The infinitesimal rigid displacement lemma in Lipschitz co-ordinates and application to shells with minimal regularity

MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2004; 27:1283–1299 (DOI: 10.1002/mma.501)MOS subject classi�cation: 74K 25

The in�nitesimal rigid displacement lemma in Lipschitzco-ordinates and application to shells

with minimal regularity

Sylvia Anicic1;‡, Herv�e Le Dret2;∗;† and Annie Raoult3;4;§

1MOX; Dipartimento di Matematica F. Brioschi; Politecnico di Milano; Piazza L. da Vinci 32;Milano 20133; Italy

2Laboratoire Jacques-Louis Lions; Universit�e Pierre et Marie Curie; 75252 Paris Cedex 05; France3Laboratoire TIMC=IMAG; Domaine de la Merci; 38706 La Tronche Cedex; France

4Laboratoire de Mod�elisation et Calcul=IMAG; B.P. 53; 38041 Grenoble Cedex 9; France

Communicated by R. P. Gilbert

SUMMARY

We establish a version of the in�nitesimal rigid displacement lemma in curvilinear Lipschitz co-ordinates. We give an application to linearly elastic shells whose midsurface and normal vector areboth Lipschitz. Copyright ? 2004 John Wiley & Sons, Ltd.

KEY WORDS: in�nitesimal rigid displacement lemma; bilipschitz mappings; tubular neighbourhood;shell theory

1. INTRODUCTION

The in�nitesimal rigid displacement lemma in Cartesian co-ordinates states that a distributionu on a connected open subset � of Rn with values in Rn, whose gradient is skew-symmetric,is in point of fact an a�ne function of the form x �→ a + Wx, where W is a n × n skew-symmetric matrix and a a vector in Rn. This result immediately follows from the fact that allsecond order partial derivatives of u can be recovered as linear combinations of �rst orderderivatives of the symmetric part of the gradient of u. If the latter is null, so are the secondorder derivatives.We present here a version of the in�nitesimal rigid displacement lemma written in non-

smooth curvilinear co-ordinates, namely Lipschitz co-ordinates, that is valid for H 1 functions.We thus consider a Lipschitz mapping � on � whose Jacobian is bounded below almost

∗Correspondence to: Herv�e Le Dret, Laboratoire Jacques-Louis Lions, Universit�e Pierre et Marie Curie, 75252 ParisCedex 05, France.

†E-mail: [email protected]‡E-mail: [email protected]§E-mail: [email protected]

Published online 11 May 2004Copyright ? 2004 John Wiley & Sons, Ltd. Received 12 March 2003

1284 S. ANICIC, H. LE DRET AND A. RAOULT

everywhere by a strictly positive constant. Under this hypothesis, we show that if a function Uin H 1(�;Rn) is such that ∇U T∇�+∇�T∇U =0 almost everywhere, which is the algebraiccondition replacing the skew-symmetry of the gradient in the Cartesian case, then U is locallyof the form a+W�. The vector a and the skew-symmetric matrix W may not be globallyconstant. However, if � is in addition assumed to be locally bilipschitz, then both are constanton �, which is the in�nitesimal rigid displacement lemma in our context.The need for an in�nitesimal rigid displacement lemma in Lipschitz co-ordinates stems

from several issues arising in linearly elastic thin shell theory, pertaining in particular to theKoiter model and its variants, in the context of shells of minimal regularity.The regularity of the midsurface �rst plays a role in the derivation of the Koiter model

[1]. The idea is to write the linearized elasticity equations in a curvilinear system adaptedto the geometry of the shell, namely the tubular neighbourhood mapping of the midsurfaceassociated with a given chart. Several approximations—Kirchho�–Love and plane stress hy-potheses in particular—and integrations across the thickness of the shell eventually lead tothe Koiter model. From the modelling point of view, both in terms of geometry and in termsof mechanics, it is important that the tubular neighbourhood mapping in question be globallyinjective when the thickness is small enough. This follows classically from the inverse func-tion theorem and a compactness argument when the midsurface chart is of class C2, see forinstance Reference [2].Regularity of the midsurface also came into play in the �rst proof of existence and unique-

ness for the Koiter model given by Bernadou and Ciarlet [3], who assumed the midsurfacechart to be of class C3. This assumption was necessary to make sense of a speci�c termcontaining derivatives of the second fundamental form that appears in the classical expressionfor the change of curvature tensor of a shell displacement in terms of covariant derivatives.The assumption is, however, unduly restrictive from the point of view of the applications. Inparticular, very commonplace shells such as a square plate continued by a circular cylinderare excluded. Moreover, the exact status of what it means for an actual midsurface to be C3

can be questioned from the point of view of modelling, since the C3 character does not de-scribe a geometric feature of the midsurface. In addition, existence for the three-dimensionallinearized elasticity problem only requires a Lipschitz boundary. Since the boundary of thethree-dimensional shell includes images of the midsurface through the tubular neighbourhoodmapping, the jump from Lipschitz to C3, when passing from the three-dimensional problemto the shell problem, certainly cannot be optimal.It is therefore important on several accounts to try and lower the required regularity for

