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ABSTRACT. The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a αβ ) of order two and a field of symmetric matrices (b αβ ) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a connected and simply-connected open subset ω of R 2 , then there exists an immersion θ : ω → R 3 such that these fields are the first and second fundamental forms of the surface θ(ω) and this surface is unique up to proper isometries in R 3 .In this paper, we identify new compatibility conditions, expressed again in terms of the functions a αβ and b αβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the formwhere A 1 and A 2 are antisymmetric matrix fields of order three that are functions of the fields (a αβ ) and (b αβ ), the field (a αβ ) appearing in particular through its square root. The unknown immersion θ : ω → R 3 is found in the present approach in function spaces "with little regularity", viz.,Une nouvelle approche du théorème fondamental de la théorie des surfaces.RÉSUMÉ. Le théorème fondamental de la théorie des surfaces affirme classiquement que, si un champ de matrices (a αβ ) symétriques définies positives d'ordre deux et un champ de matrices (b αβ ) symétriques d'ordre deux satisfont ensemble leséquations de Gauss et Codazzi-Mainardi dans un ouvert ω ⊂ R 2 connexe et simplement connexe, alors il existe une immersion θ : ω → R 3 telle que ces deux champs soient les première et deuxième formes fondamentales de la surface θ(ω), et cette surface est unique aux isométries propres de R 3 près.Dans cet article, nous identifions de nouvelles conditions de compatibilité, exprimées a nouveauà l'aide des fonctions a αβ et b αβ , qui conduisent aussià un théorème analogue d'existence et d'unicité. Ces conditions sont de la formeoù A 1 et A 2 sont des champs de matrices antisymétriques d'ordre trois, qui sont des fonctions des champs (a αβ ) et (b αβ ), le champ (a αβ ) apparaissant en particulier par l'intermédiaire de sa racine carrée. L'immersion inconnue θ : ω → R 3 est trouvée dans cette approche dans des espaces fonctionnels "de faible régularité",à savoir W 2,p loc (ω; R 3 ), p > 2.

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We proposed a new approach to the existence theory for quadratic minimization problems that arise in linear shell theory. The novelty consists in considering the linearized change of metric and change of curvature tensors as the new unknowns, instead of the displacement vector field as is customary.Such an approach naturally yields a constrained minimization problem, the constraints being ad hoc compatibility relations that these new unknowns must satisfy in order that they indeed correspond to a displacement vector field. Our major objective is thus to specify and justify such compatibility relations in appropriate function spaces. Interestingly, this result provides as a corollary a new proof of Korn's inequality on a surface. While the classical proof of this fundamental inequality essentially relies on a basic lemma of J. L. Lions, the keystone in the proposed approach is instead an appropriate weak version of a classical theorem of Poincaré.The existence of a solution to the above constrained minimization problem is then established, also providing as a simple corollary a new existence proof for the original quadratic minimization problem.

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