c 2001 Society for Industrial and Applied Mathematics
SIAM J. MATH. ANAL.
Vol. 33, No. 4, pp. 877–895
AN UP-TO-THE BOUNDARY VERSION OF FRIEDRICHS’S LEMMA
AND APPLICATIONS TO THE LINEAR KOITER SHELL MODEL∗
ADEL BLOUZA† AND HERVÉ LE DRET‡
Abstract. In this work, we introduce a variant of the standard mollifier technique that is valid
up to the boundary of a Lipschitz domain in Rn . A version of Friedrichs’s lemma is derived that gives
an estimate up to the boundary for the commutator of the multiplication by a Lipschitz function
and the modified mollification. We use this version of Friedrichs’s lemma to prove the density of
smooth functions in the new function space introduced in our earlier work concerning the linear
Koiter shell model for shells with little regularity. The density of smooth functions is in turn used
to prove continuous dependence of the solution of Koiter’s model on the midsurface. This provides
a complete justification of our new formulation of the Koiter model.
Key words. Friedrichs’s lemma, shell theory, Koiter’s model
AMS subject classifications. 35A99, 74K25
PII. S0036141000380012
1. Introduction. Mollification is a basic technique in analysis. It is classically
performed by convolution with a compactly supported mollifier. In order for the
convolution to be defined, it is necessary either to work on the whole of Rn or in a
compactly contained subdomain ω of the domain of interest Ω. For many classical
function spaces on a domain Ω, approximation by smooth functions is quite straightforward if the boundary of the domain Ω is regular enough. In effect, in the latter
case, there usually is a continuous extension operator that reduces the case of the domain to that of Rn . It suffices to perform the mollification on the extended function
and then restrict the mollifed function to the domain. See [1], [13], [14], [22], and [24]
among others.
Our main field of application here is the linear Koiter shell model in elasticity (see
[21]). In [5], [6], we introduced a new formulation of this model that makes sense and
is well-posed for midsurfaces of class W 2,∞ instead of C 3 , as was customarily assumed
earlier; see also [9] and [11] for shell models in the same context of regularity. The
simplest and most natural examples of W 2,∞ -shells are given by globally C 1 - and
piecewise C 3 -midsurfaces. Consider, for instance, a shell consisting of a planar part
that is connected to a circular cylinder part or an egg-shaped shell made of a quarter
of a sphere and a quarter of an ellipsoid glued together along a circle. Our new
formulation entails the introduction of a new functional setting. The new function
space involves multiplication of distributional partial derivatives of the functions by
given Lipschitz functions related to the geometry of the midsurface. It is required
that such quantities be square-integrable. To the best of our knowledge, this specific
kind of function space was not studied before regarding such fundamental issues as
the density or nondensity of smooth functions. There are relatively close ideas in
∗ Received by the editors October 24, 2000; accepted for publication (in revised form) May 21,
2001; published electronically December 28, 2001.
http://www.siam.org/journals/sima/33-4/38001.html
† Laboratoire d’Analyse et Modélisation Stochastique, Université de Rouen, 76821 Mont-St-Aignan
Cedex, France ([email protected]).
‡ Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, Boı̂te Courrier 187, 75252
Paris Cedex 05, France ([email protected]). The work of this author was completed during a stay
at the Texas Institute for Computational and Applied Mathematics, University of Texas at Austin,
with the support of the TICAM Visiting Fellowship program.
877
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ADEL BLOUZA AND HERVÉ LE DRET
transport equation theory (see [2], [12], [10]) although the techniques used therein do
not apply in our case. For the function space introduced in [5], there is no obvious
extension operator, and it is not natural to work on the whole of R2 . Thus different
ideas were needed to address the density question.
One such idea was first put forth in [23], without mention of regularity for the
domain, and was then rediscovered simultaneously later on by the authors and by [15]
in a slightly different form. The idea consists in defining a new mollification technique
in which the mollifier is simultaneously scaled and translated inside the domain. For
the technique to work, it is necessary that the domain satisfy a uniform cone condition, which is practically equivalent to being Lipschitz; see [1], [16]. This simple but
powerful idea yields mollified functions defined on the whole domain without any need
for an extension operator to provide values for the function outside of its domain of
definition.
It is a straightforward matter to reproduce the proof of Friedrichs’s lemma using
the above convolution-translation in place of the convolution itself. This gives an
Lp estimate on the whole domain for the commutator of the convolution-translation
and the multiplication by a Lipschitz function applied to partial derivatives of an Lp
function.
This version of Friedrichs’s lemma is the main tool in proving the density of
smooth functions in the new function space for Koiter’s model. This density has
important consequences. For example, it shows that standard finite element methods
actually approximate the solution of the newly formulated variational problem for
Koiter’s model; see [19] and [20]. Another consequence that we develop here is that if
we consider a sequence of W 2,∞ -midsurfaces that converge in a natural sense toward
a given midsurface and a sequence of loads that also converge, then the corresponding
sequence of solutions to Koiter’s model converges in a natural sense too. Since the
new model coincides with the classical model for C 3 -midsurfaces, taking a sequence
of such C 3 -midsurfaces converging to a midsurface that is only W 2,∞ shows that our
new formulation is an appropriate extension of the classical formulation to less regular
midsurfaces from the mathematical point of view.
2. A modified mollification technique. In this section, we develop the convolution-translation technique introduced in [23] (see also [15]) that allows for up-tothe-boundary mollification without requiring an extension operator. It is well known
that the density of smooth functions in, for example, Sobolev spaces, may fail if the
domain under consideration is not regular. It is thus natural that the regularity of
the boundary should come into play.
First of all, let us recall the uniform cone property for a domain Ω in Rn . We
refer the reader to [1], [13], [16], and [24] for details.
Definition 2.1. An open set Ω ⊂ Rn is said to satisfy the cone property if there
exists an open cone
C = {x = (x , xn ) ∈ Rn ; 0 < xn < h, |x | < xn tan(θ/2)},
with h > 0 and 0 < θ < π, such that for every point x in Ω̄ there is a rotation Rx
such that C̄x = x + Rx C̄ ⊂ Ω̄. In other words, any point x is the vertex of a cone
congruent to C and included in Ω̄ (or Ω). Here | · | denotes the standard Euclidean
norm either on Rn−1 or on Rn .
The set Ω is said to satisfy the uniform cone property if there exists a locally
finite open covering {Ui }i≥1 of ∂Ω, and a corresponding sequence {Ci } of cones, each
congruent to some fixed cone C, such that the following hold.
UP-TO-THE-BOUNDARY VERSION OF FRIEDRICHS’S LEMMA
879
(i) There exists M such that every Ui has diameter less
∞than M .
(ii) For some δ > 0, Ωδ = {x
∈
Ω;
dist(x;
∂Ω)
<
δ}
⊂
i=1 Ui .
(iii) For every i ∈ N, Qi = x∈Ω∩Ui (x + Ci ) ⊂ Ω.
(iv) There exists N ∈ N such that every collection of N + 1 of the sets Qi has an
empty intersection.
Remark 2.2. Note that a bounded domain of Rn satisfies the uniform cone
property if and only if its boundary is Lipschitz; see [7] or [16, Theorem
1.2.2.2].
