Differential Geometry
Approximating W2,2 isometric immersions
Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 189-192.

Let SR2 be a bounded Lipschitz domain and set Wiso2,2(S;R3)={uW2,2(S;R3):(u)T(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 W2,2 closure of Wiso2,2(S;R3)C(S¯;R3) agrees with Wiso2,2(S;R3).

Soient SR2 un domaine lipschitzien borné et Wiso2,2(S;R3) l'ensemble Wiso2,2(S;R3)={uW2,2(S;R3):(u)T(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 W2,2 de Wiso2,2(S;R3)C(S¯;R3) est Wiso2,2(S;R3).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.01.001
Hornung, Peter 1

1 Fachbereich Mathematik, Universität Duisburg-Essen, 47048 Duisburg, Germany
@article{CRMATH_2008__346_3-4_189_0,
     author = {Hornung, Peter},
     title = {Approximating $ {W}^{2,2}$ isometric immersions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {189--192},
     publisher = {Elsevier},
     volume = {346},
     number = {3-4},
     year = {2008},
     doi = {10.1016/j.crma.2008.01.001},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2008.01.001/}
}
TY  - JOUR
AU  - Hornung, Peter
TI  - Approximating $ {W}^{2,2}$ isometric immersions
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 189
EP  - 192
VL  - 346
IS  - 3-4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2008.01.001/
DO  - 10.1016/j.crma.2008.01.001
LA  - en
ID  - CRMATH_2008__346_3-4_189_0
ER  - 
%0 Journal Article
%A Hornung, Peter
%T Approximating $ {W}^{2,2}$ isometric immersions
%J Comptes Rendus. Mathématique
%D 2008
%P 189-192
%V 346
%N 3-4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2008.01.001/
%R 10.1016/j.crma.2008.01.001
%G en
%F CRMATH_2008__346_3-4_189_0
Hornung, Peter. 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. http://www.numdam.org/articles/10.1016/j.crma.2008.01.001/

[1] Ball, J.M. Singularities and computation of minimizers for variational problems, London Math. Soc. Lecture Note Ser., vol. 284, 2001, pp. 1-20

[2] Conti, S.; Dolzmann, G. Derivation of a plate theory for incompressible materials, C. R. Acad. Sci. Paris, Ser. I, Volume 344 (2007), pp. 541-544

[3] Friesecke, G.; James, R.D.; Müller, S. Rigorous derivation of nonlinear plate theory and geometric rigidity, C. R. Acad. Sci. Paris, Ser. I, Volume 334 (2002), pp. 173-178

[4] Friesecke, G.; James, R.D.; Müller, S. A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. Pure Appl. Math., Volume 55 (2002), pp. 1461-1506

[5] P. Hornung, A density result for W2,2 isometric immersions, preprint, available under http://analysis.math.uni-duisburg.de/publications/preprints/Hornung4.pdf

[6] B. Kirchheim, Geometry and rigidity of microstructures, Habilitation thesis, University of Leipzig, 2001

[7] Müller, S.; Pakzad, M.R. Regularity properties of isometric immersions, Math. Z., Volume 251 (2005) no. 2, pp. 313-331

[8] Pakzad, M.R. On the Sobolev space of isometric immersions, J. Differential Geom., Volume 66 (2004) no. 1, pp. 47-69

[9] Pantz, O. Une justification partielle du modèle de plaque en flexion par Γ-convergence, C. R. Acad. Sci. Paris, Ser. I, Volume 332 (2001), pp. 587-592

[10] Pantz, O. On the justification of the nonlinear inextensional plate model, Arch. Ration. Mech. Anal., Volume 167 (2003), pp. 179-209

Cited by Sources: