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varifolds.

ESAIM: Control, Optimisation and Calculus of Variations (1)

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The nonlinear membrane model: a Young measureand varifold formulation
Med Lamine Leghmizi, Christian Licht, Gérard Michaille
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 11 / Issue 3 / July 2005

Published online by Cambridge University Press:

15 July 2005, pp. 449-472

Print publication:

July 2005
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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 energyfunctional 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. Theinterest of the first formulation is to encode the oscillation informations on the gradients minimizing sequences related to the classicalformulation. The second formulation moreover accounts for concentration effects.