Let us now introduce some notation for the convolution-translation operator,
which is the basic tool for subsequent developments.
Definition 2.3. Let e be a unit vector in Rn and τ > 0. For all u, v ∈ L1 (Rn ),
we define their convolution-translation (of amount τ in the direction e) u τ,e v by
u τ,e v(x) = u v(x − τ e),
(2.1)
that is,
(2.2)
u τ,e v(x) =
Rn
u(x − τ e − y)v(y) dy.
Obviously, u τ,e v ∈ L1 (Rn ), u τ,e v = v τ,e u, and if v is C ∞ , so is u τ,e v with
∂ (u τ,e v) = u τ,e ∂ α v for any multi-index α ∈ Nn .
The interesting feature of this slightly modified convolution is that it can be used
to define a mollification technique for Sobolev spaces that is valid up to the boundary
of any domain in Rn that satisfies the uniform cone condition, without using any
extension operator. Let us now describe how this can be achieved.
Let ρ be a standard mollifier, i.e., a positive C ∞ function on Rn supported in the
unit ball and such that Rn ρ(x) dx = 1. Let Ω be a domain in Rn that satisfies the
uniform cone condition, and denote by Ui the locally finite covering of the boundary
from Definition 2.1. To entirely cover Ω, we let U0 = {x ∈ Ω; dist(x; ∂Ω) > δ/2}. We
denote by (ϕi )i∈N an associated C ∞ partition of unity.
Theorem 2.4. For all u ∈ W m,p (Ω), there exists a sequence uε in C ∞ (Ω̄) such
that
α
(2.3)
uε → u
in
W m,p (Ω)
when
ε → 0.
Proof. This is of course a very classical result. We only include it here to show how
it can be proved using the convolution-translation instead of standard mollification
together with an extension operator.
We begin by localizing u = i∈N ui with ui = ϕi u. Each ui has compact support
in Ui ∩ Ω̄. The “interior” part u0 does not pose any problem and can be approximated
by standard mollification. Let us concentrate on what happens near the boundary.
From now on, we may thus assume that u has support in, say, U1 ∩ Ω̄ = U without
loss of generality. As far as cones are concerned, we may as well assume that C = C1 .
We consider an open subset Ω of Rn such that U1 ∩ Ω̄ = U and that satisfies
conditions (i), (ii), and (iii) of Definition 2.1 with just one cone equal to C. Such a
set clearly exists. Indeed, in view of [16, Theorem 1.2.2.2], ∂Ω ∩ U is the graph of a
Lipschitz function Φ from a compact subset K of a hyperplane in Rn into R (using
an appropriate coordinate system (x , xn )). If M denotes the Lipschitz constant of Φ,
the standard McShane, or Whitney, extension of Φ to the whole hyperplane defined
by
Φ̃(x ) = min(Φ(y) + M |x − y|)
y∈K
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ADEL BLOUZA AND HERVÉ LE DRET
provides a globally defined Lipschitz extension of Φ with the same Lipschitz constant
M . It is thus sufficient to set Ω = {x ∈ Rn ; xn < Φ̃(x )}. Now, the extension of u by
zero to Ω \ Ω clearly yields a function in W m,p (Ω ).
It follows from the previous considerations that we may assume that Ω satisfies
the uniform cone condition with just one cone equal to C. Let θC , eC , and hC be,
respectively, the cone’s angle, outward unit axis vector, and height.
For all 0 < ε, we now define
η(ε) = ε sin(θC /2)
(2.4)
and
ρε (y) = η(ε)−n ρ(y/η(ε)).
(2.5)
Let εC =
hC
1+sin (θC /2) .
For all 0 < ε < εC and all x ∈ Ω̄, we then let
(2.6)
uε (x) =
B(0,η(ε))
u(x − εe − y)ρε (y) dy.
By construction, we have x − εe − B(0, η(ε)) ⊂ x + C ⊂ Ω; therefore, uε (x) is well
defined up to the boundary. Moreover, since ρε has support in B̄(0, η(ε)), we have
that
uε (x) =
ũ(x − εe − y)ρε (y) dy = ũ ε,e ρε (x),
Rn
i.e., uε = (ũε,e ρε )|Ω̄ , where ũ is any extension of u to Rn —for instance, the extension
by 0. It follows that uε ∈ C ∞ (Ω̄). Moreover, it is easy to see that, for any multi-index
α and all x ∈ Ω̄, we also have
α
α u(x − εe − y)ρ (y) dy = ∂
αu ∂
∂ uε (x) =
ε
ε,e ρε (x).
Rn
To conclude, it is thus sufficient to show that uε → u in Lp (Ω) when ε → 0. This
follows from the same argument that is used in standard mollification by approximating u in Lp (Ω) by a compactly supported continuous function and performing the
convolution-translation on this function.
3. A generalized version of Friedrichs’s lemma. Friedrichs’s lemma was
introduced to deal with partial differential equations with varying coefficients. There
are many different versions of the lemma. We are concerned here with the version
that estimates the commutator of the multiplication by a Lipschitz function and the
convolution by a mollifier on Rn or on a compactly contained subset of the domain Ω;
see [17] and [18]. We replace here the usual convolution by the previously introduced
up-to-the-boundary convolution-translation.
In what follows, Ω is a domain in Rn that satisfies the uniform cone condition
with just one cone, as before. For all v ∈ Lp (Ω), we denote by ∂α v its distributional
partial derivative with respect to xα . (We thus do not use the multi-index notation.)
Therefore, ∂α v ∈ W −1,p (Ω). We need to define the convolution-translation for such
distributions:
∂α v ε,e ρε (x) =
(3.1)
v(x − εe − y)∂α ρε (y) dy,
B(0,η(ε))
UP-TO-THE-BOUNDARY VERSION OF FRIEDRICHS’S LEMMA
881
where η(ε) and ρε are defined as in formulas (2.4) and (2.5). Clearly, this definition
agrees with (2.6) when v is C ∞ . Moreover, the resulting function is in C ∞ (Ω̄). Finally,
if we take a sequence vk ∈ C ∞ (Ω̄) such that vk → v in Lp (Ω) when k → +∞, then
∂α vk ε,e ρε → ∂α v ε,e ρε in Lp (Ω).
Our version of Friedrichs’s lemma is as follows.
Lemma 3.1. Let v ∈ Lp (Ω) and a ∈ W 1,∞ (Ω); then there exists a constant M
which depends only on ρ and on the cone angle such that
(3.2)
(a∂α v) ε,e ρε − a(∂α v ε,e ρε )Lp (Ω) ≤ M aW 1,∞ (Ω) vLp (Ω) .
Proof. Note first that if v ∈ Lp (Ω) and a ∈ W 1,∞ (Ω), then a∂α v ∈ W −1,p (Ω) so
that all terms are well defined. Moreover, in view of the above remarks, it is sufficient
to establish estimate (3.2) for functions v in D(Ω), which is a dense subset of Lp (Ω).
In this case, a∂α v belongs to W 1,∞ (Ω) with compact support.
Let us estimate the commutator. For all x ∈ Ω̄, we have
[(a∂α v)ε,e ρε −a(∂α vε,e ρε )](x) =
[a(x−εe−y)−a(x)]∂α v(x−εe−y)ρε (y) dy.
