The infinitesimal rigid displacement lemma in Lipschitz coordinates and application to shells with minimal regularity

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1 The infinitesimal rigid displacement lemma in Lipschitz coordinates and application to shells with minimal regularity Sylvia Anicic, Hervé Le Dret, Annie Raoult Abstract We establish a version of the infinitesimal rigid displacement lemma in curvilinear Lipschitz coordinates. We give an application to linearly elastic shells whose midsurface and normal vector are both Lipschitz. Résumé On établit une version du lemme du mouvement rigide infinitésimal en coordonnées curvilignes lipschitziennes. On en donne une application aux coques linéairement élastiques dont la surface moyenne et le vecteur normal sont lipschitziens. 1 Introduction The infinitesimal rigid displacement lemma in Cartesian coordinates states that a distribution u on a connected open subset Ω of R n with values in R n whose gradient is skew-symmetric, is in point of fact an affine function of the form x a + W x, where W is a n n skew-symmetric matrix and a a vector in R n. This result immediately follows from the fact that all second order partial derivatives of u can be recovered as linear combinations of first order derivatives of the symmetric part of the gradient of u. If the latter is null, so are the second order derivatives. MOX, Dipartimento di Matematica F. Brioschi, Politecnico di Milano, Piazza L. da Vinci 32, Milano, Italy, sylvia.anicic@mate.polimi.it Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris Cedex 05, France, ledret@ccr.jussieu.fr Laboratoire TIMC/IMAG, Domaine de la Merci, La Tronche Cedex and Laboratoire de Modélisation et Calcul/IMAG, B.P. 53, Grenoble Cedex 9, France, annie.raoult@imag.fr 1

2 We present here a version of the infinitesimal rigid displacement lemma written in nonsmooth curvilinear coordinates, namely Lipschitz coordinates, that is valid for H 1 functions. We thus consider a Lipschitz mapping Ψ on Ω whose Jacobian is bounded below almost everywhere by a strictly positive constant. Under this hypothesis, we show that if a function U in H 1 (Ω; R n ) is such that U T Ψ+ Ψ T U = 0 almost everywhere, which is the algebraic condition replacing the skew-symmetry of the gradient in the Cartesian case, then U is locally of the form a + W Ψ. The vector a and the skew-symmetric matrix W may not be globally constant. However, if Ψ is in addition assumed to be locally bilipschitz, then both are constant on Ω, which is the infinitesimal rigid displacement lemma in our context. The need for an infinitesimal rigid displacement lemma in Lipschitz coordinates stems from several issues arising in linearly elastic thin shell theory, pertaining in particular to the Koiter model and its variants, in the context of shells of minimal regularity. The regularity of the midsurface first plays a role in the derivation of the Koiter model [25]. The idea is to write the linearized elasticity equations in a curvilinear system adapted to the geometry of the shell, namely the tubular neighborhood mapping of the midsurface associated with a given chart. Several approximations Kirchhoff-Love and plane stress hypotheses in particular and integrations across the thickness of the shell eventually lead to the Koiter model. From the modeling point of view, both in terms of geometry and in terms of mechanics, it is important that the tubular neighborhood mapping in question be globally injective when the thickness is small enough. This follows classically from the inverse function theorem and a compactness argument when the midsurface chart is of class C 2, see for instance [21]. Regularity of the midsurface also came into play in the first proof of existence and uniqueness for the Koiter model given by Bernadou and Ciarlet [6], who assumed the midsurface chart to be of class C 3. This assumption was necessary to make sense of a specific term containing derivatives of the second fundamental form that appears in the classical expression for 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. In particular, very commonplace shells such as a square plate continued by a circular cylinder are excluded. Moreover, the exact status of what it means for an actual midsurface to be C 3 can be questioned from the point of view of modeling, since the C 3 character does not describe a geometric feature of the midsurface. In addition, existence for the three-dimensional linearized elasticity problem only requires a Lipschitz boundary. Since the boundary of the three-dimensional shell includes images of the midsurface through the tubular neighborhood mapping, the jump from Lipschitz to C 3 when passing from the three-dimensional problem to the 2

3 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 first achieved by Blouza and Le Dret in [8] for the infinitesimal rigid displacement lemma for a midsurface of class W 2,. Existence and uniqueness for the Koiter model was obtained in this context in [9], with several generalizations worked out in detail in [10]. The key idea to lower the regularity requirements is to forsake the use of covariant components and derivatives and to express the quantities of interest directly in terms of vector unknowns. It turns out that this approach radically simplifies the expressions of the change of metric and change of curvature tensors. In particular, the term that formerly forced the chart to be C 3 no longer appears. This is due to the fact that, since covariant components are scalar products of the vector displacement with the covariant basis, regularity of the chart and regularity of the displacement are entangled in the classical formulation. The vector approach also made it possible to study the continuous dependence of the solution of Koiter s model on the midsurface [11] and also applies to more general models such as Naghdi s model [12]. Let us also mention that an alternate approach of W 2, or C 1,1 shells was developed in [18], 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 above works. In effect, the change of curvature tensor still contained Christoffel symbols which prevented the regularity of being lowered further. It was noticed by Anicic in [3] that the vector idea could be pursued even beyond the W 2, case by introducing the infinitesimal rotation vector in the definition of the function space for Koiter s model. The obtained model then continued to make sense for shells with a midsurface chart of class W 1, only, possessing a unit normal vector almost everywhere and such that this normal vector is itself of class W 1,. This generalization is especially interesting since it includes some G 1 surfaces arising in CAD after reparametrization. Such surfaces are defined via smooth patches with matching normal vectors on their interfaces, see [30]. In Anicic s formulation of Koiter s model and its variants, there is no need for matching normal derivatives of the charts across the patch interfaces, which makes for great versatility in practice. Several questions were however not entirely settled in [3]. First of all, the injectivity of the tubular neighborhood 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 introduced in [3] by assuming that the midsurface chart is bilipschitz with a Lipschitz normal vector. In this case, the tubular neighborhood mapping is 3

4 obviously Lipschitz. We prove 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 infinitesimal rigid displacement lemma for the midsurface. We introduce the Kirchhoff-Love displacement associated with a displacement of the midsurface and check that the vanishing of its change of metric and change of curvature tensors is equivalent to the algebraic condition for the 3D infinitesimal rigid displacement lemma in Lipschitz coordinates. This particular point was worked out in the present context of regularity in [3] and in [1] in the C 3 case. Existence and uniqueness for the Koiter model then follow along the lines of [10]. The idea of introducing the three-dimensional Kirchhoff-Love displacement associated with a two-dimensional shell displacement and using known threedimensional results to deduce two-dimensional results was already mentioned in [20]. It was implemented in [1], to establish the ellipticity of Koiter s model directly from the 3D Korn inequality and also to prove the infinitesimal rigid displacement lemma, for shells of class C 3, and in [16] to establish Korn inequalities on a surface also of class C 3. It is a legitimate question to wonder whether it possible to further reduce the regularity of the midsurface while retaining the injectivity of the tubular neighborhood mapping and the existence and uniqueness results. Some kind of continuity of the tangent plane is obviously required, otherwise the context would rather be that of folded shells or junctions of shells, an altogether different topic, see [1], [3], [14], [23], [26], [27], [28] and [32], among others. It is interesting to note that one of the intermediary results that we obtain in the course of the proof of the bilipschitz character of the tubular neighborhood mapping is one such geometrical continuity condition. 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 surfaces with Lipschitz normal vectors (or actually, only continuous normal vectors for this particular property) do possess the kind of regularity that differentiate them from folded or even more irregular Lipschitz surfaces. It seems to be difficult to find a less regular framework than the one used here that would still be adequate for shell theory, hence the admittedly slightly optimistic minimal regularity advertised in the title of this article. 4

5 2 A Lipschitz version of the n-dimensional infinitesimal rigid displacement lemma Let A n denote the set of n n skew-symmetric matrices. The main result of this section is the following. Theorem 2.1 Let Ω be a bounded connected open subset of R n and let Ψ W 1, (Ω; R n ) be such that det Ψ γ > 0 almost everywhere. Let U H 1 (Ω; R n ) satisfy U T Ψ + Ψ T U = 0 almost everywhere in Ω. (1) Then there exist a dense open subset O of Ω and locally constant mappings a and W from O into R n and A n respectively such that U(x) = a(x) + W (x)ψ(x) in O. We first recall some well-known results concerning the relationship between Lipschitz functions and W 1, functions. Theorem 2.2 Let Ω be a bounded, connected, open, Lipschitz subset of R n. Then we have W 1, (Ω) = C 0,1 ( Ω) algebraically and topologically, and there exists a constant C Ω such that for all x, y Ω and all u W 1, (Ω), u(x) u(y) C Ω u L (Ω) x y. This result is not easy to find in the literature, for instance it is mentioned in passing and without proof in [24]. Therefore, we include a proof in appendix for the reader s convenience. Another classical result that will be of use is Rademacher s theorem stating that a locally Lipschitz function on R n is almost everywhere differentiable, see [7] for a recent proof. It follows that its distributional differential coincides with its almost everywhere differential. Next, we recall an almost everywhere inverse function theorem for Sobolev mappings given in [22]. Theorem 2.3 Let Ψ W 1,n (Ω; R n ) be such that det Ψ > 0 almost everywhere. There exists a set of zero measure N Ω such that, for all x 0 / N, there exists a scalar r(x 0 ) > 0, an open set x 0 D x0 Ω and a mapping w x0 from B x0 = B(Ψ(x 0 ), r(x 0 )) into D x0 such that i) w x0 W 1,1 (B x0 ; R n ), ii) w x0 Ψ(x) = x almost everywhere in D x0, iii) Ψ w x0 (y) = y almost everywhere in B x0, iv) w x0 (y) = ( Ψ(w x0 (y))) 1 almost everywhere in B x0. 5

6 The exceptional set N that appears in the proof of Theorem 2.3 is the set of points where Ψ fails to be differentiable, viz. Rademacher s theorem, or where det Ψ 0. Corollary 2.4 If Ψ W 1, (Ω; R n ) is such that det Ψ γ > 0 almost everywhere, then w x0 W 1, (B x0 ; R n ) and equalities ii) and iii) hold everywhere on their respective domains. Proof. This is clear since ( Ψ(x)) 1 = (det Ψ(x)) 1 (cof Ψ(x)) T. Proof of Theorem 2.1. Let x 0 be a point in Ω \ N. By Theorem 2.2 and Corollary 2.4, the function w x0 is a bilipschitz homeomorphism between B x0 and D x0. It follows from [29], page 16, that V = U w x0 belongs to H 1 (B x0 ; R N ) with V (y) = U(w x0 (y))( Ψ(w x0 (y))) 1 almost everywhere in B x0. We thus obtain V + V T = U Ψ 1 + Ψ T U T = Ψ T ( Ψ T U + U T Ψ) Ψ 1 = 0 almost everywhere in B x0. By the classical infinitesimal rigid displacement lemma, there exists a vector a(x 0 ) R N and a skew-symmetric matrix W (x 0 ) A N such that V (y) = a(x 0 ) + W (x 0 )y in B x0, that is to say U(x) = a(x 0 ) + W (x 0 )Ψ(x) in D x0. Let us now set O = x/ N D x. It is an open subset of Ω and since Ω \ N O, it follows that Ω \ O N is of zero measure. Therefore O is dense in Ω. To conclude, consider two points x and x such that B x B x, or in other words D x D x. Clearly, w x = w x on B x B x, thus the affine functions a(x) + W (x)y and a(x ) + W (x )y coincide on the nonempty open set B x B x. 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 Ψ that defines a local bilipschitz change of coordinates, we have a stronger alternate version. Theorem 2.5 Under the previous hypotheses, if Ψ is in addition locally bilipschitz, then there exist a vector a and a skew-symmetric matrix W such that U(x) = a + W Ψ(x) in Ω. (2) Proof. Same proof, but working directly on Ω, without requiring the use of Theorem

7 Remark. It is not clear whether there are examples in which the mappings a and W of Theorem 2.1 actually are non constant on Ω. However, Theorem 2.5 seems to indicate that possible non constancy is more connected with the lack of local invertibility of Ψ than with its lack of differentiability (as indicated by the exceptional set N). We describe below a C example with non constant W. Unfortunately, this example does not satisfy the hypothesis on the bound below for the Jacobian, but only the less demanding hypothesis det Ψ > 0 almost everywhere. The example is as follows. Let Ω = {x = (x 1, x 2 ); x 1 < 1, x 2 < 1}, Ω + = Ω {x 2 > 0} and Ω = Ω {x 2 < 0}. We define Ψ(x) = ( x1 x 2 2 x 2 We have det Ψ(x) = x 2 2 > 0 almost everywhere and Ψ is not locally injective on the line {x 2 = 0}. The complement of this line has two connected components, Ω and Ω +. Let now ( ) x2 U(x) = 0 in Ω, U(x) = x 1 x 2 in Ω +. 2 Clearly U H 1 (Ω; R 2 ) and U T Ψ + Ψ T U = 0 almost everywhere in Ω. Moreover, ( ) ( ) U(x) = Ψ(x) in Ω 0 0 and U(x) = Ψ(x) in Ω It should also be noted that there are mappings Ψ satisfying all the hypotheses of Theorem 2.1 and that are not locally bilipschitz, but such that a and W are nonetheless constant on Ω. One such example is the example of Ball, see [5], where Ω is the unit disk of R 2 and ( ) 1 x 2 Ψ(x) = 1 x (x x 2 2) 2x 1 x 2 This mapping is such that det Ψ(x) = 1 almost everywhere and Ψ is not locally injective, hence not locally bilipschitz, and not differentiable at 0. So in this case, the exceptional set is N = {0}. However, the complement of N is connected, therefore a and W are constant by Theorem 2.1. ). 3 Shells of minimal regularity: Geometrical aspects The classical geometrical setting for a thin shell is as follows. Let ω be a bounded open connected Lipschitz subset of R 2 and ϕ: ω R 3 a sufficiently regular chart 7

8 for the midsurface of the shell. Let a 3 = 1 ϕ 2 ϕ/ 1 ϕ 2 ϕ be the unit normal vector to the midsurface in the chart ϕ. The tubular neighborhood mapping of the midsurface is defined by Φ(x, x 3 ) = ϕ(x) + x 3 a 3 (x). (3) It is globally injective on ω [ h, h] for h > 0 small enough when ϕ is of class C 2. This injectivity results from the inverse function theorem for local injectivity followed by a compactness argument, see for instance [21] or [15]. The global injectivity of the tubular neighborhood mapping is a crucial, although sometimes overlooked, element of thin shell modeling. It could even be said that it is the distinguishing factor for surfaces that may qualify as midsurface of a thin shell. In attempting to lower the regularity of the midsurface, it is thus important to retain the injectivity of the tubular neighborhood mapping. For the W 2, or C 1,1 case, we refer to [18] for a possible approach based on the signed distance function. We are interested here in an even lower regularity framework, basically introduced in [3] and slightly modified here. Let ω be as above and consider a mapping ϕ: ω R 3 that will describe the midsurface of a shell. Hypothesis 3.1 We assume that ϕ is bilipschitz, i.e., there exist two constants 0 < α β such that x, y ω, α y x ϕ(y) ϕ(x) β y x. (4) Let us now turn to the normal vector. By Rademacher s theorem, ϕ is almost everywhere differentiable. As a consequence of Hypothesis 3.1, we first note that the tangent vectors are almost everywhere uniformly linearly independent, in the following sense. Lemma 3.2 At all points x of differentiability of ϕ, we have 1 ϕ(x) 2 ϕ(x) α 2. (5) Proof. Let x ω be a point of differentiability of ϕ. For all y R 2 and s R sufficiently small, we have by Hypothesis 3.1 ϕ(x + sy) ϕ(x) α s y. Dividing by s and letting s tend to 0, we obtain ϕ(x)(y) α y. 8

