We establish two new formulations of the membrane problem by working in the space of ${W}_{{\Gamma}_{0}}^{1,p}(\Omega ,{\mathbf{R}}^{3})$-Young measures and ${W}_{{\Gamma}_{0}}^{1,p}(\Omega ,{\mathbf{R}}^{3})$-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.

Classification: 74K15, 35B05, 49J45

Keywords: membrane, Young measures, varifolds

@article{COCV_2005__11_3_449_0, author = {Leghmizi, Med Lamine and Licht, Christian and Michaille, G\'erard}, title = {The nonlinear membrane model : a Young measure and varifold formulation}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {11}, number = {3}, year = {2005}, pages = {449-472}, doi = {10.1051/cocv:2005014}, zbl = {1081.74027}, mrnumber = {2148853}, language = {en}, url = {http://www.numdam.org/item/COCV_2005__11_3_449_0} }

Leghmizi, Med Lamine; Licht, Christian; Michaille, Gérard. The nonlinear membrane model : a Young measure and varifold formulation. ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 3, pp. 449-472. doi : 10.1051/cocv:2005014. http://www.numdam.org/item/COCV_2005__11_3_449_0/

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