcal{P}_{\kappa}(\varepsilon;\Omega)$
and repeating the same computations as in part (iv) gives:

$0\leq\limsup_{\varepsilon\to
0}\Lambda(\varepsilon)\leq-\int_{\omega}a^{\alpha\beta\sigma\tau}\Upsilon_{\sigma\tau}\Upsilon_{\alpha\beta}\sqrt{a}\,\mathrm{d}y\leq
0.$

In conclusion, by the uniform positive-definiteness of the two-dimensional
fourth-order elasticity tensor (Lemma 4.1) and the asymptotic behavior of the
sequence of vector fields $(\tilde{\bm{w}}(\varepsilon))_{\varepsilon>0}$ in
(17), we obtain $\Upsilon_{\alpha\beta}=0$ and

$\lim_{\varepsilon\to 0}\Lambda(\varepsilon)=0.$

These relations in turn imply that the strong convergence

$\dfrac{1}{\varepsilon}e_{i\|j}(\varepsilon)\to e_{i\|j}^{1}\quad\textup{ in
}L^{2}(\Omega)$

holds. The functions $e_{\alpha\|\beta}^{1}$ are uniquely determined, since
they are given by

$e_{\alpha\|\beta}^{1}=-x_{3}\rho_{\alpha\beta}(\overline{\bm{u}})$

and the vector field $\overline{\bm{u}}$ is uniquely determined as the unique
solution of Problem $\mathcal{P}_{F}(\omega)$. That the functions
$e_{i\|3}^{1}$ are uniquely determined then follows from the relations
established in part (iii). Therefore, the whole sequence
$(\varepsilon^{-1}e_{i\|j}(\varepsilon))_{\varepsilon>0}$ strongly converges
to the functions $e_{i\|j}^{1}$ in $L^{2}(\Omega)$ and the proof is complete.
∎

Observe that the conclusion of Theorem 5.1 and Theorem 4.2-1(b) of [8] give
that for each $\delta>0$ and for each $1\leq i\leq 3$ there exists a number
$\varepsilon_{2}=\varepsilon_{2}(\delta)>0$ such that for all
$0<\varepsilon<\varepsilon_{2}$ it results

(18)
$\left\|\dfrac{1}{2}\int_{-1}^{1}u_{i,\kappa}(\varepsilon)(\cdot,x_{3})\,\mathrm{d}x_{3}-\zeta_{i}\right\|_{H^{1}(\omega)}<\dfrac{\delta}{2},$

where $\bm{u}_{\kappa}(\varepsilon)=(u_{i,\kappa}(\varepsilon))$ is the unique
solution of Problem $\mathcal{P}_{\kappa}(\varepsilon;\Omega)$, and
$\bm{\zeta}=(\zeta_{i})$ is the unique solution of Problem
$\mathcal{P}_{F}(\omega)$, and $\kappa$ is as in (4).

Therefore, the convergence as $\varepsilon\to 0$ of the solutions
$\bm{u}(\varepsilon)$ of Problem $\mathcal{P}(\varepsilon;\Omega)$ to the
solution $\bm{\zeta}$ of Problem $\mathcal{P}_{F}(\omega)$ can thus be
established as a direct corollary of Theorem 5.1.

######  Corollary 5.2.

Let $\omega$ be a domain in $\mathbb{R}^{2}$, let
$\bm{\theta}\in\mathcal{C}^{3}(\overline{\omega};\mathbb{E}^{3})$ be the
middle surface of a _flexural shell_ , let $\gamma_{0}$ be a non-zero length
portion of the boundary $\gamma$ (cf. section 4) and let
$\bm{q}\in\mathbb{E}^{3}$ be a non-zero vector given once and for all. Let us
consider the _non-trivial_ space (cf. section 4)

$\bm{V}_{F}(\omega):=\\{\bm{\eta}=(\eta_{i})\in H^{1}(\omega)\times
H^{1}(\omega)\times H^{2}(\omega);\gamma_{\alpha\beta}(\bm{\eta})=0\textup{ in
}\omega\textup{ and }\eta_{i}=\partial_{\nu}\eta_{3}=0\textup{ on
}\gamma_{0}\\},$

and let us define the set

$\displaystyle\bm{U}_{F}(\omega)$
$\displaystyle:=\\{\bm{\eta}=(\eta_{i})\in\bm{V}_{F}(\omega);\big{(}\bm{\theta}(y)+\eta_{i}(y)\bm{a}^{i}(y)\big{)}\cdot\bm{q}\geq
0\textup{ for a.a. }y\in\omega\\}.$

Let there be given a family of linearly elastic flexural shells with the same
middle surface $\bm{\theta}(\overline{\omega})$ and thickness
$2\varepsilon>0$, and let $\bm{u}(\varepsilon)\in\bm{U}(\varepsilon;\Omega)$
denote, for each $\varepsilon>0$, the unique solution of Problem
$\mathcal{P}(\varepsilon;\Omega)$.