9 The 3 2 matrix A = ϕ(x) is thus such that Ay α y for all y R 2, that is to say y T A T Ay α 2 y T y. If v 1 v 2 denote the singular values of A, this implies that α 2 v1 2 v2. 2 In particular, v 1 v 2 α 2. Let a 1 = 1 ϕ(x) and a 2 = 2 ϕ(x) be the two column vectors of A and let us choose a third vector A 3 such that a 1 A 3 = a 2 A 3 = 0 and A 3 = 1. Let à be the 3 3 matrix composed of these three vectors. We have det à = A 3 (a 1 a 2 ), so that by our choice of A 3, det à = a 1 a 2. By construction, we have ( ) A à T à = T A 0, 0 1 written as a block-matrix, so that (det Ã)2 = det(ãt Ã) = det(a T A) = v1v Hence, det à = v 1v 2 α 2. Lemma 3.2 shows that ϕ( ω) is a surface in R 3 in the sense that it has a tangent plane almost everywhere. Moreover, we can define a unit normal vector almost everywhere by a 3 (x) = 1ϕ(x) 2 ϕ(x) 1 ϕ(x) 2 ϕ(x). (6) By construction, we have a 3 L (ω; R 3 ). We now make an additional regularity hypothesis that was introduced in [3]. Hypothesis 3.3 We assume that a 3 W 1, (ω; R 3 ). (7) In other words, we assume that the vector a 3 that was a priori only defined almost everywhere, coincides with a Lipschitz function defined on ω. Remark. It is important to note that only the unit normal vector is assumed to be Lipschitz, whereas the vector 1 ϕ 2 ϕ remains solely in L (ω; R 3 ). A concrete example of this situation is given by the piecewise W 2, surfaces considered in [3]. We next define the tubular neighborhood mapping by formula (3) as usual. Under hypotheses 3.1 and 3.3, the mapping Φ is clearly Lipschitz on ω [ h, h] for all h > 0. Our next goal is to prove that it is locally bilipschitz for h small enough. The proof proceeds in several steps. First is the result for the restriction of Φ to all sheaves ω {x 3 }, x 3 h. Recall that C ω denotes the constant depending on ω of Theorem 2.2. Lemma 3.4 For h = α(2c ω a 3 L (ω)) 1, we have for all x 3 h and x, y ω. Φ(x, x 3 ) Φ(y, x 3 ) α x y (8) 2 9

10 Proof. Let L a3 = C ω a 3 L (ω). By the triangle inequality, we obtain Φ(x, x 3 ) Φ(y, x 3 ) ϕ(x) ϕ(y) x 3 a 3 (x) a 3 (y) (α hl a3 ) x y, hence the result. Remark. The value α/2 in formula (8) is of course arbitrary. Any value of h strictly less than α/l a3 will do. Note also that this result shows that the boundary of the three-dimensional shell is Lipschitz, 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 3.5 For all x 0 ω, we have ϕ(x) ϕ(x 0 ) x x 0 a 3 (x 0 ) 0 when x x 0. (9) Remark. The geometric meaning of Lemma 3.5 is that the Bouligand contingent cone to the surface (see [31]) always remains orthogonal to the normal vector. Note that the result is obvious at every point x of differentiability of ϕ, that is to say almost everywhere in ω. However, we need it not only almost everywhere, but everywhere, which we now proceed to prove. Proof. Let us fix a point x 0 ω. The function x P (x) = (ϕ(x) ϕ(x 0 )) a 3 (x 0 ) is Lipschitz on ω and its distributional/almost everywhere gradient is given by P (x) = ϕ(x) T a 3 (x 0 ). Therefore, for all x ω, P (x) = P (x) P (x 0 ) C ω ϕ T a 3 (x 0 ) L ( B(x 0, x x 0 )) ω x x 0. Now, almost all points of B(x 0, x x 0 ) ω are points of differentiability of ϕ. Hence, at any such point y, a 3 (y) is orthogonal to the image of ϕ(y) by construction, that is to say ϕ(y) T a 3 (y) = 0. Consequently, we have that, almost everywhere in B(x 0, x x 0 ) ω, so that ϕ(y) T a 3 (x 0 ) = ϕ(y) T a 3 (x 0 ) ϕ(y) T a 3 (y), ϕ(y) T a 3 (x 0 ) ϕ(y) T a 3 (x 0 ) a 3 (y) almost everywhere. Therefore, C ω a 3 L (ω) ϕ(y) T y x 0 ϕ T a 3 (x 0 ) L ( B(x 0, x x 0 )) ω C ω a 3 L (ω) ϕ L (ω) x x 0, hence the result. 10

