C. R. Acad. Sci. Paris, Ser. I 336 (2003) 959–966
Mathematical Problems in Mechanics
On rigid displacements and their relation to the infinitesimal rigid
displacement lemma in shell theory
Philippe G. Ciarlet a , Cristinel Mardare b
a Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
b Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
Received and accepted 16 April 2003
Presented by Robert Dautray
Abstract
Let ω be an open connected subset of R2 and let θ be an immersion from ω into R3 . It is established that the set formed by
all rigid displacements of the surface θ (ω) is a submanifold of dimension 6 and of class C ∞ of the space H 1 (ω). It is shown
that the infinitesimal rigid displacements of the same surface θ(ω) span the tangent space at the origin to this submanifold. To
cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 336 (2003).
 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
Résumé
Déplacements rigides et leur relation au lemme du mouvement rigide infinitésimal en théorie des coques. Soit ω un
ouvert connexe de R2 et θ une immersion de ω dans R3 . On établit que l’ensemble formé par les déplacements rigides de la
surface θ (ω) est une sous-variété de dimension 6 et de classe C ∞ de l’espace H 1 (ω). On montre aussi que les déplacements
rigides infinitésimaux de la même surface θ(ω) engendrent le plan tangent à l’origine à cette sous-variété. Pour citer cet
article : P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 336 (2003).
 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
Version française abrégée
Les notations sont définies dans la version anglaise.
Le lemme du déplacement rigide infinitésimal sur une surface, qui joue un rôle important dans l’analyse du
modèle linéaire de Koiter [12], et plus généralement des coques linéairement élastiques (cf. [6, Chapitre 2]),
s’énonce ainsi : soit ω un ouvert connexe de R2 , soit θ une immersion suffisamment régulière de ω dans un
espace euclidien tri-dimensionnel E3 , et soit η̃ ∈ H 1 (ω) un champ de vecteurs vérifiant
γαβ (η̃) = 0
p.p. dans ω
et ραβ (η̃) = 0 dans H −1 (ω),
où
E-mail addresses: [email protected] (P.G. Ciarlet), [email protected] (C. Mardare).
1631-073X/03/$ – see front matter  2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights
reserved.
doi:10.1016/S1631-073X(03)00205-X
960
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 959–966
1
γαβ (η̃) = (∂α η̃ · a β + ∂β η̃ · a α ),
2
σ
∂σ η̃ · a 3 ,
ραβ (η̃) = ∂αβ (η̃ · a 3 ) − ∂α η̃ · ∂β a 3 − ∂β (η̃ · ∂α a 3 ) − Γαβ
2
les vecteurs a α = ∂α θ sont tangents à la surface θ (ω), le vecteur unitaire a 3 = |aa11 ∧a
∧a 2 | est normal à θ (ω), et les
σ sont les symboles de Christoffel. Alors il existe des vecteurs c ∈ E3 et d ∈ E3 tels que
fonctions Γαβ
η̃(y) = c + d ∧ θ(y)
pour presque tout y ∈ ω.
Une première démonstration de ce lemme a été donnée dans [4, Théorème 5.1-1] pour des champs de vecteurs
η̃ ∈ H 1 (ω) tels que η̃ · a 3 ∈ H 2 (ω), sous les hypothèses suivantes : l’ouvert ω est borné, sa frontière est
lipschitzienne, et θ ∈ C 3 (
ω) ; on en trouve une démonstration plus simple dans [6, Théorème 2.6-2]. Diverses
généralisations, correspondant à des hypothèses de régularité nettement moins restrictives sur l’application θ , ont
été obtenues ; cf. [3] et [5].
En théorie des coques, la surface θ (ω) ⊂ E3 est la configuration de référence de la surface moyenne d’une coque
élastique et le champ η̃ est un champ de déplacements de la surface θ (ω). Les fonctions γαβ (η̃) et ραβ (η̃) sont les
composantes covariantes du tenseur linéarisé de changement de métrique et du tenseur linéarisé de changement
de courbure associés au champ η̃. Un champ de déplacement de la forme ci-dessus η̃ = c + d ∧ θ est appelé un
déplacement rigide infinitésimal de la surface θ (ω).
