$\mathrm{\Gamma }$-Convergence for plane to wrinkles transition problem

We consider a variational problem modeling transition between flat and wrinkled region in a thin elastic sheet, and identify the $\mathrm{\Gamma }$-limit as the sheet thickness goes to 0, thus extending the previous work of the first author [Bella, ARMA 2015]. The limiting problem is scalar and convex, but constrained and posed for measures. For the $\mathrm{\Gamma }$-liminf inequality we first pass to quadratic variables so that the constraint becomes linear, and then obtain the lower bound using Reshetnyak’s theorem. The construction of the recovery sequence for the $\mathrm{\Gamma }$- limsup inequality relies on mollification of quadratic variables, and careful choice of multiple construction parameters. Eventually for the limiting problem we show existence of a minimizer and equipartition of the energy for each frequency.