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The nonlinear membrane model: a Young measure and varifold formulation

Published online by Cambridge University Press:  15 July 2005

Med Lamine Leghmizi ,
Christian Licht  and
Gérard Michaille
Med Lamine Leghmizi
Affiliation:
Centre Universitaire de Médéa, Institut des Sciences de l'Ingénieur, CC 151, Quartier Ain-D'Heb, Médéa (26000), Algeria.
Christian Licht
Affiliation:
Laboratoire de Mécanique et de Génie Civil, UMR-CNRS 5508, Université Montpellier II, Case courier 048, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France.
Gérard Michaille
Affiliation:
ACSIOM et EMIAN, UMR-CNRS 5149, Université Montpellier 2 et CUFR de Nîmes, Case courier 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France.
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Abstract

We establish two new formulations of the membrane problem by working in the space of $W^{1,p}_{\Gamma_0}(\Omega,\mathbf R^3)$-Young measures and $W^{1,p}_{\Gamma_0}(\Omega,\mathbf R^3)$-varifolds. The energy functional related to these formulations is obtained as a limit of the 3d formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. The interest of the first formulation is to encode the oscillation informations on the gradients minimizing sequences related to the classical formulation. The second formulation moreover accounts for concentration effects.

Keywords

MembraneYoung measuresvarifolds.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 3 , July 2005 , pp. 449 - 472
Copyright
© EDP Sciences, SMAI, 2005

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