11 Remark. In [4] we announced a proof using Clarke s generalized gradients, see [17]. However, in the meantime we found the above much simpler and more elementary proof. Note that it is enough for a 3 to be continuous for Lemma 3.5 to hold true. Next we have a similar result for the normal vector. Lemma 3.6 For all x 0 ω, we have a 3 (x) a 3 (x 0 ) x x 0 a 3 (x 0 ) 0 when x x 0. (10) Proof. Let x n ω be a sequence that tends to x 0. The vectors a 3(x n ) a 3 (x 0 ) x n x 0 remain in the ball B(0, La3 ). We can thus extract a subsequence x n such that a 3 (x n ) a 3 (x 0 ) x n x 0 τ for some vector τ B(0, L a3 ). Now we have (a 3 (x n ) a 3 (x 0 )) a 3 (x 0 ) = a 3 (x n ) (a 3 (x 0 ) a 3 (x n )). Dividing by x n x 0 and passing to the limit, we thus obtain that τ a 3 (x 0 ) = a 3 (x 0 ) τ, hence the result by uniqueness of the limit. Remark. Geometrically speaking, the result is due to the fact that a 3 is S 2 -valued. Therefore, the Bouligand contingent cone of its image at a given point a 3 (x 0 ) is obviously included in the tangent plane to the sphere at that same point. We now are in a position to prove the main result of this section. Theorem 3.7 Let h be as above. Then, the tubular neighborhood mapping Φ is locally bilipschitz on ω [ h, h]. Proof. We argue by contradiction. Let us thus assume that Φ is not locally bilipschitz at a point (x 0, x 3 ) ω [ h, h]. There exists a sequence (x n, x n 3) (x 0, x 3 ) in ω [ h, h] such that Φ(x n, x n 3) Φ(x 0, x 3 ) 2 < 1 n ( xn x x n 3 x 3 2 ). In view of the above inequality, we can be assured that x n 3 x 3 or x n x 0. We observe that Φ(x n, x n 3) Φ(x 0, x 3 ) = Φ(x n, x n 3) Φ(x 0, x n 3) + Φ(x 0, x n 3) Φ(x 0, x 3 ) = Φ(x n, x n 3) Φ(x 0, x n 3) + (x n 3 x 3 )a 3 (x 0 ). 11

12 Therefore, Φ(x n, x n 3) Φ(x 0, x 3 ) 2 = Φ(x n, x n 3) Φ(x 0, x n 3) 2 + x n 3 x (x n 3 x 3 )(Φ(x n, x n 3) Φ(x 0, x n 3)) a 3 (x 0 ) α2 4 xn x x n 3 x (x n 3 x 3 )(Φ(x n, x n 3) Φ(x 0, x n 3)) a 3 (x 0 ) by Lemma 3.4. Let β = min(α 2 /4, 1). We deduce from the above inequalities that ( ) 2(x n 3 x 3 ) x n ϕ(x x 0 n ) ϕ(x 0 )+x n 3 (a 3(x n ) a 3 (x 0 )) x n x 0 a 3 (x 0 ) 1 x n x x n 3 x 3 2 n β. Now, for all couples of nonzero real numbers (A, B) we have 1 2AB 1. A 2 +B 2 Therefore, the left-hand side of the above inequality tends to 0 when n + by Lemmas 3.5 and 3.6, whereas the right-hand side tends to β < 0, which is a contradiction. Corollary 3.8 The mapping Φ is locally injective on ω [ h, h]. We can then follow [21] or [15] to obtain the global tubular neighborhood property. Theorem 3.9 There exists 0 < h < h such that the mapping Φ is globally injective on ω [ h, h ]. In other words, a midsurface satisfying (4) (7) allows for a correct geometrical modeling of a thin shell of thickness 2h. 4 Shells of minimal regularity: The infinitesimal rigid displacement lemma Most existence and uniqueness proofs for linear shell models rest on an infinitesimal rigid displacement lemma for the midsurface. Such a lemma is useful in proving the ellipticity of the bilinear form of the variational formulation of the equilibrium problem, [6], [10], [12]. See however [1] for an ellipticity proof using directly the three-dimensional Korn inequality, hence implicitly the threedimensional infinitesimal rigid displacement lemma, in the C 3 regularity setting. 12

