Differential Geometry/Mathematical Problems in Mechanics
On the fundamental theorem of surface theory under weak regularity assumptions
Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 71-76.

We consider a symmetric, positive definite matrix field of order two and a symmetric matrix field of order two that together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply connected open subset of 2 . If these fields are of class C2 and C1 respectively, the fundamental theorem of surface theory asserts that there exists a surface immersed in the three-dimensional Euclidean space with the given matrix fields as its first and second fundamental forms. The purpose of this Note is to prove that this theorem still holds true under the weaker regularity assumptions that these fields are of class W1,∞loc and Lloc respectively, the Gauss and Codazzi–Mainardi equations being then understood in a distributional sense.

On considère un champ de matrices symétriques définies positives d'ordre deux et un champ de matrices symétriques d'ordre deux qui satisfont ensemble les équations de Gauss et de Codazzi–Mainardi dans un ouvert connexe et simplement connexe de 2 . Si ces champs sont respectivement de classe C2 et C1, alors le théorème fondamental de la théorie des surfaces affirme qu'il existe une surface plongée dans l'espace Euclidean tridimensionnel dont ces champs sont les première et deuxième formes fondamentales. L'objet de cette Note est d'établir que ce théorème reste vrai sous les hypothèses de régularités affaiblies selon lesquelles ces champs sont respectivement de classe W1,∞loc et Lloc, les équations the Gauss et de Codazzi–Mainardi étant alors satisfaites aux sens des distributions.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2003.10.027
Mardare, Sorin 1

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
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Mardare, Sorin. On the fundamental theorem of surface theory under weak regularity assumptions. Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 71-76. doi : 10.1016/j.crma.2003.10.027. http://www.numdam.org/articles/10.1016/j.crma.2003.10.027/

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