The nonlinear membrane model: a Young measure and varifold formulation

Med Lamine Leghmizi; Christian Licht; Gérard Michaille

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 11, Issue: 3, page 449-472
  • ISSN: 1292-8119

Abstract

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We establish two new formulations of the membrane problem by working in the space of W Γ 0 1 , p ( Ω , 𝐑 3 ) -Young measures and W Γ 0 1 , p ( Ω , 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.

How to cite

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Leghmizi, Med Lamine, Licht, Christian, and Michaille, Gérard. "The nonlinear membrane model: a Young measure and varifold formulation." ESAIM: Control, Optimisation and Calculus of Variations 11.3 (2010): 449-472. <http://eudml.org/doc/90772>.

@article{Leghmizi2010,
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. },
author = {Leghmizi, Med Lamine, Licht, Christian, Michaille, Gérard},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Membrane; Young measures; varifolds.; energy functional; variational convergence},
language = {eng},
month = {3},
number = {3},
pages = {449-472},
publisher = {EDP Sciences},
title = {The nonlinear membrane model: a Young measure and varifold formulation},
url = {http://eudml.org/doc/90772},
volume = {11},
year = {2010},
}

TY - JOUR
AU - Leghmizi, Med Lamine
AU - Licht, Christian
AU - Michaille, Gérard
TI - The nonlinear membrane model: a Young measure and varifold formulation
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 3
SP - 449
EP - 472
AB - 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.
LA - eng
KW - Membrane; Young measures; varifolds.; energy functional; variational convergence
UR - http://eudml.org/doc/90772
ER -

References

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  9. H. Le Dret and A. Raoult, The nonlinear membrane model as Variational limit in nonlinear three-dimensional elasticity. J. Math. Pures Appl., IX. Ser.74 (1995) 549–578.  Zbl0847.73025
  10. P. Pedregal, Parametrized measures and variational Principle. Birkhäuser (1997).  
  11. M.A. Sychev, A new approach to Young measure theory, relaxation and convergence in energy. Ann. Inst. Henri Poincaé16 (1999) 773–812.  Zbl0943.49012
  12. M. Valadier, Young measures. Methods of Nonconvex Analysis, A. Cellina Ed. Springer-Verlag, Berlin. Lect. Notes Math.1446 (1990) 152–188.  
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