Then, we have that

$\dfrac{1}{2}\int_{-1}^{1}\bm{u}(\varepsilon)\,\mathrm{d}x_{3}\to\bm{\zeta},\quad\textup{
in }\bm{H}^{1}(\omega)\textup{ as }\varepsilon\to 0,$

where $\bm{\zeta}$ is the unique solution to the two-dimensional variational
Problem $\mathcal{P}_{F}(\omega)$.

###### Proof.

We now show that for each $\delta>0$ and for all $1\leq i\leq 3$ there exists
a number $\tilde{\varepsilon}>0$ such that for all
$0<\varepsilon<\tilde{\varepsilon}$ it results

(19)
$\left\|\dfrac{1}{2}\int_{-1}^{1}u_{i}(\varepsilon)(\cdot,x_{3})\,\mathrm{d}x_{3}-\zeta_{i}\right\|_{H^{1}(\omega)}<\delta,$

where $\bm{u}(\varepsilon)=(u_{i}(\varepsilon))$ is the unique solution of
Problem $\mathcal{P}(\varepsilon;\Omega)$ and $\bm{\zeta}=(\zeta_{i})$ is the
unique solution of Problem $\mathcal{P}_{F}(\omega)$.

In order to prove (19), fix $\delta>0$ and take
$\tilde{\varepsilon}:=\varepsilon_{2}=\varepsilon_{2}(\delta)$. For each
$0<\varepsilon<\varepsilon_{2}$ we have that an application of (3), the
triangle inequality, Theorem 4.2-1 (b) of [8], and (18) gives

(20)
$\displaystyle\left\|\dfrac{1}{2}\int_{-1}^{1}u_{i}(\varepsilon)\,\mathrm{d}x_{3}-\zeta_{i}\right\|_{H^{1}(\omega)}\leq\left\|\dfrac{1}{2}\int_{-1}^{1}u_{i,\kappa}(\varepsilon)(\cdot,x_{3})\,\mathrm{d}x_{3}-\zeta_{i}\right\|_{H^{1}(\omega)}+\|u_{i}(\varepsilon)-u_{i,\kappa}(\varepsilon)\|_{H^{1}(\Omega)}$
$\displaystyle<\dfrac{\delta}{2}+\dfrac{\delta}{2}=\delta,$

for all $0<\kappa<\kappa_{0}(\delta)$. Observe that if we take $\kappa$ as in
(4) and it resulted $\kappa_{0}(\delta)<\sqrt{\varepsilon}=\kappa$, then it
would suffice to reduce $\varepsilon_{2}$ until the requirement for the
penalty parameter is met. In doing so, the fact that $\kappa_{0}$ is
independent of $\varepsilon$ plays a critical role in establishing the sought
convergence.

The estimate (20) means that the solutions $\bm{u}(\varepsilon)$ of Problem
$\mathcal{P}(\varepsilon;\Omega)$ converge to the solution of Problem
$\mathcal{P}_{F}(\omega)$ as $\varepsilon\to 0$, as it had to be proved. ∎

