A generalization of the classical Cesàro Volterra path integral formula

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1 C. R. Acad. Sci. Paris, Ser. I 347 (2009) Mathematical Problems in Mechanics A generalization of the classical Cesàro Volterra path integral formula Philippe G. Ciarlet a, Liliana Gratie a, Cristinel Mardare b a Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong b Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005, Paris, France Received and accepted 24 February 2009 Available online 3 April 2009 Presented by Philippe G. Ciarlet Abstract If a symmetric matrix field e of order three satisfies the Saint Venant compatibility conditions in a simply-connected domain in R 3, there then exists a displacement field u of such that e = 2 ( ut + u) in. Ifthefielde is sufficiently smooth, the displacement u(x) at any point x can be explicitly computed as a function of e and CURL e by means of a Cesàro Volterra path integral formula inside with endpoint x. We assume here that the components of the field e are only in L 2 (), in which case the classical path integral formula of Cesàro and Volterra becomes meaningless. We then establish the existence of a Cesàro Volterra formula with little regularity, which again provides an explicit solution u to the equation e = 2 ( ut + u) in this case. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009) Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. Résumé Une généralisation de la formule classique de l intégrale curviligne de Cesàro Volterra. Si un champ e de matrices symétriques d ordre trois vérifie les conditions de compatibilité de Saint Venant dans un domaine simplement connexe de R 3, alors il existe un champ u de déplacements de tel que e = 2 ( ut + u) dans. Silechampe est suffisamment régulier, le déplacement u(x) peut être calculé explicitement en tout point x comme une fonction de e et de CURL e, au moyen d une intégrale curviligne de Cesàro Volterra le long d un chemin contenu dans et d extrémité x. On suppose ici que les composantes du champ e sont seulement dans L 2 (), auquel cas la formule intégrale de Cesàro Volterra n a pas de sens. On établit alors l existence d une «formule de Cesàro Volterra avec peu de régularité», qui donne à nouveau dans ce cas une solution explicite u de l équation e = 2 ( ut + u). Pour citer cet article : P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009) Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. addresses: mapgc@cityu.edu.hk (P.G. Ciarlet), mcgratie@cityu.edu.hk (L. Gratie), mardare@ann.jussieu.fr (C. Mardare) X/$ see front matter 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. doi:0.06/j.crma

2 578 P.G. Ciarlet et al. / C. R. Acad. Sci. Paris, Ser. I 347 (2009) Introduction Latin indices range in the set {, 2,...,n} for some n 2 and the summation convention with respect to repeated Latin indices is used in conjunction with this rule. The sets of all real matrices of order n, of all real symmetric matrices of order n, and of all real antisymmetric matrices of order n, are respectively denoted M n, S n, and A n. It is well known that, if is a simply-connected open subset of R n and if a matrix field e = (e ij ) C 2 (; S n ) satisfies the Saint Venant compatibility conditions lj e ik + ki e jl li e jk kj e il = 0 inc 0 (), () then there exists a vector field u = (u i ) C 3 (; R n ) that satisfies the equations 2 ( j u i + i u j ) = e ij in. Besides, all other solutions ũ = (ũ i ) C 3 (; R n ) to the equations 2 ( j ũ i + i ũ j ) = e ij in are of the form (2) ũ(x) = u(x) + a + Aox, x, for some a R n and A A n. (3) It is less known (Gurtin 3] constitutes an exception) that an explicit solution u = (u i ) to the equations (2) can be given in the form of the following Cesàro Volterra path integral formula, so named after Cesàro 4] and Volterra 5]: Let be any path of class C contained in and joining a point x 0 (considered as fixed) to any point x. Then { u i (x) = eij (y) + ( k e ij (y) i e kj (y) ) (x k y k ) } dy j, x. (4) It can then be verified that each value u i (x) computed by formula (4) is independent of the path chosen for joining x 0 to x, thanks to the compatibility conditions (). If n = 3, the Cesàro Volterra path integral formula (4) can be equivalently rewritten in vector-matrix form, as u(x) = e(y) dy + yx ( CURL