Γ-Convergence for Plane to Wrinkles Transition Problem

Abstract

We consider a variational problem modeling transition between flat and wrinkled region in a thin elastic sheet and identify the Γ-limit as the sheet thickness goes to 0, thus extending the previous work of the first author (Bella et al. in Arch Ration Mech Anal 218(1):163–217, 2015). The limiting problem is scalar and convex, but constrained and posed for measures. For the Γ-lim inf 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 Γ-lim sup 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.