Figure 1. _Geometrical interpretation of Corollary 5.2 and criticality of the
condition (4)_. Let $\delta>0$ be fixed and let
$0<\tilde{\varepsilon}<\varepsilon_{2}$, where $\varepsilon_{2}$ is the
parameter found in Corollary 5.2. Let $\bm{\zeta}$ denote the solution to
Problem $\mathcal{P}_{F}(\omega)$. Let $\tilde{\kappa}$ be a “penalty”
parameter in the form (4) associated with $\tilde{\varepsilon}$ for which the
distance between the averaged solution of Problem
$\mathcal{P}_{\tilde{\kappa}}(\tilde{\varepsilon};\Omega)$ (i.e., the vector
field $\bm{\zeta}_{\tilde{\kappa}}(\tilde{\varepsilon})$) and the solution
$\bm{\zeta}$ of Problem $\mathcal{P}_{F}(\omega)$ with respect to the
$\bm{H}^{1}(\omega)$ norm is less than $\delta/2$.
If $\tilde{\kappa}>\kappa_{0}=\kappa_{0}(\delta)$, it might happen, as a
consequence of (3), that the distance between the averaged solution of Problem
$\mathcal{P}_{\tilde{\kappa}}(\tilde{\varepsilon};\Omega)$ (i.e., the vector
field $\bm{\zeta}_{\tilde{\kappa}}(\tilde{\varepsilon})$) and the averaged
solution of Problem $\mathcal{P}(\tilde{\varepsilon};\Omega)$ (i.e., the
vector field $\bm{\zeta}(\tilde{\varepsilon})$) is, with respect to the
$\bm{H}^{1}(\omega)$ norm, strictly greater than the fixed number $\delta$.
If, however, we reduce the parameter $\tilde{\varepsilon}$ until
$\tilde{\kappa}<\kappa_{0}$, then we can find an element
$\bm{\zeta}_{\kappa}(\tilde{\varepsilon})$, solution of Problem
$\mathcal{P}_{\kappa}(\tilde{\varepsilon};\Omega)$, whose distance (once again
with respect to the $\bm{H}^{1}(\omega)$ norm) from both
$\bm{\zeta}_{\tilde{\kappa}}(\tilde{\varepsilon})$ and
$\bm{\zeta}(\tilde{\varepsilon})$ is strictly less than $\delta/2$.

It remains to “ _de-scale_ ” the results of Theorem 5.1 and Corollary 5.2,
which apply to the solutions $\bm{u}(\varepsilon)$ of the _scaled_ problem
$\mathcal{P}(\varepsilon;\Omega)$. This means that we need to convert these
results into ones about the _unknown_
$u^{\varepsilon}_{i}\bm{g}^{i,\varepsilon}:\overline{\Omega^{\varepsilon}}\to\mathbb{E}^{3}$,
which represents the physical _three-dimensional vector field_ of the actual
reference configuration of the shell. As shown in the next theorem, this
conversion is most conveniently achieved through the introduction of the
_averages_
$\displaystyle\frac{1}{2\varepsilon}\int^{\varepsilon}_{-\varepsilon}u^{\varepsilon}_{i}\bm{g}^{i,\varepsilon}\,\mathrm{d}x^{\varepsilon}_{3}$
across the thickness of the shell, a procedure which also clearly highlights
the striking _difference_ (in terms of function spaces) between the asymptotic
behaviors of the _tangential_ and _normal_ components of the displacement
field of the middle surface of the shell.

######  Theorem 5.3.

Let the assumptions on the data be as in _section 4_ and let the assumptions
on the immersion
$\bm{\theta}\in\mathcal{C}^{3}(\overline{\omega};\mathbb{E}^{3})$ be as in
_Theorem 4.3_. Let
$\bm{u}^{\varepsilon}=(u^{\varepsilon}_{i})\in\bm{U}(\Omega^{\varepsilon})$
denote for each $\varepsilon>0$ the unique solution of the variational Problem
$\mathcal{P}(\Omega^{\varepsilon})$ and let $\bm{\zeta}\in\bm{U}_{F}(\omega)$
denote the unique solution to the variational inequalities in Problem
$\mathcal{P}_{F}(\omega)$. Then

$\displaystyle\frac{1}{2\varepsilon}\int^{\varepsilon}_{-\varepsilon}u^{\varepsilon}_{\alpha}\bm{g}^{\alpha,\varepsilon}\,\mathrm{d}x^{\varepsilon}_{3}$
$\displaystyle\to\zeta_{\alpha}\bm{a}^{\alpha},\quad\textup{ in
}\bm{H}^{1}(\omega)\textup{ as }\varepsilon\to 0,$
$\displaystyle\frac{1}{2\varepsilon}\int^{\varepsilon}_{-\varepsilon}u^{\varepsilon}_{3}\bm{g}^{3,\varepsilon}\,\mathrm{d}x^{\varepsilon}_{3}$
$\displaystyle\to\zeta_{3}\bm{a}^{3},\quad\textup{ in
}\bm{H}^{1}(\omega)\textup{ as }\varepsilon\to 0.$

###### Proof.

The proof is analogous to that of Theorem 6.4-1 in [8] and for this reason is
omitted. ∎