L’objet de cette Note est d’établir que le lemme du déplacement rigide infinitésimal sur une surface n’est autre
que la version linéarisée (dans un sens précisé au Théorème 4.1) du théorème de rigidité bien connu de la théorie
des surfaces, une fois celui-ci convenablement étendu à l’espace de Sobolev H 1 (ω).
Cette extension (Théorème 2.1) est établie à partir d’une extension, récemment établie dans [8] et rappelée dans
le Théorème 1.1, du théorème de rigidité pour un ouvert tri-dimensionnel.
On établit ensuite (Théorème 3.1 et son corollaire) que l’ensemble M rig formé par les déplacements rigides (au
sens du Théorème 2.1) de la surface θ(ω) est une sous-variété de dimension 6 et de classe C ∞ de l’espace H 1 (ω).
On montre enfin (Théorème 4.1) que l’espace vectoriel engendré par les déplacements rigides infinitésimaux de
la surface θ (ω) n’est autre que l’espace tangent à l’origine à la variété M rig . Ce dernier résultat est démontré à
partir d’une généralisation, également établie dans [8] et rappelée dans le Théorème 1.2, du lemme du déplacement
rigide infinitésimal en coordonnées curvilignes sur un ouvert tri-dimensionnel.
Les énoncés des théorèmes mentionnés ci-dessus se trouvent dans la version anglaise. On trouvera les
démonstrations complètes des Théorèmes 2.1, 3.1, et 4.1 dans [9].
1. Preliminaries
Complete proofs of Theorems 2.1, 3.1, and 4.1 are found in [9].
All spaces, matrices, etc., considered are real. The notations M3 , O3 , O3+ , and A3 respectively designate the
sets of all square matrices of order 3, of all orthogonal matrices of order 3, of all matrices Q ∈ O3 with det Q = 1,
and of all antisymmetric matrices of order 3.
Latin indices range over the set {1, 2, 3} except when they are used for indexing sequences, and the summation
convention with respect to repeated indices is used in conjunction with this rule.
√
The notation E3 designates a three-dimensional Euclidean space and a · b, a ∧ b, and |a| = a · a respectively
3
designate the Euclidean inner product and the exterior product of a, b ∈ E , and the Euclidean norm of a ∈ E3 .
Let Ω be an open subset of R3 , let xi denote the coordinates of a point x ∈ R3 , and let ∂i := ∂/∂xi . Let
Θ ∈ C 1 (Ω; E3) be an immersion. The metric tensor field (gij ) ∈ C 0 (Ω; M3 ) of the set Θ(Ω) is then defined by
means of its covariant components
gij (x) := g i (x) · g j (x),
x ∈ Ω,
where g i (x) := ∂i Θ(x). The following result, established in [8], is an extension of the classical rigidity theorem
for an open set, to mappings in the Sobolev space H 1 (Ω) (this theorem is usually established for mappings in the
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 959–966
961
class C 1 (Ω); see, e.g., [7, Theorem 3]). Note that this extension itself relies on a crucial extension of the classical
Liouville theorem, originally due to Reshetnyak [13] and recently given a particularly concise and elegant proof
by Friesecke, James and Müller [11]. The notation ∇Θ(x) designates the matrix whose columns are the vectors
g i (x), x ∈ Ω.
Theorem 1.1. Let Ω be a connected open subset of R3 and let Θ ∈ C 1 (Ω) := C 1 (Ω; E3) be a mapping that
∈ H 1 (Ω) := H 1 (Ω; E3) that satisfies
satisfies det ∇Θ > 0 in Ω. Assume that there exists a vector field Θ
> 0 a.e. in Ω
det ∇Θ
and g̃ij = gij
a.e. in Ω
(with self-explanatory notations). Then there exist a vector c ∈ E3 and a matrix Q ∈ O3+ such that
Θ(x)
= c + QΘ(x)
for almost all x ∈ Ω.
The covariant components of the linearized change of metric tensor associated with a displacement field ṽ of
the set Θ(Ω) are defined by
1
eij (ṽ) = (∂i ṽ · g j + ∂j ṽ · g i ),
2
where g i := ∂i Θ.
A displacement field ṽ ∈ H 1 (Ω) that satisfies eij (ṽ) = 0 a.e. in Ω is called an infinitesimal rigid displacement.