the midsurface of a shell as much as possible. This was �rst achieved by Blouza and LeDret in Reference [4] for the in�nitesimal rigid displacement lemma for a midsurface ofclass W 2;∞. Existence and uniqueness for the Koiter model was obtained in this context inReference [5], with several generalizations worked out in detail in Reference [6]. The keyidea to lower the regularity requirements is to forsake the use of covariant components andderivatives and to express the quantities of interest directly in terms of vector unknowns. Itturns out that this approach radically simpli�es the expressions of the change of metric andchange of curvature tensors. In particular, the term that formerly forced the chart to be C3 nolonger appears. This is due to the fact that, since covariant components are scalar productsof the vector displacement with the covariant basis, regularity of the chart and regularity ofthe displacement are entangled in the classical formulation. The vector approach also madeit possible to study the continuous dependence of the solution of Koiter’s model on the

Copyright ? 2004 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2004; 27:1283–1299

THE INFINITESIMAL RIGID DISPLACEMENT LEMMA IN LIPSCHITZ CO-ORDINATES 1285

midsurface [7] and also applies to more general models such as Naghdi’s model [8]. Let usalso mention that an alternate approach of W 2;∞ or C1;1 shells was developed in Reference[9], leading to a whole hierarchy of shell models.Even though the W 2;∞ framework includes such shells of interest as the planar/cylindrical

shell mentioned above, the approach was not conducted to its logical conclusion in the aboveworks. In e�ect, the change of curvature tensor still contained Christo�el symbols which pre-vented the regularity of being lowered further. It was noticed by Anicic in Reference [10] thatthe vector idea could be pursued even beyond the W 2;∞ case by introducing the in�nitesimalrotation vector in the de�nition of the function space for Koiter’s model. The obtained modelthen continued to make sense for shells with a midsurface chart of class W 1;∞ only, possess-ing a unit normal vector almost everywhere and such that this normal vector is itself of classW 1;∞. This generalization is especially interesting since it includes some G1 surfaces arisingin CAD after reparametrization. Such surfaces are de�ned via smooth patches with matchingnormal vectors on their interfaces, (see Reference [11]). In Anicic’s formulation of Koiter’smodel and its variants, there is no need for matching normal derivatives of the charts acrossthe patch interfaces, which makes for great versatility in practice.Several questions were however not entirely settled in Reference [10]. First of all, the

injectivity of the tubular neighbourhood mapping of such midsurfaces remained an assumption.Secondly, existence and uniqueness for Koiter’s model was only proved for piecewise W 2;∞

midsurfaces. Our second purpose in this article is to address both of these issues.Interestingly, these issues are connected to each other. We slightly modify the setting intro-

duced in Reference [10] by assuming that the midsurface chart is bilipschitz with a Lipschitznormal vector. In this case, the tubular neighbourhood mapping is obviously Lipschitz. Weprove that it is globally injective when the thickness is small enough, by showing directly,without an inverse function theorem, that it is in fact locally bilipschitz near the midsurface.This injectivity result allows us to deduce an in�nitesimal rigid displacement lemma for the

midsurface. We introduce the Kirchho�–Love displacement associated with a displacement ofthe midsurface and check that the vanishing of its change of metric and change of curvaturetensors is equivalent to the algebraic condition for the three-dimensional in�nitesimal rigiddisplacement lemma in Lipschitz co-ordinates. This particular point was worked out in thepresent context of regularity in Reference [10] and in Reference [12] in the C3 case. Existenceand uniqueness for the Koiter model then follow along the lines of Reference [6].The idea of introducing the three-dimensional Kirchho�–Love displacement associated with

a two-dimensional shell displacement and using known three-dimensional results to deducetwo-dimensional results was already mentioned in Reference [13]. It was implemented inReference [12], to establish the ellipticity of Koiter’s model directly from the three-dimensionalKorn inequality and also to prove the in�nitesimal rigid displacement lemma, for shells ofclass C3, and in Reference [14] to establish Korn inequalities on a surface also of class C3.It is a legitimate question to wonder whether it is possible to further reduce the regularity

of the midsurface while retaining the injectivity of the tubular neighbourhood mapping and theexistence and uniqueness results. Some kind of continuity of the tangent plane is obviouslyrequired, otherwise the context would rather be that of folded shells or junctions of shells,an altogether di�erent topic, see References [10,12,15–20] among others. It is interesting tonote that one of the intermediary results that we obtain in the course of the proof of thebilipschitz character of the tubular neighbourhood mapping is one such geometrical continuitycondition. Namely, we show that our midsurfaces are such that their contingent cone of



Bouligand remains everywhere normal to the normal vector. In this sense, Lipschitz surfaceswith Lipschitz normal vectors (or actually, only continuous normal vectors for this particularproperty) do possess the kind of regularity that di�erentiate them from folded or even moreirregular Lipschitz surfaces. It seems to be di�cult to �nd a less regular framework thanthe one used here that would still be adequate for shell theory, hence the admittedly slightlyoptimistic ‘minimal regularity’ advertized in the title of this article.