B(0,η(ε))
(3.3)
Integrating the right-hand side of (3.3) by parts in the ball, we obtain
[(a∂α v) ε,e ρε −a(∂α v ε,e ρε )](x) =
∂α a(x − εe − y)v(x − εe − y)ρε (y) dy
B(0,η(ε))
+
(3.4)
[a(x−εe−y)−a(x)]v(x−εe−y)∂α ρε (y) dy.
B(0,η(ε))
Note that both integral quantities vanish for x outside of a compact neighborhood K
˜ the McShane
of the support of v that is independent of ε for ε ≤ εC . We denote by ã
extension of the restriction of a to K, and by ṽ the extension of v by zero. It is then
clear that for all x ∈ Ω̄,
(3.5)
[(a∂α v) ε,e ρε − a(∂α v ε,e ρε )](x) = f1 (x) + f2 (x),
where
f1 (x) =
and
Rn
f2 (x) =
Rn
˜ − εe − y)ṽ(x − εe − y)ρε (y) dy
∂α ã(x
˜
˜
[ã(x−εe−y)
− ã(x)]ṽ(x−εe−y)∂
α ρε (y) dy
for all x ∈ Rn . It is thus enough to estimate f1 and f2 in Lp separately.
For the first term, we have
˜ − εe − y)||ṽ(x − εe − y)|ρε (y) dy
|f1 (x)| ≤
|∂α ã(x
Rn
˜ L∞ (Rn )
≤ ∂α ã
Rn
|ṽ(x − εe − y)|ρε (y) dy
≤ aW 1,∞ (Ω) (|ṽ| ρε )(x − εe)
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ADEL BLOUZA AND HERVÉ LE DRET
for all x ∈ Rn . Therefore,
f1 Lp (Ω) ≤ f1 Lp (Rn )
≤ aW 1,∞ (Ω) (|ṽ| ρε )Lp (Rn )
≤ aW 1,∞ (Ω) ṽLp (Rn )
= aW 1,∞ (Ω) vLp (Ω) .
Similarly, for the second term,
˜
˜
|f2 (x)| ≤
|ã(x−εe−y)
− ã(x)|
|ṽ(x−εe−y)| |∂α ρε (y)| dy
Rn
˜ L∞ (Rn )
≤ ∇ã
Rn
|ṽ(x−εe−y)| |εe − y| |∂α ρε (y)| dy
≤ aW 1,∞ (Ω) (|ṽ| gε )(x − εe)
for all x ∈ Rn , where
gε (y) = |εe − y| |∂α ρε (y)|.
In view of definitions (2.4) and (2.5), it is easy to see that
1
+ 1 ∂α ρL1 (Rn ) .
gε L1 (Rn ) ≤
sin(θC /2)
Therefore,
1
+ 1 ∂α ρL1 (Rn ) aW 1,∞ (Ω) vLp (Ω) ,
sin(θC /2)
hence the result with M = 1 + sin(θ1C /2) + 1 ∂α ρL1 (Rn ) .
The following corollary will be the basic tool for our density results in the context
of the Koiter shell model.
Corollary 3.2. For all v ∈ Lp (Ω) and a ∈ W 1,∞ (Ω),
f2 Lp (Ω) ≤
(3.6)
(a∂α v) ε,e ρε − a(∂α v ε,e ρε )Lp (Ω) → 0
when
ε → 0.
Proof. Proceed as in [18] by approximating v in Lp (Ω) by a sequence of functions
in D(Ω).
4. Application to the Koiter shell model.
4.1. Formulation of the problem. In this section, we briefly recall the formulation of the linear Koiter shell model introduced in [5] and [6]. This formulation is
much simpler than the classical formulation and is, furthermore, valid for midsurfaces
that can have discontinuous curvatures. We refer to [3] and [8] for general elastic shell
theory.
In what follows, Greek indices and exponents always belong to the set {1, 2},
while Latin indices and exponents belong to the set {1, 2, 3}. We use the Einstein
summation convention unless otherwise specified.
Let ω denote a Lipschitz domain of R2 . We consider a shell with midsurface S =
ϕ(ω̄), where ϕ ∈ W 2,∞ (ω; R3 ) is a one-to-one mapping such that the two vectors aα =
∂α ϕ are linearly independent at each point x ∈ ω̄. We let a3 = a1 ∧ a2 /|a1 ∧ a2 | be the
UP-TO-THE-BOUNDARY VERSION OF FRIEDRICHS’S LEMMA
883
unit normal vector on the midsurface at point ϕ(x). The vectors ai define the covariant
basis at point ϕ(x). The regularity of the midsurface chart and the hypothesis of
linear independence on ω̄ imply that the vectors ai belong to W 1,∞ (ω; R3 ). The
contravariant basis ai is defined by the relations
ai (x) · aj (x) = δji ,
where δji is the Kronecker symbol. In particular, a3 (x) = a3 (x). As before, ai ∈
√
W 1,∞ (ω; R3 ). We let a(x) = |a1 (x) ∧ a2 (x)|2 so that a is the area element of the
midsurface in the chart ϕ.
The first fundamental form of the surface is given in covariant components by
aαβ = aα · aβ ∈ W 1,∞ (ω). The Christoffel symbols of the midsurface are given by
Γραβ = Γρβα = aρ · ∂β aα , and we have Γραβ ∈ L∞ (ω).
Let us recall the new expressions for the various strain tensors that were introduced in [5] and [6]. Let u ∈ H 1 (ω; R3 ) be a displacement of the midsurface,
i.e., a regular mapping from ω̄ into R3 . Its linearized strain tensor is given by
γ(u) = γαβ (u)aα ⊗ aβ with
(4.1)
γαβ (u) =
1
(∂α u · aβ + ∂β u · aα ) ∈ L2 (ω),
2
and its linearized change of curvature tensor is given by Υ(u) = Υαβ (u)aα ⊗ aβ with
(4.2)
Υαβ (u) = (∂αβ u − Γραβ ∂ρ u) · a3 ∈ H −1 (ω).
See [6] for a comparison with the classical approach, in which the displacement is
identified with the triple of its covariant components, and an explanation of why our
new approach does not require ϕ to be of class at least C 3 , which includes many
interesting cases, as we mentioned earlier.
In [5] and [6], we introduced the function space
W = v ∈ H 1 (ω; R3 ), ∂αβ v · a3 ∈ L2 (ω)
(4.3)
for shell displacements. Note that if v ∈ H 1 (ω; R3 ), then ∂αβ v · a3 is a priori in
H −1 (ω). In view of formulas (4.1) and (4.2), it is apparent that displacements in
W are such that their linearized strain and change of curvature tensors are squareintegrable. When equipped with its natural norm

(4.4)
vW = v2H 1 (ω;R3 ) +
1/2
∂αβ v · a3 2L2 (ω) 
,
α,β
the space W is a Hilbert space. To formulate an equilibrium problem for the shell, we
consider e > 0 to be the thickness of the shell and an elasticity tensor aαβρσ ∈ L∞ (ω),
which we assume to satisfy the usual symmetries and to be uniformly strictly positive.