The next theorem, due to [8], is an extension of the infinitesimal rigid displacement lemma in curvilinear
coordinates found in [6, Theorem 1.7-3].
Theorem 1.2. Let Ω be a connected open subset of R3 and let Θ ∈ C 1 (Ω) ∩ H 1 (Ω) be a mapping that satisfies
det ∇Θ > 0 in Ω. Then a vector field ṽ ∈ H 1 (Ω) satisfies eij (ṽ) = 0 a.e. in Ω if and only if there exist a vector
c ∈ E3 and a matrix A ∈ A3 such that
ṽ(x) = c + AΘ(x)
for almost all x ∈ Ω.
2. The rigidity theorem on a surface and its extension to Sobolev spaces
Greek indices range over the set {1, 2} and the summation convention for Latin indices also applies to these.
Let ω be an open subset of R2 , let yα denote the coordinates of a point y ∈ R2 , and let ∂α := ∂/∂yα and
∂αβ := ∂ 2 /∂yα ∂yβ .
Let θ ∈ C 1 (ω) := C 1 (ω; E3 ) be an immersion. The first fundamental form of the surface θ (ω) ⊂ E3 is defined
by means of its covariant components
aαβ (y) := a α (y) · a β (y),
y ∈ ω,
1
2 (y)
where a α (y) := ∂α θ (y). Let a 3 (y) := |aa11 (y)∧a
(y)∧a2 (y)| , y ∈ ω. If a 3 ∈ C (ω), the second fundamental form of the
surface is defined by means of its covariant components
bαβ (y) := −a α (y) · ∂β a 3 (y),
y ∈ ω.
The classical rigidity theorem on a surface (cf., e.g., [7, Theorem 6]) asserts that, if two immersions θ̃ ∈
C 2 (ω) := C 2 (ω; E3 ) and θ ∈ C 2 (ω) have the same first and second fundamental forms, i.e., if ãαβ = aαβ and
b̃αβ = bαβ in ω (with self-explanatory notations) and ω is connected, then there exist a vector c ∈ E3 and a matrix
Q ∈ O3+ such that
θ̃(y) = c + Qθ (y) for all y ∈ ω.
The following result shows that a similar result holds under the assumptions that θ̃ ∈ H 1 (ω) := H 1 (ω; E3 ) and
ã 2
1
ã 3 := |ãã 1 ∧
∧ã | ∈ H (ω) (again with self-explanatory notations). As shown below, the vector field ã 3 , which is not
1
2
962
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 959–966
necessarily well defined a.e. in ω for an arbitrary mapping θ̃ ∈ H 1 (ω), is nevertheless well defined a.e. in ω for
those mappings θ̃ that satisfy the assumptions of the next theorem.
Theorem 2.1. Let ω be a connected open subset of R2 and let θ ∈ C 1 (ω) be an immersion that satisfies a 3 ∈ C 1 (ω).
Assume that there exists a vector field θ̃ ∈ H 1 (ω) that satisfies
ãαβ = aαβ
a.e. in ω,
ã 3 ∈ H 1 (ω),
and
b̃αβ = bαβ
a.e. in ω.
Then there exist a vector c ∈ E3 and a matrix Q ∈ O3+ such that
θ̃(y) = c + Qθ (y) for almost all y ∈ ω.
Sketch of proof. (i) To begin with, several technical preliminaries are established: first, we observe that the
relations ãαβ = aαβ a.e. in ω and the assumption that θ ∈ C 1 (ω) is an immersion together imply that
|ã 1 ∧ ã 2 | = det(ãαβ ) = det(aαβ ) > 0 a.e. in ω,
so that the vector field ã 3 , and thus the functions b̃αβ , are well defined a.e. in ω. Second, we claim that
bαβ = bβα
in ω
and b̃αβ = b̃βα
a.e. in ω,
i.e., that a α · ∂β a 3 = a β · ∂α a 3 in ω and ã α · ∂β ã 3 = ã β · ∂α ã 3 a.e. in ω. To prove this, we note that either the
assumptions θ ∈ C 1 (ω) and a 3 ∈ C 1 (ω), or the assumptions θ ∈ H 1 (ω) and a 3 ∈ H 1 (ω), imply that a α · ∂β a 3 =
∂α θ · ∂β a 3 ∈ L1loc (ω), hence that ∂α θ · ∂β a 3 ∈ D (ω). It is then shown, after some intermediate computations relying
on distribution theory, that the expression D (ω) ∂α θ · ∂β a 3 , ϕD(ω) is symmetric with respect to α and β, for any
ϕ ∈ D(ω). Hence ∂α θ · ∂β a 3 = ∂β θ · ∂α a 3 in L1loc (ω). Third, let
c̃αβ := ∂α ã 3 · ∂β ã 3
and cαβ := ∂α a 3 · ∂β a 3 .