2. A LIPSCHITZ VERSION OF THE n-DIMENSIONAL INFINITESIMALRIGID DISPLACEMENT LEMMA

Let An denote the set of n × n skew-symmetric matrices. The main result of this section isthe following.

Theorem 2.1Let � be a bounded connected open subset of Rn and let �∈W 1;∞(�;Rn) be such thatdet∇�¿�¿0 almost everywhere. Let U ∈H 1(�;Rn) satisfy

∇U T∇�+∇�T∇U =0 almost everywhere in � (1)

Then there exist a dense open subset O of � and locally constant mappings a and W fromO into Rn and An, respectively, such that U (x)= a(x) +W (x)�(x) in O.

We �rst recall some well-known results concerning the relationship between Lipschitz func-tions and W 1;∞ functions.

Theorem 2.2Let � be a bounded, connected, open, Lipschitz subset of Rn. Then we have W 1;∞(�)=C0;1( ��) algebraically and topologically, and there exists a constant C� such that for all x; y∈ ��and all u∈W 1;∞(�),

|u(x)− u(y)|6C�‖∇u‖L∞(�)‖x − y‖This result is not easy to �nd in the literature, for instance it is mentioned in passing and

without proof in Reference [21]. Therefore, we include a proof in the appendix for the reader’sconvenience. Another classical result that will be of use is Rademacher’s theorem stating thata locally Lipschitz function on Rn is almost everywhere di�erentiable, see Reference [22] for arecent proof. It follows that its distributional di�erential coincides with its almost everywheredi�erential.Next, we recall an almost everywhere inverse function theorem for Sobolev mappings given

in Reference [23].

Theorem 2.3Let �∈W 1;n(�;Rn) be such that det∇�¿0 almost everywhere. There exists a set of zeromeasure N ⊂� such that, for all x0 =∈N , there exists a scalar r(x0)¿0, an open set x0 ∈Dx0b�and a mapping wx0 from Bx0 =B(�(x0); r(x0)) into Dx0 such that

(i) wx0 ∈W 1;1(Bx0 ;Rn),(ii) wx0 ◦�(x)= x, almost everywhere in Dx0 ,



(iii) � ◦ wx0 (y)=y, almost everywhere in Bx0 ,(iv) ∇wx0 (y)= (∇�(wx0 (y)))−1, almost everywhere in Bx0 .The exceptional set N that appears in the proof of Theorem 2.3 is the set of points where

� fails to be di�erentiable, viz. Rademacher’s theorem, or where det∇�60.Corollary 2.4If �∈W 1;∞(�;Rn) is such that det∇�¿�¿0 almost everywhere, then wx0 ∈W 1;∞(Bx0 ;Rn)and equalities (ii) and (iii) hold everywhere on their respective domains.

ProofThis is clear since (∇�(x))−1 = (det∇�(x))−1(cof ∇�(x))T.Proof of Theorem 2.1Let x0 be a point in �\N . By Theorem 2.2 and Corollary 2.4, the function wx0 is a bilipschitzhomeomorphism between Bx0 and Dx0 . It follows from Reference [24, p. 16], that V =U ◦wx0belongs to H 1(Bx0 ;RN ) with

∇V (y)=∇U (wx0 (y))(∇�(wx0 (y)))−1

almost everywhere in Bx0 . We thus obtain

∇V +∇V T =∇U∇�−1 +∇�−T∇U T

=∇�−T(∇�T∇U +∇U T∇�)∇�−1 = 0

almost everywhere in Bx0 . By the classical in�nitesimal rigid displacement lemma, there existsa vector a(x0)∈RN and a skew-symmetric matrix W (x0)∈AN such that

V (y)= a(x0) +W (x0)y in Bx0

that is to say

U (x)= a(x0) +W (x0)�(x) in Dx0

Let us now set O=⋃x =∈N Dx. It is an open subset of � and since �\N ⊂O, it follows that

�\O⊂N is of zero measure. Therefore O is dense in �.To conclude, consider two points x and x′ such that Bx ∩Bx′ = ∅, or in other words

Dx ∩Dx′ = ∅. Clearly, wx=wx′ on Bx ∩Bx′ , thus the a�ne functions a(x)+W (x)y and a(x′)+W (x′)y coincide on the non-empty open set Bx ∩Bx′ . Therefore a(x)= a(x′) and W (x)=W (x′).It follows that the functions a and W are constant on each connected component of O.

In the case of a mapping � that de�nes a local bilipschitz change of coordinates, we havea stronger alternate version.

Theorem 2.5Under the previous hypotheses, if � is in addition locally bilipschitz, then there exist a vectora and a skew-symmetric matrix W such that

U (x)= a+W�(x) in � (2)



ProofSame proof, but working directly on �, without requiring the use of Theorem 2.3.