13 In the sequel, we will use the summation convention unless otherwise specified. Let a α = α ϕ and a α be the covariant and contravariant basis vectors on the midsurface. Note that the covariant vectors are only in L (ω; R 3 ) and are uniformly almost everywhere linearly independent in the sense of Lemma 3.2. The contravariant vectors are uniquely defined as elements of L (ω; R 3 ) by their belonging to the plane spanned by a 1 and a 2 and the relations a α a β = δ α β almost everywhere. It is a simple matter to verify that they satisfy the same properties as the covariant vectors, see [3]. The function space for Koiter s model introduced in [3] that generalizes that initially proposed in [8]-[9] is as follows. Let u H 1 (ω; R 3 ) be a midsurface displacement (vector-valued, but seen through the chart ϕ). The infinitesimal rotation vector associated with u is defined as θ(u) = ( α u a 3 )a α. (11) It is a priori in L 2 (ω; R 3 ). The appropriate function space is then endowed with its natural Hilbert norm V = {u H 1 (ω; R 3 ), θ(u) H 1 (ω; R 3 )}, (12) u V = ( u 2 H 1 (ω;r 3 ) + θ(u) 2 H 1 (ω;r 3 ) )1/2. (13) Note that if u V, then the vector θ(u) is in H 1, but each term α u a 3 is only in L 2. In this context, the covariant components of the change of metric tensor read γ αβ (u) = 1 2 ( αu a β + β u a α ) (14) and the covariant components of the change of curvature tensor read Υ αβ (u) = α θ(u) a β β u α a 3. (15) Note that in spite of appearances, formula (15) is symmetric in α and β, as it should indeed be, see [2] or [3]. Clearly, for u V, the change of metric and change of curvature tensors γ(u) = γ αβ (u)a α a β and Υ(u) = Υ αβ (u)a α a β are both square-integrable. Naturally, in Koiter s model, the bilinear form is the sum of a positive definite quadratic form in γ(u) corresponding to membrane energy and a positive definite quadratic form in Υ(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 13

14 of the displacement and various covariant derivatives when ϕ is of class C 3, see [10], [3]. There is also a canonical correspondence between the function spaces of each formulation in this case and the solutions coincide. Moreover, in [11], it is shown that the simpler vector-valued formalism in the W 2, case arises as a natural limit of the classical C 3 case, by means of a result 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 infinitesimal rigid displacement lemma for the midsurface. Theorem 4.1 Let ϕ be a chart satisfying (4) (7) and u V be such that γ(u) = Υ(u) = 0. Then there exist two vectors a and b of R 3 such that u(x) = a + b ϕ(x) in ω. Proof. The idea is to use the Kirchhoff-Love displacement associated with u U(x, x 3 ) = u(x) x 3 θ(u)(x), (16) on Ω = ω ] h, h[, with h given by Lemma 3.4 such that Φ is locally bilipschitz on Ω. Since u V, it is clear that U H 1 (Ω; R 3 ) (and conversely). It is known, see [1], [3] and [19], that Kirchhoff-Love displacements are characterized by the relations ( U T Φ + Φ T U) i3 = 0, with i = 1, 2, 3. In fact, since U = ( 1 u x 3 1 θ(u) 2 u x 3 2 θ(u) θ(u)) and Φ = (a 1 + x 3 1 a 3 a 2 + x 3 2 a 3 a 3 ), we have ( U T Φ) 33 = θ(u) a 3, ( U T Φ) α3 = α u a 3 x 3 α θ(u) a 3, ( Φ T U) α3 = θ(u) a α x 3 θ(u) α a 3. Now, clearly θ(u) a 3 = 0 and since θ(u) H 1 and a 3 W 1,, Leibniz s formula is valid and α θ(u) a 3 + θ(u) α a 3 = 0. Moreover, θ(u) a α = α u a 3 by definition of θ(u). Performing the same computations for the tangential components, we obtain ( U T Φ) αβ = ( α u x 3 α θ(u)) (a β + x 3 β a 3 ), = α u a β + x 3 ( α u β a 3 α θ(u) a β ) x 2 3 α θ(u) β a 3, ( Φ T U) αβ = β u a α + x 3 ( β u α a 3 β θ(u) a α ) x 2 3 β θ(u) α a 3, 14

15 so that ( U T Φ + Φ T U) αβ = 2γ αβ (u) 2x 3 Υ αβ (u) x 2 3( α θ(u) β a 3 + β θ(u) α a 3 ). (17) Since a 3 is a unit vector, α a 3 is orthogonal to a 3 almost everywhere. Therefore, we can write α a 3 = b µ αa µ with b µ α = α a 3 a µ L (ω) (which are the mixed components of the second fundamental form when ϕ is regular). It follows that so that α θ(u) β a 3 = b µ β αθ(u) a µ = b µ β (Υ αµ(u) + µ u α a 3 ) = b µ β (Υ αµ(u) + b ν α µ u a ν ), α θ(u) β a 3 + β θ(u) α a 3 = b µ β Υ αµ(u) + b µ αυ µβ (u) + 2b µ αb ν βγ µν (u). (18) If we assume now that γ αβ (u) = Υ αβ (u) = 0, then we conclude from formulas (17) and (18) that ( U T Φ + Φ T U) αβ = 0. We can therefore use Theorem 2.5 to deduce that there exist two vectors a and b such that U(x, x 3 ) = a + b Φ(x, x 3 ). Setting x 3 = 0 in the above equality yields the result. Remark. The algebra leading to formulas (17) and (18) is well-known, although not exactly under this form. It was worked out as shown here under the same regularity hypotheses in [3]. The idea of using the three-dimensional Kirchhoff- Love displacement associated with a shell displacement was also used in [1] to establish the infinitesimal rigid displacement lemma for a surface in the C 3 case. The same idea was also used in [16] to prove Korn inequalities on a surface of class C 3. Remark. In [3], [2], Anicic introduced a new change of curvature tensor whose covariant components are χ αβ (u) = Υ αβ (u) b σ αγ σβ (u) b σ β γ σα(u). This tensor is better adapted to measuring the variations of the principal curvatures of the midsurface. It is then clear that a shell displacement such that γ(u) = χ(u) = 0 is also an infinitesimal rigid displacement. Once the infinitesimal rigid displacement lemma for a surface is established, it is a simple matter to reproduce the argument of [9] [10] and prove the ellipticity of the bilinear form in Koiter s model, hence the existence and uniqueness of its solution, for midsurfaces satisfying the minimal regularity hypotheses 3.1 and