Then we claim that c̃αβ = cαβ a.e. in ω. To see this, we note that the matrix fields (ã αβ ) := (ãαβ )−1 and
(a αβ ) := (aαβ )−1 are well defined and equal a.e. in ω since θ is an immersion and ãαβ = aαβ a.e. in ω. The
formula of Weingarten can thus be applied a.e. in ω, showing that c̃αβ = ã σ τ b̃σ α b̃τβ a.e. in ω. The assertion then
follows from the assumptions b̃αβ = bαβ a.e. in ω.
(ii) Starting from the set ω and the mapping θ (as given in the statement of Theorem 2.1), we next construct a
set Ω and a mapping Θ that satisfy the assumptions of Theorem 1.1. More precisely, let
Θ(y, x3) := θ(y) + x3 a 3 (y)
for all
(y, x3) ∈ ω × R,
and let ωn , n 0, be open subsets of
such that ωn is a compact subset of ω and ω = n0 ωn . Then it is easily
seen that, for each n 0, there exists 0 < εn 1 such that the mapping Θ ∈ C 1 (Ω) satisfies det ∇Θ > 0 over the
connected open set
ωn × ]−εn , εn [ ⊂ R3 .
Ω :=
R2
n0
Note that the covariant components gij ∈ C 0 (Ω) of the metric tensor field associated with the mapping Θ are
then given by (the symmetries bαβ = bβα established in (i) are used here)
gαβ = aαβ − 2x3 bαβ + x32 cαβ ,
gα3 = 0,
g33 = 1.
that
(iii) Starting with the mapping θ̃ (as given in the statement of Theorem 2.1), we construct a mapping Θ
3
satisfies the assumptions of Theorem 1.1. To this end, we define a mapping Θ : Ω → E by letting
Θ(y,
x3) := θ̃(y) + x3 ã 3 (y)
for all (y, x3 ) ∈ Ω,
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 959–966
963
∈ H 1 (Ω), since Ω ⊂ ω × ]−1, 1[. Besides, det ∇ Θ
= det ∇Θ a.e.
where the set Ω is defined as in (ii). Hence Θ
βσ
in Ω since the functions ã b̃ασ , which are well defined a.e. in ω, are equal a.e. in ω to the functions a βσ bασ .
satisfy g̃ij = gij
Likewise, the components g̃ij ∈ L1 (Ω) of the metric tensor field associated with the mapping Θ
a.e. in Ω since ãαβ = aαβ and b̃αβ = bαβ a.e. in ω by assumption and c̃αβ = cαβ a.e. in ω by part (i).
(iv) By Theorem 1.1, there exist a vector c ∈ E3 and a matrix Q ∈ O3+ such that
θ̃(y) + x3 ã 3 (y) = c + Q θ (y) + x3 a 3 (y) for almost all (y, x3) ∈ Ω.
Differentiating with respect to x3 in this equality between functions in H 1 (Ω) shows that ã 3 (y) = Qa 3 (y) for
almost all y ∈ ω. Hence θ̃ (y) = c + Qθ (y) for almost all y ∈ ω. ✷
3. The submanifold of rigid displacements on a surface
All the results needed below about submanifolds in infinite-dimensional Banach spaces are found in [1]. The
tangent space at a point m of a submanifold M of a Banach space X is denoted Tm M. If f : X → Y is a Fréchetdifferentiable mapping into a Banach space Y , the tangent map at m is denoted Tm f .