RemarkIt is not clear whether there are examples in which the mappings a and W of Theorem 2.1actually are non-constant on �. However, Theorem 2.5 seems to indicate that possible non-constancy is more connected with the lack of local invertibility of � than with its lack ofdi�erentiability (as indicated by the exceptional set N ). We describe below a C∞ examplewith non-constant W . Unfortunately, this example does not satisfy the hypothesis on the boundbelow for the Jacobian, but only the less demanding hypothesis det∇�¿0 almost everywhere.The example is as follows.Let �= {x = (x1; x2); |x1|¡1; |x2|¡1}, �+ =�∩{x2¿0} and �−=�∩{x2¡0}. We de�ne

�(x) =

(x1x22

x2

)

We have det∇�(x)= x22¿0 almost everywhere and � is not locally injective on the line{x2 = 0}. The complement of this line has two connected components, �− and �+. Let now

U (x)=0 in �−; U (x)=(

x2−x1x22

)in �+

Clearly U ∈H 1(�;R2) and ∇U T∇�+∇�T∇U =0 almost everywhere in �. Moreover,

U (x)=

(0 0

0 0

)�(x) in �− and U (x)=

(0 1−1 0

)�(x) in �+

It should also be noted that there are mappings � satisfying all the hypotheses of Theo-rem 2.1 and that are not locally bilipschitz, but such that a and W are nonetheless constanton �. One such example is the example of Ball, see Reference [25], where � is the unit diskof R2 and

�(x) =1√

2(x21 + x22)

(x21 − x222x1x2

)

This mapping is such that det∇�(x)=1 almost everywhere and � is not locally injective,hence not locally bilipschitz, and not di�erentiable at 0. So in this case, the exceptional setis N = {0}. However, the complement of N is connected, therefore a and W are constant byTheorem 2.1.

3. SHELLS OF MINIMAL REGULARITY: GEOMETRICAL ASPECTS

The classical geometrical setting for a thin shell is as follows. Let ! be a bounded openconnected Lipschitz subset of R2 and ’ : �!→R3 a su�ciently regular chart for the midsurfaceof the shell. Let a3 = @1’ ∧ @2’=‖@1’ ∧ @2’‖ be the unit normal vector to the midsurface inthe chart ’. The tubular neighbourhood mapping of the midsurface is de�ned by

�(x; x3)=’(x) + x3a3(x) (3)



It is globally injective on �!× [−h; h] for h¿0 small enough when ’ is of class C2. This injec-tivity results from the inverse function theorem for local injectivity followed by a compactnessargument, see for instance Reference [2] or Reference [26].The global injectivity of the tubular neighbourhood mapping is a crucial, although some-

times overlooked, element of thin shell modelling. It could even be said that it is the dis-tinguishing factor for surfaces that may qualify as midsurface of a thin shell. In attemptingto lower the regularity of the midsurface, it is thus important to retain the injectivity of thetubular neighbourhood mapping.For the W 2;∞ or C1;1 case, we refer to Reference [9] for a possible approach based on

the signed distance function. We are interested here in an even lower regularity framework,basically introduced in Reference [10] and slightly modi�ed here.Let ! be as above and consider a mapping ’ : �!→R3 that will describe the midsurface of

a shell.

Hypothesis 3.1We assume that ’ is bilipschitz, i.e. there exist two constants 0¡�6� such that

∀x; y∈ �!; �‖y − x‖6‖’(y)− ’(x)‖6�‖y − x‖ (4)

Let us now turn to the normal vector. By Rademacher’s theorem, ’ is almost everywheredi�erentiable. As a consequence of Hypothesis 3.1, we �rst note that the ‘tangent’ vectors arealmost everywhere uniformly linearly independent, in the following sense.

Lemma 3.2At all points x of di�erentiability of ’, we have

‖@1’(x) ∧ @2’(x)‖¿�2 (5)

ProofLet x∈! be a point of di�erentiability of ’. For all y∈R2 and s∈R su�ciently small, wehave by Hypothesis 3.1

‖’(x + sy)− ’(x)‖¿�|s| ‖y‖Dividing by s and letting s tend to 0, we obtain

‖∇’(x)(y)‖¿�‖y‖The 3 × 2 matrix A=∇’(x) is thus such that ‖Ay‖¿�‖y‖ for all y∈R2, that is to say

yTATAy¿�2yTy. If v16v2 denote the singular values of A, this implies that �26v216v22.

In particular, v1v2¿�2.Let a1 = @1’(x) and a2 = @2’(x) be the two column vectors of A and let us choose a

third vector A3 such that a1 · A3 = a2 · A3 = 0 and ‖A3‖=1. Let A be the 3× 3 matrixcomposed of these three vectors. We have det A=A3 · (a1 ∧ a2), so that by our choice ofA3; |det A|= ‖a1 ∧ a2‖.By construction, we have

ATA=(ATA 00 1

)



written as a block-matrix, so that (det A)2 = det(ATA)=det(ATA)= v21v22. Hence, |det A|=

v1v2¿�2.