16 5 Appendix We give here a proof of Theorem 2.2. Recall first that the geodesic distance between two points in Ω is the infimum of all lengths of paths that connect the two points in Ω. Proposition 5.1 Let Ω be a bounded, open, connected, Lipschitz subset of R n and d Ω denote the geodesic distance in Ω. There exists a constant C Ω such that x, y Ω, d Ω (x, y) C Ω x y. (19) Proof. We first note that the geodesic distance is bounded from above. Indeed, since Ω is compact and Ω is Lipschitz, there exists r > 0 and a finite number of open cubes C i, i = 1,..., k, of edge 2r that cover Ω and such that C i Ω is the hypograph of a Lipschitz function θ i : R N 1 R in a coordinate system attached to the cube. We may assume that the centers c i of these cubes lie in Ω. We cover the rest of Ω with more cubes, possibly smaller but still in finite number i = k + 1,..., p, that do not intersect Ω. Obviously, since Ω is open and connected, it is arcwise connected, and any couple of centers can be connected by a path of finite length. Therefore max 1 i,j p d Ω (c i, c j ) = M < +. Moreover, for all x, y Ω, there exists i and j such that x C i and y C j. Therefore, we have d Ω (x, y) d Ω (x, c i ) + d Ω (c i, c j ) + d Ω (c j, y) M + d Ω (x, c i ) + d Ω (y, c j ). We just have to estimate the last two terms. If k + 1 i n, then C i Ω and is convex, hence d Ω (x, c i ) = x c i < nr. If 1 i k, let Mi be the Lipschitz constant of θ i. We denote by (z, z n ) the coordinate system adapted to the cube, in the sense that C i Ω = {z = (z, z n ) C i ; z n < θ i (z )}. Let ε > 0 be such that (z, θ i (z ) ε) C i for all z C i. Such an ε obviously exists. We connect x to c i via a three-legged path in Ω as follows: Let γ 1 be the vertical segment {(x, (1 λ)x n + λ(θ i (x ) ε)), λ [0, 1]}. Then let γ 2 be the curve {((1 λ)x + λc i, θ i ((1 λ)x + λc i) ε), λ [0, 1]}, which follows the boundary inside Ω, and draw a final vertical segment γ 3 given by {(c i, λc i,n + (1 λ)(θ i (c i) ε)), λ [0, 1]}. It is fairly clear that the length of γ 1 and γ 3 is bounded from above by 2r and that the length of γ 2 is bounded by 2r(1 + M 2 i ) 1/2. Putting all these estimates together, we obtain the global bound d Ω (x, y) M + 2 max( nr, 2r(2 + max 1 i k (1 + M 2 i ) 1/2 )). Let us now turn to the equivalence. We argue by contradiction. Assume that there is a sequence x j, y j Ω such that d Ω (x j, y j ) j x j y j. (20) 16

17 Extracting a subsequence, we may assume that x j x Ω and y j y Ω. Due to the remark above, we have x = y. If x Ω, there exists a ball B(x, r ) Ω such that x j and y j eventually end up in this ball. Therefore, d Ω (x j, y j ) = x j y j for j large enough in this case. It follows that x Ω. In this case, a similar argument as above with a slightly different path yields an estimate of the form d Ω (x j, y j ) 2((1 + Mi 2 ) 1/2 + M i ) x j y j for j large enough, for some i such that x C i, hence a contradiction. Ω C i 1 x 2 3 c i Ω x j x y j Ω The paths used in the proof of Proposition 5.1. Lemma 5.2 Let u W 1, (Ω). Then u is a Lipschitz function on Ω such that for all x, y Ω, u(x) u(y) C Ω u L (Ω) x y. (21) Proof. We know that for all u W 1, (Ω) and all x, y Ω, u(x) u(y) u L (Ω)d Ω (x, y), see [13]. Since Ω is Lipschitz, we deduce estimate (21) from Lemma 5.1 for all x, y Ω. It follows that u is uniformly continuous on Ω, hence it has a unique continuous extension to Ω which still satisfies (21). Lemma 5.3 Let u C 0,1 ( Ω). Then u W 1, (Ω), its distributional differential coincides with its almost everywhere differential and { u(x) u(y) u L (Ω) sup, x, y x y Ω, } x y. (22) Proof. Let L = sup { u(x) u(y) x y, x, y Ω, x y } be the Lipschitz constant of u. For all Ω Ω, the translates τ h u(x) = u(x + h) are well-defined for h small enough and obviously τ h u u L (Ω ) L h. Therefore, u W 1, (Ω) and u L (Ω) L. In addition, for h = se i, s R and e i a basis vector, it is trivial that (τ h u u)/s i u in D (Ω ). Since the differential quotients also converge 17