In terms of boundary conditions, the simplest case is that of a shell clamped on
all of its boundaries. This corresponds to the space
(4.5)
V1 = v ∈ W ; v = ∂α v · a3 = 0 on ∂ω ,
which is a closed subspace of W , endowed with the norm of W . In [6], we proved the
following existence and uniqueness result for Koiter’s model.
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ADEL BLOUZA AND HERVÉ LE DRET
Theorem 4.1. Let f ∈ L2 (ω; R3 ) be a given force resultant density. Then there
exists a unique solution to the variational problem: Find u ∈ V1 such that
√
√
e2
∀v ∈ V1 ,
eaαβρσ γαβ (u)γρσ (v) + Υαβ (u)Υρσ (v)
a dx =
f ·v a dx.
12
ω
ω
(4.6)
Remark 4.2. 1. The proof of this theorem relies on the new version of the rigid
displacement lemma and the Korn inequality for surfaces with W 2,∞ regularity; see
[6] for details.
2. Also in [6], it is proved that the space W defines an extension of the classical
framework of [4] to our case. Indeed, when ϕ ∈ C 3 , the function space of [4] is
canonically isomophic to the space V1 . Moreover, the new and classical expressions for
the linearized strain and change of curvature tensors coincide under this isomorphism.
Consequently, the solution given by Theorem 4.1 is in this case equal, modulo the
isomorphism, to the solution found in [4].
3. The case of a shell clamped on a part of its boundary γ0 ⊂ ∂ω and submitted
to tractions and moments on the remaining part is also treated in [6]. The relevant
function space is
V1,γ0 = v ∈ W ; v = ∂α v · a3 = 0 on γ0 ,
(4.7)
which is also a closed subspace of W . For a simply supported shell, the relevant space
is simply
(4.8)
V0 = {v ∈ W ; v = 0 on ∂ω}.
We thus also obtain existence and uniqueness results in these cases.
4.2. Density results. One fundamental issue that was not addressed in [6] is
the density of smooth functions in the various function spaces introduced in our new
formulation of the Koiter model. The density of smooth functions is, for instance,
required in order to make sure that standard finite element methods will actually
approximate the solution of the continuous problem. Another use of this density will
be to show the continuous dependence of the solution of the model on the midsurface, in an appropriate sense. In [6], the consistency of our formulation with the
classical formulation is an a priori consistency: we know that our formulation is more
general than and coincides with the classical formulation when both are applicable.
Continuous dependence is a way to prove a posteriori consistency via a convergence
result.
It should be noted that such a density result cannot be taken for granted since
these spaces are not of a standard kind. Similar questions arise in transport theory;
see [2], [12], and [15], for example. In the case of the transport equation, the definition
of the relevant function spaces involves a directional derivative of the form a∇u, with
a a vector field and u a scalar unknown, that is required to satisfy some integrability
condition. Although formally slightly reminiscent of this situation, our function space
setting is different, since the quantities of interest in shell theory, ∂αβ u · a3 , are not
directional derivatives—a3 does not “live” in the same space as u—and we cannot
adapt techniques based on this special structure to our case.
It should also be noted that if ω = R2 , then the density of smooth functions in W
follows more or less readily from the classical version of Friedrichs’s lemma (see [18]),
UP-TO-THE-BOUNDARY VERSION OF FRIEDRICHS’S LEMMA
885
as will be made clear in the ensuing proofs. However, from the point of view of the
applications, a shell whose midsurface was described by a chart over R2 would be of
little interest. Prescribing tractions and moments on the boundary would be difficult,
and the shell would be diffeomorphic to an open disk, which is restrictive in terms of
the topology of the shell since multiply connected shells would not be allowed. Thus
there is no escaping the difficulties that arise at the boundary.
As we mentioned earlier, the standard way of performing mollification up to
the boundary consists in using an extension operator and then mollifying over R2 .
This does not seem to be of much help here. Indeed, if E1 (u) and E2 (ϕ) denote,
respectively, an H 1 -extension operator for the displacement u and a W 2,∞ -extension
operator for the chart ϕ to R2 , there does not seem to be an easy way of devising E1
and E2 in a way that ensures that ∂αβ (E1 (u)) · ã3 will belong to L2 (R2 ), where ã3
denotes the corresponding extended normal vector (assuming it is defined), whenever
∂αβ u · a3 belongs to L2 (ω). The classical techniques using reflections or integral
operators do not seem to work very well because of the product of two quantities.
The same remark applies if we try to extend a3 itself without any reference to the
geometrical underpinnings of the situation.
It is this failure that prompted us to look for an alternative and eventually rediscover a mollification technique essentially already put forth in [23].
Let us start with the larger function space, without boundary conditions.
Theorem 4.3. Assume that ω satisfies the uniform cone condition. Then the
space C ∞ (ω̄; R3 ) is dense in W .
Proof. First, it is clear that C ∞ (ω̄; R3 ) ⊂ W . Let u ∈ W . We want to construct
a sequence uε of C ∞ (ω̄; R3 )-functions that converges to u in the norm of W , i.e., such
that uε → u in H 1 (ω; R3 ) and ∂αβ uε · a3 → ∂αβ u · a3 in L2 (ω) for all indices α, β.
It is not difficult to check that the space W can be localized using a partition
of unity that is adapted to the uniform cone condition satisfied by ω. We can thus
assume that u is compactly supported in one of the sets Ui ∩ ω̄ introduced in the proof
of Theorem 2.4. We leave the case of the “interior” part in U0 ∩ ω̄ aside for the time
being.
Let U = U1 ∩ ω̄. As in the proof of Theorem 2.4, we can assume that ω satisfies
the uniform cone condition with just one cone and that u is compactly supported in
U . Then introducing
uε = ũ ε,e ρε ,
Theorem 2.4 shows that
uε → u
in
H 1 (ω; R3 ).
Let ui , uε,i , and a3,i denote the Cartesian components of u, uε , and a3 , respectively, so that ∂αβ u · a3 = (∂αβ ui )a3,i . Applying Corollary 3.2 to ∂β ui ∈ L2 (ω), we
obtain
(a3,i ∂αβ ui ) ε,e ρε − a3,i ((∂αβ ui ) ε,e ρε )L2 (ω) −→ 0 when ε → 0.
Now, since u ∈ W , we also have that (a3,i ∂αβ ui ) ∈ L2 (ω). Therefore, by Theorem
2.4, it follows that
a3,i ∂αβ ui − (a3,i ∂αβ ui ) ε,e ρε L2 (ω) −→ 0 when ε → 0
886
ADEL BLOUZA AND HERVÉ LE DRET
as well. Since
∂αβ ui,ε = (∂αβ ui ) ε,e ρε ,
we have thus shown that
a3,i ∂αβ ui − a3,i ∂αβ ui,ε L2 (ω) −→ 0 when ε → 0,
which concludes the proof near the boundary.
Concerning the interior part of u, it is apparent that the same proof works using
standard mollification and the classical Friedrichs lemma.
Let us now consider the case of various boundary conditions. As before, we
assume that ω satisfies the uniform cone condition.
Theorem 4.4. For a totally clamped shell, D(ω; R3 ) is dense in V1 .