Lemma 3.2 shows that ’( �!) is a surface in R3 in the sense that it has a tangent planealmost everywhere. Moreover, we can de�ne a unit normal vector almost everywhere by

a3(x)=@1’(x)∧ @2’(x)‖@1’(x)∧ @2’(x)‖ (6)

By construction, we have a3 ∈L∞(!;R3). We now make an additional regularity hypothesisthat was introduced in [10].

Hypothesis 3.3We assume that

a3 ∈W 1;∞(!;R3) (7)

In other words, we assume that the vector a3 that was a priori only de�ned almost everywhere,coincides with a Lipschitz function de�ned on �!.

RemarkIt is important to note that only the unit normal vector is assumed to be Lipschitz, whereasthe vector @1’∧ @2’ remains solely in L∞(!;R3). A concrete example of this situation isgiven by the piecewise W 2;∞ surfaces considered in [10].

We next de�ne the tubular neighbourhood mapping by formula (3) as usual. Under Hy-potheses 3.1 and 3.3, the mapping � is clearly Lipschitz on �!× [−h; h] for all h¿0. Ournext goal is to prove that it is locally bilipschitz for h small enough. The proof proceeds inseveral steps. First is the result for the restriction of � to all sheaves �!×{x3}, |x3|6h. Recallthat C! denotes the constant depending on ! of Theorem 2.2.

Lemma 3.4For h= �(2C!‖∇a3‖L∞(!))−1, we have

‖�(x; x3)−�(y; x3)‖¿�2‖x − y‖ (8)

for all |x3|6h and x; y∈ �!.

ProofLet La3 =C!‖∇a3‖L∞(!). By the triangle inequality, we obtain

‖�(x; x3)−�(y; x3)‖¿‖’(x)− ’(y)‖ − |x3| ‖a3(x)− a3(y)‖¿(�− hLa3)‖x − y‖hence the result.

RemarkThe value �=2 in formula (8) is of course arbitrary. Any value of h strictly less than �=La3will do. Note also that this result shows that the boundary of the three-dimensional shell isLipschitz, hence the three-dimensional linearized elasticity problem is well-posed.



The next result can be construed as a kind of geometric regularity for the midsurface.

Lemma 3.5For all x0 ∈ �!, we have

’(x)− ’(x0)‖x − x0‖ · a3(x0)→ 0 when x→ x0 (9)

RemarkThe geometric meaning of Lemma 3.5 is that the Bouligand contingent cone to the surface(see Reference [27]) always remains orthogonal to the normal vector. Note that the resultis obvious at every point x of di�erentiability of ’, that is to say almost everywhere in !.However, we need it not only almost everywhere, but everywhere, which we now proceed toprove.

ProofLet us �x a point x0 ∈ �!. The function x �→P(x)= (’(x)−’(x0)) ·a3(x0) is Lipschitz on �! andits distributional=almost everywhere gradient is given by ∇P(x)=∇’(x)Ta3(x0). Therefore,for all x∈ �!,

|P(x)| = |P(x)− P(x0)|6C!‖∇’Ta3(x0)‖L∞( �B(x0 ;‖x−x0‖))∩!‖x − x0‖Now, almost all points of �B(x0; ‖x − x0‖)∩! are points of di�erentiability of ’. Hence, atany such point y, a3(y) is orthogonal to the image of ∇’(y) by construction, that is to say∇’(y)Ta3(y)=0. Consequently, we have that, almost everywhere in �B(x0; ‖x − x0‖)∩!,

∇’(y)Ta3(x0)=∇’(y)Ta3(x0)−∇’(y)Ta3(y)so that

‖∇’(y)Ta3(x0)‖6 ‖∇’(y)T‖ ‖a3(x0)− a3(y)‖

6C!‖∇a3‖L∞(!)‖∇’(y)T‖ ‖y − x0‖

almost everywhere. Therefore,

‖∇’Ta3(x0)‖L∞( �B(x0 ;‖x−x0‖))∩!6C!‖∇a3‖L∞(!)‖∇’‖L∞(!)‖x − x0‖hence the result.

RemarkIn Reference [28] we announced a proof using Clarke’s generalized gradients, see Reference[29]. However, in the meantime we found the above much simpler and more elementary proof.Note that it is enough for a3 to be continuous for Lemma 3.5 to hold true.

Next, we have a similar result for the normal vector.

Lemma 3.6For all x0 ∈ �!, we have

a3(x)− a3(x0)‖x − x0‖ · a3(x0)→ 0 when x→ x0 (10)



ProofLet xn ∈ �! be a sequence that tends to x0. The vectors (a3(xn)−a3(x0))=‖xn−x0‖ remain in theball �B(0; La3). We can thus extract a subsequence x

n′ such that (a3(xn′)−a3(x0))=‖xn′ −x0‖→ �

for some vector �∈ �B(0; La3). Now we have

(a3(xn′)− a3(x0)) · a3(x0)= a3(xn′) · (a3(x0)− a3(xn′))

Dividing by ‖xn′ − x0‖ and passing to the limit, we thus obtain that � · a3(x0) = −a3(x0) · �,hence the result by uniqueness of the limit.