In view of the scalings, viz.,
$u^{\varepsilon}_{i}(x^{\varepsilon})=u_{i}(\varepsilon)(x)$ at each
$x^{\varepsilon}\in\Omega^{\varepsilon}$, and of the assumption on the data,
viz., $f^{i,\varepsilon}(x^{\varepsilon})=f^{i}(x)$ at each
$x^{\varepsilon}\in\Omega^{\varepsilon}$, made in section 4, it is natural to
also “de-scale” the unknown appearing in the _limit two-dimensional problem_
found in Theorem 4.3, by letting

$\zeta^{\varepsilon}_{i}(y):=\zeta_{i}(y)\textup{ at each }y\in\omega,$

and by using in its formulation the contravariant components

$p^{i,\varepsilon}:=\varepsilon^{3}\left\\{\int_{-1}^{1}f^{i}\,\mathrm{d}x_{3}\right\\}=\int^{\varepsilon}_{-\varepsilon}f^{i,\varepsilon}\,\mathrm{d}x^{\varepsilon}_{3}$

instead of their scaled counterparts
$p^{i}=\int^{1}_{-1}f^{i}\,\mathrm{d}x_{3}$. In this fashion, it is
immediately found that
$\bm{\zeta}^{\varepsilon}=(\zeta^{\varepsilon}_{i})\in\bm{U}_{F}(\omega)$ _is
the unique solution to the variational inequalities_

$\displaystyle\dfrac{\varepsilon^{3}}{3}\int_{\omega}a^{\alpha\beta\sigma\tau}\rho_{\sigma\tau}(\bm{\zeta}^{\varepsilon})\rho_{\alpha\beta}(\bm{\eta}-\bm{\zeta}^{\varepsilon})\sqrt{a}\,\mathrm{d}y\geq\int_{\omega}p^{i,\varepsilon}(\eta_{i}-\zeta^{\varepsilon}_{i})\sqrt{a}\,\mathrm{d}y,$

for all $\bm{\eta}=(\eta_{i})\in\bm{U}_{F}(\omega)$. These inequalities now
display the _factor_ $\dfrac{\varepsilon^{3}}{3}$, which always appears in the
left-hand sides of equations modelling _flexural_ shells.

## Conclusion and final remarks

In this paper we identified a set of two-dimensional variational inequalities
that model the displacement of a linearly elastic flexural shell subjected to
a confinement condition, expressing that all the points of the admissible
deformed configurations remain in a given half-space.

The starting point of the rigorous asymptotic analysis we carried out is a set
of variational inequalities based on the classical equations of three-
dimensional linearized elasticity, and posed over a non-empty, closed and
convex subset of a suitable Sobolev space. These variational inequalities
govern the displacement of a three-dimensional flexural shell subject to a
confinement condition like the one recalled beforehand.

By means of the penalized version of the aforementioned problem (i.e., the set
of variational inequalities based on the classical equations of three-
dimensional linearized elasticity), we managed to show that, as the thickness
parameter approaches zero, the average across the thickness of the solution of
the original three-dimensional model converges to the solution of a _ad hoc_
two dimensional model. In this regard, it is worth recalling that the rigorous
asymptotic analysis (Theorem 5.1) hinged on an _ad hoc_ scaling of the penalty
coefficient with respect to the shell thickness.

The two-dimensional model we recovered in this paper coincides with the two-
dimensional model recovered as a result of a rigorous asymptotic analysis
carried out on Koiter’s model in the case where the linearly elastic shell
under consideration is subjected to an obstacle (viz. [16] and [17]).

It is worth mentioning that, unlike the case where the linearly elastic shell
under consideration was a linearly elastic elliptic membrane shell (viz. [19],
[20] and [30]), the rigorous asymptotic analysis carried out in this paper
does not resort to any additional assumption apart from those already made in
[18] (see also Chapter 6 of [8]).

Indeed, when the linearly elastic shell under consideration was a linearly
elastic elliptic membrane shell, the authors had to resort to a _ad hoc_
“density property” in order to carry out the asymptotic analysis leading to
the identification of a suitable two-dimensional set of variational
inequalities. The “density property” we mentioned is ensured only under
certain geometrical assumptions, which restrict the applicability of the
recalled result. This very “density property” was recently exploited in [31]
(see also [29]) to show that the solution of a certain obstacle problem in
linearised elasticity enjoys, at least locally, higher regularity properties.

## Acknowledgements

The author is greatly indebted to Professor Philippe G. Ciarlet for his
encouragement and guidance.