Proof. Localize as before near the boundary. We claim that the extension ũ of u
˜3 ∈ L2 (R2 ). Indeed,
by zero to the whole of R2 is such that ũ ∈ H 1 (R2 ; R3 ) and ∂αβ ũ· ã
1
2
3
˜3 ∈ H 1 (R2 ). Since
both conditions are equivalent to having ũ ∈ H (R ; R ) and ∂α ũ· ã
1
both functions are piecewise H and have no jump on ∂ω, the claim is true.
Instead of translating inside the domain as earlier, we translate here outside and
let uε = ũ ε,−e ρε . The same proof as in Theorem 4.3 shows that (uε )|ω → u in W
and that uε ∈ C ∞ (R2 ; R3 ). Moreover, since ũ is identically zero outside of ω, it is
clear that uε has compact support in ω. It may be necessary to change the cone in
such a way that the exterior cone condition is also satisfied to achieve this, which is
possible since ω is locally Lipschitz. Hence the result.
Theorem 4.4 above can be seen as an intermediary step for the following density
result.
Theorem 4.5. Assume that γ0 consists of a finite union of open arcs in ∂ω,
∞
and let Cc,γ
(ω̄; R3 ) denote the set of functions in C ∞ (ω̄; R3 ) that are equal to 0 in a
0
∞
(ω̄; R3 ) is dense in V1,γ0 .
neighborhood of γ0 . Then Cc,γ
0
Proof. We localize as before around γ0 , the interior of its complement in ∂ω, and
the endpoints of γ0 . Clearly, for the parts localized around γ0 , the same argument as
in the proof of Theorem 4.4 applies. Equally clearly, for the parts localized around
the interior of the complement, the argument of the proof of Theorem 4.3 applies.
What remains are the parts that are localized around the endpoints of γ0 .
Let us thus assume that 0 is such an endpoint, and let us localize u in a ball
of radius ε around this point. To this end, we introduce a function θ ∈ D(B(0, 1))
such that θ(x) = 1 if |x| ≤ 1/2 and θ(x) = 0 for |x| ≥ 3/4, and let θε (x) = θ(x/ε).
We want to show that θε u tends to zero strongly in W when ε → 0, so that we can
approximate u by 0 in B(0, ε/2).
Since u and ∂α u · a3 vanish on an arc that has 0 as an endpoint, we can apply
Poincaré’s inequality to both quantities to deduce that
(4.9)
u2L2 (B(0,ε);R3 ) ≤ ε2 ∇u2L2 (B(0,ε);M32 )
and
(4.10)
∂α u · a3 2L2 (B(0,ε)) ≤ ε2 ∇(∂α u · a3 )2L2 (B(0,ε);R2 ) .
By estimate (4.9), we see that θε u → 0 in H 1 (ω; R3 ). Indeed, ∂α (θε u) = (∂α θε )u+
θε ∂α u, and
(∂α θε )u2L2 (B(0,ε);R3 ) ≤ ∇u2L2 (B(0,ε);M32 ) → 0 when ε → 0.
UP-TO-THE-BOUNDARY VERSION OF FRIEDRICHS’S LEMMA
887
We now note that u · a3 also vanishes on the same arc so that by Poincaré’s
inequality
(4.11)
u · a3 2L2 (B(0,ε)) ≤ ε2 ∇(u · a3 )2L2 (B(0,ε);R2 ) .
Now ∂α (u · a3 ) = ∂α u · a3 + u · ∂α a3 , and therefore
∂α (u·a3 )2L2 (B(0,ε)) ≤ 2 ∂α u·a3 2L2 (B(0,ε)) +u·∂α a3 2L2 (B(0,ε)) ≤ Cε2 u2W (B(0,ε)) ,
using estimate (4.10) for the first term and estimate (4.9) and the fact that ∂α a3 ∈
L∞ (ω) for the second term, where · W (B(0,ε)) denotes the local W -norm on B(0, ε).
Consequently, by estimate (4.11), we obtain
(4.12)
u · a3 2L2 (B(0,ε)) ≤ Cε4 u2W (B(0,ε)) .
We have
∂αβ (θε u) · a3 = (∂αβ θε )u · a3 + (∂α θε )(∂β u) · a3 + (∂β θε )(∂α u) · a3 + θε (∂αβ u) · a3 .
Therefore, putting estimates (4.10) and (4.12) together, we see that
∂αβ (θε u) · a3 L2 (B(0,ε)) ≤ CuW (B(0,ε)) → 0 when ε → 0,
which shows that θε u → 0 in W .
Finally, it is fairly clear that the elements of the sequence uε , which are recon∞
structed by patching together all the local approximations of u, belong to Cc,γ
(ω̄; R3 ),
0
and the theorem is proved.
∞
Remark 4.6. Note that the space Cc,γ
(ω̄; R3 ) does not depend on the chart ϕ,
0
whereas the space V1,γ0 does. This is useful since one of the applications we have in
mind is the dependence of the solution of Koiter’s model on the midsurface. The space
∞
Cc,γ
(ω̄; R3 ) is a common dense subspace of all possible V1,γ0 spaces for all possible
0
midsurfaces.
The case of a simply supported shell actually seems to be more difficult. We only
solve it here for a domain ω of class C ∞ and by resorting to classical techniques.
Theorem 4.7. Assume that ω is of class C ∞ . Then C ∞ (ω̄; R3 ) ∩ H01 (ω; R3 ) is
dense in V0 .
Proof. We proceed in a classical fashion. First localize as before. For the parts
near the boundary, we can thus assume that ω = {(x1 , x2 ) ∈ R2 ; x2 < Ψ(x1 )}, where
Ψ: R → R is of class C ∞ . Next we flatten the boundary using the C ∞ -diffeomorphism
Θ1 (x) = x1 ,
Θ2 (x) = x2 − Ψ(x1 ).
This obviously induces an isomorphism on the associated V0 spaces so that we are
reduced to the case ω = R × R∗− . We now extend u and a3 for x2 > 0 by
ũ(x1 , x2 ) = −u(x1 , −x2 ),
ã3 (x1 , x2 ) = a3 (x1 , −x2 ),
respectively. Clearly, ũ ∈ H 1 (R2 ; R3 ), ã3 ∈ W 1,∞ (R2 ; R3 ), and ũ is odd with respect
to x2 .
888
ADEL BLOUZA AND HERVÉ LE DRET
Let us show that ∂αβ ũ · ã3 belongs to L2 (R2 ), or, equivalently, that ∂α ũ · ã3 is in
H (R2 ). For x2 > 0, we have
∂1 ũ · ã3 (x1 , x2 ) = −∂1 u · a3 (x1 , −x2 ),
∂2 ũ · ã3 (x1 , x2 ) = ∂2 u · a3 (x1 , −x2 ),
1
so that (∂α ũ · ã3 )|R×R∗+ belongs to H 1 (R × R∗+ ). It thus suffices to prove that the jump
of both quantities across x2 = 0 is zero. This is clear for the second one as the traces
at x2 = 0 of both sides of the equality obviously coincide. It can also be shown with
a little more work that ∂1 u · a3 considered as a function in the variable x2 belongs to
C 0 (R− ; H −1 (R)) with ∂1 u · a3 (0) = 0, and thus there is no jump either.