RemarkGeometrically speaking, the result is due to the fact that a3 is S2-valued. Therefore, theBouligand contingent cone of its image at a given point a3(x0) is obviously included in thetangent plane to the sphere at that same point.

We now are in a position to prove the main result of this section.

Theorem 3.7Let h be as above. Then, the tubular neighbourhood mapping � is locally bilipschitz on�!× [−h; h].ProofWe argue by contradiction. Let us thus assume that � is not locally bilipschitz at a point(x0; x3)∈ �!× [−h; h]. There exists a sequence (xn; xn3)→ (x0; x3) in �!× [−h; h] such that

‖�(xn; xn3)−�(x0; x3)‖2¡1n(‖xn − x0‖2 + |xn3 − x3|2)

In view of the above inequality, we can be assured that xn3 = x3 or xn = x0. We observe that

�(xn; xn3)−�(x0; x3) =�(xn; xn3)−�(x0; xn3) + �(x0; xn3)−�(x0; x3)

=�(xn; xn3)−�(x0; xn3) + (xn3 − x3)a3(x0)

Therefore,

‖�(xn; xn3)−�(x0; x3)‖2 = ‖�(xn; xn3)−�(x0; xn3)‖2 + |xn3 − x3|2

+ 2(xn3 − x3)(�(xn; xn3)−�(x0; xn3)) · a3(x0)

¿�2

4‖xn − x0‖2 + |xn3 − x3|2

+ 2(xn3 − x3)(�(xn; xn3)−�(x0; xn3)) · a3(x0)

by Lemma 3.4.



Let �=min(�2=4; 1). We deduce from the above inequalities that

2(xn3 − x3)‖xn − x0‖((’(xn)− ’(x0) + xn3(a3(xn)− a3(x0)))=‖xn − x0‖) · a3(x0)‖xn − x0‖2 + |xn3 − x3|2

61n− �

Now, for all couples of non-zero real numbers (A; B) we have −162AB=(A2 +B2)61. There-fore, the left-hand side of the above inequality tends to 0 when n→+∞ by Lemmas 3.5 and3.6, whereas the right-hand side tends to −�¡0, which is a contradiction.Corollary 3.8The mapping � is locally injective on �!× [−h; h].We can then follow Reference [24] or Reference [26] to obtain the global tubular neigh-

bourhood property.

Theorem 3.9There exists 0¡h′¡h such that the mapping � is globally injective on �!× [−h′; h′].In other words, a midsurface satisfying (4)–(7) allows for a correct geometrical modelling

of a thin shell of thickness 2h′.

4. SHELLS OF MINIMAL REGULARITY: THE INFINITESIMALRIGID DISPLACEMENT LEMMA

Most existence and uniqueness proofs for linear shell models rest on an in�nitesimal rigiddisplacement lemma for the midsurface. Such a lemma is useful in proving the ellipticityof the bilinear form of the variational formulation of the equilibrium problem, [3,6,8]. Seehowever Reference [12] for an ellipticity proof using directly the three-dimensional Korninequality, hence implicitly the three-dimensional in�nitesimal rigid displacement lemma, inthe C3 regularity setting.In the sequel, we will use the summation convention unless otherwise speci�ed. Let a�= @a’

and a� be the covariant and contravariant basis vectors on the midsurface. Note that the covari-ant vectors are only in L∞(!;R3) and are uniformly almost everywhere linearly independentin the sense of Lemma 3.2. The contravariant vectors are uniquely de�ned as elements ofL∞(!;R3) by their belonging to the plane spanned by a1 and a2 and the relations a� ·a�= ��almost everywhere. It is a simple matter to verify that they satisfy the same properties as thecovariant vectors, see Reference [10].The function space for Koiter’s model introduced in Reference [10] that generalizes that

initially proposed in References [4,5] is as follows. Let u∈H 1(!;R3) be a midsurface dis-placement (vector-valued, but seen through the chart ’). The in�nitesimal rotation vectorassociated with u is de�ned as

�(u)= (@�u · a3) a� (11)

It is a priori in L2(!;R3). The appropriate function space is then

V = {u∈H 1(!;R3); �(u)∈H 1(!;R3)} (12)



endowed with its natural Hilbert norm

‖u‖V =(‖u‖2H 1(!;R3) + ‖�(u)‖2H 1(!;R3))1=2 (13)

Note that if u∈V , then the vector �(u) is in H 1, but each term @�u · a3 is only in L2.In this context, the covariant components of the change of metric tensor read

��(u)= 12(@�u · a� + @�u · a�) (14)

and the covariant components of the change of curvature tensor read

��(u)= @��(u) · a� − @�u · @�a3 (15)

Note that in spite of appearances, formula (15) is symmetric in � and �, as it shouldindeed be, see References [10] or [30]. Clearly, for u∈V , the change of metric and change ofcurvature tensors �(u)= ��(u) a�⊗ a� and (u)=��(u) a�⊗ a� are both square-integrable.Naturally, in Koiter’s model, the bilinear form is the sum of a positive de�nite quadraticform in �(u) corresponding to membrane energy and a positive de�nite quadratic form in(u) corresponding to bending energy, so the functional setting is well-suited in this respect.It can of course be checked that formulas (14) and (15) coincide with the lengthy and

cumbersome classical expressions in terms of covariant components of the displacement andvarious covariant derivatives when ’ is of class C3, see References [6,10]. There is also acanonical correspondence between the function spaces of each formulation in this case andthe solutions coincide. Moreover, in Reference [7], it is shown that the simpler vector-valuedformalism in the W 2;∞ case arises as a natural limit of the classical C3 case, by means of aresult of continuity with respect to the midsurface.We now use the results of the previous sections to prove the main goal of this section—the

in�nitesimal rigid displacement lemma for the midsurface.