e(y) ] dy ), x, (5) where designates the vector product in R 3, and CURL designates the matrix curl operator. The sufficiency of the Saint Venant compatibility conditions () was recently shown to hold under substantially weaker regularity assumptions on the given tensor field e = (e ij ), according to the following result, due to Ciarlet and Ciarlet, Jr. 5]: Let be a bounded and simply-connected open subset of R n with a Lipschitz-continuous boundary, and let there be given functions e ij = e ji L 2 () that satisfy the Saint Venant compatibility conditions with little regularity, viz., lj e ik + ki e jl li e jk kj e il = 0 in H 2 (). (6) Then there exists a vector field (u i ) H (; R 3 ) that satisfies 2 ( j u i + i u j ) = e ij in L 2 (). Besides, all the other solutions ũ = (ũ i ) H (; R 3 ) to the equations 2 ( j ũ i + i ũ j ) = e ij are again of the form (3) (this result has since then been extended in various ways; see Geymonat and Krasucki 9], Ciarlet, Ciarlet, Jr., Geymonat and Krasucki 6], and Amrouche, Ciarlet, Gratie and Kesavan 2]). Clearly, the classical Cesàro Volterra path integral formula (4) becomes meaningless when the functions e ij satisfying (6) are only in the space L 2 (). The question then naturally arises as to whether there exists any Cesàro Volterra formula with little regularity, which (i) would again provide an explicit solution to the equations (7) when the functions e ij are only in L 2 () and (ii) would in some way resemble (4). The purpose of this Note is to provide a positive answer to this question. Complete proofs will be found in 8]. (7)

3 P.G. Ciarlet et al. / C. R. Acad. Sci. Paris, Ser. I 347 (2009) A Poincaré lemma with little regularity A domain in R n is an open, bounded, connected subset of R n, with a Lipschitz-continuous boundary. The mapping T = (T i ) defined in the next theorem (for a proof, see Corollaries 2.3 and 2.4, Chapter, of Girault and Raviart 2]; or Theorem 2 of Bourgain and Brezis 3] for the extension to L p spaces, <p< ) plays a key role in our approach. Theorem. Let be a domain in R n. Then there exists a linear and continuous operator } T = (T i ) : L 2 0 {v () := L 2 (); v dx = 0 H0 ( ; R n ), (8) such that div(t v) = v for all v L 2 0 (). Our approach for finding a Cesàro Volterra formula with little regularity also relies on the following Poincaré lemma with little regularity, due to Ciarlet and Ciarlet, Jr. 5] (for recent extensions of this result, see Amrouche, Ciarlet and Ciarlet, Jr. ], Geymonat and Krasucki 0,] and, especially, S. Mardare 4]). Theorem 2. Let be a simply-connected domain in R n, and let f i H () be distributions that satisfy i f j j f i = 0 in H 2 (). Then there exists a function u L 2 (), unique up to an additive constant, such that i u = f i in H (). We first show that, even under the weak regularity assumptions of Theorem, there is a way to compute a solution u L 2 () to the equations i u = f i in H (). In what follows,, denotes the duality pairing between a topological space and its dual space. Theorem 3. Let be a simply-connected domain in R n, let the space D 0 () be defined as { } D 0 () := ϕ D(); ϕ dx = 0, and let f i H () be distributions that satisfy i f j j f i = 0 in H 2 (). Then a function u L 2 () satisfies i u = f i in H () if and only if u, ϕ = f i,t i ϕ for all ϕ D 0 (), () where T = (T i ) : L 2 0 () H 0 (; Rn ) is the continuous linear operator defined in Theorem. Interestingly, the solution to the equations i u = f i in H () can also be found by solving a variational problem (cf. (2) below), which clearly satisfies all the assumptions of the Lax Milgram lemma: Theorem 4. Let be a simply-connected domain in R n, let the space L 2 0 () be defined as in (8), and let there be given distributions f i H () that satisfy i f j j f i = 0 in H 2 (). Then the variational problem: Find a function u L 2 0 () such that (9) (0) u, v = f i,t i v for all v L 2 0 (), has a unique solution, which is also a solution to the equations i u = f i in H (). (2) 3. A Cesàro Volterra formula with little regularity Given functions e ij = e ji L 2 () that satisfy the compatibility conditions (6), the classical Cesàro Volterra path integral formula (4) becomes meaningless. But we nevertheless show that there is still a way in this case to compute a solution u = (u i ) H (; R n ) to Eqs. (7).