We can now introduce a radial mollifier ρ, define ρε = ε−2 ρ(·/ε), and let uε =
ũ ρε . Clearly, uε|ω ∈ C ∞ (ω̄; R3 ), uε|ω → u in W as ε → 0 by the classical Friedrichs
lemma, and uε (x1 , 0) = 0 by the imparity of u and parity of ρ with respect to x2 .
We can go back to the original domain by composition with the C ∞ -diffeomorphism Θ−1 .
Remark 4.8. 1. Since we are mainly interested in finding a dense subspace that
does not depend on the midsurface, the above proof shows that if ω is of class W 2,∞ ,
then W 2,∞ (ω; R3 ) ∩ H01 (ω; R3 ) is dense in V0 .
2. The above proof also works for piecewise C ∞ domains satisfying the uniform
cone condition, for instance polygons, by performing adequate reflections at the angles.
In this case, C ∞ (ω̄; R3 ) ∩ H01 (ω; R3 ) is also dense in V0 .
4.3. Continuous dependence on the midsurface. Our goal in this section
is to show that the formulation of Koiter’s model we proposed in [5] and [6] provides an adequate extension of the classical formulation for C 3 -midsurfaces to W 2,∞ midsurfaces, at least from the mathematical point of view. The density results of the
previous section were formulated for a domain that satisfies the uniform cone condition. For practical purposes in the case of shells, we will from now on consider only
bounded Lipschitz domains.
Let us thus consider a sequence of midsurface charts ϕn that approximate a
given chart ϕ in the sense that ϕn → ϕ in W 2,p (ω; R3 ) strong for all 1 < p < +∞ and
∗
ϕn / ϕ in W 2,∞ (ω; R3 ) weak-∗. (Note that for any ϕ ∈ W 2,∞ (ω; R3 ), it is possible to
construct such a sequence with ϕn of class C 3 .) All corresponding geometric quantities
will from now on be indicated by an n superscript. For instance, the covariant basis
vectors are denoted by ani , the covariant
components of the first fundamental form
√
and the area element by anαβ and an , respectively, and the Christoffel symbols by
Γn,ρ
αβ . We assume for simplicity that all shells have the same thickness and the same
Lamé moduli µ and λ and denote by an,αβρσ the contravariant components of the
elasticity tensor
(4.13)
an,αβρσ = 2µ(an,αβ an,ρσ + an,ασ an,βσ ) +
4λµ n,αβ n,ρσ
a
a
,
λ + 2µ
where an,αβ denote the contravariant components of the first fundamental form. Finally, for all displacements v of the shells, we denote by
n
γαβ
(v) =
1
(∂α v · anβ + ∂β v · anα )
2
and
n
Υnαβ (v) = (∂αβ v − Γn,ρ
αβ ∂ρ v) · a3
UP-TO-THE-BOUNDARY VERSION OF FRIEDRICHS’S LEMMA
889
the covariant components of the strain and change of curvature tensors with explicit
dependence on the charts.
Let us collect all the information on convergence properties of the various geometric and mechanical quantities that we will need later on in one lemma.
Lemma 4.9. Let ϕn be a sequence of charts such that ϕn → ϕ in W 2,p (ω; R3 )
∗
strong for all 1 < p < +∞ and ϕn / ϕ in W 2,∞ (ω; R3 ) weak-∗. Then
(4.14)
(4.15)
ani → ai strongly in W 1,p (ω; R3 ) ∀ 1 < p < +∞ and weakly-∗ in W 1,∞ (ω; R3 ),
√
√
anαβ → aαβ , an → a, and an,αβρσ → aαβρσ in C 0 (ω̄),
and
(4.16)
ρ
p
∞
Γn,ρ
αβ → Γαβ strongly in L (ω) ∀ 1 < p < +∞ and weakly-∗ in L (ω).
Proof. The proof is clear, using Morrey’s theorem and the fact that the covariant
tangent vectors are assumed to be linearly independent.
In what follows, we will concentrate on the case of a totally clamped shell submitted only to force resultants for brevity. All results remain true—with appropriate
modifications—for a simply supported shell and a partially clamped shell submitted
to edge tractions and moments on the free part of the boundary. Let us rewrite
the spaces
on the charts. For all n, we thus let
involved with explicit dependence
W n = v ∈ H 1 (ω; R3 ), ∂αβ v·an3 ∈ L2 (ω) , equipped with their natural norm vW n =
1/2
, and V1n = v ∈ W n ; v = ∂α v·an3 = 0 on ∂ω ,
v2H 1 (ω;R3 ) + α,β ∂αβ v·an3 2L2 (ω)
which is a closed subspace of W n for all n.
For f n ∈ L2 (ω; R3 ) we let un be the unique solution to the variational formulation
of Koiter’s model: Find un ∈ V1n such that
√
e2 n
n
n
n,αβρσ
n
n n
n
n
n
n
γαβ (u )γρσ (v ) + Υαβ (u )Υρσ (v )
ea
an dx
∀v ∈ V1 ,
12
ω
√
=
f n · v n an dx.
(4.17)
ω
Our main result is the following.
Theorem 4.10. Let ϕn be a sequence of charts such that ϕn → ϕ in W 2,p (ω; R3 )
∗
strong for all 1 < p < +∞ and ϕn / ϕ in W 2,∞ (ω; R3 ) weak-∗, and let f n be a
sequence of force resultant densities such that f n → f in L2 (ω; R3 ). Then
(4.18)
un → u in H 1 (ω; R3 )
and
Υnαβ (un ) → Υαβ (u) in L2 (ω),
where u is the solution to Koiter’s model for a clamped shell with midsurface chart ϕ
and applied force resultant density f .
The proof is comprised of a series of lemmas.
Lemma 4.11. If v n ∈ H 1 (ω; R3 ) and ϕn ∈ W 2,∞ (ω; R3 ) are two sequences such
that v n / v weakly in H 1 (ω; R3 ), ϕn → ϕ strongly in W 2,p (ω; R3 ) for all p < +∞,
∗
n
and ϕn / ϕ weakly-∗ in W 2,∞ (ω; R3 ), then γαβ
(v n ) / γαβ (v) weakly in L2 (ω) and
n
n
−1
Υαβ (v ) / Υαβ (v) weakly in H (ω).
Proof. First, since anα → aα strongly in C 0 (ω̄; R3 ), it follows clearly that
n
(v n ) =
γαβ
1
(∂α v n · anβ + ∂β v n · anα ) / γαβ (v) in L2 (ω).
2
890
ADEL BLOUZA AND HERVÉ LE DRET
The case of the change of curvature tensor is more intricate. We know that
∂αβ v n / ∂αβ v in H −1 (ω; R3 ). Let θ be a test-function in H01 (ω). By the Sobolev
embedding theorem, θ ∈ L4 (ω), and, as ∂α an3 → ∂α a3 strongly in L4 (ω; R3 ) by (4.14)
with p = 4, it follows that
θan3 → θa3 ,
strongly in L2 (ω; R3 )
∂α (θan3 ) = (∂α θ)an3 + θ∂α an3 → ∂α (θa3 ),
so that
θan3 → θa3 strongly in H01 (ω; R3 ).