## References

  * Bernadou and Ciarlet [1976] M. Bernadou and P. G. Ciarlet.  Sur l’ellipticité du modèle linéaire de coques de W. T. Koiter.  _Computing methods in applied sciences and engineering (Second Internat. Sympos., Versailles, 1975), Part 1. Lecture Notes in Econom. and Math. Systems_ , 134:89–136, 1976.
  * Bernadou et al. [1994] M. Bernadou, P. G. Ciarlet, and B. Miara.  Existence theorems for two-dimensional linear shell theories.  _J. Elasticity_ , 34:111–138, 1994.
  * Bernard [2011] J. M. E. Bernard.  Density results on Sobolev spaces whose elements vanish on a part of the boundary.  _Chinese Ann. Math., Ser B_ , 32:823–846, 2011.
  * Caillerie and Sánchez-Palencia [1995a] D. Caillerie and E. Sánchez-Palencia.  A new kind of singular stiff problems and application to thin elastic shells.  _Math. Models Methods Appl. Sci._ , 5(1):47–66, 1995a.
  * Caillerie and Sánchez-Palencia [1995b] D. Caillerie and E. Sánchez-Palencia.  Elastic thin shells: asymptotic theory in the anisotropic and heterogeneous cases.  _Math. Models Methods Appl. Sci._ , 5(4):473–496, 1995b.
  * Ciarlet [1988] P. G. Ciarlet.  _Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity_.  North-Holland, Amsterdam, 1988.
  * Ciarlet [1989] P. G. Ciarlet.  _Introduction to Numerical Linear Algebra and Optimisation_.  Cambridge University Press, 1989.
  * Ciarlet [2000] P. G. Ciarlet.  _Mathematical Elasticity. Vol. III: Theory of Shells_.  North-Holland, Amsterdam, 2000.
  * Ciarlet [2005] P. G. Ciarlet.  _An Introduction to Differential Geometry with Applications to Elasticity_.  Springer, Dordrecht, 2005.
  * Ciarlet [2013] P. G. Ciarlet.  _Linear and Nonlinear Functional Analysis with Applications_.  Society for Industrial and Applied Mathematics, Philadelphia, 2013.
  * Ciarlet and Destuynder [1979] P. G. Ciarlet and P. Destuynder.  A justification of the two-dimensional linear plate model.  _J. Mécanique_ , 18:315–344, 1979.
  * Ciarlet and Lods [1996a] P. G. Ciarlet and V. Lods.  On the ellipticity of linear membrane shell equations.  _J. Math. Pures Appl._ , 75:107–124, 1996a.
  * Ciarlet and Lods [1996b] P. G. Ciarlet and V. Lods.  Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations.  _Arch. Rational Mech. Anal._ , 136(2):119–161, 1996b.
  * Ciarlet and Lods [1996c] P. G. Ciarlet and V. Lods.  Asymptotic analysis of linearly elastic shells. III. Justification of koiter’s shell equations.  _Arch. Rational Mech. Anal._ , 136(2):191–200, 1996c.
  * Ciarlet and Lods [1996d] P. G. Ciarlet and V. Lods.  Asymptotic analysis of linearly elastic shells: “generalized membrane shells”.  _J. Elasticity_ , 43(2):147–188, 1996d.
  * Ciarlet and Piersanti [2019a] P. G. Ciarlet and P. Piersanti.  Obstacle problems for Koiter’s shells.  _Math. Mech. Solids_ , 24:3061–3079, 2019a.
  * Ciarlet and Piersanti [2019b] P. G. Ciarlet and P. Piersanti.  A confinement problem for a linearly elastic Koiter’s shell.  _C.R. Acad. Sci. Paris, Sér. I_ , 357:221–230, 2019b.
  * Ciarlet et al. [1996] P. G. Ciarlet, V. Lods, and B. Miara.  Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations.  _Arch. Rational Mech. Anal._ , 136(2):163–190, 1996.
  * Ciarlet et al. [2018] P. G. Ciarlet, C. Mardare, and P. Piersanti.  Un problème de confinement pour une coque membranaire linéairement élastique de type elliptique.  _C.R. Acad. Sci. Paris, Sér. I_ , 356(10):1040–1051, 2018.
  * Ciarlet et al. [2019] P. G. Ciarlet, C. Mardare, and P. Piersanti.  An obstacle problem for elliptic membrane shells.  _Math. Mech. Solids_ , 24(5):1503–1529, 2019.
  * Duvaut and Lions [1976] G. Duvaut and J. L. Lions.  _Inequalities in Mechanics and Physics_.  Springer, Berlin, 1976.