The next theorem shows that the set M formed by all the mappings θ̃ ∈ H 1 (ω) that satisfy the assumptions
of the rigidity theorem on a surface (Theorem 2.1) is a finite-dimensional submanifold of the space H 1 (ω). The
tangent space to M at θ is also identified. Another equally important characterization of the same tangent space,
involving this time the linearized change of metric and linearized change of curvature tensors, will be given in
Theorem 4.1.
Theorem 3.1. Let ω be a connected open subset of R2 and let θ ∈ C 1 (ω) ∩ H 1 (ω) be an immersion that satisfies
a 3 ∈ C 1 (ω) ∩ H 1 (ω). Then the set
M := θ̃ ∈ H 1 (ω); ãαβ = aαβ a.e. in ω, ã 3 ∈ H 1 (ω), b̃αβ = bαβ a.e. in ω
is a submanifold of class C ∞ and of dimension 6 of the space H 1 (ω). Its tangent space at θ is given by
Tθ M = η̃ ∈ H 1 (ω); ∃c ∈ E3 , ∃A ∈ A3 , η̃ = c + Aθ a.e. in ω .
Sketch of proof. (i) Define the linear mapping f : (c, F ) ∈ E3 × M3 → f (c, F ) = c + F θ ∈ H 1 (ω). By the
rigidity theorem (Theorem 2.1), the above set M may be equivalently defined as M = f (E3 × O3+ ). If the mapping
f : E3 × M3 → H 1 (ω) is injective, in which case f is a C ∞ -diffeomorphism from E3 × M3 onto f (E3 × M3 ), the
proof that M = f (E3 × O3+ ) is a submanifold of H 1 (ω) is simple. Since submanifolds of class C ∞ are preserved
by C ∞ -diffeomorphisms, M is a submanifold of class C ∞ and of dimension 6 of f (E3 × M3 ). As a closed subspace
of the Hilbert space H 1 (ω), the image f (E3 × M3 ) has a closed complement, i.e., f (E3 × M3 ) is “split” in H 1 (ω).
The set M is thus also a submanifold of class C ∞ and of dimension 6 of H 1 (ω) (this conclusion follows from the
definition of a submanifold; see [1, Definition 3.2.1]).
If the mapping f : E3 × M3 → H 1 (ω) is not injective, some care has to be exercised. More specifically, one
needs to prove that the restriction f % of the mapping f to the set E3 × O3+ is an embedding, in the sense that
the following two properties are satisfied: first, for each (c, Q) ∈ E3 × O3+ , the tangent map T(c,Q) f is injective,
with a closed range having a closed complement in H 1 (ω). Second, the restriction f % of the mapping f to the
submanifold E3 × O3+ is a homeomorphism, hence a C ∞ -diffeomorphism since f is linear, from E3 × O3+ onto
the image f (E3 × O3+ ) equipped with the relative topology induced by that of H 1 (ω).
Once these two properties are established (by means of arguments that are essentially technical, but too lengthy
to be sketched here), it can be concluded that the set M = f % (E3 × O3+ ) is a submanifold of dimension 6 of H 1 (ω),
since E3 × O3+ is a submanifold of dimension 6 of E3 × M3 (see [1, Section 3.5]). Since the manifolds E3 × O3+
and H 1 (ω) are of class C ∞ and the mapping f % is of class C ∞ , the submanifold M is also of class C ∞ .
964
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 959–966
(ii) Since f is linear and TI O3+ = A3 , the tangent space to M at θ is given by
Tθ M = Tf (0,I ) f E3 × O3+ = f T(0,I ) E3 × O3+ = f E3 × A3
= η̃ ∈ H 1 (ω); ∃c ∈ E3 , ∃A ∈ A3 , η̃ = c + Aθ a.e. in ω .
✷
Remark. Interestingly, one can establish that the mapping f : E3 × M3 → H 1 (ω) is injective if and only if the
surface θ (ω) is not contained in a plane.
In shell theory, the surface θ (ω) is the reference configuration of the middle surface of an elastic shell (under the
additional assumption that the immersion θ is injective, but this assumption is irrelevant for our present purposes).