Therefore,
∂αβ v n · an3 , θ = ∂αβ v n , θan3 → ∂αβ v · a3 , θ;
hence
∂αβ v n · an3 / ∂αβ v · a3 weakly in H −1 (ω).
Let us now deal with the other part of Υnαβ (v n ). We have that ∂ρ v n / ∂ρ v weakly
ρ
p
n
1,p
in L2 (ω; R3 ), Γn,ρ
(ω; R3 ) for
αβ → Γαβ strongly in L (ω), and a3 → a3 strongly in W
all p < +∞. It follows easily from this and Hölder’s inequality that
ρ
n
n
q
Γn,ρ
αβ ∂ρ v · a3 / Γαβ ∂ρ v · a3 weakly in L (ω) ∀ 1 < q < 2.
Now, by the two-dimensional Sobolev embedding theorem, we have Lq (ω) 4→ H −1 (ω)
for all 1 < q < 2, hence the result.
Let us now establish some uniform norm equivalence results.
Lemma 4.12. There exist two constants 0 < c < C < +∞ independent of n such
that for all v n ∈ V1n ,
(4.19) cv n W n ≤



αβ
1/2

n
γαβ
≤ Cv n W n .
(v n )2L2 (ω;R3 ) + Υnαβ (v n )2L2 (ω;R3 )

Proof. The proof is essentially identical to that of Lemma 11 in [6] for a single
chart.
We also need uniform positive definiteness of the elasticity tensors. By assumption, for each midsurface,
there exists a constant ηn > 0 such that for all symmetric
tensors τ = ταβ and almost all x ∈ ω, an,αβρσ (x)ταβ τρσ ≥ ηn ταβ ταβ . For instance,
in the case of an isotropic material, this is a statement concerning the Lamé moduli
µ and λ and not the geometry of the midsurface. We will concentrate here on the
isotropic case.
Lemma 4.13. There is constant η > 0 independent of n such that ηn ≥ η for all
n.
Proof. We know that an,αβρσ converge uniformly to aαβρσ . As ηn is the infimum
of the quadratic form an,αβρσ (x)ταβ τρσ on the Cartesian product of the unit sphere
of the space of symmetric tensors with ω̄, the result is clear.
We now are in a position to establish uniform bounds for the various quantities
of interest.
UP-TO-THE-BOUNDARY VERSION OF FRIEDRICHS’S LEMMA
891
Lemma 4.14. There is constant M independent of n such that
(4.20)
un H 1 (ω;R3 ) ≤ M,
∂αβ un · an3 L2 (ω) ≤ M.
Proof. Let us take v n = un as a test-function in the variational formulation of
Koiter’s problem (4.17). In view of Lemmas 4.12 and 4.13, we obtain
√
√
cη δ min(e, e3 /12)un 2W n ≤ an f n L2 (ω;R3 ) un L2 (ω;R3 ) ,
where δ is a uniform lower bound for an (x). The lemma easily follows from the above
estimate.
Lemma 4.15. There exists a subsequence (still denoted by) un and u ∈ V1 such
that
(4.21) un / u weakly in H 1 (ω; R3 )
and
∂αβ un · an3 / ∂αβ u · a3 weakly in L2 (ω).
Proof. Because of the previous bounds, we can find a subsequence un , a function
u ∈ H01 (ω; R3 ), and functions καβ ∈ L2 (ω) such that
un / u weakly in H 1 (ω; R3 )
and ∂αβ un · an3 / καβ weakly in L2 (ω).
As in the proof of Lemma 4.11, we see that καβ = ∂αβ u · a3 so that u ∈ W .
Moreover, ∂α un · an3 / ∂α u · a3 in H 1 (ω) so that ∂α u · a3 ∈ H01 (ω) and therefore
u ∈ V1 .
Our next task is to identify the weak limit u of the above subsequence of solutions
un as being the solution to the Koiter problem corresponding to the limit midsurface
and loads.
Lemma 4.16. The limit u is the unique solution to the following: Find u ∈ V1
such that
√
√
e2
∀v ∈ V1 ,
eaαβρσ γαβ (u)γρσ (v) + Υαβ (u)Υρσ (v)
a dx =
f ·v a dx.
12
ω
ω
(4.22)
The whole sequence is convergent.
Proof. By Theorem 4.4, we know that the spaces V1n and V1 all share a common
dense subspace, namely here, D(ω; R3 ). Therefore, any ψ ∈ D(ω; R3 ) is a legitimate
test-function for all n, as well as for the eventual limit problem (4.22). Now
√
√
n
an,αβρσ γρσ
(ψ) an → aαβρσ γρσ (ψ) a strongly in L2 (ω)
and
√
√
an,αβρσ Υnρσ (ψ) an → aαβρσ Υρσ (ψ) a strongly in L2 (ω)
by Lemma 4.9. On the other hand,
n
(un ) / γαβ (u) weakly in L2 (ω)
γαβ
and
Υnαβ (un ) / Υαβ (u) weakly in L2 (ω)
by Lemma 4.11 and since Lemmas 4.12 and 4.14 imply that Υnαβ (un ) is bounded in
L2 (ω). Therefore, we can pass to the limit as n → +∞ in problem (4.17) and obtain
892
ADEL BLOUZA AND HERVÉ LE DRET
problem (4.22) for all test-functions ψ ∈ D(ω; R3 ). The identification of u then follows
from the fact that D(ω; R3 ) is dense in V1 , viz. Theorem 4.4.
The solution of problem (4.22) is unique; therefore, the standard uniqueness argument shows that the whole sequence un converges, and not just a subsequence
thereof.
The final step in the proof of Theorem 4.10 consists in showing that all weak
convergences are actually strong. This would be a straightforward matter if the testfunction spaces did not depend on n. We would just take u − un as a test-function.
There is a slight twist here since u ∈ V1n and un ∈ V1 so that u − un is a legitimate
test-function neither for problem (4.17) nor for problem (4.22). We recast the problem
in abstract form to circumvent this difficulty.
Let us be given a family of Hilbert spaces (H n , · n )n∈N and a Hilbert space
(H, · ) with the following properties:
(i) There is a subspace D common to all H n and H such that D is dense in H.
(ii) For all y ∈ D, yn → y as n → +∞.
Let us also be given a corresponding family of continuous (on their respective
spaces) symmetric bilinear forms an and a and a family of continuous linear forms ln
and l with the following properties:
(iii) There exists a constant η independent of n such that
∀y n ∈ H n ,
an (y n , y n ) ≥ ηy n 2n .
(iv) For all y, z in D, an (y, z) → a(y, z) and ln (y) → l(y) when n → +∞.
Lemma 4.17. Assume that hypotheses (i)–(iv) are satisfied, and let xn ∈ H n and
x ∈ H be the solutions of the variational problems
∀y n ∈ H n ,
an (xn , y n ) = ln (y n )
and
∀y ∈ H,
a(x, y) = l(y).
If, in addition, ln (xn ) → l(x), then
xn n → x
(4.23)
when
n → +∞.
Proof. For all y ∈ D, we have
an (xn − y, xn − y) = ln (xn − 2y) + an (y, y).
By assumption (iii), it follows that
ηxn − y2n ≤ ln (xn ) − 2ln (y) + an (y, y).
