Soit ω un ouvert connexe et simplement connexe de , muni d'une métrique riemannienne. Sous une certaine hypothèse de régularité sur la frontière de ω, on établit d'abord l'existence et l'unicité aux isométries près d'une immersion isométrique de ω dans l'espace euclidien , « jusqu'au bord » de ω. Lorsque ω est borné, on montre aussi que l'application qui associe aux données géométriques prescrites la sous-variété ainsi reconstruite est localement lipschitzienne pour les topologies usuelles des espaces de Banach .
Let ω be a connected and simply connected open subset of endowed with a Riemannian metric. Under a smoothness assumption on the boundary of ω, we first establish the existence and uniqueness up to isometries of an isometric immersion of ω into the Euclidean space , ‘up to the boundary’ of ω. When ω is bounded, we also show that the mapping that associates with the prescribed geometrical data the reconstructed submanifold is locally Lipschitz-continuous with respect to the topology of the Banach spaces .
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@article{CRMATH_2004__339_4_265_0, author = {Szopos, Marcela}, title = {On the recovery and continuity of a submanifold with boundary in higher dimensions}, journal = {Comptes Rendus. Math\'ematique}, pages = {265--270}, publisher = {Elsevier}, volume = {339}, number = {4}, year = {2004}, doi = {10.1016/j.crma.2004.05.022}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2004.05.022/} }
TY - JOUR AU - Szopos, Marcela TI - On the recovery and continuity of a submanifold with boundary in higher dimensions JO - Comptes Rendus. Mathématique PY - 2004 SP - 265 EP - 270 VL - 339 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2004.05.022/ DO - 10.1016/j.crma.2004.05.022 LA - en ID - CRMATH_2004__339_4_265_0 ER -
%0 Journal Article %A Szopos, Marcela %T On the recovery and continuity of a submanifold with boundary in higher dimensions %J Comptes Rendus. Mathématique %D 2004 %P 265-270 %V 339 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2004.05.022/ %R 10.1016/j.crma.2004.05.022 %G en %F CRMATH_2004__339_4_265_0
Szopos, Marcela. On the recovery and continuity of a submanifold with boundary in higher dimensions. Comptes Rendus. Mathématique, Tome 339 (2004) no. 4, pp. 265-270. doi : 10.1016/j.crma.2004.05.022. http://www.numdam.org/articles/10.1016/j.crma.2004.05.022/
[1] Differential geometric aspects of continuum mechanics in higher dimensions, Differential Geometry: The Interface between Pure and Applied Mathematics, Contemp. Math., vol. 68, 1987, pp. 23-37
[2] Riemannian Geometry, Birkhäuser, Boston, 1992
[3] Recovery of a manifold with boundary and its continuity as a function of its metric tensor, C. R. Acad. Sci. Paris, Ser. I, Volume 338 (2004), pp. 333-340
[4] Ph.G. Ciarlet, C. Mardare, A surface as a function of its two fundamental forms, in press
[5] The Gauss–Codazzi equations, Tensor (N.S.), Volume 39 (1982), pp. 15-22
[6] M. Szopos, On the recovery and continuity of a submanifold with boundary, in press
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