Then, for each θ̃ ∈ H 1 (ω), the surface θ̃(ω) is a deformed configuration of the middle surface and the field
η̃ ∈ H 1 (ω) defined by θ̃ = θ + η̃ is a displacement field of the reference configuration θ (ω). If in particular θ̃ ∈ M,
the field η̃ defined in this fashion is called a rigid displacement, and the subset M rig of H 1 (ω) defined by
M = θ + M rig
is called the manifold of rigid displacements. We now recast Theorem 3.1 in terms of the manifold M rig .
Corollary to Theorem 3.1. Let ω be a connected open subset of R2 , and let θ ∈ C 1 (ω) ∩ H 1 (ω) be an immersion
that satisfies a 3 ∈ C 1 (ω) ∩ H 1 (ω). Then the manifold of rigid displacements of the surface θ (ω), viz.,
M rig := η̃ ∈ H 1 (ω); ãαβ = aαβ a.e. in ω, ã 3 ∈ H 1 (ω), b̃αβ = bαβ a.e. in ω ,
is a submanifold of class C ∞ and of dimension 6 of the space H 1 (ω) and its tangent space at 0 is given by
T0 M rig = Tθ M = η̃ ∈ H 1 (ω); ∃c ∈ E3 , ∃A ∈ A3 , η̃ = c + Aθ a.e. in ω .
4. The infinitesimal rigid displacement lemma on a surface revisited
The covariant components of the linearized change of metric tensor and linearized change of curvature
tensor associated with a smooth enough displacement field η̃ of the surface θ (ω) are defined by γαβ (η̃) :=
1
lin and ρ (η̃) := [b̃
lin
αβ
αβ − bαβ ] , where aαβ and ãαβ , and bαβ and b̃αβ , respectively designate the
2 [ãαβ − aαβ ]
covariant components of the first, and second, fundamental forms of the surfaces θ (ω) and θ̃(ω) where θ̃ := θ + η̃,
and [·]lin denotes the linear part with respect to η̃ in the expression [·]. A formal computation immediately gives
1
γαβ (η̃) = (∂α η̃ · a β + ∂β η̃ · a α ),
2
where a α := ∂α θ.
This expression thus shows that γαβ (η̃) ∈ L2loc (ω) if η̃ ∈ H 1 (ω) and θ ∈ C 1 (ω).
Another formal, but substantially less immediate, computation shows that (see, e.g., [6, Theorem 2.5-1])
σ
ραβ (η̃) = ∂αβ (η̃ · a 3 ) − ∂α η̃ · ∂β a 3 − ∂β (η̃ · ∂α a 3 ) − Γαβ
∂σ η̃ · a 3 ,
σ := a σ τ a · ∂ a are the Christoffel symbols of the surface θ (ω). This expression thus
where the functions Γαβ
τ
α β
shows that ραβ (η̃) ∈ H −1 (ω) if η̃ ∈ H 1 (ω) and θ ∈ C 2 (ω) and a 3 ∈ C 2 (ω).
Under these assumptions on the mapping θ and the field a 3 , a displacement field η̃ ∈ H 1 (ω) that satisfies
γαβ (η̃) = 0 a.e. in ω and ραβ (η̃) = 0 in H −1 (ω) is called an infinitesimal rigid displacement of the surface θ (ω).
Accordingly, the infinitesimal rigid displacement lemma on a surface consists in identifying the vector space V lin
rig
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 959–966
965
formed by such displacements. This is the object of the next theorem, which shows that the space V lin
rig has a remarkably simple interpretation in terms of the manifold M rig of rigid displacements introduced at the end of Section 3.
Theorem 4.1. Let ω be a connected open subset of R2 and let θ ∈ C 2 (ω) ∩ H 1 (ω) be an immersion that satisfies
a 3 ∈ C 2 (ω) ∩ H 1 (ω). Then the space of infinitesimal rigid displacements of the surface θ (ω), viz.,
1
−1
V lin
(ω) ,
rig := η̃ ∈ H (ω); γαβ (η̃) = 0 a.e. in ω and ραβ (η̃) = 0 in H
is given by
V lin
rig = T0 M rig ,
where the tangent space T0 M rig has been identified in the Corollary to Theorem 3.1.
Sketch of proof. The proof is reminiscent of that used in [2] or [10] for establishing the Korn inequality on a
surface as a consequence of its three-dimensional counterpart in curvilinear coordinates.
