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We establish two new formulations of the membrane problem by working in the space of $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-Young measures and $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-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...

We establish two new formulations of the membrane problem by working in the space of $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-Young measures and $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-varifolds. The energy functional related to these formulations is obtained as a limit of the 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...