  * Glowinski [1984] R. Glowinski.  _Numerical Methods for Nonlinear Variational Problems_.  Springer-Verlag, New York, 1984.
  * Kikuchi and Oden [1988] N. Kikuchi and J. T. Oden.  _Contact problems in elasticity: a study of variational inequalities and finite element methods_ , volume 8.  Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988.
  * Léger and Miara [2008] A. Léger and B. Miara.  Mathematical justification of the obstacle problem in the case of a shallow shell.  _J. Elasticity_ , 90:241–257, 2008.
  * Léger and Miara [2018] A. Léger and B. Miara.  A linearly elastic shell over an obstacle: The flexural case.  _J. Elasticity_ , 131:19–38, 2018.
  * Lions [1969] J. L. Lions.  _Quelques méthodes de résolution des problèmes aux limites non linéaires_.  Dunod; Gauthier-Villars, Paris, 1969.
  * Miara and Sanchez-Palencia [1996] B. Miara and E. Sanchez-Palencia.  Asymptotic analysis of linearly elastic shells.  _Asymptotic Anal._ , 12(1):41–54, 1996.
  * Piersanti [2021] P. Piersanti.  On the justification of the frictionless time-dependent Koiter’s model for thermoelastic shells.  _J. Differential Equations_ , 296:50–106, 2021.
  * Piersanti [2022a] P. Piersanti.  On the improved interior regularity of the solution of a fourth order elliptic problem modelling the displacement of a linearly elastic shallow shell lying subject to an obstacle.  _Asymptot. Anal._ , 127(1-2):35–55, 2022a.
  * Piersanti [2022b] P. Piersanti.  _J. Differential Equations_ , 320:114–142, 2022b.  URL https://doi.org/10.1016/j.jde.2022.02.053.
  * Piersanti [In press] P. Piersanti.  On the improved interior regularity of the solution of a second order elliptic boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle.  _Discrete Contin. Dyn. Syst._ , In press.
  * Piersanti et al. [a] P. Piersanti, K. White, B. Dragnea, and R. Temam.  Modelling virus contact mechanics under atomic force imaging conditions.  _Applicable Anal._ , a.  doi: 10.1080/00036811.2022.2044027.  URL https://www.tandfonline.com/doi/abs/10.1080/00036811.2022.2044027.
  * Piersanti et al. [b] P. Piersanti, K. White, B. Dragnea, and R. Temam.  A three-dimensional discrete model for approximating the deformation of a viral capsid subjected to lying over a flat surface.  b.  URL https://arxiv.org/abs/2202.05625.
  * Piersanti et al. [2021] R. Piersanti, P. C. Africa, M. Fedele, C. Vergara, L. Dedè, A. F. Corno, and A. Quarteroni.  Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations.  _Comput. Methods Appl. Mech. Engrg._ , 373:113468, 33, 2021\.
  * Regazzoni et al. [2021] F. Regazzoni, L. Dedè, and A. Quarteroni.  Active force generation in cardiac muscle cells: mathematical modeling and numerical simulation of the actin-myosin interaction.  _Vietnam J. Math._ , 49(1):87–118, 2021.
  * Rodríguez-Arós [2018] A. Rodríguez-Arós.  Mathematical justification of the obstacle problem for elastic elliptic membrane shells.  _Applicable Anal._ , 97:1261–1280, 2018.
  * Scholz [1984] R. Scholz.  Numerical solution of the obstacle problem by the penalty method.  _Computing_ , 32(4):297–306, 1984.
  * Telega and Lewiński [1998a] J. J. Telega and T. Lewiński.  Homogenization of linear elastic shells: $\Gamma$-convergence and duality. Part I. Formulation of the problem and the effective model.  _Bull. Polish Acad. Sci., Techincal Sci._ , 46:1–9, 1998a.
  * Telega and Lewiński [1998b] J. J. Telega and T. Lewiński.  Homogenization of linear elastic shells: $\Gamma$-convergence and duality. Part II. Dual homogenization.  _Bull. Polish Acad. Sci., Techincal Sci._ , 46:11–21, 1998b.
  * Zingaro et al. [2021] A. Zingaro, L. Dedè, F. Menghini, and A. Quarteroni.  Hemodynamics of the heart’s left atrium based on a Variational Multiscale-LES numerical method.  _Eur. J. Mech. B Fluids_ , 89:380–400, 2021.