(i) Starting from the set ω and the mapping θ , we begin by constructing a set Ω and a mapping Θ as in parts
(ii) and (iii) of the proof of Theorem 2.1, with the additional requirements that the open sets ωn be connected and
that they satisfy ωn ⊂ ωn+1 for any n 0.
(ii) Given any displacement field η̃ ∈ V lin
rig , let
ṽ(y, x3 ) := η̃(y) − x3 ∂α (η̃ · a 3 ) − η̃ · ∂α a 3 a α (y)
for almost all (y, x3 ) ∈ Ω, where Ω is defined as in part (i) and a α := a αβ a β . The vector field ṽ defined in this
fashion is such that ṽ ∈ H 1 (Ωn ) for all n 0, where Ωn = ωn × ]−εn , εn [.
A careful computation then shows that, for any n 0, the covariant components eij (ṽ) ∈ L2 (Ωn ) of the
linearized change of metric tensor (see Section 1) associated with the above displacement field ṽ vanish a.e. in Ωn .
Theorem 1.2 can thus be applied, showing that, for each n 0, there exist a vector cn ∈ E3 and a matrix An ∈ A3
such that ṽ(x) = cn + An Θ(x) for almost all x ∈ Ωn . This last relation in turn implies that η̃(y) = cn + An θ (y)
for almost all y ∈ ωn . That the vectors cn and An are in fact independent of n 0 is then a consequence of the
inclusions ωn ⊂ ωn+1 , n 0. ✷
By Theorem 4.1, the infinitesimal rigid displacements of the surface θ (ω) thus span the tangent space at the
origin to the manifold formed by the rigid displacements of θ(ω).
References
[1] R. Abraham, J.E. Marsden, T. Ratiu, Manifolds, Tensor Analysis, and Applications, Second edition, Springer-Verlag, New York, 1988.
[2] J.L. Akian, A simple proof of the ellipticity of Koiter’s model, Anal. Appl. 1 (2003) 1–16.
[3] S. Anicic, H. Le Dret, A. Raoult, Lemme du mouvement rigide infinitésimal en coordonnées lipschitziennes et application aux coques de
régularité minimale, C. R. Acad. Sci. Paris, Ser. I 336 (2003) 365–370.
[4] M. Bernadou, P.G. Ciarlet, Sur l’ellipticité du modèle linéaire de coques de W.T. Koiter, in: R. Glowinski, J.-L. Lions (Eds.), Computing
Methods in Applied Sciences and Engineering, in: Lecture Notes in Econom. Math. Systems, Vol. 134, Springer-Verlag, Heidelberg, 1976,
pp. 89–136.
[5] A. Blouza, H. Le Dret, Existence and uniqueness for the linear Koiter model for shells with little regularity, Quart. Appl. Math. 57 (1999)
317–337.
[6] P.G. Ciarlet, Mathematical Elasticity, Vol. III: Theory of Shells, North-Holland, Amsterdam, 2000.
[7] P.G. Ciarlet, F. Larsonneur, On the recovery of a surface with prescribed first and second fundamental forms, J. Math. Pures Appl. 81
(2002) 167–185.
[8] P.G. Ciarlet, C. Mardare, On rigid displacements and their relation to the infinitesimal rigid displacement lemma in three-dimensional
elasticity, C. R. Acad. Sci. Paris, Ser. I 336 (2003) 873–878.
[9] P.G. Ciarlet, C. Mardare, On rigid and infinitesimal rigid displacements in shell theory, to appear.
966
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 959–966
[10] P.G. Ciarlet, S. Mardare, On Korn’s inequalities in curvilinear coordinates, Math. Models Methods Appl. Sci. 11 (2001) 1379–1391.
[11] G. Friesecke, R.D. James, S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional
elasticity, Comm. Pure Appl. Math. 55 (2002) 1461–1506.
[12] W.T. Koiter, On the foundations of the linear theory of thin elastic shells, Proc. Kon. Ned. Akad. Wetensch. Ser. B 73 (1970) 169–195.
[13] Y.G. Reshetnyak, Liouville’s theory on conformal mappings under minimal regularity assumptions, Siberian Math. J. 8 (1967) 69–85.