Letting n tend to +∞, we obtain
lim sup xn − y2n ≤
n→+∞
1
(l(x) − 2l(y) + a(y, y))
η
by assumption (iv). Therefore,
lim supxn n − yn ≤
n→+∞
1
(l(x) − 2l(y) + a(y, y)).
η
Now
n
x n − x ≤ xn n − yn + yn − y + y − x
UP-TO-THE-BOUNDARY VERSION OF FRIEDRICHS’S LEMMA
893
so that, letting n tend to +∞, we obtain
1
lim supxn n − x ≤
(l(x) − 2l(y) + a(y, y)) + y − x
η
n→+∞
for all y ∈ D, by assumption (ii). The lemma then results from assumption (i) and
the fact that a and l are continuous on H.
We can now apply Lemma 4.17 to our shell problem to complete the proof of
Theorem 4.10.
Lemma 4.18. We have
(4.24) un → u strongly in H 1 (ω; R3 )
and
Υnαβ (un ) → Υαβ (u) strongly in L2 (ω).
Proof. By the weak convergence result of Lemma 4.15, we know that
lim inf un H 1 (ω;R3 ) ≥ uH 1 (ω;R3 )
n→+∞
and
lim inf ∂αβ un ·an3 L2 (ω) ≥ ∂αβ u·a3 L2 (ω) .
n→+∞
The Hilbert spaces V1n and V1 and the bilinear and linear forms associated with
the Koiter problems clearly satisfy the hypotheses of Lemma 4.17 with D = D(ω; R3 ).
Therefore,
un 2H 1 (ω;R3 ) +
∂αβ un · an3 2L2 (ω) = un 2V1n
αβ
→ u2V1 = u2H 1 (ω;R3 ) +
∂αβ u · a3 2L2 (ω) ,
αβ
which, together with the previous estimates, implies that
un H 1 (ω;R3 ) → uH 1 (ω;R3 )
and ∂αβ un · an3 L2 (ω) → ∂αβ u · a3 L2 (ω) .
The first convergence implies that un → u strongly in H 1 (ω; R3 ), and the second
convergence implies that ∂αβ un · an3 → ∂αβ u · a3 strongly in L2 (ω). Both facts imply
that ∂α un · an3 → ∂α u · a3 strongly in H 1 (ω); therefore, by the Sobolev embedding
ρ
n
n
2
theorem and Lemma 4.9, Γn,ρ
α,β ∂ρ u · a3 → Γα,β ∂ρ u · a3 strongly in L (ω), which
completes the proof.
Remark 4.19. Note that the convergences established in Theorem 4.10 are quite
natural in the sense that they imply the convergence of the displacements and of the
associated strain and change of curvature tensors in their respective natural spaces.
This in turn implies the strong L2 convergence of the various stress resultants.
Let us close the article with the final comparison between the classical formulation
of Koiter’s model and our formulation. Let us be given a sequence of charts ϕn as in
Theorem 4.10. For any displacement v ∈ V1n , we denote by vin = v n · ani the covariant
components of v so that
v n (x) = vin (x)an,i (x).
Note that in some sense, in considering the covariant components as the basic unknown
as is classically done, one mixes regularity issues concerning the displacement with
regularity issues concerning the chart. This leads to the restrictive C 3 assumption
that is made in the classical formulation. This remark may be illustrated by the
following result.
894
ADEL BLOUZA AND HERVÉ LE DRET
Theorem 4.20. Let ϕn be a sequence of C 3 -charts such that ϕn →ϕ in W 2,p (ω; R3 )
∗
strong for all 1 < p < +∞ and ϕn / ϕ in W 2,∞ (ω; R3 ) weak-∗ with ϕ piecewise C 3
and ϕ ∈ C 3 (ω; R3 ). Let f n be a sequence of force resultant densities such that f n → f
in L2 (ω; R3 ). Let (un1 , un2 , un3 ) ∈ H01 (ω)×H01 (ω)×H02 (ω) be the solution of the classical
formulation of Koiter’s problem as in [4]. Then, for all n ∈ N,
un (x) = uni (x)an,i (x),
(4.25)
un tends to u in the sense of Theorem 4.10, but un3 is generically unbounded in H 2 (ω).
Proof. The fact that (4.25) holds was already noted in [6]. Clearly, un3 → u3 in
2
L (ω) by Theorem 4.10. If un3 was bounded in H 2 (ω), this would thus imply that
u3 ∈ H 2 (ω). This is not the case. As was already noted in [6], for a piecewise
C 3 -midsurface, the derivatives of the second fundamental form contain Dirac masses
concentrated on the interfaces between the smooth parts of the shell. The condition
∂αβ u · a3 ∈ L2 (ω) is equivalent to ∂α u · a3 ∈ H 1 (ω), which means that the jump of
∂α u·a3 vanishes on each interface. In covariant components, this reads [∂α u3 +bρα uρ ] =
0, or, equivalently, [∂α u3 ] = −[bρα ]uρ , on each interface (with u3 piecewise H 2 ), where
bρα denote the mixed components of the second fundamental form. Since the jump of
bρα is nonzero for some components, this will generically induce a jump on ∂α u3 . The
normal component u3 thus cannot be in H 2 (ω).
Remark 4.21. The previous result indicates that a continuous dependence analysis similar to the present one would be difficult to carry out in the classical formulation.
Some extra conditions would need to be imposed on the sequence of midsurfaces in
order to obtain a uniform H 2 -bound on un3 .
Example 4.22. Let us consider the example of a W 2,∞ -shell made of a plane part
and a circular cylindrical part. We take ω = {−π/2, π/2} × {0, 1} and
ϕ(x) =
(x1 , x2 , 0)T
(sin x1 , x2 , 1 − cos x1 )T
for x1 ≤ 0,
for x1 > 0.
The midsurface of this shell is of class W 2,∞ and has a curvature discontinuity across
x1 = 0. In particular, it is not C 3 and the classical formulation of Koiter’s model is
not applicable. It is easy to construct an explicit sequence of C 3 -midsurfaces ϕn that
converge in all W 2,p and in W 2,∞ weak-∗ by using Hermite interpolation polynomials
in the strip {0 ≤ x1 ≤ 1/n}. The sequence of solutions associated with the sequence
of interpolated midsurfaces falls into the classical framework. The limit midsurface
is not of class C 3 ; thus the limit displacement requires our new formulation. The
interface between the smooth parts is the line {x1 = 0}. On this interface, all mixed
components of the second fundamental form are continuous except b11 , and it is easy
to see that [b11 ] = 1. Therefore, we obtain that [∂1 u3 ] = −u1 (and [∂2 u3 ] = 0). Since,
in general, u1 will not vanish on {x1 = 0}, we see that u3 is not in H 2 (ω). This is
not surprising if we remember that u3 is a covariant component of u, hence a scalar
product of u with the covariant basis vector a3 . The (lack of) regularity of a3 is thus
necessarily reflected in the degree of regularity of u3 .
Acknowledgments. The authors would like to thank V. Girault and L. R. Scott
for fruitful discussions concerning Friedrichs’s lemma and its extension up to the
boundary.
UP-TO-THE-BOUNDARY VERSION OF FRIEDRICHS’S LEMMA
895
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