Approximating $ {W}^{2,2}$ isometric immersions

Peter Hornung

doi:10.1016/j.crma.2008.01.001

Differential Geometry

Approximating

W^{2, 2}

isometric immersions

Presented by: John M. Ball

Peter Hornung ¹

¹Fachbereich Mathematik, Universität Duisburg-Essen, 47048 Duisburg, Germany

Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 189-192

Abstract (Original language)
Résumé

Let $S \subset R^{2}$ be a bounded Lipschitz domain and set $W_{iso}^{2, 2} (S; R^{3}) = {u \in W^{2, 2} (S; R^{3}) : {(\nabla u)}^{T} (\nabla u) = Id a.e.}$ . Under an additional regularity condition on the boundary ∂S (which is satisfied if it is piecewise continuously differentiable) we prove that the $W^{2, 2}$ closure of $W_{iso}^{2, 2} (S; R^{3}) \cap C^{\infty} (\bar{S}; R^{3})$ agrees with $W_{iso}^{2, 2} (S; R^{3})$ .

Soient $S \subset R^{2}$ un domaine lipschitzien borné et $W_{iso}^{2, 2} (S; R^{3})$ l'ensemble $W_{iso}^{2, 2} (S; R^{3}) = {u \in W^{2, 2} (S; R^{3}) : {(\nabla u)}^{T} (\nabla u) = Id p.p.}$ . Sous une hypothèse supplémentaire de régularité sur la frontière ∂S (qui est satisfaite dans le cas où ∂S est continument différentiable par morceaux), nous prouvons que l'adhérence $W^{2, 2}$ de $W_{iso}^{2, 2} (S; R^{3}) \cap C^{\infty} (\bar{S}; R^{3})$ est $W_{iso}^{2, 2} (S; R^{3})$ .

Received: 2007-04-13
Accepted: 2008-01-07
Published online: 2008-01-28

DOI: 10.1016/j.crma.2008.01.001

Author's affiliations:

Peter Hornung ¹

¹ Fachbereich Mathematik, Universität Duisburg-Essen, 47048 Duisburg, Germany

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Peter Hornung. Approximating $ {W}^{2,2}$ isometric immersions. Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 189-192. doi: 10.1016/j.crma.2008.01.001

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