Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singular patterns. We aim here at providing a comprehensive and geometric (i.e., coordinate-free) framework. First, we determine the minimal assumptions required on the metric tensor in order to give a rigorous meaning to the spacetime curvature within the framework of distribution theory. This leads us to a direct derivation of the jump relations associated with singular parts of connection and curvature operators. Second, we investigate the induced geometry on a hypersurface with general signature, and we determine the minimal assumptions required to define, in the sense of distributions, the curvature tensors and the second fundamental form of the hypersurface and to establish the Gauss-Codazzi equations.2000 Mathematics Subject Classification. Primary : 35L65. Secondary : 76L05, 76N.

Abstract. These notes are intended to provide a thorough introduction to the mathematical theory of elastic shells.The main objective of shell theory is to predict the stress and the displacement arising in an elastic shell in response to given forces. Such a prediction is made either by solving a system of partial differential equations or by minimizing a functional, which may be defined either over a three-dimensional set or over a two-dimensional set, depending on whether the shell is viewed in its reference configuration as a three-dimensional or as a two-dimensional body (the latter being an abstract idealization of the physical shell when its thickness is "small").The first part of this article is devoted to the three-dimensional theory of elastic bodies, from which the three-dimensional theory of shells is obtained simply by replacing the reference configuration of a general body with that of a shell. The particular shape of the reference configuration of the shell does not play any rôle in this theory.The second part is devoted to the two-dimensional theory of elastic shells. In contrast to the three-dimensional theory, this theory is specific to shells, since it essentially depends on the geometry of the reference configuration of a shell.For a more comprehensive exposition of the theory of elastic shells, we refer the reader to Ciarlet [18] and the references therein for the first part of the article, and to Ciarlet [20] and the references therein for the second part. Une introduction a la théorie des coquesResumé. Ces notes sont destinéesà fournir une introduction détailléeà la théorie mathématique des coquesélastiques.L'objectif principal de la théorie des coques est de prédire les contraintes et les dépla-cements survenant dans une coqueélastique en réponseà des forces appliquées. Une telle prédiction est faite soit en résolvant un système d'équations aux dérivées partielles, soit en minimisant une fonctionnelle, qui peuventêtre définis soit dans un ensemble tridimensionnel, soit dans un ensemble bidimensionnel, selon que la coque est vue dans sa configuration de référence comme un corps tridimensionnel ou comme un corps bidimensionnel (ce dernierétant alors une idéalisation de la coque physique lorsque sonépaisseur est "petite").La première partie de cet article est consacréeà la théorie tridimensionnelle des corpś elastiques,à partir de laquelle la théorie tridimensionnelle des coques est obtenue en remplaçant simplement la configuration de référence d'un corpsélastique général par celle d'une coque. La forme particulière de la configuration de référence d'une coque ne joue aucun rôle dans cette théorie.La deuxième partie de cet article est consacréeà la théorie bidimensionnelle des coqueś elastiques. Contrairementà la théorie tridimensionnelle, cette théorie est spécifique aux coques, puisqu'elle dépend de façon essentielle de la géométrie de la configuration de référence de la coque.Pour un exposé plus complet de la théorie de coquesélastiques, nous renvoyons le lecteurà Ciarlet [18] et ses réf...

International audienceLet Ω be a domain in R^N, i.e., a bounded and connected open subset of R^N with a Lipschitz-continuous boundary ∂Ω, the set Ω being locally on the same side of ∂Ω. A fundamental lemma, due to Jacques-Louis Lions, provides a characterization of the space L^2(Ω), as the space of all distributions on Ω whose gradient is in the space H^{−1}(Ω). This lemma, which provides in particular a short proof of a crucial inequality due to J. Necas, is also a key for proving other basic results, such as, among others, the surjectivity of the divergence operator acting from H^1_0(Ω) into L^2_0(Ω), a “weak” form of the Poincare lemma or a “simplified version” of de Rham theorem, each of which provides sufficient conditions insuring that a vector field in H^{−1}(Ω) is the gradient of a function in L^2(Ω). The main objective of this paper is to establish an “equivalence theorem”, which asserts that J.L. Lions lemma is in effect equivalent to a number of other fundamental properties, which include in particular the ones mentioned above. The key for proving this theorem is a specific “approximation lemma”, itself one of these equivalent results, which appears to be new to the best of our knowledge. Some of these equivalent properties can be given an independent, i.e., “direct”, proof, such as for instance the constructive proof by M.E. Bogovskii of the surjectivity of the divergence operator. Therefore, the proof of any one of such properties provides, by way of our equivalence theorem, a means of proving J.L. Lions lemma, the known “direct” proofs of which for a general domain are notoriously difficult