4 580 P.G. Ciarlet et al. / C. R. Acad. Sci. Paris, Ser. I 347 (2009) This objective is achieved by means of an explicit expression in terms of the data e ij L 2 () of the duality pairings u, ϕ := u i,ϕ i = u iϕ i dx for all vector fields ϕ = (ϕ i ) D(; R n ) that satisfy ϕ i dx = (x j ϕ i x i ϕ j ) dx = 0. By reference with the classical Cesàro Volterra path integral formula, we will say that relations (4) constitute the Cesàro Volterra formula with little regularity (this terminology will be further substantiated in Theorem 7). Theorem 5. Let be a simply-connected domain in R n, let the space D (; R n ) be defined as ( D ; R n ) { := ϕ = (ϕ i ) D ( ; R n) } ; ϕ i dx = (x j ϕ i x i ϕ j ) dx = 0, (3) and let there be given a matrix field e = (e ij ) L 2 (; S 3 ) whose components e ij = e ji L 2 () satisfy the Saint Venant compatibility conditions with little regularity (6). Then a vector field u = (u i ) H (; R n ) satisfies Eqs. (7) if and only if u i,ϕ i = e ij,t i ϕ j + k Ti (T j ϕ k T k ϕ j ) ] for all ϕ = (ϕ i ) D ( ; R n ), (4) where T = (T i ) : L 2 0 () H 0 (; Rn ) is the continuous linear operator defined in Theorem. Sketch of proof. (i) Assume first that a vector field u = (u i ) H (; R n ) satisfies 2 ( j u i + i u j ) = e ij in L 2 (), and let there be given a vector field ϕ = (ϕ i ) D (; R n ). Define the functions a ij = a ij := 2 ( j u i i u j ) L 2 (), so that j u i = e ij +a ij. Since each component ϕ i of the vector field ϕ belongs to the space L 2 0 (), Theorem shows that each vector field T ϕ i = (T j ϕ i ) H0 (; Rn ) satisfies j (T j ϕ i ) = ϕ i in L 2 (). Consequently, u i,ϕ i = e ij,t i ϕ j + 2 a ij,t j ϕ i T i ϕ j. We next prove that each function (T j ϕ i T i ϕ j ) H 0 () also belongs to the space L2 0 (). Consequently, T j ϕ i T i ϕ j = k T k (T j ϕ i T i ϕ j ). We also note that k a ij = 2 ( jku i ik u j ) = i e kj + j e ki in H (), so that we obtain a ij,t j ϕ i T i ϕ j =2 e ij, k Ti (T j ϕ k T k ϕ j ) ]. Therefore, relations (4) are established. (ii) Assume next that a vector field u = (u i ) H (; R n ) satisfies relations (4). Let then a matrix field ψ = (ψ ij ) D(; S n ) be given. We first prove that ( j ψ ij ) n i= D (; R n ) and that, by (4), 2 j u i + i u j,ψ ij = e ij,t i ( k ψ jk ) + ( k e ij j e ik,t i Tj ( l ψ kl ) ). We next observe that the Saint Venant compatibility conditions with little regularity (6) may be rewritten as l h jki = i h jkl in H 2 (), where h jki = h kj i := k e ji j e ki H (). The Poincaré lemma with little regularity (Theorem 2) therefore shows that there exist functions p jk L 2 (), each one being unique up to an additive constant, such that i p jk = h jki = k e ij j e ik in H (). Since i (p jk +p kj ) = h jki + h kj i = 0, these additive constants can be adjusted in such a way that p jk + p kj = 0inL 2 (). Consequently, ( k e ij j e ik,t i Tj ( l ψ kl ) ) = ( pjk, i Ti Tj ( l ψ kl ) T k ( l ψ jl ) )]. 2 We then show that each function (T j ( l ψ kl ) T k ( l ψ jl )) (i.e., for each j =,...,n and each k =,...,n)also belongs to the space L 2 0 (). As a result, ( k e ij j e ik,t i Tj ( l ψ kl ) ) = p jk,t j ( l ψ kl ),

5 P.G. Ciarlet et al. / C. R. Acad. Sci. Paris, Ser. I 347 (2009) so that 2 j u i + i u j,ψ ij = e ij,ψ ij + p jk e jk,ψ jk + T j ( l ψ kl ), since p jk,ψ jk =0. Noting that the functions q jk := p jk e jk L 2 () satisfy l q jk = j q lk in H (),weagain resort to the Poincaré lemma with little regularity (Theorem 2) to conclude that there exist functions v k H (), each one being unique up to an additive constant, such that q jk = j v k = p jk e jk in L 2 (). Consequently, pjk e jk,ψ jk + T j ( l ψ kl ) = v k, j ψ jk + j T j ( l ψ kl ), since (ψ jk + T j ( l ψ kl )) H0 (). But the definition of the operators T j and the symmetries ψ kl = ψ lk together imply that j T j ( l ψ kl ) = l ψ kl = j ψ jk. Combining the above relations, we are thus left with 2 j u i + i u j,ψ ij = e ij,ψ ij. Since this relation holds for any matrix field ψ = (ψ ij ) D(; S n ), it follows that 2 ( j u i + i u j ) = e ij in L 2 (), as announced. We also show that the solution u = (u i ) to the equations 2 ( j u i + i u j ) = e ij in L 2 () can be found by solving a variational problem (cf. (6) below), which satisfies all the assumptions of the Lax Milgram lemma (as is easily seen). Note that