Theorem 4.1Let ’ be a chart satisfying (4)–(7) and u∈V be such that �(u)=(u)=0. Then there existtwo vectors a and b of R3 such that

u(x)= a+ b ∧ ’(x) in �!

ProofThe idea is to use the Kirchho�–Love displacement associated with u

U (x; x3)= u(x)− x3�(u) (x) (16)

on �=!× ] − h, h[, with h given by Lemma 3.4 such that � is locally bilipschitz on �.Since u∈V , it is clear that U ∈H 1(�;R3) (and conversely).It is known, see References [10,12,31], that Kirchho�–Love displacements are characterized

by the relations (∇U T∇�+∇�T∇U )i3 = 0, with i=1; 2; 3. In fact, since

∇U =(@1u− x3@1�(u)|@2u− x3@2�(u)| − �(u))



and

∇�=(a1 + x3@1a3|a2 + x3@2a3|a3)we have

(∇U T∇�)33 =−�(u) · a3(∇U T∇�)�3 = @�u · a3 − x3@��(u) · a3(∇�T∇U )�3 =−�(u) · a� − x3�(u) · @�a3

Now, clearly �(u) · a3 = 0 and since �(u)∈H 1 and a3 ∈W 1;∞, Leibniz’s formula is valid and@��(u) · a3 + �(u) · @�a3 = 0. Moreover, �(u) · a�= @�u · a3 by de�nition of �(u).Performing the same computations for the tangential components, we obtain

(∇U T∇�)�� = (@�u− x3@��(u)) · (a� + x3@�a3)

= @�u · a� + x3(@�u · @�a3 − @��(u) · a�)− x23@��(u) · @�a3(∇�T∇U )�� = @�u · a� + x3(@�u · @�a3 − @��(u) · a�)− x23@��(u) · @�a3

so that

(∇U T∇�+∇�T∇U )�� =2��(u)− 2x3��(u)

− x23(@��(u) · @�a3 + @��(u) · @�a3) (17)

Since a3 is a unit vector, @�a3 is orthogonal to a3 almost everywhere. Therefore, we canwrite @�a3 = b

�� a� with b

�� = @�a3 ·a� ∈L∞(!) (which are the mixed components of the second

fundamental form when ’ is regular). It follows that

@��(u) · @�a3 = b�� @��(u) · a�

= b�� (��(u) + @�u · @�a3)

= b�� (��(u) + b��@�u · a�)

so that

@��(u) · @�a3 + @��(u) · @�a3 = b��(u) + b��(u) + 2b�� b��(u) (18)

If we assume now that ��(u)=��(u)=0, then we conclude from formulas (17) and (18)that (∇U T∇�+∇�T∇U )��=0.



We can therefore use Theorem 2.5 to deduce that there exist two vectors a and b such that

U (x; x3)= a+ b ∧�(x; x3)Setting x3 = 0 in the above equality yields the result.

RemarkThe algebra leading to formulas (17) and (18) is well-known, although not exactly under thisform. It was worked out as shown here under the same regularity hypotheses in Reference [10].The idea of using the three-dimensional Kirchho�–Love displacement associated with a shelldisplacement was also used in Reference [12] to establish the in�nitesimal rigid displacementlemma for a surface in the C3 case. The same idea was also used in Reference [14] to proveKorn inequalities on a surface of class C3.

RemarkIn References [10,30], Anicic introduced a new change of curvature tensor whose covariantcomponents are ��(u)=��(u) − b��(u) − b��(u). This tensor is better adapted to mea-suring the variations of the principal curvatures of the midsurface. It is then clear that a shelldisplacement such that �(u)= (u)=0 is also an in�nitesimal rigid displacement.

Once the in�nitesimal rigid displacement lemma for a surface is established, it is a simplematter to reproduce the argument of [3,18] and prove the ellipticity of the bilinear form inKoiter’s model, hence the existence and uniqueness of its solution, for midsurfaces satisfyingthe minimal regularity hypotheses 3.1 and 3.3.

APPENDIX A

We give here a proof of Theorem 2.2. Recall �rst that the geodesic distance between twopoints in � is the in�mum of all lengths of paths that connect the two points in �.