18 almost everywhere by Rademacher s theorem and are dominated by a constant L which is integrable on Ω bounded, they converge in L 1 (Ω ) to their almost everywhere limit. Therefore, the distributional and almost everywhere derivatives coincide. Remark. Note that Ω does not need to be Lipschitz for Lemma 5.3. References [1] J.-L. Akian, Analyse asymptotique de jonctions de coques en flexion, Doctoral dissertation, Université de Poitiers, [2] S. Anicic, Mesure des variations infinitésimales des courbures principales d une surface, C. R. Acad. Sci. Paris, t. 335, Série I, 2002, [3] S. Anicic, Du modèle de Kirchhoff-Love exact à un modèle de coque mince et à un modèle de coque pliée, Doctoral dissertation, Université Joseph Fourier, [4] S. Anicic, H. Le Dret, A. Raoult, Lemme du mouvement rigide en coordonnées lipschitziennes et application aux coques de régularité minimale, C. R. Acad. Sci. Paris, Série I, 2003, to appear. [5] J.M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. Edinburgh, 88A, 1981, [6] M. Bernadou, P.G. Ciarlet, Sur l ellipticité du modèle linéaire de coques de W.T. Koiter, in Computing Methods in Sciences and Engineering, R. Glowinski, J.-L. Lions (eds.), Lecture Notes in Economics and Math. Systems, vol. 134, Springer-Verlag, Berlin, pp , [7] D.N. Bessis, F.H. Clarke, Partial subdifferentials, derivates and Rademacher s theorem, Trans. Amer. Math. Soc., 351, 1999, [8] A. Blouza, H. Le Dret, Sur le lemme du mouvement rigide, C. R. Acad. Sci. Paris, t. 319, Série I, 1994, [9] A. Blouza, H. Le Dret, Existence et unicité pour le modèle de Koiter pour une coque peu régulière, C. R. Acad. Sci. Paris, t. 319, Série I, 1994, [10] A. Blouza, H. Le Dret, Existence and uniqueness for the linear Koiter model for shells with little regularity, Quart. Appl. Math., LVII, 1999, [11] A. Blouza, H. Le Dret, An up-to-the-boundary version of Friedrichs s lemma and applications to the linear Koiter shell model, SIAM J. on Math. Anal., 33, 2001,

19 [12] A. Blouza, H. Le Dret, Nagdhi s shell model: existence, uniqueness and continuous dependence on the midsurface, J. Elasticity, 64, 2001, [13] H. Brezis, Analyse fonctionnelle. Théorie et applications, Masson, Paris, [14] D. Choi, On geometrical rigidity of surfaces. Application to the theory of thin linear elastic shells, Math. Models Methods Appl. Sci., 7, 1997, [15] P.G. Ciarlet, Mathematical elasticity. Vol. III: Theory of shells, Studies in Mathematics and its Applications, 29, North-Holland Publishing Co., Amsterdam, [16] P.G. Ciarlet, S. Mardare, Sur les inégalités de Korn en coordonnées curvilignes, C. R. Acad. Sci. Paris, t. 331, Série I, 2000, [17] F.H. Clarke, Optimization and Nonsmooth Analysis, SIAM Classics in Applied Mathematics, 5, 1990 (second printing). [18] M.C. Delfour, Intrinsic differential geometric methods in the asymptotic analysis of linear thin shells, in Boundaries, Interfaces, and Transitions (Banff, AB, 1995), 19 90, CRM Proc. Lecture Notes 13, AMS, Providence, [19] P. Destuynder, Modélisation des coques minces élastiques, Masson, Paris, [20] P. Destuynder, M. Salaün, A mixed finite element for shell model with free edge boundary conditions, Part 1. The mixed variational formulation, Comput. Methods Appl. Mech. Engrg. 120, 1995, [21] M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Englewood Cliffs, Prentice-Hall, [22] I. Fonseca, W. Gangbo, Degree Theory in Analysis and Applications, Oxford Lecture Series in Mathematics and its Applications 2, Oxford University Press, [23] P. Gérard, É. Sanchez-Palencia, Sensitivity phenomena for certain thin elastic shells with edges, Math. Methods Appl. Sci., 23, 2000, [24] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Berlin, New York, Heidelberg, Springer-Verlag, 1983 (second printing). [25] W.T. Koiter, A consistent first approximation in the general theory of thin elastic shells, in Proc. IUTAM Symposium on the Theory of Thin Elastic Shells, August 1959, 12 32, North Holland, Amsterdam, [26] H. Le Dret, Modeling of a folded plate, Comput. Mech, 5, 1990, [27] H. Le Dret, Folded plates revisited, Comput. Mech, 5, 1989,

20 [28] H. Le Dret, Problèmes variationnels dans les multi-domaines : Modélisation des jonctions et applications. RMA 19, Masson, Paris, [29] V.G. Maz ja, Sobolev Spaces, Springer-Verlag, Berlin, New York, Tokyo, [30] J.-J. Risler, Méthodes mathématiques pour la CAO, RMA 18, Masson, Paris, [31] T.R. Rockafellar, La théorie des sous-gradients et ses applications à l optimisation, Les Presses de l Université de Montréal, [32] I. Titeux, É. Sanchez-Palencia, Junction of thin plates, Eur. J. Mech. A Solids, 19, 2000,