both Theorems 5 and 6 have direct applications to intrinsic elasticity; cf. 7]. Theorem 6. Let be a simply-connected domain in R n, let the space L 2 (; Rn ) be defined as L 2 ( ; R n ) { } := v = (v i ) L 2 (; R n ); v i dx = (x j v i x i v j ) dx = 0, (5) and let there be given functions e ij = e ji L 2 () that satisfy the Saint Venant compatibility conditions with little regularity (6). Then the variational problem: Find a vector field (u i ) L 2 (; Rn ) such that u i,v i = e ij,t i v j + k Ti (T j v k T k v j ) ] for all (v i ) L 2 ( ; R n ), (6) has a unique solution. Besides, (u i ) is in fact in the space H (; R n ) and is a particular solution to the equations 2 ( j u i + i u j ) = e ij in L 2 (). Finally, we show that, when the data are smooth enough, the Cesàro Volterra formula with little regularity reduces to the classical Cesàro Volterra formula. Note that the proof of relation (7) below, which only involves the functions e ij, does not use that its left-hand side is also given by u i,ϕ i, by Theorem 5 (otherwise this information would immediately provide a proof of (7), through the expression of u i (x) given by the classical Cesàro Volterra formula). Theorem 7. Let the assumptions be those of Theorem 5, the functions e ij = e ji L 2 () being in addition assumed to be in the space C () H (), and let the operator (T i ) : L 2 0 () H 0 (; Rn ) be that defined in Theorem. Fix a point x 0, and, given any point x, let be any path of class C contained in and joining x 0 to x. Then the right-hand side of the Cesàro Volterra formula with little regularity (4) can be rewritten in this case as eij,t i ϕ j + k Ti (T j ϕ k T k ϕ j ) ] = { eij (y) + ( k e ij (y) i e kj (y) ) (x k y k ) } ] dy j } ϕ i (x) dx (7)

6 582 P.G. Ciarlet et al. / C. R. Acad. Sci. Paris, Ser. I 347 (2009) for all (ϕ i ) D (; R n ). Relations (7) in turn imply that any vector field (u i ) H (; R n ) that satisfies the Cesàro Volterra formula with little regularity (4) is also given by { u i (x) = eij (y) + ( k e ij (y) i e kj (y) ) (x k y k ) } dy j, x, (8) up to the addition of a vector field of the form x a + Aox for some a R n and A A n. Besides, (u i ) C 2 (; R n ) in this case. Acknowledgements The work described in this paper was supported by a Strategic Research Grant from City University of Hong Kong (Project No ). References ] C. Amrouche, P.G. Ciarlet, P. Ciarlet Jr., Vector and scalar potentials, Poincaré theorem and Korn s inequality, C. R. Acad. Sci. Paris, Ser. I 345 (2007) ] C. Amrouche, P.G. Ciarlet, L. Gratie, S. Kesavan, On the characterization of matrix fields as linearized strain tensor fields, J. Math. Pures Appl. 86 (2006) ] J. Bourgain, H. Brezis, On the equation div Y = f and application to control of phases, J. Amer. Math. Soc. 6 (2002) ] E. Cesàro, Sulle formole del Volterra, fondamentali nella teoria delle distorsioni elastiche, Rend. Napoli 2 (906) ] P.G. Ciarlet, P. Ciarlet Jr., Direct computation of stresses in planar linearized elasticity, Math. Models Methods Appl. Sci. (2009), in press. 6] P.G. Ciarlet, P. Ciarlet Jr., G. Geymonat, F. Krasucki, Characterization of the kernel of the operator CURL CURL, C. R. Acad. Sci. Paris, Ser. I 344 (2007) ] P.G. Ciarlet, L. Gratie, C. Mardare, Intrinsic methods in elasticity: A mathematical survey, Discrete and Continuous Dynamical Systems 23 (2009) ] P.G. Ciarlet, L. Gratie, C. Mardare, A Cesàro Volterra formula with little regularity, J. Math. Pures Appl. (2009), in press. 9] G. Geymonat, F. Krasucki, Some remarks on the compatibility conditions in elasticity, Rend. Accad. Naz. Sci. XL 23 (2005) ] G. Geymonat, F. Krasucki, Beltrami s solutions of general equilibrium equations in continuum mechanics, C. R. Acad. Sci. Paris, Ser. I 342 (2006) ] G. Geymonat, F. Krasucki, Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains, Comm. Pure Appl. Anal. 8 (2009) ] V. Girault, P.A. Raviart, Finite Element Methods for Navier Stokes Equations, Springer, Heidelberg, ] M.E. Gurtin, The linear theory of elasticity, in: S. Flügge, C. Truesdell (Eds.), Handbuch der Physik, vol. VIa/2, Springer-Verlag, 972, pp ] S. Mardare, On Poincaré and de Rham s theorems, Rev. Roumaine Math. Pures Appl. 53 (2008) ] V. Volterra, Sur l équilibre des corps élastiques multiplement connexes, Ann. Ecole Normale 24 (907)