Proposition A.1Let � be a bounded, open, connected, Lipschitz subset of Rn and d� denote the geodesicdistance in �. There exists a constant C� such that

∀x; y∈ ��; d�(x; y)6C�‖x − y‖ (A1)

ProofWe �rst note that the geodesic distance is bounded from above. Indeed, since @� is compactand � is Lipschitz, there exists r¿0 and a �nite number of open cubes Ci; i=1; : : : ; k,of edge 2r that cover @� and such that Ci ∩� is the hypograph of a Lipschitz function�i : RN−1→R in a co-ordinate system attached to the cube. We may assume that the centresci of these cubes lie in �. We cover the rest of � with more cubes, possibly smaller butstill in �nite number i= k + 1; : : : ; p, that do not intersect @�. Obviously, since � is openand connected, it is arcwise connected, and any couple of centres can be connected by a pathof �nite length. Therefore max16i; j6p d�(ci; cj)=M¡+∞. Moreover, for all x; y∈ ��, thereexists i and j such that x∈Ci and y∈Cj. Therefore, we have d�(x; y)6d�(x; ci)+d�(ci; cj)+d�(cj; y)6M + d�(x; ci) + d�(y; cj). We just have to estimate the last two terms.



If k + 16i6n, then Ci ⊂ � and is convex, hence d�(x; ci)= ‖x − ci‖¡√nr. If 16i6k,

let Mi be the Lipschitz constant of �i. We denote by (z′; zn) the co-ordinate system adaptedto the cube, in the sense that Ci ∩�= {z=(z′; zn)∈Ci; zn¡�i(z′)}. Let �¿0 be such that(z′; �i(z′)− �)∈Ci for all z′ ∈C ′

i . Such an � obviously exists. We connect x to ci via a three-legged path in � as follows: Let �1 be the ‘vertical’ segment {(x′; (1− �)xn + �(�i(x′)− �)),�∈ [0; 1]}. Then let �2 be the curve {((1−�)x′+�c′i ; �i((1−�)x′+�c′i)− �), �∈ [0; 1]}, whichfollows the boundary inside �, and draw a �nal vertical segment �3 given by {(c′i ; �ci;n +(1− �)(�i(c′i)− �)), �∈ [0; 1]}. It is fairly clear that the length of �1 and �3 is bounded fromabove by 2r and that the length of �2 is bounded by 2r(1+M 2

i )1=2. Putting all these estimates

together, we obtain the global bound

d�(x; y)6M + 2max(√nr; 2r

(2 + max

16i6k(1 +M 2

i )1=2))

Let us now turn to the equivalence. We argue by contradiction. Assume that there is asequence x j, yj ∈ �� such that

d�(x j; y j)¿j‖x j − yj‖ (A2)

Extracting a subsequence, we may assume that x j→ x∈ �� and yj→y∈ ��. Due to the remarkabove, we have x=y. If x∈�, there exists a ball B(x; r′)⊂� such that x j and yj eventuallyend up in this ball. Therefore, d�(x j; y j)= ‖x j−yj‖ for j large enough in this case. It followsthat x∈ @�. In this case, a similar argument as above with a slightly di�erent path yields anestimate of the form d�(x j; y j)6

√2((1+M 2

i )1=2 +Mi)‖x j−yj‖ for j large enough, for some

i such that x∈Ci, hence a contradiction.

A.1. The paths used in the proof of Proposition A.1

Lemma A.2Let u∈W 1;∞(�). Then u is a Lipschitz function on �� such that for all x; y∈ ��,

|u(x)− u(y)|6C�‖∇u‖L∞(�)‖x − y‖ (A3)

ProofWe know that for all u∈W 1;∞(�) and all x; y∈�,

|u(x)− u(y)|6‖∇u‖L∞(�)d�(x; y)

see Reference [32]. Since � is Lipschitz, we deduce estimate (A3) from Lemma 5.1 for allx; y∈�. It follows that u is uniformly continuous on �, hence it has a unique continuousextension to �� which still satis�es (A3).

Lemma A.3Let u∈C 0;1( ��). Then u∈W 1;∞(�), its distributional di�erential coincides with its almosteverywhere di�erential and

‖∇u‖L∞(�)6 sup{ |u(x)− u(y)|

‖x − y‖ ; x; y∈ ��; x =y}

(A4)



ProofLet L= sup{|u(x) − u(y)|=‖x − y‖; x; y∈ ��; x =y} be the Lipschitz constant of u. For all�′b�, the translates �hu(x)= u(x+ h) are well-de�ned for ‖h‖ small enough and obviously

‖�hu− u‖L∞(�′)6L‖h‖Therefore, u∈W 1;∞(�) and ‖∇u‖L∞(�)6L.In addition, for h= sei, s∈R∗ and ei a basis vector, it is trivial that (�hu − u)=s→ @iu in

D′(�′). Since the di�erential quotients also converge almost everywhere by Rademacher’stheorem and are dominated by a constant L which is integrable on �′ bounded, they convergein L1(�′) to their almost everywhere limit. Therefore, the distributional and almost everywherederivatives coincide.

RemarkNote that � does not need to be Lipschitz for Lemma A.3.

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The infinitesimal rigid displacement lemma in Lipschitz co-ordinates and application to shells with minimal regularity